Seepage Hydraulics

Seepage Hydraulics DEVELOPMENTS IN WATER SCIENCE, 10 advisory editor VEN TE C H O W Professor of Civil and Hydrosyste...

Author: G. Kovács

400 downloads 2383 Views 35MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Seepage Hydraulics

DEVELOPMENTS IN WATER SCIENCE, 10 advisory editor VEN TE C H O W Professor of Civil and Hydrosystems Engineering Hydrosystems Laboratory University of Illinois Urbana, IL.. U. S. A. OTHER TITLES IN THIS SERIES I G. BUGLIARELLO A N D F. GUNTER COMPUTER SYSTEMS A N D WATER RESOURCES 2 H. L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY

3 Y. Y. HAIMES, W. A. H A L L and H. T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF METHOD

4 J. J. FRIED GROUNDWATER POL LUTlON 5 N. RAJARATNAM TURBULENT JETS

6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS

7 V. HALEK A N D J. SVEC GROUNDWATER HYDRAU LlCS 8 J. BALEK HYDROLOGY A N D WATER RESOURCES IN TROPICAL AFRICA 9 T. A. McMAHON A N D R. G. MElN RESERVOIR CAPACITY A N D YIELD 10 G. KOVACS SEEPAGE HYDRAULICS

11 W. H. GRAF A N D C. H. MORTIMER (EDITORS) HYDRODYNAMICS OF LAKES

12 W. BACK A N D D. A. STEPHENSON (EDITORS) CONTEMPORARY HYDROGEOLOGY

Seepage Hydraulics Gyorgy K o v k s Corresponding Member of the Hungarian Academy of Sciences Director-General of the Research Centre of Water Resources Development (VITUKI) Budapest, Hungary

Elsevier Scientific Pubtishing Company Amsterdam

Oxford

New York i981

This book is the revised English version of the "A sziv6rgds hidraulikdja" published by Akademiai Kiad6. Budapest Translated by Katalin Kovdcs The distribution of this book i s being handled by the following publishers and Canada Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York. New York 10017, U.S.A.

for the U.S.A.

for the East European countries, Democratic People's Republic of Korea, People's Republic of China. People's Republic of Mongolia, Republic of Cuba and Socialist Republic of Vietnam

Akademiai Kiad6, The Publishing House of the Hungarian Academy of Sciences, Budapest for a11 remaining areas

Elsevier Scientific Publishing Company 335 Jan van Galenstraat P. 0. Box 211, lo00 AE Amsterdam, The Netherlands Library of Congress Cataloging in Publication Data Kovdcs. Gyorgy, 1925Seepage hydraulics. (Developments in water science ; 10) Translation of A szivdrgds hidraulikdja. Bibliography: p. Includes index. 1. Seepage. 2. Groundwater flow. 1. Title. II. Series. TCI 76.K6813 627'.17 ISBN 0 -444- 99755 5 ISBN 0-444-41669-2 (Series)

-

0 Akadimiai

80-20236

Kiad6, Budapest 1981

Joint edition published by Elsevier Scientific Publishing Company, Amsterdam, The Netherlands and A k a d h i a i Kiad6, The Publishing House of the Hungarian Academy of Sciences, Budapest, Hungary Printed in Hungary

Preface

Seepage hydraulics, a relatively small branch of hydro- and aeromechanics, deals in general with the properties, the behaviour and the motion of liquid and gmeous media. Its methodology differs basically, however, from that applied to investigating other transport processes of fluids or gases. The main differences result from the special structure of the flow-domain in which the seepage develops. Most of this flow-domain is filled with solid particles; the movement of water or other fluid is only possible through randomly interconnected channels. These channels are of a size which varies considerably and are composed either of the pores surrounded by the grain3 or of fissures and fractures. From this definition of seepage i t can be seen that seepage hydraulics covers bordering fields of the earth sciences, physics and technical sciences. The role of the earth sciences is to determine the properties of the solid matrix. The principles of physics are utilized to describe the transport and storage processes effected by means of the versatile system of water-conveying interstices and to derive the basic relationships between the kinematic parameters by using applied mathematics. The solution of practical problems by simultaneously satisfying the kinematic equations and the boundary conditions developing along the perimeter of the flow-domain (which condition may represent either natural effects or man-induced actions) belongs to the technical sciences. There are numerous handbooks already published concerning the problems of flow through porous media. Some of them give excellent summaries of applied mathematics related to seepage. Others give similarly good physical and mathematical treatments of flow theory. I n another group, the geological approach ie emphasized by dealing mostly with the characterization of the solid matrix and with the description of its behaviour. The new feature of this book is that i t provides an integrated system covering the entire field of seepage hydraulics and i t presents solutions for practising engineers without neglecting the scientific background. It is hoped that the different approaches (theoretical and practical; mathematical and physical) and the various scientific aspects (hydraulics, hydrology, geology, soil science) a m presented in a balanced form in relation to:

6

the characterization of porous media, the dynamic analysis of the acting processes, and the kinematic description of the transport of fluids. It is also hoped that the book will provide guidance towards the complete solution of the problems, ensuring thereby not only a firm theoretical basis for the investigations but also their practical application. Large numbers of experimental and field data are summarized in the tables; these should be of value to those who wish to use the book for further research. This manualwaa written primarily forhydraulic engineers; apart from them it can be used also by others who need to solve various problems related to the movement o i subsurface water in geology, mining, civil and geotechnical engeneering, agricultural sciences (land reclamation, irrigation and drain&ge) etc. The book was first published in Hungarian in 1972. For the English version the original version has been completely revised and up-dated to include the latest developments, which have been extremely rapid in the first half of the 1970s. Important new parts concerning seepage through fractured rocks and unsaturated media have been added. -

Dr. Gyiirgy Kovdcs

Contents

Part 1 Fundamentals for the investigation of seepage

Chapkr 1 . 1 General characterization of subsurface water and its movement 1.1.1 Classificationof the various types of subsurface water 1.1.2 Characterization of flow through various aquifers 1.1.3 T h e definition of seepage References to Chapter 1.1

....

........ ........... ................................ ........................................

Chapter 1.2 Physical and mineralogical parameters of loose clastic sediments influencing permeability 1.2.1 The size and shape of grains forming a layer 1.2.2 Investigation of grain-size distribution 1.2.3 Mineralogical composition of sediments and its influence on the determination of the effective diameter 1.2.4 Relationship between porosity and the geometrical parameters of grains 1.2.5 Physical model representing the irregular network of pores and channels between grains References to Chapter 1.2

...................................... ................. .................... ..................

............................................. ................................

.......................................

.................

Chapter 1.3 Dynamics of soil moisture above the water table 1.3.1 General characterization of soil moisture and processes acting in the soil-moisture zone 1.3.2 Parameters characterizingthe various moisture content8 of soil samples 1.3.3 Interpretation of field capacity, gravitational porosity and soil-moisture retention curve 1.3.4 Description and determination of the soil-moisture retention curve References to Chapter 1.3

................................ .............................................. ............................

................................................ ....................................... Chapter 1.4 Investigation of the balance of the ground-water space ............ 1.4.1 General hydrological characterization of water exchange between soil moisture and ground water ....................

13 13 20 28

37 38 39 43

60 76

87 94 98 99 104 109 123 146 147 148

8

Contents

............................. .................... ............................... .......................................

1.4.2 Storage capacity of aquifers 1.4.3 Analyeis of horizontal ground-water flow for the determination of the vertical water exchange 1.4.4 Interpretation and determination of the characteristic curve of the ground-water balance References to Chapter 1.4

153 178 188 199

Part 2 Dynamic interpretation and determination of hydraulic conductivity in homogeneous loose clastic sediments

.................................. ................. ....................................... ................... ........................................

Chapter 2.1 Dynamic analysis of seepage 202 203 2.1.1 Forces influencing the flow between grains 2.1.2 Dimensionless numbers characterizing the various validity zones of seepage 226 2.1.3 Numerical limits of the zones of seepage 230 References to Chapter 2.1 238

..................... ........... ...........................

Chapter 2.2 Hydraulic conductivity of saturated layers 239 2.2.1 Determination of D ~ C Y ’hydraulic B conductivity 240 2.2.2 Investigation of the turbulent and transition zones of seepage 251 2.2.3 Investigation of microseepage 269 References to Chapter 2.2 281

.......................................

........................... 283 284 ............ 293

Chapter 2.3 Seepage through unsaturated layers 2.3.1 Movement equations characterizing unsaturated flow ........ 2.3.2 Physical soil paremeters used in diffusion theory 2.3.3 Theoretical analysis of hydraulic conductivity in unsaturated porousmedia References to Chapter 2.3

.......................................... .......................................

301 314

Part 3 Permeability of natural layers and processes influencing its change in time

Chapter 3.1 Characterization of special behaviour of hydraulic conductivity in loose claatic sediments 3.1.1 Laboratory and field methods to determine permeability .... 3.1.2 Evaluation of pumping tests with more observation wells 3.1.3 Characterization. determination and practical consideration of anisotropy References to Chapter 3.1

.......................................

.... ............................................

........................................

318 319 326 338 348

........ 349 ...................... 351 ............................. 362 ................. 369 ........................................ 378

Chapter 3.2 The motion of grains in cohesionless loose claatic sediments 3.2.1 The motion of f h e grains (suffusion) 3.2.2 T h e liquidization of the layer 3.2.3 Design of protective filters and well screens References to Chapter 3.2

9

Contents Chapter 3.3 Investigation of clogging .................................... 3.3.1 The change of concentration of the percolating water depending on time and place ................................... 3.3.2 Application of the capillary tube model of the porous medium to characterize the clogging process ...................... References to Chapter 3.3

380

381 387

........................................ 395

Chapter 3.4 Hydraulic conductivity and intrinsic permeability of fissured and fractured rocks ........................................ 3.4.1 Characterization of various non.carbonate, water.bearing, fissuredandfracturedrocks .............................. 3.4.2 Hydraulic properties of carbonate rocks ................... 3.4.3 Models for the characterization of flow through the openings of solid rocks ............................................ 3.4.4 A conceptual model for the determination of hydraulic conductivity of fissured rocks .............................. References to Chapter 3.4

........................................

396 397 417 437 454 470

Part 4 Kinematic characterization of seepage

Chapter 4.1 Kinematic relationships characterizing laminar seepage ........... 475 4.1.1 Interpretation of velocity-potential and potential water 476 movement ............................................ 4.1.2 Interrelation between stream-function and potentia1.function Theflownet ........................................... 482 4.1.3 Geometrical and kinematic classification of seepage .......... 493 References to Chapter 4.1 ....................................... 504

.

Chapter 4.2 Boundary and initial conditions of potential flow through porous media ............................................... 505 4.2.1 Characterization of external boundary conditions ........... 506 4.2.2 T h e investigation of the layered seepage field ............... 519 4.2.3 Application of hodograph image and other special transformations for the characterization of boundary conditions ........ 529 4.2.4 Consideration of initial conditions ........................ 539 References to Chapter 4.2 ........................................ 541 Chapter 4.3 Kinematic characterization of non-laminar seepage .............. 4.3.1 Differential equations equivalent t o Laplece’s equations for the various zones of seepage ............................. 4.3.2 Consideration of the continuous change of the flow condition within the seepage field ................................. References to Chapter 4.3

........................................

542 543 549 555

10

Contents Part 5 Solution of movement equations describing seepage

............ ....................

Chapter 6.1 Characterization of two-dimensionalpotential seepage 660 6.1.1 Complex potential Conjugate velocity 663 6.1.2 Solution of seepage problems by applying mapping .......... 667 6.1.3 Basic mapping functions applied most frequently ........... 676 6.1.4 Application of Schwartz-Christoffel’s mapping ............. 686 References to Chapter 6.1 ........................................ 695

.

Chapter 5.2 Combined application of various mapping functions .............. 6.2.1 Application of a series of mapping functions within the flow plane of a two-dimensional seepage 6.2.2 Combination of mapping functions applied on two different planes of the flow space ................................. 6.2.3 Combined application of hodogreph and conformal mappings 6.2.4 Characterization of unconfined field by the application of Zhukovsky’s function Referencesto Chapter 6.2 ........................................

.......................

696 697 610

. . 616 ................................... 626

Chapter 6.3 Horizontal unconfined steady seepage (Dupuit’s equations and the limits of their application) 6.3.1 Derivation of Dupuit’s equations ......................... 6.3.2 The influence of the capillary water conveyance 6.3.3 Characterization of horizontal unconfined seepage influenced by accretion 6.3.4 Local resistance occurring in the vicinity of the entry and exit faces References to Chapter 6.3

626

627 ............................... 628 ............. 648 .......................................... 661 ................................................. 670

........................................

686

.................... 687 ........................ 688 ............... 696 ............ 704 ........................................ 712 Chapter 6.6 Model laws for sand box models ............................... 713 6.6.1 General derivation of model laws for hydraulic models ....... 714 6.6.2 Geometrically distorted sand box models ................... 719 722 References to Chapter 6.6 ....................................... Subject index ................................................... 723

Chapter 6.4 Investigation of horizontal unsteady seepage 6.4.1 Derivation of Boussinesq’s equations and the problems in connection with their linearization 6.4.2 Application of the differentialequation of unsteady flow to the characterization of seepage in an infinite field 6.4.3 Unsteady seepage in a horizontally limited field References to Chapter 6.4

Part 1

Fundamentals for the investigation of seepage

The greatest amount of water on the globe is stored in the basins of oceans and seas. This water however, is salty, and therefore, unsuitable for direct congumption. Fresh water comprises only 2.8% of the total amount of water (i.e. about 37 million cubic kilometres) and is stored on and below the surface of the earth. It occurs in several different forms: ice and snow in polar areas and mountainous regions, ground water in the pores of aquifers, water in the beds of lakes and rivers, etc. Water stored below the surface of the earth in the form of soil moisture and ground water makes u p more than 20 % of the earth’s total fresh water resources. It is distributed almost equally over the continents, and its exploitation is a fundamental basis of water management in every country. Although only the smaller part of the total amount is available for direct use, (because the exploitation of water stored in clay, or at great depth is not yet economical) even this amount would be enough to meet the increased world demand for a very long period. It is not permissible however, to use up large quantities of stored water, because it is one of the basic elements of the biosphere, and some very serious changes would be caused by its absence (e.g. heat balance would be disturbed by the decrease in evaporation, or vegetation would be destroyed by lowering the water table over large areas). The tltored water - the so-called stutic water resource - haa t o be considered therefore, as a safety reservoir, from which water may be exploited on a limited scale, when its recharge is expected. For this reason, the amount of water replenished per annum on the surface of the continents in the form of precipitation is more important from the aspect of water reaources development, than the stored quantity. The detailed investigation of these dynamic resources involves the determination of parameters characterizing the subsurface branch of the hydrological cycle. This topic belongs to hydrology, especially ground-water hydrology. Another baaic requirement for this investigation is the study of the movement of water within the cycle, namely hydrodynamics. The investigation of the subterranean part of the hydrological cycle - the dynamics of flow through porous media - is also a fundamental part of the scientific background of water resources development. Apart from the analysis of this general flow system, many hydrodynamical problems have to be solved when designing the structures (wells, shafts, or

12

1 Fundamentals for the investigation of seepage

drains) through which the exploitation of ground water becomes possible. I n other cases, hydraulic structures (reservoirs, dams, canals) raise the level of surface waters above thenatural ground-water table, and thus, create seepage. Sometimes the position of the water table, and the moisture content of the upper soil have t o be controlled, either t o ensure higher agricultural production (land drainage, irrigation), or to create better environmental conditions (marsh reclamation). Similarly, the permanent or temporary modification of the position of the phreatic ground-water surface and/or the pressure condition of ground water might be the aim of special operations (dewatering of construction pits, protection of mines against water intrusion). The solution of all the problems related either to the design or the operation of the hydraulic structures listed, requires the investigation of water movement in various flow domains, the boundaries of which are determined by the geological structure of the layers. The boundary conditions are influenced by hydrological and hydrogeological phenomena, and the flow conditions are governed by the soil- or rock-mechanical properties of the beds in question. This short summary of the problems already indicates the multidisciplinary character of seepage hydraulics. It is basically part of hydromechanics, starting with the theoretical discussion of the kinematics and dynamics of water movement, and also including the practical application of the theoretical results. The fundamental topics of hydromechanics (e.g. interpretation and determination of mass, density, specific weight and viscosity of fluids, or that of hydrostatic pressure and its distribution) may be regarded as wellknown, and need not be repeated here. The reader is referred to existing manuals dealing with these basic concepts of hydrodynamics (e.g. NBmeth, 1963; Bear, 1972). At the same time, because of the multidisciplinary character of seepage hydraulics, i t is necessary to deal with some aspects of the interrelated sciences as well (e.g. hydrology, hydrogeology, mineralogy, soil phyeics, soil sciences, etc.), whichcan be regarded as fundamental to our discussion. Naturally, the same basic philosophy as mentioned in connection with hydromechanics is followed here, that is, that the principles of these sciences are not explained - only some special points having close connection with seepage hydraulics are considered. Those aspects are emphasized, which may influence the discussion of flow through porous media. Sometimes the method of discussion of topics taken from other scientific fields and summarized in Part 1 does not follow that generally used in the original literature. Slight modifications are made, or ideas further developed, to facilitate the easy application of the results in seepage hydraulics. It is hoped that these changes do not alter the basic concepts of the related sciences, and that contradictions are not caused by them.

1.1 General characterization of subsurface water

13

Chapter 1.1 General characterization of subsurface water and i t s movement The first problem which hau to be solved before starting the detailed discussion of the various scientific topics is the establishment of a common terminology, so that misunderstandings can be avoided. The clarification of the definitions of terms used generally is necessary not only because the meanings of some expressions differ in different languages, but also because differences can be found in the literature of the various sciences written in the same language. To create a suitable framework for the comparison of the various expressions, and to clarify the defhitions used in this book, a short summary of basic hydrogeological concepts, and of expressions concerning the flow of water through porous media, will be given here as an introduction. This chapter includes, therefore, the analysis of the various effects influencing the character of water movement below the surface, and finally, the defhition of seepage itself. When establishing the proposed terminology, attempts were made to follow the definitions accepted in recently published international manuals and glossaries (UNESCO, 1972;FAO, 1972;FAO-UNESCO, 1973;IAHS, 1974; UNESCO-WMO, 1974). Changes - or more precisely supplementation were made only in a few cases, where the special character of some aspects of seepage hydraulics required them.

1.1.1 Classification of the various types of subsurface water

A considerable amount of the water falling on the continents in the form of precipitation infiltrates through the surface. This infiltrating water becomes part of the subsurface water, which is stored for either a short or a long period in the various interstices of the crust (pores, hollows, joints, fissures and fractures), and moves along the channels composed of these pores and openings. When investigating the hydrological processes occurring under the surface of the earth, the most important aspects to be considered are the part of the subsurface water plays in the entire hydrological cycle, and the characterization of both flow and storage in the layers as influenced by their structure, and physical characteristics. The subsurface waters can be classified in many different ways, taking their various properties into consideration as the basis of the classification (e.g. temperature, chemical composition, origin, character of movement, etc.). As explained earlier, the two most important aspects from the hydrological point of view are the character of the water-bearing layers and their connection with the meteorological and hydrological processes on and above the surface. According to the claasification based on these two parameters (Table 1.1-l),three different levels of subsurface water can be distinguished (soil moisture, shallow ground water and deep ground water). Each level can

14

1 Fundamentals for the investigation of seepage Table 1.1-1. Hydrological classification of the various types of subsurface water B

L o w elastic sediment

I

Position

of aquifer

-(limestone, (dolomite) Karstio

gravel, send, silt, clay

Zone of aeration

I

Nan Laretic

moisture content of rocks

i Zone of eaturation

M i d rocka

cohive

2

3

-

shallow ground water

shallow karstic water

-~ deep ground water

deep kerstic water

Iw

~ ~ ~ ~ Of t e n t

in shallow position in deep position

be further divided into two main groups, considering those characteristics of the water-bearing layers, which influence the flow of water, and the process of its storage to the greatest extent (loose clastic sediments, or fissured and fractured solid rocks). Soil moisture is the water content of the pores in the soil above the water table, retained in various forms against gravity. The most important forces influencing the development of soil moisture are adhesion and capillarity. Adhesion creates a thin fllm of water covering the wall of the pores (adhesive water) while capillarity completely fills some of the pores with water (or all the pores, depending on the height above the water table of the point considered) (capillary water). In fissured and fractured solid rocks, the same type of water is called the moisture content of rocks. For the complete clarification of this definition, it is necessary to analyze the term water table as well. According t o the generally accepted meaning, it is the surface of unconfined ground water bodies dividing the ground water and the soil moisture zones. Another characteristic feature of the water table is that the pressure of the water included in the pores is equal here to that of the atmosphere. It is necessary t o note that the water table is not a real surface, because there is no rapid change in the saturation of pores at this level. Both the ground-water zone and the lower part of the capillary zone are completely saturated. The decrease in water content related to porosity starts gradually only in the upper reaches of the closed capillary zone, and it is continued in the open capillary zone. Thus the level of the water table is indicated only by the change in the sign of the water pressure - it is positive in the ground-water zone and negative (suction, tension) above the water table. It should be noted here that water pressure is always expressed in the form of excess pressure (p),i.e. the difference between the total ( p l ) and the atmospheric ( p o )value:

P = Pt

- Po.

(1.1-1)

This parameter becomes negative where and when the total pressure is lower than that caused by the atmosphere a t the time and place of the


15

investigation. The change in the sign of the excess pressure is naturally a gradual quantitative modification, and no qualitative difference occurs a t the boundary. Hence, the most important common characteristic of any type of soil moisture (and of the moisture content of rocks) is the negative pressure governing the behaviour and the movement of water in this zone. (A more detailed explanation of the various subzones of the soil moisture zone, and the description of the forces acting there, will be given in Chapter 1.3.) Having established a definition of “water table”, i t is relatively eaay to define the term most commonly used in this book. Qround water is that part of the subsurface water which is stored (or moves) in the interstices of the crust below the water table. Hence its total pressure is higher than the atmospheric one. This pressure may be hydrostatic, or influenced by the movement of water (hydraulic pressure), but its excess value is always positive. In the ground-water zone, all the pores and other types of interstices are completely filled with water (or with other fluids, and in exceptional cases, with gases under pressure). Considering the connection of ground water with the hydrological and meteorological processes on and above the surface, two groups may be distinguished: shallow and deep ground water. Shallow ground water is stored near the surface, below the water table and above the first, largely extended, continuous, impervious formation. It is directly influenced by meteorological and hydrological events (recharged by precipitation, drained by evaporation and transpiration, having interactions with surface waters). Deep ground water can be found in aquifers, lying below continuous, impervious beds, which hinder the direct contact between surface and ground water over a very large area, as well aa the influence of atmospheric prosesses on ground water. It hm no direct recharge from precipitation or surface waters, and is drained only through the shallow ground water. It is very difficult to make clear distinctions between the two types of ground water. Shallow ground water may be partly covered by impervious strata, but if the covering layers are not continuous or large enough, the effects of the meteorological and hydrological events can be clearly observed in the regime of the ground water. On the other hand, i t is also possible, that the aquifer covered by impervious formations over a large area may occur near the surface far from the place of investigation. Thus, the same groundwater body may be regarded aa deep within the investigated area, and shallow elsewhere. It is not possible therefore, to draw a sharp line between the two types. There are several transition forms, and the distinction can be baaed only on the investigation of the dominating factors influencing the ground-water regime. The means of this investigation are the observed ground-water data. Considering the rough sketch of the vertical section of a sedimentary basin given in Fig. 1.1-1, deep ground water may be divided into several subgroups: e.g. closed ground water, when the aquifer is completely surrounded by impervious layers; artesian water, which hm a recharge area where the water table is above the artesian area, and thus, the water level in wells tapping the artesian layer rises above the surface. The adjective

16


Fig. 1.1-1. Rough vertical section of a sedimentary basin for representing the Various types of aquifers

confined (or unconfined) also indicates whether the water-bearing layer is covered by impervious formations or not. This term does not however, express the extension of the cover. Hence, it is suillcient for characterizing the behaviour of water movement in the aquifer, but not for that of its hydrological contact. It can well be used therefore, in seepage hydrtGulics without providing su5cient information for hydrogeologists. It can be seen from the foregoing that such expressions aa confined, closed or artesian aquifer have special meanings, and they do not completely characterize the hydrological behaviour of the water-bearing layers. It is advisable, therefore, to use the adjectives of shallow and deep for distinguishing ground water directly influenced by meteorological and hydrological phenomenon from that recharged or drained only through another part of the ground-water space. Several expressions have already been used to characterize the layers containing water, or hindering its movement: e.g. aquifer or water-bearing layer, impervious formation etc. Four terms are generally used t o describe the behaviour of strata with respect to the flow of water. These are as follows:


17

(a) Aquifers - permeable geological formations having interconnected interstices, which permit an appreciable quantity of water t o move through them under ordinary field conditions. Ground-water reservoir, waterbearing or permeable layer are commonly used synonyms. impermeable strata, which may contain a great quan(b) Aquicludes tity of water (in some cmes more than aquifers, e.g. clay). Some aquicludes do not transmit water at all, and others only a very small quantity. (c) Aquifuges - also impermeable formations but neither contain nor transmit water. These rocks have no interconnected pores or fissures and cannot therefore, either absorb water or allow it to pass through. Aquicludes and aquifuges are the two opposite types of impermeable geological formations. (d) Aquitard - a transition form between aquifers and aquicludes. This type of layer contains interconnected pores, but the water-conveying capacity of the channels made up of pores is relatively small compared to that in aquifers. However, if the direction of flow is perpendicular to the large bordering surface of such a layer, the amount of water conveyed by seepage is not negligible, because of the great extent of the area. These strata are often called semi-pervious or leaky formations. The hydrogeological terms listedin the previous paragraphs are not entirely sufficient for clwsifying the layer hydrologically, or from the point of view of seepage hydraulics. They indicate only the existence, or the lack of interconnected pores, and thus, the possibility of water movement, but do not describe the structure of the network composed of the interstices, which is one of the most important factors influencing the character of flow. The latter information is basically required t o classify the layers from the aspects of hydrology and hydraulics. Considering the origin of the various rocks, the geological classification distinguishes the following types:

-

Sediments - of mechanical origin: loose clastic sediments or cemented sediments; - of chemical origin; - of biological origin; Igneous rocks - effusive ( h e grained) volcanic rocks; - intermediate (porphyriiic) rocks; - intrusive (coarse grained) plutonic rocks; Metamorphic rocks. Because the water-conveying network in some of these formations (and hence, the character of the movement in or through the layers) are very similar, several geological groups may be combined when deciding the h a 1 form of the hydrological classification. The most important difference is between the network of pores in loose clastic sediments, and that of the fissures and fractures of solid rocks. Hence, these two main groups were selected as the basis of the hydrological classification. Within both groups, two further subgroups were distinguished: the non-cohesive loose clastic sediments are pervious formations, and the cohesive ones may be regarded 2

18

1 Fundament& for the investigation of seepage

as impervious, or semi-pervious layers, although there is a continuous transition between the two subgroups without any sharp borderline. In the group of solid rocks, it is reasonable to deal with the karstic formations separately. In these, the water-conveying openings are generally enlarged by chemical solution and mechanical erosion cawing high permeability.The other remaining subgroup is composed of non-karstic solid rocks. It can easily be seen that the structure of the basic classification summarized in Table 1.1-1 is also suitable to determine the relationship between

Fig.

absorbed fossil ,juvenile water water water _.Grouping c vsriouS types of mbsurface water according to their O L ~ J in he system of h~dmlogicalc l d o a t i q n

the hydrological c l d c a t i o n of subsurface water, and some other types of grouping baaed on different parameters of the water, or those of the layers containing it. The origin of water can be used as an example to show this (Fig. 1.1-2). It is well known that generally three types of subsurface water can be distinguished amording to their origin: internal water (juvenile water) is derived from the interior of the earth as anew resource; external water originates from atmospheric or surface water and may be trapped in rocks when the constituent material was deposited (fossil or connate water), or i t may be absorbed into the interstices of the layers some time after deposition, even quite recently (absorbed water) (UNESCO,1972). The amount of juvenile water is negligible from the point of view of water resources development. Coming from very great depths, its occurence in deep rather than shallow ground water is more probable. Because the upward movement of juvenile water generdy follows faults, larger amounts may be expected in solid rocks, where it sometimes rises into shallow positions. I n most cases, fossil water preserves its original characteristics (e.g. chemical composition) derived from the water-bearing layers in which it formed or into which it idltrated during their development. Changes may occur aa a result of water exchange between aquifers, for instanoe by migration and diffusion. For this reason, a higher ratio of fossil water may be expected


19

in layers where the movement of water is considerably hindered. Thus, the highest amount is probable in cohesive, loose, clastic sediments. Adsorbed water indicates the recent influence of meteorological and hydrological processes. It is quite evident that soil moisture is entirely, and shallow ground water almost completely composed of this type of water. Similar to the previous example, the interest of the various water oriented sciences in the multidisciplinary field of subsurface water can also be clearly represented in the proposed system of classification (Fig. 1.1-3).

nydrohydrology hydropedology mechanics geologi Fig. 1.1-3. The grouping of various types of subsurface water according to the sciences involved in the system of hydrological claasifkation

Soil moisture and part of shallow ground water ensuring the recharge of the former are the most important sources of water supply to plants, and they take part in the development of the various types of soils. Thus, the investigation of this part of subsurface water is an important field of soil science: i.e. hydropedology. The most important taak of hydrology is to explore the interrelationships between the various forms of water on the globe and to analyse its continUOUE movement along the hydrological cycle. When investigating subsurface water from the hydrological point of view, its contact with atmospheric and surface water aa well as the flow of water along the subsurface branch of the cycle have to be studied. The field of interest of hydrology is concentrated, therefore, mostly on soil moisture and shallow ground water, these having direct contact with hydrological and meteorological processes on and above the surface. This interest also extends partly t o deep ground water, which may join the hydrological cycle in the form of water exchange between deep and more shallow layers. The pores and other interstices of strata developed in geological ages are m e d with water. The presence, pressure, chemical composition and movement of water also influence the behaviour of the layers. Thus, the water itself is also part of the crust, and a topic of geology, a science which investigates the structure and composition of the crust. That part of geology which investigates the extension of water-bearing layers, their parameters in con2*

20


nection with the storage and movement of water, and the characteristics of ground-water is called hydrogeology. All three sciences mentioned in the previous paragraphs require the investigation of the movement of various types of subsurface water (infiltration; flow of water and air through unsaturated pores; confined or unconfined seepage of both shallow and deep ground water, etc.). The analysis of movement is part of physics, and, within this large field, that of mechanics. When dealing with the flow of fluids it is called hydromechanics. This branch of science - together with hydraulics which is a form of hydromechanics simplified and applied in engineering practice - supply important supplementary information for soil science and geology, as well as hydrology, and covers the entire domain of subsurface water.

1.1.2 Characterization of flow through various aquifers

As explained in the previous section, hydromechanics and hydraulics include the investigation of movement of all types of subsurface water. When further distictions have to be made according to the different forms of flow, it is necessary to analyse the parameters characterizing the movement of the previously selected groups of subsurface water. It has also been mentioned that the character of the flow is basically determined by the structure of the water-conveying network composed of the interconnected interstices of the layers. Apart from its structure, the instantaneous conditions of the network at the time of the movement (e.g. the rate of saturation, or the prevailing pressure condition) also provide important aspects for the classification of flow. The greater part of subsurface water (at least that water available for practical utilization, and economical exploitation) is stored in loose clastic sediments. The flow of ground water is, therefore, closely related t o the physical character of these aquifers. The pores between grains form randomly interconnected, complicated pipe- or channel-systems in such a layer. The resistance of these continuous but very tortuous channels has t o be overcome when water flows through the layer. The network of the waterconducting channels is generally equally distributed in these sediments, the contact between the pores being ensured in each direction with equal probability. Hence, in most cases these aquifers can be regarded as the homogeneous flow space of a porous medium limited by the neighbouring impervious formations. In connection with the previous statement, it can easily be seen from the description of the character of the channel network composed of the pores between grains, that the loose clmtic sediments satisfy the requirements set up in the definition of porous media by various authors (Bear et al., 1968; Bear, 1972), mostly in the form of relative terms. The solid matrix surrounds the pores, the significant majority of which are interconnected. The pores are filled with liquid and/or gases, and the pore size is relatively small compared to the whole aquifer. The specific surface (the internal surface dividing the different phases of the heterogeneous medium

1 . 1 General characterization of subsurface water

21

related to its total volume) is relatively high, and the phases of which the medium is composed, are present in each representative elementary volume. [The detailed interpretation of the representative elementary volume is based on the continuum approach of porous media (Bear, 1972). This is defined as the limit of the size of an elementary cube or sphere around an arbitrarily chosen point of the domain, above which the numerical parameters describing the structure of the multiphase medium (e.g. ratio of pores to the total volume, or specific surface) do not change considerably as a function of the size of the investigated body, but only have random fluctuation. Below this limit, the same parameters determined for a volume of the same shape may change in magnitude depending on the size in question (see Fig. 1.1-8).] The constancy of characteristic values determined for bodies larger than the representative elementary volume makes the elementary volume values characteristic of the whole aquifer. This reasoning, which is generally valid for loose clastic sediments, is the basis of the practice, which regards the aquifers as unified flow spaces. It has also been mentioned that in most cases, the pores are equally distributed in these formations, and the development of channels composed of the interconnected pores may be observed in all directions with equal probability. This is the basis of the theory which states that loose clastic sediments usually form homogeneous flow domains. Solid rocks having equally distributed pores, or a dense network of fine fissures, form unified flow domains similar to that of loose clastic sediments (sandstone, fractured dolomite). This is because the structure of the channels built up by these pores and fissures never or rarely differs from the network composed of the interconnected pores between individual grains. However, some differences may be caused by the different size of the pores (e.g. in sandstones the cementation decreases the size of the pores), or in fissured rocks the closed and open fissures may be oriented according to the main directions of geomechanical forces. Inhomogeneity may also result from porous solid rocks (sandstone, basaltic lava flows) because the permeability of the fractured zones is generally higher, than that of the whole mass of the formation. The development of fractures causes not only inhomogeneity of the flow domain, but also changes some of the characteristics of the formation. If the characteristics of the solid matrix are the function of the direction of flow, the domain is called anisotropic. Orientation of the interstices is more dominant in beds where water movement develops along the contact planes of thin layers. In this way, the flow in laminated rocks (e.g. slate, marl) may become two-dimensional, even if the movement extends to the total volume of the layer. Sometimes, the flow in the space of the aquifer may be further simplified and investigated as a one-dimensional movement in solid rocks, where flow can develop only along large openings or dissolved channels (e.g. karstic limestone), when the channels are not interconnected. Naturally these types of movement may be present simultaneously as well. For instance in limestone in addition to one-dimensional flow, or in sandstone in addition to the three-dimensional movement, there may develop two-dimensional flow along fractured zones. The latter may even become dominant. As another example, loose clastic

22


sediments can be mentioned, which may have anisotropic properties caused by varying conditions during the process of their deposition. In the previous paragraphs the structure of the flow domain, or that of the water conveying network was described only qualitatively. Even when describing porous media, relative terms were used and not numerical parameters. Naturally, quantitative values are also applied t o characterize the various aquifers, e.g. specific surface which has already been mentioned, or porosity, which is the most commonly used parameter. Porosity (or volumetric porosity) is defined a,s the ratio of the volume of void space to the bulk volume of the porous medium. When determining this parameter, one of the basic requirements is, that the investigated sample should be larger than the representative elementary volume (Bear, 1972). Another important aspect is t o distinguish the pores acting as part of the interconnected water-conveying network from those not taking part in water transport. The pores may be inactive, either because they are completely separated from the interconnected channels, or the same pores may form dead ends of the network, or the opening may be so small that the fluids contained in them are rendered essentially static by the close proximity of the force fields at the surface of the solid matrix (De Wiest et al., 1969). Taking into account the difference between the numerical values of the pore volume used as the basis of the calculation of porosity, different terms may be distinguished such as: (a) Total porosity, when the volume of all pores are related to that of the sample; (b) E8ective porosity considering only the pores available for the transmission of fluid in the porous medium; (c) Isolccted porosity due to separated and inactive interstices. The detailed analysis of porosity in loose clastic sediments will be given in Chapter 1.2, and some special aspects of determining porosity in solid rocks (e.g. interpretation of primary and secondary porosity; statistical evaluation of porosity in karstic formations) will be discussed in Part 3. It is also necessary to note that there are other interpretations of porosity which differ from that given previously as the ratio of pore volume to total volume, and which is called volumetric porosity. The other possible msthods of determining porosity are the calculation of linear and areal values (Fig. 1.1-4).

Analysing the porosity of a sample along a line, the lengths of the pores are related to the total length of the sample. The calculated ratio is the parameter called linear porosity: k

n , =-1-1

Ax

(1.1-2) *

Similarly, the sum of the arew of the pores in a given section has to be related t o the total area of ths cross section t o obtain areal porosity:


23

(1.1-3)

Special relationships exist between the various parameters of porosity, on the basis of which they can be substituted by each other. The average areal porosity can be calculated aa the mean value of the same parameters

------Fig. 1.1-4. Interpretation of linear, areal, and volumetric porosities

determined for several parallel sections. If the distances between the sections are not equal t o each other, the areal porosities measured in various sections have to be adjusted according to the length of the sample belonging to the section in question. Assuming that the element distance is small ( d z ) and the change in areal porosity in the direction of the z axis is given by a continuous function m(z) the calculation of the average value can be expressed in the form of an integral:

where S is the total area of the cross section (S = Ax Ay in the figure) and the s h ( 2 ) fundion expresses the sum of the area of pores in the sections depending on its position [thus, sh(z)= Sm(z)].It is evident, therefore, that the length of the sample multiplied by the area of the cross section is equal to the volume of the sample ( V ) ,in prismatic or cylindrical samples. The integration of the s&)function dong the length of the sample similarly gives the total pore volume of the sample (V,,).Because the ratio of these

24


two volumes is equal to the volumetric porosity, i t can be stated, that the average areal porosity is equal to the volumetric parameter in the case of randomly distributed pores [the latter hypothesis originates from the condition where the m(z) function is continuous]. The equality of the average linear porosity and areal porosity can similarly be proved. Hence, it can be

Y

Fig. 1. 1-5. Ih e a r , areal and volumetric porosities of a structure having interstices composed of equidistant spaces between cubes

assumed, that the three different interpretations of porosity calculated for samples larger than the representative elementary volume, pro\-ide the same numerical value. A condition of the previous derivation was the random distribution of pores. In the case of a regular structure of fissures [when the m(z) function is not continuous], 8ome complications are caused by the difference between total and effective porosity. The orthogonal system of equidistant spaces between cubes can be used as an example (Fig. 1.1-5) for investigating the problems arising. Two different values of linear porosity can be calculated parallel to one of the main axes of the structure. Depending on the position of the line in question, it may cross the cubes or run inside the spaces: (1.1-5)

Similarly, two parameters characterize areal porosity as well:

25


while total volumetric porosity is determined by the following equation:

(a + b)3- a3 3ab2+ 3a2 b + b3 (a b)3 (a b)3.

n=

+

+

(1.1-7)

Comparing the parameters, which are not equal to unity, an apparent contradiction occurs because these values are not equal, although equality would be expected for randomly distributed pores. At the same time, it is evident that the areal porosity expressed by Eq. (1.1-6) assumes a flow, perpendicular to one of the main planes of the structure lying along two axes of the coordinate system. In this case, however, a considerable portion of the pores (those spaces perpendicular to the direction of flow) do not transmit water, because there is no pressure difference within them. After decreasing the pore volume by the volume of the inactive spaces, effective porosity can be calculated [(a b ) 2 - u'](u b ) 2ab + b2 neff= = n;. (1.1-8) ( a bI3 (a"b)2

+

~

+

+

It can be proved in this way, that the equality of areal and volumetric porosity is valid also in the case of a regular network of fissures, but effective porosity has to be determined for such systems, considering only interstices taking part in water transport. After describing the structure of water conveying networks, the acting forces have also to be analysed for the complete characterization of flow. According to Newton, movement can only develop as a result of a driving force or forces, and, if there is movement in the system, forces are also acting as a reaction to the accelerating forces to stop, or slow down the motion. Steady movement can only develop, if the entire system of forces (both those accelerating and retarding the motion) is in a balanced condition. The most important accelerating force in the case of ground-water flow is gravity. Apart from this force, the pressure of the upper layers may be taken into account. These cause compression and force water to flow out of the pores due to their reduced volume. Sometimes there may be a further accelerating force as well: the pressure of vapour and gases enclosed in the layers. These are generally only significant at greater depth. Their effects become important, therefore, in mining of hydrocarbons, and also in the case of water exploitation from deep aquifers, and should be taken into consideration, when determining hydrodynamic relationships related to deep wells. There is a special case due to gravity caused by the Werent specific weights of water stored in various layers. The specific weights of water can differ from each other, because of differences in the amount of dissolved salts and/or temperature. These effects may be important, when investigating water exchange between deep and shallow aquifers and when studying sea-water intrusion. Soil moisture adheres to the wall of the solid matrix and is kept in an elevated pobition above the water table against gravity by adhebive forces.

26


The dynamics of this type of water differs from the water movement in the saturated zone, because the adhesive forces created by the interaction of the solid particles and water molecules are also accelerating forces apart from gravity. On the basis of the hydrodynamic aspects described above, the movement of subsurface waters can be divided into the following main groups:

( A ) Flow through saturated porous media, wherein the network of the pores is equally distributed and connected at random This and the following groups can be subdivided according to the main accelerating forces: (a) Gravity which is the solely dominant accelerating force; (b) The pressure of overlying layers, the difference i n the specific weights of water, or the pressure of vapour and gases also have important roles apart from gravity. Further clrtssification within this group considers the three possible retarding forces, and their combined effects related to one another. These forces are: inertia, friction and the adhesive forces between solid grains and water molecules. This type of subdivision can of course be applied in both previously mentioned subgroups baaed on the weight of the accelerating forces. Thus, the following secondary subgroups can be distinguished:

( i ) All other retarding forces can be neglected except inertia, the flow being turbulent ; ( i i ) There are two retarding forces to be taken into consideration, inertia and friction. The condition of flow can be regarded m a transition from the turbulent to the laminar zone; ( i i i ) The solely dominant retarding force is friction; the flow is laminar (the greater part of ground-water movement pertains t o this group, which is the so-called Darcy’s flow condition); ( i v )Adhesive forces createdby tension, and acting between the solid walls of pores and water molecules, also have an important role in the system as retarding forces apart from friction. This type of flow is called micro-seepage. Ground-water movement in loose clmtic sediments can mainly be characterized by the types of flow through the saturated porous medium previously described. There are, however, some solid rocks aa well (sandstone, fractured dolomite, basaltic lava, etc.), the water movement in which can be considered in this main group. The importance of the latter is, however, s m d e r , than that in sedimentary baains filled with loose claetic sediments, which are generally the major possibilities for the large-scale use of ground-water resources. For this remon, the determination of the hydrodynamic parameters of flow through porous media is baaed on the investigation of samples built up from grains and saturated pores.

( B )Flow through saturated, fracturated and fissured r o c b In solid rocks, water movement can develop mostly along the contact planes of layers, fractured zones and fbsures, or through dissolved openings


27

and channels. The flow is usually turbulent, or belongs to the transition zone, because the size of the water conducting channels is generally larger, than that of the pores between the grains. The interstices are not equally distributed through the whole volume of the layer, but are concentrated in fractured zones. Their interconnections do not occur at random, but are directed in most cases by the structural planes of the strata. A classification based on the accelerating forces, is unreasonable for this group. There are only very rare cases, when the second most important accelerating force, the pressure of overlying layers, may have any role in addition t o gravity. Adhesion can be neglected in almost every cme, because the ratio of the internal specific surface, or the surface of the solid matrix related to the volume of water contained in the pores, is considerably smaller than that in loose cla.stic sediments. There are many cams, when inertia has a dominant role in the relatively largeopenings. Hence, Darcy’s equation can hardly be applied, or only in a modified form. I n other cases, where the interstices are narrow, the supposition of laminar flow is acceptable, but the non-evenly, or non-randomly distributed pores make the application of the hypothesis of united flow domain more difficult. There is another aspect, according to which some distinction between the various types can be made. This is the character of the water conducting channels or structures: (a) One-dimensional flow in openings, channels, and conduits connected in only a few places, the investigation of which is similar to the hydraulics of water movement in networks of pipes. This type of movement is characteristic of karstic formations, most important of all in limestones. (b) Two-dimensional flow along the contact planes of layers, structural planes, and in fractured zones. This type of movement can develop both in karstic and non-karstic formations. The most important types of rocks in the latter group are mark and slates. (c) Flow through the interstices of solid rocks where no definite relationship exists between the material of the layer and the structure, or character of the water conducting network.

( C ) Flow through unsaturated porous layers

M fractured rocks Among the accelerating forces creating and maintaining this type of water movement, two have a dominant role: gravity and the tension diflerence between two points on the surface of water films created by adhesive forces. The size of the effective area of the flow cross section is essentially influenced by the amount of air entrapped in the sample. Some subgroups within this type of flow can be distinguished according to, whether the air bubbles in the flow space are in contact with the atmosphere, or not. In the former case, it is advisable to distinguish between porous sediments and fractured rocks. As a result, the following classification can be given:

(a) In the case of flow through unsaturated porous sediments above the water table, the air included in the layer is in direct contact with the atmosphere, its pressure is determined by the atmospheric pressure and it may also be

28


influenced by the air-permeability of the formation (Morel-Seytoux, 1973; Morel-Seytoux and Khanji, 1974).The surface of the solid grains is relatively large, compared to the volume of the moving water, and the influence of adhesive forces as a retarding effect is, therefore, considerable. The decrease in these two parameters (e.g. in coarse-grained gravel) forms a transition to the second subgroup. (b) In fractured zones or conduits of solid rocks above the water table, contact with the atmosphere is similar t o that mentioned in the previous case, but the flow is less affected by adhesion. Infiltration is relatively rapid, storage capacity is small, and the evaporation of ground water through this unsaturated zone can be neglected. As a special form of this type of water movement, the open channel flow in karstic conduits can he mentioned, the hydraulic description of which is similar t o that of water courses on the surface. (c) Unsaturated layers can develop at great depths as well, where the gaseous phase originates from vapours and dissolved gases in the water, when the temperature is raised considerably. The pressure of this phase is determined by the local conditions prevailing at depth.

1.1.3 The definition of seepage

On the basis of the clausification of subsurface water and that of the hydrodynamical aspects explained in connection with the analysis of groundwater flow, those types of subsurface water whose detailed investigation is the objective of seepage hydraulics can easily be selected. It wm clearly demonstrated that the scope of hydrodynamics may include the investigation of the movement of any kind of ground water or soil moisture. Sometimes the whole section of hydraulics dealing with subsurface flow is called seepage hydraulics. This use of the term seepage is, however, not quite correct, because some examples of this water movement (e.g. turbulent flow in large karstic conduits either under pressure or with a free water surface) are far from the idea generally connected with the word: seepage. The best way to understand the clear concepts of seepage is to go back into the past and analyse the results of the original experiments by Darcy, who established the scientific study of seepage hydraulics. He investigated the resistance to flow of sand filters, and on the basis of the empirical evaluation of the measurements, in his famous book (Darcy, 1856) he evolved a relationship between the hydraulic parameters of flow through a sandy layer, thought to be generally valid for the characterization of water movement in porous media. The sketch of equipment used for repeating Darcy’s baaic experiments is shown in Fig. 1.1-6. A tube of constant cross-sectional area A (cylindrical or prizmatic) is flled with the porous medium. The sample is bordered at both ends by planes, perpendicular to the axis of the tube, and its length (the distance between the two borders) is L. Flow through the sample is created by a constant difference ( A H ) between the pressure heads of headand tail-water, maintained by recharging the upper space of the system,

29


reference level Fig. 1.1-6. Sketch of equipment used for repeating Darcy's experiments

and draining the lower one, ensuring a constant water level at both places. The sample is in a horizontal position in the figure, although the original experiments were executed with vertically standing samples, and the literature also describes equipment generally using vertical samples. This difference, however, does not cause any discrepancy, because the potentials at any point of 110th ends of the sample are calculated as the sum of potential- and pressure-energy ( z y p ) , or, when expressed as the height of an equivalent water column, that of the height of the point above an arbitrarily chosen reference level, and the height of the water level above the point (z h ) . The reistance against the flow before and behind the sample is negligible because the cross section of the tube is relatively very large. The pressure is proportional to the difference in level, and hence independent of the po,4tions of the points of meaurement, depending on the vertical difference between the head- and tail-water levels only. The constant pressure head difference creates steady flow, the discharge (&) of which can be determined as the amount of water flowing through the system. This is measured in unit time at the spill way of the tail water. Darcy found, that the specific discharge (the ratio of the measured discharge to the total area of a section of the sample perpendicular to the direction of flow which is equal to the cross section of the tube containing the sample) is linearly proportional to the difference between the pressure heads and inversely proportional to the length of the sample:

+

+

(1.1-9)

The dimeiision of the specific discharge is equal to that of velocity [LT-l], and, therefore, Darcy named this parameter as seepage velocity. This is,

30


however, an apparent value, because a considerable part of the cross section of the tube is occupied by the solid matrix, and the area which is free for conveying water, is only a portion of the total area ( Z a ) .The ratio of free and total surfaces is equal t o areal porosity ( n l ) , which, considering Eq. ( 1 . 1 4 ) , can be substituted by effective porosity ( n ) .The actual mean velocity of water in the pores can be calculated as the quotient of the discharge and the sum of the free areas of the pores. This parameter is called eflective velocity and is proportional to Darcy’s seepage velocity, being l / n , the factor of proportionality: & & 1 V,ff = -= --- - v ; because Za = n A . (1.1-10) Za n A n

(To avoid misinterpretation of seepage velocity and effective mean velocity, the former is often called flux or specific flux in the literature, and is indicated by q instead of v.) Both terms in the quotient on the right handsideof Eq. (1.1-9) (difference between pressure heads and the length of the sample) have a dimension of [L], and hence their ratio I is dimensionless. This parameter, the hydraulic gradient expresses in an equivalent water column the energy loss necessary t o overcome the resistance against the flow along a unit length of the porous medium. In the case of steady flow, it is constant in time, and independent of the position of the investigated section, if the flow domain is homogeneous, and the area of the cross section does not change. Since the publication of Darcy’s book, Eq. (1.1-9) became the basic law of seepage hydraulics, often referred to in the literature as Darcy’s law. It states that seepage velocity is linearly proportional to the hydraulic gradient, and the factor of proportionality ( K D , hydraulic conductivity) is independent of either velocity or gradient, being a material constant which summarizes all the parameters of both solid matrix and moving,fluid, influencing the resistance against seepage. Hydraulic conductivity has a dimentjion of velocity [LT-l], and according to the dehition, i t is equal to seepage velocity created by a hydraulic gradient of unity in a given porous medium and a given fluid. Research workers, having further developed Darcy’s basic ideas, determined the dependence of conductivity on the parameters of the transported fluid. They found that hydraulic conductivity is proportional to Che ratio of specific weight ( y) and dynamic viscosity ( p ) of the fluid, or t o acceleration due to gravity (9) divided by the kinematic viscosity ( v ) of the fluid, which is equivalent to this quotient. The remaining part of the parameter i.e. E , the factor of proportionality in the mentioned relationship, depends only on the properties of the solid matrix of the porous medium, and is called intrinsic permeability, matrix permeability, rock permeability or sometimes only permeability. The dimension of k is consequently [L2]:

K D -- k - =Y k - . 9 P

(1.1-11)

V

Since the establishment of Darcy’s law, many research workers have tried to prove its accuracy and validity on the basis of their own measurements


31

and theoretical studies (Hazen, 1893, 1895; Carman, 1956; Koieny, 1953; Zamarh, 1928; Zauberei, 1932; Zunker, 1930). Others - on the basis of experimental data - have denied the accuracy and applicability of this equation, and proposed other formulae instead of Darcy’s law (Forchheimer, 1886,1924; Lindquist, 1933; Lovaas, 1954). On the baais of the dynamic analysis of the movement of fluids through porous medium, recent investigations have proved that there is a large zone of flow conditions (i.e. laminar seepage), where Darcy’s equation is acceptable aa a good approximation As a further result of these investigations, (Kovhcs, 1969s; 1969b; 1969~). it can be stated that the formal application of Eq.(1.1-9) can be extended to non-laminar flow conditions aa well, if Darcy’s hydraulic conductivity is multiplied by a function depending on either seepage velocity or hydraulic gradient, and the non-linear proportionality between the two variables can be followed in this way: v =KI; where

R = R D @ I ( v ) = k -@9I

( v );

V

or

R

= KD@2(I)= k -9a 2 ( I ) .

(1.l-12)

V

Consequently, the generalized hydraulic conductivity is composed of three factors: (a) Intrinsic permeability, including all the parameters of the solid matrix influencing the development of seepage; (b) Properties of the flowing fluid, expressed a,a the ratio of acceleration due to gravity and kinematic viscosity of the fluid in question; (c) Function depending on either seepage velocity or hydraulic gradient for characterizing the mtml flow condition. A further expansion of the concept of hydraulic conductivity is its application t o the description of flow through unsaturated porous media (Averjanov, 1949a, 1949b; Bear et al., 1968; Bear, 1972; Irmay, 1954; Kovhcs, 1971a, 1971b). It is a generally accepted assumption that the specific flux through an unsaturated sample is smaller but proportional to that flowing through the same porous matrix in a saturated condition, assuming that hydraulic gradients are equal in both systems. It follows from this relationship that the apparent hydraulic conductivity of an unsaturated sample, generally called unsaturated conductivity can be determined by multiplying Darcy’s hydraulic conductivity, or more precisely, the generalized hydraulic conductivity, by a factor depending on the rate of saturation of the sample. Unsaturated conductivity includes not only the resistance of the solid matrix, but also the effect of the decrease in area available for the transport of fluid, when part of the pores is occupied by air. In the equation

32


below the rate of saturation s is the quotient of volumetric water content W and effective porosity n:

K,

9

= K f ( s ) = K D @ z ( I ) f ( s ) = k -@ 2 ( 1 )

f(8).

(1.1.13)

V

It is necessary to note here that in the caae of unsaturated flow, apart from friction, the molecular forces become dominant as retarding forces. I n this zone, the modification of Darcy’s parameter can be better expressed aa the function of hydraulic gradient, than that of seepage velocity. For this reason, the second form of the generalized hydraulic conductivity waa used in Eq. (1.1-13). In connection with the validity of Darcy’s law in most cases only the interpretation of hydraulic conductivity and the applicability of the linear relationship is queried in the literature. As summarized in the previous paragraphs and aa will be explained in detail in Part 2 of the book, the solution of these problems can eaaily be found in the form of the dynamic interpretation of the generalized hydraulic conductivity, and the relationships between the various parameters can be determined. In Part 2, the practical application of certain factors will be discussed, together with their numerical determination from known physical parameters of soil. There is, however, a further hypothesis in the derivation of Eq. (1.1-9) which has to be considered when applying either Darcy’s relationship, or any other type of seepage laws. This supposition is included in Eq. (1.1-9) in the form of Darcy’s seepage velocity, although this fact is not explicitly explained either in Darcy’s publication or by other research workers having followed his experiments. The exact wording of the condition is aa follows: any type of seepage law can be applied only if the water-conveying openings in the cross sections, perpendicular to the main direction of flow, are randomly distributed, and small, compared to the entire area of the crosa section. If this condition is fulflled, the effective mean velocity in the pores can be substituted by seepage velocity (specific flux) calculated for any subarea of the cross section. Here a further condition arises: the subarea used as a unit should be larger in magnitude, than the representative elementary unit. In thitj caae, the ratio of seepage and effective velocities can be regarded aa equal to areal porosity, and the porous medium, bordered by pervious or impervious boundaries having geometrically fixed positions, can be dealt with aa a continuous flow domain. The basic physical concept of this special approach t o the investigation of seepage summarized in the previous paragraph is explained in various ways in recently published manuals dealing generally with hydromechanics (NBmeth, 1963), or especially with the hydrodynamics of flow through porous media (De Wiest, 1969). One of the latest and most comprehensive explanations is that given by Bear and discussed as a continuum approach in his book (Bear, 1972). For this reason, this main line of thought is pursued in the following paragraphs. To achieve the objective, of the characterization of the porous aquifer as a continuous flow field, the double application of statistical averaging has


33

to be performed. The flowing water has to be described first of all as a continuum, instead of a mass of numerous, separately moving molecules. The molecular structure of water is substituted by water particles of a conveniently chosen size, ensuring that the material behaviour of the maas and the average parameters of movement can be regarded as homogeneous within the particles. Using these particles as parts of the water continuum, their dynamic and kinematic analysis provides the investigators with all the necessary information and means to solve the movement equations, and to determine in this way the parameters of flow if the boundary conditions of the flow domain are known. I n a porous matrix however, the actual flow field is composed of the very complicated and ramifying channels formed by the interstices. It is almost impossible to apply to seepage hydraulics the methods of solution, generally accepted in hydrodynamics and easily applicable to relatively simple forms of flow domains, because the consideration of the boundary conditions in the microscopic network of channels results in a large group of complex movement equations which cannot be dealt with mathematically. A second statistical averaging enables investigators to neglect the microscopic flow pattern of the actual water movement, and describe the entire domain of the porous medium as a unified and continuous flow space, the behaviour of which is influenced by both the solid matrix and the transported fluid. On this macroscopic scale the simplified parameters of flow can be easily determined by applying the basic movement equations for the new continuum, substituting the statistically averaged macroscopic properties of the domain, and considering the boundary conditions of this complex system. Thus any investigation of seepage requires the substitution of the molecular structure of the water with the Continuum of water particles, and the microswpic flow pattern followed by these particles with the macroscopic continuum of the porous medium. The application of the continuum approach is based on the theory of random variables, where the expected value of a quantity scattered randomly around a given value can be characterized by the average of the observed data, if the number of observations is large enough for statistical evaluation. As an initial example, the investigated random variable may be the velocity vector of the water molecules. It is well known that they have random movement (Brownian movement), even if the mass of water is in a static state. The resultant average of the velocity vectors scatters around zero in this case, and the variance can be lowered below a given limit, assuming, that the number of molecules included in the investigation is large enough. Ahove this limit the variance is practically constant, but below it the scattering gradually increases with the decreasing number of molecules, and achieves its highest value, equal to the fluctuation of the velocity of one molecule, if a single molecule is considered. Because the velocity of the Brownian movement depends on temperature, it is evident, that the smallest possible number of molecules, or the smdlest volume of water including this number of molecules - which is called the representative elementary unit - will change with temperature, and also with any other variables influencing the Brownian movement. Therefore, the size of the water par-

34


ticles has t o be determined, so, that they are hrger than the largest representative elementary unit occurring under the conditions of the investigation. The expected value of the resultant of Brownian velocities determined for such particles is zero, Therefore, the velocity vector of a particle characterizes the actual flow in the continuum. There is an upper limit to the particle size as well. The purpose of the concept of continuum approach is to describe some behaviour of the flow domain: e.g. the movement in the present caBe. The parameter to be determined may continuously change from point to point and i t may be represented by a series of discrete data calculated for each particle. Thus, the approximation of the continuum would be very rough, when applied to a particle of large size. The average value of a parameter determined for the particles still haa a random variance, compared to the expected value. It is necessary, therefore, to evduate statistically the number of particles for a reliable calculation of the expected value. Hence, a set of some tens of particles are regarded a~ a unit. These units can then characterize the behaviour of the flow space, a condition which requires the particle size to be smaller in magnitude, than the investigated domain. The transformation of a set of molecules into homogeneous particles ia the basis of the interpretation of most of the physical parameters of water e.g. the mass or density can be described by the number of molecules within a representative elementary volume. The latter is also a random variable, which gives an approximate value only, when a particle larger than a given limit is analyzed. The ratio of the jointed dipole molecules and the freely moving ones within a particle also may be statistically evaluated as a random variable influenced by Brownian movement. This property is viscosity. As already explained, the use of a regular system of discrete continuous functions for describing the properties and behaviour of the domain, instead of individual molecules randomly filling in the space, makes possible the application of hydrodynamic movement equations. These equations can be solved however, only in the case of relatively simple boundary conditions, and it is hardly regarded a~ a likely method to determine the parameters of flow through the microscopic channels of porous media, even if the capacity of today’s computers could be considerably enlarged. The microscopic view should be changed to the macroscopic characterization of the flow domain, following the same general concepts a~ previously used. It is necessary to note that the parameters calculated in this way are only statistical averages for a unit area or volume (larger than the representative elementary

Fig. 1 .l-7. Actual and average flow direction of watm particles t,hroughporous medium

35


unit), and do not represent the actual extreme values: e.g. a continously changing velocity characterizes the flow domain although the effective velocity in a channel changes rapidly and periodically, following the random variation of the cross section. I n Fig. 1.1-7, the possible paths of water particles are shown through a porous matrix. Two particles following each other, may choose different paths at a bifurcation. Thus this process itself is a random variable, and the effective velocity at a given point may fluctuate with time. It can be stated, therefore, that the movement is time variable, even if the boundary conditions are time invariant, and theoretically, an unsteady flow is created by steady conditions, the average parameters of which (mean direction of flow, average flow rate through a unit area, etc.) are constant, and the movement in the flow domain can be regarded zw a quasi steady flow. The interpretation of porosity is a good example of the application of continuum approach for the determination of the parameters of a macroscopic flow domain. Similar to the process explained in connection with the averaging of molecular velocities, the porosity may also be determined for gradually increasing volumes. The extremes occur when using infinitesimal volumes. They are one or zero depending on whether the centre of the investigated volume is in a pore or within the solid matrix, and are independent of the actual character of the porous medium. When increwing the volume, the scattering decreases and achieves a constant minimum at the representative elementary volume. The calculated porosity fluctuates randomly around the expected value when the volume is above this limit. Naturally, the size of the representative elementary unit is strongly influenced by the structure of the solid matrix of the porous medium in question.

i

R

. . . -, ,..-. - .. .....,. ... .. : ,.... ... . -.,.'>.L< . ,/: ..;. ' .

t.;:

.

,.TTfVT S T ? r n f f T '

'

. : 4 .

f 7 ffff

,

I .

3;

,

,

of a homogenpors layer

0

maximum hoinoge~eausvolume Fig. 1.1-8. Interpretation of representative elementary unit using the example of determination of porosity representative elementary volume

3*

36


Porosity represents a good example of this statement as well. In a loose clastic sediment, the limit above which porosity fluctuates randomly, may be expected to be low, especially in sand and gravel. If the formation is layered, a slow change in the average porosity is probable after crossing an upper limit, indicating that above a given size, the investigated volume becomes inhomogeneous (it includes also a layer having properties different to the basic one), and the average parameter is more and more influenced by the second layer. Figure 1.1-8 shows the interpretationof porosity asa function of the volume size, based on the figures in Bear (1972). In fissured and fractured rocks, the distribution of the interstices is more uneven than that in a loose clastic sediment. The representative elementary length is generally longer in such formations and greatly varies with the structure of the rock type. To substantiate this statement, the linear porosity of a dolomite covered with dense fine joints is compared to the same parameter in karstic limestone. Four lines of various lengths in the dolomite were investigated, and the two extremes are represented in the figure. 1.6

(a) dolomite

a.9

( 2 samples of 50 rm) 0.8 0.05

n.7 ~

0.6

& B

$

a5

o

L

s

5

IO

{5

za

30

25

length af the separafely enalysed part of the fine rcm3

RJ

0.4

(b) limesfone

a3

( I sample of 25 m)

$& 2%

frequency disfribufian of the parasity of samples

0.2

8s -* & p 3 sz

a.i

a

-._._._

b.

I

I

1

1

I

I

I

2

4

6

8

ID

12

44

16

length of the separatply analysed part of the line [mi Fig. 1.1-9. Comparison of porosity distribution of limestone and dolomite


37

Although the joints are relatively well distributed, the range of porosity determined for various stretches of 10 cm is still about 30-707; of the average parameters, while it is reduced below 10-20% if the investigated length is 25 cm. In karstic limestone, where chemically dissolved openings as large aa 50 cm may be found, the length of the acceptable representative elementary unit is 50-100 times longer than in dolomite (Fig. 1.1-9). On the basis of continuum approach and considering the interpretation of the general seepage law determining the relationship between seepage velocity and hydraulic gradient, the definition of seepage can be clarified. T h e movement of any type of subsurface water i s called seepage, if seepage velocity, as a characteristic parameter of the flow space, can be calculated and applied to any point of the domain, as a function of the local hydraulic gradient. In loose clastic sediments and solid rocks having similar structures (e.g. sandstone), the flow may be called seepage almost without any restriction according to the extension of the flow domain. In the case of fissured and fractured rocks, the minimum size of the flow space above which the aquifer can be substituted by a continuous domain may be determined and thus, the movement can be investigated by applying the seepage law. However, if the size of the investigated layer is smaller than the limite, the water transport through the separate channels has to be calculated. The applicability of both the term seepage and the relationships determined for the characterization of seepage depends on the size of the flow domain, if the latter is composed of fissured and fractured rocks as will be discussed in detail in Part 3.

References to Chapter 1.1 AVERJANOV, S. V. (1949a): Relationship of Permeability of Soil with Air Content (in Russian). D.A.N., No. 2. AVERJANOV, S. V. (1949b): Approximative Evaluation of the Role of Seepage in the Capillary Fringe (in Russian). D.A.N., No. 3. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. S. (1968): Physical Principles of Water PercolaBEAR, J., ZASLAVSKY,D. and IRMAY tion and Seepage. UNESCO, Paris. CARMAN, P. C. (1966): Flow of Gases through Porous Media. Butterworth, London. DAROY,H. (1866): Publio Water Supply of Dijon (in French). Dalmont, Paris. DEWIEST,R. J. M. (1969): Flow through Porous Media. Academic Press, New York, London. F A 0 (1972): Glossary and Multilingual Equivalents of Karst Terms. Paris. PAO-UNESCO (1973): Irrigation, Drainege and Salinity. Hutohison and Co., London, Southampton. FOROEH&IMER, PE.(1886): On the Yield of Wells and Drains (in German). Hannover Z&chrift dea Archdekten und Ingenieure. Vol. 32. FOROHHEIBIER, PH. (1924): Hydraulios (in German). Teubner, Leipzig, Berlin. HAZEN,A. (1893): Some Physioal Properties of Sands and Gravels with Reference Lo Tbeir Use in Filtration. 24th Annual Report of the Massachusetts State Board of Health, Boeton. HAZEN, A. (1896): The Filtration of Publia Water Supplies. New York. IAHS (1974): Hydrologid Investigation of the Unsaturated Zone. 2nd Circular. International Glossary. Budapest.

38

1 Fundamentals for the investigation of seepege

IRMAY, S. (1964): On the Hydraulic Conductivity of Unsaturated Soil. Trunaactwn of ABU, Vol. 36, No. 1. KovAcs, G. (1969a): General Characterization of Different Types of Seepage. 13rd COngreee of IAHR, Kyoto, 1969. KOVACS, a. (1969b): Rdationahip Between Velocity of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th Congreee of IAHR, Kyoto, 1969. KovAcs, G. (19690): Seepage Law of Microseepage. 13th Cmgreaa of IAHR, Kyoto, 1969. KovAcs, G. (1971a): Seepage through Saturated and Unsaturated Layer. Bulletin of IAHS, No. 2. KovAcs, G. (1971b): Seepage through Unsaturated PorousMedie. 14th Cbngrese of IAHR, Paria, 1971. KO-, (1963): Hydraulics (in German). Springer, Wien. LINDQUIST,E. (1933): On the Elow of Water through Porous Soils.1atCbrrgreee of ICOLD, Stockholm, 1933. LOVASS,L. (1964): New Reeulta of Investigations Related to the Permeability of Soils (in Hungmian). Hidrol6giai Ko&ny, No. 9-10. MORE~SEYTOUX, H. J. (1973): On a Modified Theory of Infiltration (in French). Cahiere ORSTOM, Sene Hydrolo ie, Part 1, No. 2 and Part 2, No. 3. M O ~ E Y T O T J XH. , J. and !CE.AN~I, J. (1974): Derivation of an Equation of Infiltration. Water Reaoumes Reaearch, No. 4. N ~ T HE., (1963): Hydromeahmica (in Hungarian). Tankonyvkiad6, Budapest. UNESCO (1972): Ground-water Studies. Paris. UNESCO-WMO (1974): International Glossary for Hydrology. Geneva. ZAXARIN, J. A. (1928): Calculation of Ground-water Flow (in Ruesian). Trudey I.V.H. Twkent. ZAUBEREI, I. I. (1932): On the Problem and Determination of Hydraulio Conduotivity (in Ruesian). Iaveetia V.N.I.I.G. M n g r a d , No. 3-6. ZUNKER, F. (1930): Bebviour of Soils in Connection with Water (in Germen). Handbook of Soil Science, Springer, Berlin. Vol. VI.

Chapter 1.2 Physical and mineralogical parameters of loose clastic sediments influencing permeability During the process of seepage, the resistance of channels formed! by pores between grains connected almost continuously and distributed at random in the flow space, must be overcome by the forces accelerating and maintaining the movement. The resistance of the network depends mostly on the size and shape of the pores forming the channels. It is also influenced however, by the adhesive forces acting between the solid walls of the channels and water molecules. It is necessary, therefore, t o investigate and determine all the geometrical factors describing the size of the channels, and the other characteristics influencing the conditions of their w d s . The geometrical parameters of the network depend on the size and shape of the grains, the degree of sorting of grain sizes (grain distribution), and the porosity. Apart from the latter, the adhesive forces are affected by the mineralogical and chemical character of the grains. These factors and their influence on the flow of water are discussed in this chapter, which summarizes some physical and mineralogical fundamentals of seepage.

1.2 Physical and mineralogical parameters of clmtic sediments

39

1.2.1 The size and shape of grains forming a layer Grains and their main parameters (surface, area volume, etc.) can be characterized by one or more dimensions of the particle only in the case of regular geometrical forms. In nature, the grains forming the various layers differ considerably from regular forms, especially from that of the sphere, which is generally used in soil physics to substitute the actual shape of grains, because this is the simplest form, all the parameters of which can be determined knowing only one datum, i.e. the diameter of the sphere. For this rertson, the introduction of a shape coefficient is also necessary in addition t o the diameter of the grains to characterize the difference between the sphere and the actual form. There are two different definitions of the diameter of a sphere equivalent to the actual grain. Both are related to the methods used in soil physics for measuring the size of grains. If the grain is larger than 0.1 mm, this measurement is determined using sievee. Thus, the diameter of the equivalent sphere in this range is equal to the size of the holes in the sieve (which is the diameter for a circular hole, or the length of the side of a square), through which the grain can fall. I n the case of finer grains, whose clize is determined by sedimentation (hydrometric method), the equivalent diameter is the size of a sphere settling in the water at the same velocity as the actual grain in question. In the first case, the diameter of the grain is equal to the diameter of the clphere encircling the grain. In the smaller grain-size range such a clear relationship between the grain size and the substituting sphere cannot be determined. A further problem is caused in the latter case by the fact that the colloid grains may sink separately or joined together as a flake, according to the colloidal character of the suspension (morphology of the surface of grains, coagulation, peptization). The settling velocity of an aggregate composed of many grains is equal to that of a single grain having the same diameter rts the flake. Taking into consideration this uncertainty, it is also acceptable t o assume that a grain size measured by sedimentation is approximately equal to the diameter of the encircling sphere, and t o ensure in this way the uniformity of the interpretation of grain size. There are several shape coefficients for characterizing the difference between the actual grain and the equivalent sphere. These are generally, calculated from the three main dimensions of the grain, perpendicular to each other. A further basic requirement is that the shape coefficient should characterize the difference mentioned above from aspects relevant to the investigation in question. Thus there are shape coefficients used in geology to determine the origin of the grains and the distance they have travelled (Hagerman, 1938; MihBltz and UngBr, 1954; Sztideczky-Kardoss, 1933) and in hydraulics concerning sediment transport and sedimentation (Heywood, 1938; Ivicsics, 1957; Stelczer, 1967). To investigate seepage, a shape coefficient has to be chosen, which suits the physical description of the process in question (KovBcs, 1968a). Friction is the most important force among those retarding the velocity of seepage. Apart from this inertia and adhesion may be dominant. TWOof

40

1 Fundamentala for the investigation of seepage

these forces (friction and adhesion) are proportional to the contact surface between the solid and liquid phases of the system. Acceleration due to gravity is proportional to the volume of water. The aftect of solid grains can be accounted by using the porosity. Finally the ratio of retarding and accelerating forces, which physically, characterize the flow, can be approximated by the ratio of the surface area of a grain (A) to its volume (V). The dimension of this ratio is [L-'1. Hence it can be expressed as the quotient of a dimensionless shape coefficient (a) and a characteristic diameter ( d ): (1.2- 1)

As shown by the equation, a is a function of the diameter chosen to characterize the grain. According to the physical determination of the soil, this diameter can be that of the encircling sphere (D). The corresponding value of the shape coefficient ( a D )for some particles having regular geometrical form is as follows: Sphere a D = 6; Cube tcg = 10.4; Octahedron aD= 10.4; Tetrahedron aD= 18. There are some crystals, where the ratio of the two main axes is constant (those determining its cross section) and the third varies considerably. Thus, the grains may have any form from a laminated plate to the shape of an elongated nail (pyramid, tetragonal, or hexagonal prism, circular cylinder, tetragonal or hexagonal pyramid). The shape coefficient of such grains can be expressed as a function of the quotient calculated from the diameter of a circle encircling the main section of the grain (d,) and the height of the grain ( 1 ) (Fig. 1.2-1). The dotted lines show the values of a,,.(the shape coefficient being calculated not from the diameter of the encircling sphere but from dl). The comparison of the graphs reveals that there is practically no difference between the two types of shape coefficient (i.e. aD and adl) in the zone of the disc-shaped grains, but the two values differ considerably in the case of nail-shaped crystals. Another conclusion which can be drawn from the figure is that the shape coefficients of prismatic forms having a square and oblong cross section respectively may also differ only in the caae of elongated grains, and only where the ratio of the sides of the oblong is significant (e.g. 1 : 10). The shape coefficient depends on the mineralogical character of the grains, which in turn may relate to the size of the grains. I n gravels, the dominant material of this grain size being quartz, the partioles are generally stubby, the lengths of the main axes not differing considerably, and the quotient l/d, is nearly 1. Their form can be approximated by a sphere, or a cube, depending on the amount of abrasion whioh has occurred during transport from the place of origin to the sedimenbation location. Thus, their shape coefficient is generally about aD =

1.2 Physical end mineralogical parametere of clastic sediments

ratio of hetghf to tbe dhneief of the circle surrounding the cross section ofa gmn1

C/q

Fig. 1.2-1. Shape coefficient of grains as a function of the ratio of their length to the circle encircling their cross seotion

E

42


Table 1.2-1. Shape coefficients of clay minerals

1 Kaolinite Illite Montmorillonite Na-montmorillonite Nail-shaped grains (e.g. halloysite, attapulgite)

1

I~UI

I

4~

I

W,

aD

0.16-0.40 O.OpO.03 gO.01 n porosity, n Fig. 1.3-13. Comparison of Smith's data and capillary height calculated from Eq. ( 1.3-27)

The size of the pores can be considered as a random value. A given probability can be attached, therefore, to both d , and d,, while the ratio of the various pore-sizes to the average value can be characterized by a probability distribution curve. From the measured values of the cross-sectional area of every pore in various samples a statistical analysis was made proving that the ratio of the individual areas ( f ) to the average value ( f o ) can be well approximated by a distribution function. The frequency distribution and its integral form determining the probability of the number of pores having a given pore-size is as follows:

r

and (1.3-29)

The particular character of the function applied to the description of poresize distribution can be expressed by considering special parameters : (a) The smallest possible area tends to zero, and thus the lowest limit of the distribution curve is zero (xo70); (b) T h e random variable is the ratao of the area of pores related to their mean d u e (x = f / f o ) . The mean of this variable is, therefore, equal to unity ( G l e a n = 1); (c) I t follows from the previous condition that the two further parameters of the functional are equal to one another (k/A= m; m = 1; and k = A). Hence, the final form of the frequency distribution function is

9*

132


where

x = f/f,.

(1.3-30)

I n this special caue, direct relationships exist between the I and k parameters of the distribution function and the m and u parameters of the normal distribution

r

Thus, not only the author’s own measurements (four different samples and from each more different packing, namely, 7, 6, 8 and 4 measurements respectively) were available for further analysis but also the data of a previous publication which summarizes the mean value and the variation in the pore-size distribution measured on seven different samples and approximated with the normal distribution function (Murota and Sato, 1969). The graphical representation of the measurements in the I , vs. n coordinate system (Fig. 1.3-14) and the statistical investigation of the data have shown that the mean value of I , is 1.14 with a variance of o = 0.42 (this variance being 37 %). Considering the observed relationship between porosity and the 1, parameter, the variance can be slightly reduced (a, = 0.34), proving that the use of the equation determined by regression analysis

I,

= 2.19 - 3.75n & 0.34

(1.3-31)

can eliminate only about 20 yoof the uncertainties. The scatter of the points belonging t o the same sample with different porosities indicates that the remaining variance is caused mainly by sorne physical parameter of the soil not considered in the investigation (lines joining the points have similar slopes

4l 2.3 -

15 -

4.g -

c.5-

0-

Fig. 1.3-14. Relationship between porosity end the 1, parameter of the I’distribution function

1.3 Dynamics of soil moisture

133

to each other in the direction of increasing porosity and intersect the vertical axis at different heights). Considering the relatively small discrepancy between I, mean and unity and also the position of the line representing Eq. (1.3-31) in the range of the most probable porosity (0.3 < n < 0.4), the simplification by applying the I , = 1.0 value seems to be acceptable. In this case the r distribution function is reduced to an exponential function:

I(1.3-32)

and

Representing the P ( z ) functions belonging to ‘various I, values (0.5; 1.0; 1.5 and 2.0, Fig. 1.3-15), it can be seen that the probability of pores having a diameter of d , is between 20 % and 30 %, while the same parameter characterizing the pores with a diameter of d, is 80%. Thus, half of all the pores are those having diameter between the small and large diameters calculated from Eq. (1.2-12), the sample hawing about 30% smaller and 20% larger pores than those represented in the hydraulic model. Knowing the probable number of pores of various sizes, the total area of the pores having a smccller area than a given limit can easily be calculated. within a unit section of the sample can be The total number of pores (N) divided into m groups, according to the size of the pores. The basis of all intervak is df and the average area of the i-th interval (in which there are m

N I number of pores) is f i ;

2 N i= N lkl

1

.The specificfrequency distribution

can be characterized by dividing the number of the pores in each interval by the total number of the pores. When approximating the experimental

Fig. 1.3-16. Probable distribution of the number of pores as a funct)ionof the specific surface

134


frequency distribution by continuous function, the parameters of the latter can be calculated as the area above the horizontal axis and below the graph which is equal to the area below the curve (Fig. 1.3-16):

mAf Fig. 1.3-16. Theoretical sketch represent.ing the frequency distribution of the pore size

and

-

[ v(f; df)df = Of where

0

(if d f

-

0);

(1.3-33)

and v ( f ; df)is the continuous approximation of the frequency distribution. If the df interval is small enough, the total area of pores (which is equal t o the areal porosity where the total cross section of the sample is unity) can be expressed as the sum of the fiNi product:


135

therefore (1.3-34)

and

-

J ’ f v ( f ; A f ) d f = 1N2 A A f = f O A f ; 0

because the areal porosity can also be determined as a product of the total number of pores and their mean value (nA= Nf,). Further investigations can be simplified by introducing dimensionless variables. This goal can be achieved if the variables having an areal dimension are divided by the mean value of the measured pore sizes

The function describing the frequency distribution of the number of pores of different sizes has also to be transformed using the E(z;As)function instead of v(f; df). Introducing the new variables, the integrals in Eqs. (1.3-33) and (1.3-34) are as follows:

-

J E(s;

dx)dz = AS;

(1.3-35)

0

~ z [ ( z ;AX)& = AS.

0

Considering the results of the statistical analysis of the measured data, and accepting the simplified form of the distribution functions [see Eq. (1.3-32)], the mathematical description of the E(z;dz) function satisfying the conditions specified in Eq. (1.3-35) can be given in the following form: t ( z ; Ax) = Ax exp (-x)

.

(1.3-36)

Investigat,ing the rate of capillary saturation of a sample, the network of pores can be substituted b y vertical straight capillary tubes as a fist approximation. The distribution of the number of tubes with different diameters is the same as that determined by Eq. (1.3-30) [or by its simplified form - i.e. Eq. (1.3-32)]. In this case at an elevation of h, above the water table all the tubes having smaller cross-sectional area than f, are completely saturated, while the larger sections are sled with air. The rate of capillary saturation can be determined, therefore, by relating the sum of the area of pores having a smaller area than fc to the total free area in the cross section of the sample:

where

136


Substituting Eq. (1.3-36) and integrating the rate of capillary saturation is expressed as the function of the limiting vdue of the pore size:

+

(1.3-38) = 1 - (fdfo 1) exP ( I - f J f o ) * The ratio between the cross-sectional areaa of two circular tubes is equal to the square of the rate of their diameters. Thus, the coefficient of capillary saturation can be expressed aa the function of a given limiting value of the diameter related to the average pore diameter: sc

because

sc = 1 -

+ 11exp [-(dJdo)21;

[(dJdo)a

(1.3-39)

fdfo= (dc/~o)a. The soil-moisture retention curve describes the relationship between suction (elevation above the water table) and the water content. If it is divided into two parts (i.e. the rate of capillary and adhesive saturation, respectively) and only the effect of capillarity is investigated, the interrelation indicates which part of the pores can be saturated by capillarity at a n elevation of hc. This is the ratio of pores having a capillary height equal to or greater than he [see Eqs (1.3-12) and (1.3-13)]. Considering Eq. (1.3-26), this ratio depends only on the pore diameter and, therefore, the capillary component of the soil-moisture retention curve can also be approximated by a probability distribution function (Rbthgti, 1960). The distribution function describing the relationship between the rate of capillary saturation and capillary height can be characterized mathematically by applying a further transformation in Eq. (1.3-39). Considering that the capillary height is inversely proportional to the pore diameter, the ratio of the limiting diameter to the average value can be substituted by the quotient of the average capillary height and the height of the investigated elevation above the water table: 8,

because

=I1

-

+ 11exp [ - ( ~ c d h J 2 1 ;

( 1.3-40)

-=dc hco do hc

The curves described by Eqs (1.3-38), (1.3-39) and (1.3-40) arerepresented in Fig. 1.3-17. The figure shows that among the three capillary heights calculated from Eq. (1.3-27) one belongs to the s = 0.5 value of the rate the second to s = 0.01 0.05 (h, while the third of saturation (he (hcocalculated from do average pore diameter) gives the position of the capillary component of the soil-moisture retention curve where the rate of saturation is equal to s = 0.25. The basis of the previous derivation is a model composed of straight capillary tubes having different diameters, though the diameter of each tube is constant. The network composed of the pores is, however, a system of channels with a changing cross section and not that of straight pipes with constant diameters. The capillary height is influenced, therefore, not only by the

-


137

Fig. 1.3-17. Distribution of pore size, pore diameter and capillary height determined by applying a model composed of capillary tubes of constantfdiameters

distribution of the pore sizes in a horizontal section but by the vertical change of the pore diameters as well. For this reaaon some pores are not saturated when the dry sample is made wet from the direction of the water table, while the same pores can retain capillary water when the process starts with the complete saturation of the sample and dynamic equilibrium is achieved by drainage. This phenomenon is the hysteresis of the soil-moisture retention curve (the nearly horizontal section of the curve haa a lower position if i t is determined by wetting and a higher one in the case of drainage). In the literature, the phenomenon of hysteresis is generally explained by two different physical causes (Bear, 1972). The f i s t possible reason may be the rain drop egect, according to which the angle of contact of the meniscus with the wall of the capillary tube (8,) is larger when the water advances and the decreme in the in the tube than when in a static condition (8,) angle can be observed (0, < 0,)in the case of a receding water column (see Fig. 2.1-17). The same difference in the contact angles can be observed between the front and the rear of a rain drop moving along a sloping glass plate (the name of the phenomenon is based on this similarity). The water level may remain a t a higher position when a capillary tube with changing diameter is drained, compared to the rising condition, because the downward movement can be stopped when the meniscus reaches a small cross section (where capillarity can balance the weight of a relatively higher water column), while, similarly, the rise of the meniscus stops at a large diameter. This phenomenon is called the ink-bottle egect, because in old ink bottles this effect was used to keep the level of ink near the mouth of the bottle. From the two possible causes, the rain drop effect cannot be accepted as an explanation of hysteresis. The soil-moisture retention curve represents the vertical moisture distribution in a static condition. On the contrary, the

138


rain drop effect is a process occurring only in the case of flow in capillary tubes. Thus, a phenomenon characterizing the movement of water cannot be the cause of a phenomenon (the two possible positions of the retention curve) which describes the static condition of the moisture distribution.

i Fig. 1.3-18. Distribution of capillary height in the case of a bundle of identical wavy capillary tubes

For this rewon, the change i n diameter of the chccnnels will be used to explain the existence of hysteresis. It is necessary, therefore, to use a model composed of tubes, the longitudinal section of which is represented by two undulating lines having a minimum distance of d,, a maximum one of d2, and a wave length of 2a (Figs 1.3-18 and 1.3-19), instead of the straight capillary tubes mentioned previously. Even if the wave length is constant (all the tubes are identical with each other as in Fig. 1.3-18), the pore-size distribution previously described in a horizontal cross section can eaaily be ensured by placing the sections of tubes of different diameters at the level of the water table. The same result can be achieved with the other model, where the amplitude (the fluctuation of the diameter of the tubes) is the same as in the previous case, but the wave length varies from tube to tube (the parameter of a is also a random variable, described by a distribution function; Fig. 1.3-19). Investigation of the model represented in Fig. 1.3-18, shows that the maximum capillary height (belonging to diameter d,) can never be achieved by wetting the tubes from the direction of the water table, if the difference between the maximum and minimum capillary rise is greater than the wave h, < (h, min 2a), all the tubes length. This is because in the zone of h, and the have a section of diameter d, whose capillary rise is equal to h,

h, > (h, - 2a) because all these tubes have a section of diameter d,, where the capillary tension can balance a water column of 11, max. In the case of the more realistic model composed of diflerent wavy tubes, the wave length is a random variable (Fig. 1.3-19). In investigating the wetting process of the sample, the water can be raised by capillarity to the level of h, max in some of the tubes [the wave length of which is greater than the difference of (h, max - h, mln) and if the capillary rise is unaffected by a large section of the tube below the maximum level]. In the tubes having shorter wave lengths, the capillary rise is limited to the level (h, mln+2a)tm described in the previous' paragraph. If the wave length is long enough but has a large diameter below the highest capillary rise, the elevation where the weight of the water column is balanced by capillarity depends on the diameter at this level. In this case, the average (mostprobable) capillary height is between the minimum capillary rise and the level characterized by the sum of the former and the most probable wave length (2a). Only the vertical shifting of the nearly horizontal stretch of the soilmoisture retention curve can be explained by the ink-bottle effect. The


140

observations have shown, however, that the y vs. W relationship is not a single-valued function in the zone of adheeion either. One of the remons for the different water contents at a given elevation above the water table is the d e r a b l e time lug between the time points of the change of the position

(b)

10

-

0

I

0

1

0.1

1

I

I

0.2

1

I

0.3

water content, W Fig. 13-20. The development of the soil-moistureretention curve in time

of the water table and the development of the dynamic equilibrium. Figure 1.3-20 .

shows the refjults of an experiment illustrating the observation of the process in time. The data indicate the extremely slow development of equilibrium in the adhesive zone when the column is drained, viz. in this case, the upper part of the sample has very small hydraulic conductivity because of the low

141


water content, and thus, the excess water percolates downwards with a very small velocity. The wetting is generally a faster process, but a considerable difference can be observed between the initial wetting (when the process starts with a completely dry sample) and rewetting (when the sample waa previously drained and some soil moisture was retained in the pores). In the first caae, the development of the dynamic equilibrium in the adhesive zone has, perhaps, a greater time lag than that of the column drainage. Thus, (b)

variation of moisture content

v s. sucfion bead relatiunship with

time on the basis of fteld measurements at a given depth (Bourm, z = 40 cm)

0

0.l0

0.20

0.30

water confeflf, WI

1 '

0

0.1

water

I

I

0.2

a3

content, W

Fig. 1.3-21. Characterization of hysteresis by laboratory and field meaaurementa (after Vachaud and Thony, 1971; Royer and Vachaud, 1975)

in many cmes the difference between the water contents observed at the 8ame elevation in a wetted and drained column respectively can be explained by the insufficient length of the observations. The horizontal shifting of the vertical stretch of the retention curve is, however, also indicated by data determined directly from simultaneous measurements of tension and soil-moisture values (Fig. 1.3-21). In this case the corresponding data are not influenced by the time-dependent develop- . ment of the dynamic equilibrium. The cause of hysteresis is the water retained in the corners of the pores after draining. The explanation of the phenomenon of hysteresis indicates many uncertainties influencing the form of the soil-moisture retention curve and cause its multi-valued character. It is not expected, therefore, that the history of

142

1 Fundamentals for the investigetion of seepage

Toucbetsilty loam

a

az

0.4

06

I

1

1

1

02 n.4 06 .a3 Fig. 1.3-22. Comparison of the theoretical soil-moisture OR s

{a a

1 , -


0

0.4

D.6

143

00 s lo 0

3.0h -

3.0.

(h) sand

nco 2.5 n

=

o m

n

= 0.346

2.0 -

1.5 -

1.0 -

0.5 0

P

D-

o

02

04

06

20

retention curve with measured data

s

ID

o

02

ao

05

JB s

/c

144


the wetting and draining processes can be followed theoretically. The random position of the retention curve can be characterized only by comparing the calculated average curve and measured data. For this comparison observations published by three different laboratories supplemented by the author’s own measurements were used (Figs 1.3-22a and 1.3-22b, Johnson et d., 1963; Prill et al., 1965; Figs 1.3-22c and 1.3-22d, Ldiberte et al., 1966; Figs 1.3-220 and 1.3-22fJ Vachaud et aZ., 1975; Vauclin, 1975; Khanji, 1975; Vachaud et al., 1970; Vachaud and Thony, 1971; Figs 1.3-22g and 1.3-22h the author’s own measurements). The measured data prove the reliability of the theoretical relationships [Eqs (1.3-24) and (1.3-40)] and considering the composition of adhesive and capillary retentions (Fig. 1.3-23), the final form of the equation describing the average retention curve is as follows:

(1.3-41) To characterize the probable zone of hysteresis, it can be proposed that separate enveloping curves should be applied along the capillary and adhe-

Fig. 1.3-23. Composition of soil-moistureretention curve from adhesive and capillary retention

145


sive stretches respectively. The scattering of data in the zone of adhesion has already been discussed in connection with Eq. (1.3-24). Considering uncertainties explained there, the enveloping curves can be given in the following: form: ( 1.3-42)

In the capillary zone the enveloping curves can similarly be given by using a multiplying factor in the distribution fnnction which characterizes the rate of capillary saturation

sc2= 1 -

[i z I : 1.5-

+1

1

[[

exp - 1.5-

,

These curves are also represented in Figs 1.3-22g and 1.3-22h. Their position proves that Eqs ( 1 . 3 4 2 ) and (1.3-43) are suitable to characterize approximately the expected scattering caused by hysteresis. References to Chapter 1.3 BEAR,J. (1972) Dynamics of Fluids in Porous Media. Elsevier New York, London, .Amsterdam G. (1966): Certain Problems of Infiltration in Unsaturated BORELI,M., VACHAUD, Porous Media. Saopstenja Inetittda za VodorpriiVp.eduJaroslav Cerni,N o 38. BncKwaHAM, E. (1907): Studies in the Movement of Soil Moisture. USDA Bureau of So& Bulletin, No. 38. L)E WIEST,J. M. (1969): Flow through Porous Media. Academic Press, New York, London. DoLaov, S. I. (1948): Investigation of the Movement and the Availability of Soilmoisture (in Russian). Izd. A d . Nauk SSSR, Moscow, Leningrad. ELRICK, D. E. (1966): The Microhydrologic Characterization of Soils. I A S H Symposium on Water i n the Unaaturated Zone, Wageningen, 1966. FEKETE, Z. (1950): Comparative Investigation of Parameters Characterizing the Water Balance of Soils (in Hungarian). Hidroh5qiai Kozlony, No. 7-8. IASH (1974): Hydrological Investigation of the Unsaturated Zone (Second circular). Budapest. JOHNSON, A. I., PRILL,R. C. and MORRIS, D. A. (1963): Specific Yield-column Drainage and Centrifuge Moisture Equivalent. Qeologiccrl Survey Water Supply Paper, 1662-A. JwAsz, J. (1967): Hydrogeology (in Hungarian). Tankonyvkiad6, Budapest. KASTANEK,F. (1971): Numerical Simulation Technique for Vertical Drainage from a Soil Column. Journal of Hydrology. p. 213. KHANJI,J. D. (1976): Investigation of the Recharge of Ground Water with Free Water Table by Infiltration (in French). Doctoral Thesis Univereity o f arenoble. 10

146


CUTE, A. (1962): Some Theoretical Aspects of the Flow of Water in Unseturated So&. Proceedings of American Sod SciencS Society, Vol. 16, No. 2. KOVAOS,G. (1968): Cheraoterization of the Moleoular For- Muencing Seepage with the He1 of the p F Curve. Agrokhia b Tdajtan, (Supplementum). KOVAOS, (19718): Seepage through Unsaturated Porous Media. 14th Uongrei?s of IAHR, Pa&, 1971.

Cf

KOVACB, G. (1971b): Seepage through Saturated and Unsaturated Layers. Bdletin of IASH, No. 2. KREYBIU,L. (1961): Heat- and Water-balanoe of Soils (in Hungarian). Budapest. KWON, H. (1932): Characterization of SoilMoisture (in German). Zeitschrijt fiir Pflanzen, Berlin. LALIBERTE,G. E., COREY,A. T. and BROOKE,R. H. (1966): Properties of Unsaturated Porous Media. Hydrology Papers, Colorado State University, November, No. 17. LEBEDIEV,A. F. (1963): Sog Moisture and Ground Water (in Russian). Izd. Akad. Nauk SSSR, Moscow. LUTZ,J. F. (1966): Proceeding of American Soil Science Society, pp. 330-361. MADOS, L. (1939): Studies Related to Irrigation and Water Management in the Irrigation Scheme of T i e z a f i i (in Hungarian). hiidaiigyi Kiizikrnknyek. MADOS, L. (1941): Use of Soil Moisture Related to Surface Drainage (in Hungarian). c)ntGz&i&yi K6demdnyek. MITEIOHERLIOH, E. A. (1932): Soil Scienoe for Agriculture and Forestry (in German). Berlin. MUROTA, A. and SATO,K. (1969): Statietical Determination of Permeability by the Pore-size Distribution in Porous Media. 13th Congress of IAHR, Kyoto, 1969. P E ~A., J. (1966): Diffueivity Determination by a New Outflow Method. IASH Sympeeium on Water in the Uneaturated Zone, Wageningen, 1966. P~OZELY, T . and ZOTTEB, K. (1973): Summary of Inte mtation and Measuring Methods of p F Curve (in Hungarian). (Manuscript). V d , Budapest. PETERSON, A. E. and DIXON,R. M. (1971): Water Movement in Large Soil Pores: Validity and Utility of the Channel SMem Concept. Reaearch Report University of WieconSin, June, No. 76. A. J. and MORRIS,D. A. (1966): Specific Yield-laboratory PRILL, R. C., JOHNSON, ExDerimenta Showinn the Effect of Time on column Drainage. Geoloaical survew. "W&er Supply Paper,"l662-B. PROOEROV. A. V. and KARASEV.N. K. 11939): Land Reclamation and Water Control (in Russian): MeZior a i gidrotekhniki, No: 12. RbTHdm, L. (19603ngineering Aspects of the Capillarity of Soils (in Hungarian). Vcziisyi Kodemknyek. No. 1. RICEARDB, L. A. (1931): Capillary Conduction of Liquids through Porous Media. PhyeicS, Nov. RODIE,A. A. (1962): Soil Moisture (in Russian). Izd. Acad. Nauk SSSR, MOSCOW. ROSE, D. A. (1966): Water Transport in Soils by Evaporation and In6ltration. IASH S y m p o h m on Water in the Unsaturated Zone, Wageningen, 1966. ROYER,J. M. and VAOHILUD, Q. (1976): Field Determination of Hysteresis in Soil Science Characteristics. Proceeding8 of Soil Science Soc&ety of America, No. 2. SMITE, W. 0.(1933): Minimum capillary Rise in an Ideal Uniform soil. Phyeics, May. SMITE,W. O., FOOTE, P. D. and BUBANU, P. F. (1931): Capillary Rise in Sands of Uniform Spherical Grains. PhyeicS, July. STAKBUN,W. P. (1966): Relations Between Partiale Size and Pore Size and Hydraulio Conductivity of Sand Separates. IASH Symposium on Water in the Unsaturated Zone, Wageningen, 1966. SUDNITBIN,L. I. (1966): Soil-moisture Pressure in Some Climatic Zones. IASH Sympeeium on W&r in the Uneaturated Zone, Wageningen, 1966. VAOEAUD, G. (1966): Study on Redistribution after Stoppin Horizontal Infiltration (in French). IASH Sympoeium on Water in the Unsaturated &me, Wageningen, 1966. VACJUUD,G. (1966):Verification of Generalized Darcy's Law and the Determination of CapilLary Conductivity by Analysing Horizontal Infiltration (in French). IASH Sympoeium on Water in the Uneaturated Zone, Wageningen, 1966.

-

1.4 Balance of the ground water

147

VACHAUD, G., CISLER, J., THOM, J. L. and DE BACKER,L. (1970):Utilization of Gamma Emission of Americium-241 for Measuring the Soil-moisture Content of Unsaturated Soils (in French). Isotope Hydrology - 1970. I A E A . VACEAUD, G. and THONY,J. L. (1971): Hysteresis during Infiltration and Redistribution in a Soil Column of Different Initial Water Contents. Water Resources Reeearch. No. 1. VACHAUD, G., GAUDET, J. P. and KURAZ, V. (1974):Air and Water Flow during Ponded Infiltration in a Vertical Bounded Column of Soil. Journal of Hydrology, NO. 1-2. VACEAUD, G., VAUCHIN, M. and HAVERKAMP, R. (1976):Towards a Comprehensive Simulation of Transient Water Table Flow Problems. Seminar on Modeling and Simuldion of Water Reaourcea System, 1975. VAN BAVEL,C.H.M. (1966):Three-phase Domain in Hydrology. IASH Symposium on Water in the Uneaturated Zone, Wageningen, 1966. V-YAI, G. (1942): Experiences in Irrigation a t MAtraliget (in Hungarian). C)ntoz&igyi Kodemdnyek, No. 2. V ~ L L Y A G. I , (1974):Investigation of Water Movement Developing in Unsaturated Soil Layer (in Hungarian). Agrokkmia 6s Talajtan, Vol. 23, No. 3-4. VAUOLIN, M. (1976):Experimental and Numerical Inveatigation of the Drainage of Ground Water with Free Water Table. The Infiuence of the Unsaturated Zone (h French). University of Grenoble (Dootoral Thesis), A ril VTJ~IC, N. (1966): M u e n o e of Soil Structure on h t r a t i o n and p P Values of Chernozem and Chernozem-like Dark Meadow Soil. IASH Symposium on Water in the Uneaturated Zone, Wagewingen, 1966. WATSON,K. K. (1967):Experimental and Numerical Study of Column Drainage. P r o d i n g s of ASCE, ELY.2. WATSON,K.K. and WHIBLER, F. D. (1969):Analysis of Infiltration into Draining Porous Media. Proceedings of ASCE, IR.4. WESSELINU, J. and WIT, K. E. (1966):An Infiltration Method for the Determination of the Capillary Conductivity of Undisturbed Soil Cores. IASH Symposium on Water in the Unaaturated Zone, Wageningen 1966. WHISLIER,F. D. and BOUWER,H. (1970):Comparison of Methods for Calculating Vertical Drainage and Infiltration for Soils. Journal of Hydrology. No. 1. WIND, G. P. (1966):Capillary Conductivity Data Estimated by Simple Method. IASH Sympoeium on Water in the Unaaturated Zone, Wageningen, 1966. ZOTTER, K. (1976):An Instrument for Determining the Upper Stretch of the PP Curve (in Hungarian). (Manuscript). VITUKI, Budapest.

Chapter 1.4 Investigation of the balance of the ground-water space In the previous chapter the hydrological processes occurring in the soilmoisture zone were analyzed. It has been shown that infiltration and evapotranspiration through the surface (both modified by the change of storage in the unsaturated pores) affect also the regime of the gravitational ground water. The resultant of these influences can be summarized aa the water exchange between soil moisture and ground water (accretion). This combined effect can be divided into two vertical components. The first is directed downwards, and it is that part of infiltration which reaches the water table. The second component is the vertical natural drainage of the ground-water space through the water table, which raises water from a deeper posit,ion by capillarity and tension difference to replenish the storage capacity of the soil-moisture zone emptied by evapotranspiration. The interaction between the two zones in question can be investigated as a pure hydrodynamical process, when the instantaneous water transport has 10*

148


to be calculated, convidering the actual flow conditions. For such an analysis, detailed knowledge of the acting forces, as well as that of the resistance of the unsaturated porous medium is necessary and the prevailing boundary conditions have to be taken into account. These aspects will be summarized later on in this book; therefore, the hydrodynamic investigation of flow through an unsaturated zone will only be discussed after the explanation of all the necessary basic concepts. There is, however, a more general way of investigating the water exchange between soil moisture and ground water, when this process i s regarded as a part of the hydrological cycle, and its average numerical parameters are determined from the water balance equations of the relevant water bodies. This hydrological analysis provides important information also for solving numerous hydraulic problems; the average parameters obtained in this way can be used for checking the hydrodynamic calculations, st8 the amount of water transport integrated for a longer period has to be equal to the result determined from the water balance equations. Another important field of application of the hydrological parameters in seepage hydraulics is the investigation of the horizontal flow of the shallow ground water, when accretion influencing the flow field through the water table has to be considered. I n this case the average parameters calculated from the balance equations can be used as independent variables describing this special boundary condition. This close contact between hydrodynamic and hydrological investigations gives ground for including the hydrological analysis of the ground-water balance in a book dealing with seepage hydraulics, as it is done in this chapter.

1.4.1 General hydrological characterization of water exchange between soil moisture and ground water

The basis of the present investigation is the model of the soil-moisture zone represented in Fig. 1.3-1 ; the same symbols are used here as those indicated there (Kovbcs, 1959a, 1959b, 1971). Let us &st consider a limited soil column in the vicinity of the surface, where evapotranspiration can affect the gravitational ground-water space owing to capillarity and the suction of the roots of vegetation. The column includes a part of the ground-water space (below the phreatic ground-water surface) and the soil-moisture zone (both the unsaturated and the transition zones between the water table and the soil surface) (Fig. 1.4-1). The equation of continuity can be determined for this column by considering the recharging and draining processes. The amount of water contained within the investigated column is increased by infiltration through the surface ( I , ) . Evapotranspiration, which includes direct evaporation from the soil moisture (Em)and transpiration by plants (T,)causes losses in the column. Ground water may flow either in or out of the space observed, recharging or draining it (Diand Do). Horizontal moisture flux, either into


149

Fig. 1.4-1. Sketches representing the components of the water balance equations characterizing the hydrological processes in the soil-moist,urezone

the column (Mi), or directed out of it ( M o ) ,is usually very small compared with the other recharging and draining processes; it is, therefore, neglected in the following discussion. The sum of water transported through the boundaries of the column is equal to the change of the water content stored within a given period (T) in the soil column. The latter includes the change i n amount of both soil moisture ( A W ) and ground water ( A V ) from the beginning to the end of t,he period (0 < t < T ) : T

2 [+ I,(t) + D,(t)- E,(t) - T,(t)

-

Do(t)]At = [-+AM'& AV],T. (1.4-1)

n

The members on the left-hand side of the equation are functions of time, and they are expressed the water amount related to the surface of the column and to the unit of time. Thus their dimension is equal to that of velocity [LT-11. Infiltration through the surface originates from precipitation, and it can be betermined by subtracting from the total precipitation both surface run-off dnd the amount of water evaporated directly from the surface, or retarded ay the vegetation. Apart from climatic parameters (amount and distribution

150


of precipitation, factors affecting evaporation), this value depends on the condition of surface (degree and direction of slopes, covering vegetation), as well as on the type and condition of soil influencing the process of infiltration. One part of the infiltrating amount is stored in the unsaturated zone (Iw) filling this layer up to its water retention capacity (or sometimes, temporarily, above this limit). The other part reaches the gravitational ground-water space ( I , ) , either during the rain, or with a time lag (Fig. 1.4-lb):

Irn(t) = Iw(t)

+ Ig(t).

(1.4-2)

As it has been mentioned, evapotranspiration is composed of direct evaporation ( E m ) and transpiration through plants (T,,,).Although both draining processes reduce the moisture content in the unsaturated zone, the total amount of water subtracted from the soil may be divided into two parts: the first is the prolonged decrease of the moisture content ( E , and T,, respectively), while the second part is replenished from the water table below; thus this amount ( E g T , = ET,) is a negative component of the ground-water balance (Fig. 1.4-lb):

+

Ern(t)

+ Trn(t)= E d t ) + Tw(t) + Eg(t) + Tg(t)= ETw(t) + ETg(t)* (1.4-3)

The amount of the soil moisture evaporated and transpired depends on the potential evapotranspiration (temperature, moisture deficit of air, wind velocity), on the parameters characterizing the evaporating surface (mostly conditions of vegetation, such as depth of the root zone, total surface of leaves, water consumption of the plants) and on the type and condition of the soil. The third group of the influencing factors is the ground-water flow rechargand outflow (Do) ing or draining the gravitational space. Both inflow (Di) depend on the hydraulic gradient developing in the vicinity of the column investigated and on the transmis8ibiZity (the product of thickness and hydraulic conductivity) of the water conveying layer. The two processes may by investigated jointly, by considering their difference:

*AD(t) = Dj(t) - D,(t).

(1.44)

Although the listed components of the balance equation are time-dependent variables, when the period investigated is long enough, their average multiplied by the length of the period can be used instead of their integrated value. _

m

1 "

- J E,(t) dt = Em ; T i 1 T -J T,,,(t) dt = F,,, ;

TO


151

1 T J AD(t)dt = AD ;

TO

(1.4-5) 1 T J ET,(t) dt = EF,,, ;

TO

1 T J I,(t) dt = 1,;

TO

1 T

- J ETg(t)dt = ETg.

TO

Substituting these averages into Eq. (1 4-1), the simplified balance equation, valid for a long period, can be written in the following form:

[ ( I g- ET,)

+ ( I , - ET,)

f ADIT = [ f d V f AW):.

(1.4-6)

The two members on the right-hand side of the equation express the change of the water amount stored in the column between the beginning and the end of the period investigated, indicated by 0 and T s u f i e d outside the bracket. The first symbol (dV) refers to the change of the stored gravitational ground w t e r , while AW indicates the diflerence i n the soil moisture. These two values are closely interrelated, because if the change i n gravitational storage is calculated as the product of the change of the water level ( A H ) and t,he effective porosity of the layer ( n ) (Fig. 1.4-lc)

f AV = f d H n ,

(1.4-7)

this parameter includes a part of soil-moisture storage aa well, while the stored amount of soil moisture also depends on the position of the water table. According to its definition, the change of storage in the soil-moisture zone is equal to the difference of the amount of moisture above the water table at t w o subsequent points of time. Thus the change may be caused not oiily by the variation of the point values of moisture content, but also by the change of the position of the water table, because the thickness of the .soil-moisture zone changes in this case as well: 0

0

f A W = J J W ( t = 0 ) dA dm - J J W ( t = T)dA dm = W 1- Wz; (1.4-8) mi ( A )

m (4

where A is the area of the horizontal cross section of the investigated column (which is supposed to be unity) and dA is its elementary part. In most areas the water table shows a regular annual fluctuation, and generally still another one, with a longer period. Under natural conditions, Iwwever, a state of equilibrium can be expected, if the period considered is long

152


enough. (Here all effects caused by human activity, excluding direct artificial recharge and pumping, are regarded as “natural”; e.g. change of cultivation, control of surface run-off, etc.). This assumption is evidently correct, because these effects act continuously over a number of decades, even centuries, almost without m y considerable changes, thus providing enough time for the development of a dynamic equilibrium. The average elevation of the undisturbed water table is, therefore, at a level, where all the recharging effects are balanced by the various draining actions. The equlibrium of the natural influences (and those acting indirectly over a long period) can be characterized with a time-invariant average depth of the water table (mav). This is the reason why the hydrographs of such water tables (the graphs representing the change of the position of the water table in time) are generally closed, if the period is long and its limits were properly chosen. (The closing of the hydrograph means, that the initial and end points are at the same elevation). Assuming the soil column t o be i n a state of equilibrium, and supposing that the determination of the investigated period satisfies the conditions mentioned in the previous paragraph, the change of the amount stored i n the gravitational ground-water space is zero (AV = 0 ) , because the position of the water table at the end of the period is the same as it was at the beginning ( A H = 0). If the starting and closing time points are in the same season, the change of the stored soil moisture m y also be neglected (AW = 0). In this special case the water balance can be limited to the gravitational ground water below the average water table (mav)(Fig. 1.4-Id). The equation of ground-water balance [Eq. (1.4-1) or (1.4-6)] remains valid also for this case, but the recharge from infiltration refers to the amount of water which reaches the water table (I,), and not to that crossing the surface. Similarly, the amount drained from the system has to be considered as the water raised from the gravitational ground water (ET,) to replenish evaporated and transpired soil moisture, and not the total amount of the latter. This modified form of the balance equation expresses that the vertical water exchange between soil moisture and ground water is eqml to the total effect of ground-water flow (the difference of outflow and inflow), if the position of the water table is practically the same at the beginning and the end of the period used for the calculation of the averages of the parameters in question: (1.4-9) I , - ETg = Do - Di. There are many cases, however, when a shorter period has to be analyzed (e.g. when studying the seasonal fluctuation of water exchange between ground water and soil moisture) and, therefore, storage cannot be neglected. The investigation can yet be limited to the balance of the gravitational space in this case as well, but the changing volume of the investigated column has to be considered; this can be expressed in the form of the change of the stored amount of ground water : (1.4-10) ( I , - ET,) (Di- Do) f AVO = 0.

+

It is necessary to note here that the AVO value is not equal to A 7 in Eq. (1.4-7), because the latter gives a part of the total change of both ground


153

water and soil moisture, which can be determined arbitrarily, considering that the calculation of the other part ( A W ) is suitable to supplement the first part and to give the correct amount of the total change. If the investigation is limited to the gravitational space, the physically correct interpretation of the storage capacity of the layer has to be determined. I n the literature the storage coeficient (storistivity, S ) or specific storage (S,) is generally used to characterize this property of confined aquifers, and the specific yield (n,) is the accepted parameter for unconfined layers. In addition to the topic discussed here, the characteristics listed have an important role in many other fields of seepage hydraulics. Their detailed analysis is given, therefore, in the next section, together with the methods proposed for their numerical determination. Supposing that the column, whose balance is investigated now, is separated from an unconfined aquifer, its storage capacity will be characterized by the specific yield (n,) of the layer in this analysis; thus the change of storage as a result of the modification on the position of the water table ( A H ) can be expressed as A V O = AHn,. ( 1.4-1 1) Finally, it can be stated that the balance equations describe the relationships between two [Eq. (1.4-9)] or three [Eq. (1.4-lo)] independent processes occurring in the gravitational ground-water space (i.e. vertical water exchange, horizontal flow and change i n storage). Owing t o the numerous uncertainties generally associated with the determination of hydrological parameters, i t is advisable to measure, or calculate, all the members of the balance equations separately, and to use the formulae only for checking the reliability of the measurements, or for determination of the closing errors (use of the implicit form of balance equations). It is also possible to express one member of the equation as a function of the others (explicit form),and calculate the former from the measured data of the known parameters. It is necessary, however, to consider in this case that all errors in the measured data are burdens to the calculated value. Considering the required implicit use of balance equations, the three processes listed previously will be analyzed separately in the following, starting with the storage in both confined and unconhed layers.

1.4.2 Storage capacity of aquifers The storage capacity of a layer is characterized i n general by the volume of water released from, or taken into storage as a result of a unit change i n pressure, or piezometric head. The volume is related to the horizontal arm of the layer, and thus it has a dimension of length [L]. This amount depends on - The compressibility of solid grains; - The compressibility of water; - The change of porosity of the saturated layer; - The change of the ratio of the saturated and unsaturated thicknesses. The f i s t component is very small in comparison with the others, therefore it is negligible. In confined aquifers (theoretically, only if it is covered with

154


aa absolutely impervious formation) the system will remain always saturated, thus the fourth effect does not influence the process. In the investigation of shallow ground water with a phreatic water table, the change of the thicknesses of the saturated and unsaturated zones has the most pronounced role. The effect of consolidation and water compressibility can therefore be neglected in this case. It is the reason why the characterization of a confined aquifer differs basically from that of unconfined water-bearing layers. In the first case the parameter defined previously is called storage coepient (S), and it d e p e d on the consolidation of the layer and on the cumpressibility of water. This value represents the total storage capacity of the layer having a thickness of b. Sometimes this parameter is related to the unit volume of the aquifer (water released from, or taken into storage in a volumetric unit of the layer). This specific storage (S,) multiplied with the thickness provides the storage coefficient:

S = bS,.

(1.4-12)

The specific yield (n,) characterizing the storage capacity of an unconfined aquifer is generally determined as the volume of water drained from a completely saturated sample related to the bulk volume of the latter. It is supposed that this parameter is equal to the amount released or stored in a column of unit horizontal cross section, if the position of the water table changes by a height of unity. The process occurs at the surface of the gravitational ground-water space, therefore, this parameter cannot be related to the thickness of the aquifer. According to its definition, the storage coefiient is composed of two parts: the change of the volume of water caused by the modification of pressure, and the change of the pore volume, where the water is stored:

s = s w + s,.

(1.4-13)

The compressibility of water is expressed by the change of density with pressure, which is proportional to the density itself (1.4-14) where B ( w e f i i e n t of compressibility) is the proportionality factor; for pure water having a temperature of 2OoC at atmospheric pressure, its numerical value is /3 = 4.6 x lo-* mm2/N.The maw being unchanged, the modification of V, volume of water can be expressed from Eq. (1.4-14):

The volume of water in a saturated layer is equal to the volume of the pores; thus in a unit volume of the aquifer it is equal to the numerical value of porosity. In a column of unit area of a homogeneous layer having a thickness of b, the total change of the water volume can be calculated con-


155

sidering the total pore volume nb, because all the parameters are constant:

r31=

- Bnb ;

[dV,]:

= - Bnbdp.

(1.4-16)

The storage coefficient meaaures this change if the pressure difference, expressed in equivalent water column, is unity: dp/r = dh = 1 [L]. Combining this condition with Eq. (1.4-16), the first component of the storage coeflcient can be determined Sw = - - y n b .

(1.4-17)

The process of compression, which is well known in soil mechanics, may be used for the determination of the second component of the storage weflcient. For simplifying the mathematical description of the phenomenon, some approximative hypotheses have to be accepted. The first is the supposition of the linear vertical compression of the layer, neglecting the possible horizontal displa,cementof particles, aa well aa the shearing stress caused by unequal comprevsion along two neighbouring vertical sections. Loading the sample gradually, relationships between the load and various physical parameters of the soil (specific linear compression, dry specific

Fig. 1.4-2. Graphs representing the results of experiments investigating compression (KBzdi, 1972)

156


weight, water content, porosity, void ratio) can be observed and represented (Fig. 1.4-2) (KBzdi, 1972). (Itis necessary to note here that the pore volume is generally characterized in this investigation with the void ratio instead of porosity, because the latter is given aa the ratio of pore volume related t o the bulk volume, and thus compression modifies the basic value aa well; void ratio, on the other hand, is the ratio of pore volume and the volume of

I

-\

Fig. 1.4-3. Sketch showing the symbols used in the interpret,ationof the relationship between load and compression

solid matrix, the latter remaining unaffected by compression.) Terzaghi (1926, 1943) established an empirical relationship, according to which the change of void ratio over the change of load is inversely proportional to the value of load ( p ) increased with a surplus load ( p o ) ,determined experimentally (Fig. 1.4-3): Ae - de 1 -- - - = = a =

4

dP

C(P

+ Po)

(1.4-18)

*

The coefficient of consolidation a can be expressed as a function of the modulus of compressibility

(a):

(1.4-19)

dP Ah where M = - , and E = - is the relative change of the vertical length of de h the loaded sample (the interpretation of Ae and Ah, as well as a comparison of the two types of compression curves is shown in Fig. 1.4-4; some informative values of M are given in Table 1.4-1). The integration of Eq. (1.4-18) gives Z’erzughi’s equation of compression [Eq. (1.2-16)].


157

Table 1.4-1. Informative values of the modulus of compressibility (KBzdi, 1972) M [kp/cm'I

I

Type of the sample

1

A) Coarse grained sPdiments

Sandy gravel Sand Very fine sand (mo)

Condition of the sample

I

IOOSe

300-800 100-300 80-120

I

(B) Cohesive sedimenls

Silty very fine eand silt Light clay Heavy clay Organic silt Organic clay Peat

800-1000 300- 600 120- 200

compacted

1000-2000 600- 800 200- 300

Condition of the sample

soft

1

I

plastic

60-80 30-60 20-60 16-40

100-160 60-100 60- 80 40- 70 6- 60 6- 40 1- 20

hard

160-200 100-160 80-120 70- 120

pressure, p [ k N / m 2 ] 200 300 400

500

pressure, p [ k N / m 2 J 200 300 400

500

400

0

I

medium

1

very hard

200-400 160-300 120-200 120- 300

0.8 t

0

m I

I

I

I

I

I

I

--+-+---Fig. 1.4-4. Comparison of the representation of compression as a function of the void ratio or the change of linear size

158


The next problem to be discussed is the way of application of the listed equations to calculate storistivity. I n a loose clastic aquifer the weight of the overlying layers (the latter related to a unit horizontal area is characterized with the total vertical stress a) is transferred only partly grain by grain (this part of the total stress is the eflective stress, a);the other part is balanced by water pressure (neutral stress, u). Creating depression ( A h ) by draining the layer, water pressure decreases, and therefore the effective stress will increase, while the total stress remains unchanged: UZ=U~-YA~; a = u1

+ al = ug+ ZZ;

(1.4-20)

+ ydh;

-

az = Z1

(sufbes 1 and 2 indicate the condition before and after draining, respectively). Eq. (1.4-20) shows that the decrease of the pressure head of water increases the eflective vertical stress and this change can be regarded as a n external load causing compression : d p = da = y d h .

(1.4-21)

According to the definition, the second component of the storage coeficient is the change of the total pore volume in a column having a unit horizontal area and a height equal to the thickness b of the layer, if the pressure head is modified with unit height. In a unit volume of the layer, the pore volume is equal to porosity. Its change over the change of pressure can be expressed from Eq. (1.4-18): dn - - dnde __-

-

dP

dedp

dn =

(1-n)2

.

C(P+PO)’

(1 - n)2

+

d p;

(1.4-22)

C(P Po) or for unit change of the pressure head:

where the pressure is equal to the effective stress ( p = Zl)in the initial state. Thus the final form of S, is b

where

b


159

is a special variable, combining all the depth-dependent parameters in one factor. Considering the numerous uncertainties influencing both the process itself and the parameters, as a &st approximation it is a generally accepted supposition that this factor is independent of the depth. The form of the storage coefficientusually published in the literature (Bear et al., 1968; Bear, 1972) is based on this hypothesis: S = y b(a - np). (1.4-24) I n highly compressible layers the effect of water compressibility can be neglected, but the change of both porosity and effective stress along a vertical section has to be considered, This can be done either by integrating Eq. (1.4-23), or in tabulated form (see Table 1.4-2), supposing the constancy of the parameters within short stretches of the vertical section (Kovhcs, 1972b).

Q6

D

I

ZDO

400

EUO

BOO

,

ill00 0 20

50 /OO 200 500 U80 effective pressure, d [kiV/m2J effective pressurer B [kN/mzJ Fig. 1.4-6. Compression curves represented on arithmetic and logarithmic scales

A further problem is caused by the fact that the layer is not completely elastic. Thus the extension following the increase of neutral stress is not equal to the compression caused by the decrease of water pressure of the same size, and the first loading creates greater compression, than reloading (Fig. 1.4-5). Although the definition of the storage coefficient supposes that the amounts of water released from, or stored in the layer as a result of a unit change in the pressure head are equal t o each other, it is quite evident from the figure that this statement is only a very rough approximation. When solving practical problems in seepage hydraulics, it is impossible to analyze the whole history of loading and reloading of the layer, therefore, a.n average parameter is acceptable to characterize the storage capacity of aquifers. Detailed analysis of compression and the direct application of soil mechanical results are also hindered by the time lag occurring between loading and the total development of compression. The curves representing the relationship between changes of both load and void ratio (see Figs 1.4-2 - 1.4-5) were

160


obtained with gradual loading, waiting after all changes of the load until no further compression or extension could be observed. When investigating the time-dependent process called consolidation, many further parameters and conditions have to be considered. A mathematical treatment of the problem needs therefore the application of further simplifying approximations. For showing the complexity of the problem to be solved, the representation of the most simple example of linear vertical compression will be continued here, supposing that the a parameter in Eq. (1.4-23) is constant, and thus the relationship between the change of porosity and that of effective stress can be expressed from Eq. (1.4-22), also considering Eq. (1.4-20): dn _ _- _ _dn = a . d3 du

(1.4-25)

Accepting the validity of Darcy’s law the specific flow rate i n verticul direction crossing a horizontal area of unity can be calculated as follows: (1.4-26)

because the hydraulic gradient can be determined as the change of pressure and potential energy along the z axis (the negative sign indicates, that the flow is directed upwards, while the z axis downwards). The modification of the flow rate within an elementary dz stretch expressed from Eq. (1.4-26) is (1.4-27)

The flow rate can increase only if there is a source in the elementary volume, and in the present case only the decrease of the pore volume can be such a source. Thus the change of porosity i n time should be equal to the change of flow rate along an elementary length. This condition combined with Eq. (1.4-25) provides the following relation: aq -

K 8%

a2

y

a22

- -an - . - -ap;

au

at

at

8% c-=--.* a22

au at ’

where c=-*

K

(1.4-28)

YQ

Thus finally a differential equation is obtained for describing the change of neutral stress and consolidation, which can be solved by integration, if the

boundary conditions are known at both horizontal boundaries of the layers. This condition adds to the compleJrity of the process, indicating that the investigation of consolidation together with that of storage capacity cannot be limited to one layer, because the neighbouring formations may recharge or drain also the aquifer in question.


161

This short summary of consolidation clearly indicates that the storage coeficient, influenced by the compressibility of the layer, depends mainly on the following factors: (a) Compressibility of water (B); (b) Compressibility of the solid matrix ( a , being the function not only of materid constants, but also of parameters describing the initial state of the layer, which may change with the depth of the point investigated; thus the

time. t

Fig. 1.4-6. Characteristic curves showing the development of consolidation u = const. hypothesis is only a rough estimation);

(c) History of previous loadings of the layer; (d) Time elapsed between the start of consolidation and the time point for which the storage capacity is qletermined; (e) Hydraulic conductivity of the layer ( K , the effect of which can be characterized with the time dependency of the absolute or relative compression, the former being the change of the height of the investigated ssmple, and the latter is the same value related to the total compression after the completion of consolidation. Fig. 1.4-6); (f) Boundary conditions (hydraulic conductivity and pressure conditions of the neighbouring layers) and initial condition (pressure distribution in the layer in question) prevailing along the boundaries of and within the investigated layer [for demonstrating this influence Fig. 1.4-7 shows the dependence of relative compression on time, considering three different boundary and initial conditions; after KBzdi (1972)].

As it has been mentioned, when the storage capacity of a shallow unconfined uquifer is investigated, the effect of compressibilities of both water and the solid matrix can be neglected, compared with the amount of water released from, or stored in the pores following the change of the position of the water 11

162


time farfor 3 O ! r

0.2

0.4

I

I

0.6 I

oe

LO

I

I

T= Ct h2 1.2 14 I J

layer drained synmetrical/y at bffofh sides

x

lagers drained on& along

n.

'

e-

.? 40 h

I

\x

initial candii'ion :

triangular pressure diStribuTiffn.?empressure

zero-pressure at tbe basis Fig. 1.4-7. Relative compressionas a function of time for different boundary and initial conditions (KBzdi, 1972)

table. To justify this statement, the values of the two components were calculated, supposing that a depression of 10 m is created in a layer, 200 m deep, bordered below by an impervious, non-deformable formation (KovAcs, 1972a). Two different sediments were considered (loose fke sand, and very compressible clay) and the physical parameters of soil were selected accordingly. The results are shown in Table 1.4-2. Although it is seen that the volum e of water released from the pores immediately after drainage is less than the total amount stored above the level of depression, the numbers indicate that - even in a thick layer, such as the one investigated - the effect of consolidation is relatively small, especially in the case of pervious formations, which are important in practice.*The statement according to which the storage capacity of unconfined aquifers should be characterized with the specific yield, seems to be acceptable. Considering the definition of specific yield, this parameter can be calculated aa the difference between porosity ( n ) and the volumetric water content retained i n the sample against gravity ( W , ) [the latter used to he a1)proximated with Lebediev's maximum molecular water capacity ( Wmol),or field capacity (WfJ as well]: (1.4-29) n, = n - W,. The difficulties arising in connection with the application of this concept are caused by the incomplete physical interpretation of the water content in question, and by the uncertainties of the respective memuring methods. The most commonly applied method of determining both specific yield and water retention capacity of the sample is column drainage, when a column composed of the material to be investigated is completely saturated,


163

Table 1.4-2. Comparison of water amounts gained by water releatie from the pores and by consolidation, respectively

(a: natural condt tion

(0)

drained condfflon

surface wafer ladle

I Total available water Water stored above the level of depression

Water gained by consolidat ion

in [ml

in [ml in [yo] of the total available water

I 4.00 3.71

93.2

6.26 4.88

78. I

in [m]

0.29

1.37

in [yo]of the total available water

7.2

in [m]

0.35

21.9 __ 1.60

__

Expected land subsidence

may

and the volume of water drained from the sample is then measured, determining the water content retained by the sample as well. Schoeller (1962) collected numerous data from the literature representing the results of such measurements. This information together with some recently published materials are summarized in Table 1.4-3. Representation of the data in the form of graphs (Fig. 1.4-8) clearly indicates the tendency of the relationship between water retention capacity and grain diameter (the former slightly increases with decreasing diameter t o D , 2 0.02 em, and below this limit the increase is more rapid), although the scattering of the points is very large. The data, especially those measured by Hazen, also indicate the influence of porosity (or uniformity, which is closely related t o poroHity), but the uncertainties hinder the numerical evaluation of this relationship. 11*

Table 1.4-3. Specific yields of various samples determined by column drainage Diameter given by

Effective diameter

(loefflcierltof

Porosity

unirormity

dcalated orestimated

[=I

4 [=I

0.003

Weter retention

specie0

field

-pacity

w,

References

n

U

0.0061

0.44

2.3

0.19

0.26

0.006 0.017 0.036 0.048 0.14 0.6

0.0102 0.0272 0.0876 0.0864 0.252 0.760

0.42 0.42 0.326 0.40 0.416 0.46

2.3 2.0 7.8 2.4 9.4 1.8

0.16 0.11 0.096 0.080 0.076 0.070

0.26 0.31 0.23 0.32 0.34 0.37

0.0204 0.036 0.009 0.139

0.020 0.036 0.070 0.140

0.413 0.406 0.391 0.382

near unity

0.061

0.302 0.362 0.360 0.346

Wollny (1886)

0.0082 0.0118 0.0166 0,0186 0.0473

0.008 0.012 0.016 0.018 0.060 0.008 0,012 0.016 0.018 0.060

0.397 0.406 0.408 0.401 0.389 0.397 0.406 0.408 0.401 0.389

nearunity

King (1898)

0.069

0.331 0.366 0.367 0.366 0.366 0.208 0.268 0.294 0.330 0.320

0.014 0.03 0.07 0.14 0.30

0.404 0.406 0.418 0.404 0.401

near unity

0.061 0.048 0.040 0.028 0.032

0.363 0.367 0.378 0.370 0.369

Atterberg (1911)

Comments

na

Haeen (1892)

The characterktic diameter k gken in the form of D,,

0.0139 0.0306 0.0094

0.139 0.303

0.044

0.041 0.037 0.066 0.060 0.041 0.035 0.034 0.180 0.148 0.114

0.080

The e*perimenb were made with elmoat homodisperee samples

IF

zB

?i

t 5'

The experiments were made with almost homodisperse samples

Directly published values for specific yield and water retention capacity Parameters recalculatedfrom dstapubliehed for the ahracterimtion of timedependency of the process

The experiments were made with almost homodisperse eamples

d

6w-

g @

9

s

H

7i 9%

0.0014 0.0047 0.0067 0.022 0.046 0.072 0.112

0.0016 0.0060 0.0060 0.022 0.046 0.072 0.112

0.422 0.406 0.381 0.372 0.362 0.360 0.366

0.0436

0.36-0.38

near unity

0.282 0.174 0.134 0.062 0.043 0.040 0.038

0.140 0.292 0.347 0.320 0.309 0.310 0.317

Zunker (1930)

The experiments were made with elmoet homodisperse samples, although the smaller porosity of more coarse materials indicates thet their coefficient of uniformity was higher than three

Johnsonetd.

GMeSs beads

~~

Given in the form of grain-size distribution

0.0120

0.39-0.41

0.0016

0.40-0.42

0.0687

0.36

0.016* 0.031** 0.120*** 0.030* 0.043** 0.280** * 0.062** 0.290*** O.W* 0.003** 0.186***

0.0212

0.37

0.0114

0.40

O.OSl* 0.090** 0.142*** 0.076* 0.100** 0.213***

0.36 0.36 0.33 0.36 0.24 0.26 0.36 0.38 0.36 0.37 0.11 0.13 0.36 0.38 0.11 0.13 0.310 0.287 0.164 0.289 0.280 0.228 0.326 0.300 0.187

(1963) Prill d al. (1906)

* at the top of sample ** around the top of the open capillery Zone *** average for the column

20 Del Monte sand

Fresno medium sand ~ ~ t u ralluviei el send (Holbrook, Arizona)

166


a32 4r sF2 aze

0.24

B P J2 5 0.20 %

s

* % s

,

I

de t P r m in P d an average as the moisture mffted of of the column the dry part ofthe rofumi

+ o.--- .-o

A----A

++ .-.-.0

0

.............x

Johnson ef al. King

Hazen

x

...-..-..+

0.16 0.12

2 c2

2

y

am

L

2

0.04

0 effective diameter, --cirFig. 1.4-8. Relationship between water retention capacity and grain diameter

The most probable reason causing the large discrepancy between the meavured data, is the development of the soil-moisture retention curve i n the drained sample. The U.S. Beologicul Survey, in cooperation with the California Department of Water Resources, made a series of experiments (Johnson et al., 1963; Prill et al., 1965), the results of which are also listed in Table 1.4-3, showing the possible large differences between the parameters determined in different ways from the same observations. The soil columns were drained either by gravity or by suction. (In the latter case a semi-permeable film was placed below the column, the water level was maintained at a given height, and it was connected with the experimental column with a tube). I n Fig. 1.4-9 the results are compared, showing the correctness of the previous statement, t h a t field capacity (in this case water retention capacity, being here equal t o field capacity) is the function of the elevation of the investigated point above the water table. The development of the soilmoisture retention curve is also well demonstrated in the figure. The time-dependent character of the specific yield was also investigated in the framework of USGS’ experiments. It was found that even 100 hours were not enough for the attainment of complete dynamic equilibrium, which can be characterized with hydrostatic tension distribution (the tension being everywhere equal t o the product of the height above the water table and the specific weight of water). Figure 1.4-10 shows the change of tension as

167


p..2,?

g 220

$& 0

5

280

10 /5 20 25

moisture contentin percentayeof r o h e

5

moisture content in percentage of volume

10

i5

20

25

moisture content in percentage of mlume

Fig. 1.4-9. Development of soil-moistureretention curves in drained soil columns

time elapsed ChoursJ I 2 5 /0/53i 7099

20

40

60

80

i00

i2G

lension in cenfimeiersof wafer Fig. 1.4-10. Graphs showing the distribution of tension during drainage of 0.12 mm glass beads (Johnson el al., 1963)

168

1 Fundamentala for the investigation of seepage

a function of time, by way of example, as memured in one of the experimental columns. It can be seen that in a very f i e sand dynamic equilibrium is achieved only in the lower 80 cm after 100 hours, while at the top-of the column (about 150 cm above the water table) the hydraulic gradient remdns almost unity. At the same time, saturation of the sample is very OW at this elevation and, therefore - aa it is well known and will be proved later on in Chapter 2.3 - unsaturated conductivity is also very low. Thus the IOU

eo 60 1532rm long column

+

40

20 0

fo

130 Ma {jo time ChuursJ Fig. 1.4-11. Relationship between speoific yield and drainage time on the basic of USOS data

a

lo

20

30

40

50

70

ED

90

IOO

iiu

120

amount of water transported by this gradient is negligible. This supposition can also be substantiated by the graphs showing the rehtionship between the drained amount of water and the time elapsed from the beginning of drainage (the graphs were constructed also from the data of USGS’ experiments; Fig. 1.4-11). Among the previous investigations, King’s measurements also afforded data representing the dependem of 8pecific yield on time (Fig. 1.4-12). The length of his experiments waa two and a half years, and water wa,s still released from the sample at the end of this very long period. The result is in good agreement with the conclusions mentioned previously, showing that the process is very slow, but the amount drained at the end of the prolonged experiments is almost negligible. Another interesting experience can be gained from King’s data as well: i.e. about 13-18% of the total stored water was lost during the long experiments, probably through evaporation (the sum of the drained water and the soil moisture retained at the end of the investigation in the sample is smaller than the total pore volume). As a consequence of the numerous uncertainties, occurring in connection with the determination of both specific yield and water retention capacities, as noted previously, there have been attempts in the literature to give average figures as rough estimations for the parameters in question. Such

169


proposah are summarized in Fig. 1.4-13 (Castagny, 1967; Major, 1972b; De Wiest, 1968). Another way to avoid the uncertainties caused by both the development of the soil-moisture retention curve and the slow process of drainage, is the use of a centrifuge for dewatering the sample. The retained moisture content may be numerically different from water retention measured with column drainage, thus i t is called centrifuge moisture equivalent. This parameter is also influenced by many factors. It was necessary therefore to standardize 0.5 9,

B E?

B

R s?

0.4

porosity

0.3

specific yield

s?

0.2

2

0.1

wafer retention capacity ,

chg fine sand sandy grave/ boulder silt sand marel Fig. 1.4-13. Average porosity, water retention capacity and specific yield as a function of grain size

170


its measurement. The American Society for Testing Materials (1958) defked the centrifuge moisture equivalent, it8 follows: “water content retained by a soil, which has been first saturated with water, and then subjected to a force, equal to one thousand times the force of gravity for one hour”. The detailed investigations performed by USGS (Johnson et al., 1963) also included an analysis of the moisture content measured by centrifuging. They have found that this parameter depends on the

farce times gravity

o

300

600 goo m a 1500 force Times gravflg

i000

z

90

Fig. 1.4-14. Effect of the acting force on the centrifuge moisture content (Johnson et al., 1963)

Temperature ; Length of period of centrifuging; Size and structure (porosity) of the sample; Size of the acting force (angular velocity); and, since the iorce changes with the distance from the centre of rotation, the moisture content also changes within the sample. This latter effect can be well demonstrated if the meaaured moisture content is represented as a function of the acting force (Fig. 1.4-14). The relationship can be characterized by considering that the tension created by rotation (hCR)expressed in centimeters of the equivalent water column depends on the angular velocity ( 0 ) [T-l], it8 well aa on the distances of both the bottom of the sample ( r l )and the specified point (r2)from the centre -

171


of rotaticm measured in centimeters: (1.4-30)

In Table 1.4-4 the centrifuge moisture equivalents (data measured with the standardized method) are listed for each sample investigated, and in the further columns the range of moisture content is given within which the parameter was found, when one of the influencing factors waa modified. The data listed in Table 1.4-4 are also graphically represented in Fig. 1.4-15, similarly to the water retention capacity. The relationship with grain diameter shows the same tendency, although the numerical values are different. The advantage of the use of a centrifuge is that the process is much quicker than column drainage, although the scattering of the measured data cannot be decreased considerably. At the same time, the real physical meaning of specific yield is lost in this way. After demonstrating all the difficulties in connection with the determination of the specific yield, it is necessary to recognize that most of the problems are caused by the insufficient physical interpretation of the term. In an unconfined system, the capillary zone develops always above the water table, hence the dynamic balance i s characterized by the soil-moisture retention curve. The correct way of determining the specific yield i s , therefore, to calculate the diflerence of the amount of water stored above a reference level (chosen arbitrarily, lmt below both the original and the modified water table), if the change in the elevation of the phreatic surface is unity, according to the definition of specific yield. Redistribution of the soil moisture following a change i n the position of the water table needs some time, the length of

g

70

6 ,$D

o

8 &, 50

+ uncontrolled temperafure

%

s

. B %

standard measurement

I scattering caused bg the modification of some of the influencing condltions

40 1

B

9 39 b

1

3.% zn B

Q

3 I0

~

5

&

s o

~

8

Range of ds*r scattering under vdoas hiSki8l

Db

la1

(I

modllied conditions

ZZ?

U n C o n W d

equivalent

Mpsnave

-

Del Monte send 80 meah 30 meah 20 mesh

0.0166 0.0300 0.0687

2.42 1.40

Glese be& 0.47 mm 0.12 m m 0.036 m m

0.0436 0.012 0.0016

1.11 1.30 1.90

69 CAL 191-244

0.0212

4.0

67 CAL 122

0.0026

6.43

4.0

67 CAL 66

0.0045

2.66

4.2

67 CAL 110

0.0030

26 00

6.4

67 CAL 118

0.0022

8.00

6.0

59 CAL 268-261

0.0017

16.66

9.8

13.0

69 CAL 261-262

0.0013

22.60

10.6

13.1

11.0

1.4 0.4 0.6 1.2 1.3 3.4

1.4 0.5

1.3 1.9

N

-

1.2 0.3 0.4

1.6 0.6

1.8 0.6 0.6

1.9

0.9 N 1.3 1.0 1.7 3.1 3.9

2.2

1.6

--

6.0

7.0

13.6

69 CAL 263-264

0.00084

61.11

59 CAL 137-138

0.00071

33.00

13.2

14.4 N 16.1

67 CAL 72

0.00227

66.00

14.1

68 OKL 16

0.00067

26.66

16.6

-

Loess (Bonny Dam, Colo.)

0.00074

6.16

17.3

67 CAL 106

0.00071

30.86

17.G

18.0

-

2.0

13.6

12.4

17.3

---

14.6

8.0 21.1

Loess (Trenton, Nebr.)

0.00076

14.0

4.69

19.0 20.6 18.8

N

20.0

69 CAL 248

0.00070

30.00

20.2

68 OKL 42

0.00091

18.76

20.4

67 CAL 166

0.00092

30.40

21.6

69 G A L 130-131

0.00038

16.00

23.1

67 G A L 139

0.00149

16.66

24.8

26.6

67 CAL 162

0.00061

38.88

28.8

21.1

67 GAL 162

0.00029

66.66

30.0

20.2

;P

67 CAL 173

0.00048

13.33

32.3

27.2

F

68 ARK 146

0.00014

4.16

37.2

67 CAL 112

0.000 17

6.43

40.9

34.6

%

67 CAL 123

0.00063

11.10

43.2

32.8

67CAL

2

0.00020

8.33

46.4

41.4

f 3

67 CAL 206

0.00013

8.33

49.3

46.9

Kaolin (N.F)

0.00029

3.36

49.6

Kaolin (EPK)

0.00007

2.86

61.2

42.0

N

48.0

Fuller's mirth

0.00048

16.66

109.0

76.0

N

80.0

N

23.4 12.4

21.9

N

26.9

F

31.0 29.4

N

N

36.0

33.6

f

5a

174

1 Fundamentals for t,he investigation of seepage

which depends on the hydraulic conductivity of the 1ayer.Thus the comparison of the stored amounts has to be made after the development of the dynamic equilibrium, hence the determination of specific yield can be reduced to the calculation of the difference between the areas of two soilmoisture retention curves (Fig. 1.4-16). Let us indicate two positions of the water table with sufhes 1 and 2, respectively. It is quite indifferent, which is the original condition, because the specific yield must have the same numerical value whether lowering or raising the water table. To f d i 3 the condition required by the definition, the distance between the two water tables is equal to unity. The stored amount of water in a column having a cross section of unit area can be easily calculated above the datum. Executing the calculation for both positions of the water table, and forming the difference of the two determined values, the physically correct specific yield is obtained:

because 21

- z2 = 1.

Comparing Eq. (1.4-31) with Eq. (1.4-29), i t is readily seen that t,he difference of the two expressions to be integrated is equal to the water retention capacity. Although this concept is theoretically correct, difticulties are raised, however, when it ix applied in the practice. The t w o most important diffi-

c Fig. 1.4-16. Calculation of the specific yield from the difference of two soil-moisture retention curves


175

culties are caused by the hysteresis of the soil-moisture retention curve and by the time lag between the modification of the water table and the final development of the dynamic equilibrium. Further problems are encountered when the unsaturated zone is layered, but details of this will not be discussed now. As it was explained in Chapter 1.3, the relationship between tension and water content is not unambiguous, owing to the phenomenon called hysteresis. Dynamic equilibrium develops at a lower position, if the soil-moisture zone is wetted from the direction of the water table, while the highest curve can be achieved if the zone had been completely saturated previously and drained vertically later. Between these extreme positions the balanced condition can develop a t any elevation. In Fig. 1.4-17 four different cases are represented:

Fig. 1.4-17. Influence of hysteresis on the specific yield

176


(a) A wetting process is followed by further riee of the water table, thus both balanced conditions are characterized by the lower stretch of the hysteresis loop. The curves are pardel, and their distance is unity, equal to the modification of the water table. (b) The original wnditwn w m achieved by drainage, and the lowering of the water table is investigated. Both soil-moisture retention curves follow the upper branch of the hysteresis loop, thus their distance is equal to case (a). (c) After wetting the soil-moisture zone,the water table is lowered once again. The lower limit of hysteresis describes the initial condition, and the upper one characterizes the final state. Thus the difference between the two areas of the retention curve is smaller than that calculated from the average curves. Theoretically the difference can also be negative, but in this case the upper stretch of the hysteresis curve is never achieved; the dynamic equilibrium develops at an intermediate position, hence the lower limit of the calculated difference is zero. (d) The case opposite to (c) is the rise of the water table in a previously drained system. Now the final state is characterized with a low retention curve, and the initial condition with a high one. The result is also a decrease of the difference between the two areas, and its lower limit can be again zero.

Further special cases may also be expected (e.g. wetting of the soil-moisture zone previously, and rise of the water table afterwarrds due to infiltration from the surface) when the distance between the initial and final retention curves is greater than unity, its upper limit being the sum of the change of the water table and the thickness of the hysteresis loop. It is obviously very difficult to investigate each of the above processes separately, when the difference between the actual and average values of the specific yield is to be determined. Only one possible way can be proposed: to accept the specific yield calculated from the average soil-moisture retention curves, and to neglect the influence of hysteresis. Realizing the uncertainty introduced in this way, some further approximations can be applied to derive a formula amenable to mathematical treatment, and suitable for describing the relationship between water retention capacity and the physical parameters of soil. When analysing the soil-moisture curve, it was shown that the relationship between tension and moisture content in the adhesive zone can be well approximated with a hyperbola of sixth order [see Eq. (1.3-24)]. At this very steep stretch, the curves belonging to theinitial and the final state practically cover each other (Fig. 1.4-18). Accepting the use of the average curve (without hysteresis) for the characterization of both conditions (before and after the modification of the water table), the difference of the area8 covered by the retention curves can be calculated a,s the free gravitat i m l porosity at the adhesive zone multiplied with the change of the elevation of the water table. The latter being equal to unity, if the purpose is the determination of the specific yield or water retention capacity, the only remaining problem is to select the numerical value suitable for the characterization of both water content and gravitational porosity at high tension.


177

. surface

Fig. 1.4-18. Sketch for the interpretation and calculation of the water retention capacity

It can be proposed that the parameter considered in the calculation should be that specific amount of adhesive soil moisture, which belongs to the highest capillary rise. The water retention capacity can thus be calculated by substituting the possible highest capillary rise [see Eq. (1.3-28)] into Eq. (1.3-25):

because

h,,,,

1 - n a . = 0.11 --, n Dh

I%(

or the same parameter expressed with the rate of saturation: so = 3.6 x 10-3

;

[l n)518

-

(1.4-33)

In Fig. 1.4-19 the relationship given by Eq. (1.4-32) is compared with the measured data listed in Tables 1.4-3 and 1 . 4 4 , and represented in Figs 1.4-8 and 1.4-15. The comparison shows that the proposed equation provides an acceptable and reasonable compromisebetween the various meaaurements, giving the lower limit of data measured with column drainage (which is acceptable considering the development of the capillary zone), but running above the centrifuge moisture equivalent. At the same time it affords an easily applicable method for the calculation in practice of the water retention capacity and specific yields as the ficnction of the physical parameters (n,Dh, a) of soil. 12

178


effective diameter, Dh [em/ Fig. 1.4-19. Comparison of the proposed theoretical value of water retention capacity with meesured data

Naturally, the time lag preceding the development of the dynamic equilibrium also disturbs the direct application of the parameter calculated from Eq. (1.4-32). I n a permeable layer, the ratio of the amount of water drained or stored at the beginning of the process is relatively high; only a few percent of specific yield occurs in the later period, when the water has t o be transferred through the upper part of the column, where the unsaturated conductivity is very small owing to the low saturation. In fine grained material the hydraulic conductivity is relatively low, therefore the whole process is considerably prolonged. Figures 1.4-1 1 and 1.4-12 give sufficient information about the magnitude of the expected time lag, although it is also mentioned in the literature that the delay will be longer if the balance is achieved by wetting the unsaturated zone than in the case of drainage (Smith, 1933).

1.4.3 Analysis of horizontal ground-water flow for the determination of the vertical water exchange The second component of Eq.(1.4-10) to be investigated is the horizontal ground-water flow. For a complete study of this movement, the knowledge of some basic concepts of seepage hydraulics is necessary, which will be dis-


179

cussed only later on in this book. This very simple form of flow can be analysed, however, if some approximative hypotheses are accepted as starting conditions: 1. Seepage velocity is linearly proportional to the hydraulic gradient [Darcy’s law, see Eq. (1.1-9)]; 2. The flow is horizontal, therefore, the velocity is constant along a vertical section, hence the numerical value of velocity can be calculated as a function of the slope of the water table (Dupuit’s hypothesis). Following Kamensky’s (1940, 1943) investigations, let us consider the water balance of an elementary part of the gravitational ground-water space, whose lower boundary is a horizontal impervious layer, and which is bordered in the horizontal plane by a rectangular oblong (Fig. 1.4-20). There are five observation wells for recording the position of the water table, one inside the oblong, and four outside its borders. The wells form a rectangular cross, and the borders are fixed that way that they divide the

12*

180


distances between the internal and external wells into halvee. Let b, indicate the length of the different sides [ b , = b, = 1

L

1

1

. . . b,

(Ax, + Ax4) ;

Y

b, = b, = - ( d X , + dX,) the time dependent areas of the vertical cross 2

sections at the borders are then

where H,(t) indicates the elevation of the water level in the i-th well above the impervious lower boundary at the time t. The average slope of the water table can also be calculated from the data of the observation wells:

I , = AHlX ( t ) ; I , = AH,, ( t ) . 4 A% ’

(1.4-351

where dHi,(t) is the difference of water levels registered at time t in the i-th well and in the centre well, respectively [AH,(t) = H,(t) - H,(t); . . . * * AH444 = H4(t) - Hp(t)l. Accepting the two basic hypotheses mentioned previously, the flow rates crossing the vertical boundaries of the investigated prism can be calculated a5 follows:

-

There is, however, a H t h bordering surfme of the prism, through which a flow can develop, i.e. the waater table. Its area is

A = b, b, ;

(1.4-37)

and the flow rate can be expressed here as the product of the area and the specific vertical water exchange [ ~ ( t ) ] :

&o(t)= Ae(t)*

( 1.4-38)

Investigating a finite time period (At = t, - t,; and t, > t,), the water balance of the prism requires that the product of the elementary time unit (dt) and the sum of flow rates summarized for the entiretimeperiod, At, should be equal to the change of the stored water amount, which is equal to the change of the position of the water table [characterized with the recorded water level (or ground-water depth) in the central well AHot = HO(t3)- Ho(t,)] mul-

181


tiplied with the horizontal area of the prism and the specific yield of the layer:

%[ &o(t)+ Z & i ( t )1dt 4

tl

= m o t An,

-

(1.4-39)

4-1

[Eq. (1.4-39) is a special form of Boussinesq's differential equation (which will be discussed in detail in Chapter 5.4) expressed with bite differences.] Substituting the time-dependent variables with their averages, which can be regarded aa the values of the function in question belonging to a point of time tz within the investigated period (tl < tz < t3), and applying transmissibility [which parameter is equal to the product of the depth of flow and conductivity

the simplified form of Eq. (1.4-39) can be given as follows:

The geometrical parameters (bi and x l ) and those calculated from water level records ( H , and H o t )can be determined with relatively high accuracy. The methods applicable for the calculation of the specific yield were already discussed in the previous section. Thus transmissibility is the remaining variable the knowledge of which is necessary, if Eq. (1.4-40) is to be used for the direct calculation of the time-dependent water exchange through the water table. If the position of the impervious lower boundary is known, only hydraulic conductivity has to be determined as a basic data. The equation can be also used for the calculation of transmissibility by substituting recorded water level data of a period when the vertical flow was surely negligible, E(t) = 0, (e.g. in winter, when frozen top layer hindered both infiltration and evaporation). If the cross of the observation wells is fixed so that one line of wells (1, 0, 3) is parallel to the dip of the water table, and the other (2, 0 , 4 , being perpendicular to the former) is parallel to its strike, there is no flow in the second direction. The data of the wells with even numbers (2, 4) need not be taken into account, and the investigation can be restricted to the analysis of the water balance within a stripe of unit width:

There are several other proposals to simplify the calculation, e.g. to choose very short At periods and substitute the time-dependent variables with their values belonging to the beginning of the period (Kamensky, 1943); to use

.

182

1 Fundementah for the investigation of seepage

continuous time-dependent functions and divide the solution into three parts, because in this case the mathematical functions approximating the separated parts can be determined in closed form by integration (Szdkely, 1973); to give graphical interpretation of the solution with finite differences (Lebediev, 1963), etc. Other authors have made efforts to generalize the

Fig. 1.4-21. Map of the Koml6si Imre Ground-waterResearch Station

method, making it applicable for an existing irregular network of observation wells, when the geometrical conditions used as the basis of the derivation are not satisfied (Rowe, 1955; Major, 1972b). It is a general opinion that the solution of water balance equations of this type with computers does not raise any special problem and, therefore, the most suitable form of the relationship has to be determined always according to the local conditions (distribution of wells within the network, type and reliability of the recorded data, information concerning the depth of flow, hydraulic conductivity and transmissibility, etc.). This is the reavon why the theoretical basis of this method will not be dealt with further on, but a practical example will be presented here, to illustrate the special features of vertical water exchange between soil moisture and gravitational ground-water zones.

183


In an experimental area (KomMsi Imre Ground-water Research Station, Hungary), where long ground-water records were available from several wells, detailed investigations were made by applying the method described in the previous paragraphs (Major, 1972b). The layer containing the shallow ground water is a relatively homogeneous h e sand, the transmissibility of which is known from pumping tests. The specific yield of the sand was estimated on the basis of the water retention curves measured in the field with neutron probes. The surface is plain, covered partly with forest surrounding grassy clearings, as indicated in the map (Fig. 1.4-21). Four areas were chosen, for which the water balance of six-day intervals were determined from the records covering a period of 18 years (1954-71). Some characteristic data of the areas and the annual sums for the decade 1961-70 of the two components of vertical water exchange (i.e. I , infiltration to the groundwater and ET, evapotranspiration replenished from the ground water) are listed in Table 1.4-5. Table 1.4-5. Annual values of the components of vertical water exchange, determined for various areas by using the water balance method

I B l c

Denotation of the arem

Extension of the area [m2]

10 000

260 000

13 640

9 200

89

78

68

43

Rate of the forested area

1% 1

1961 1962 1963 1964 1966 1966 1967 1968 1969 1970

188 88 148 66 131 261 163 4 137 191

319 303 287 242 199 264 346 384 366 330

196 112 183 103 191 313 231 30 188 292

260 196 196 146 146 230 286 226 219 266

346 236 319 307 460 632 472 212 291 386

139 27 19 13 16 43 37 32 60 89

277 260 347 362 669 604 611 306 346 386

40 22 18 16 0 30 14 16 36 62

42 1 478 616 669 662 844 618 384 693 690

To characterize the seasonal fluctuation of the components of the vertical water exchange, the monthly averages of both I , and E T , were calculated from the data. As an example, the parameters determined for area A using a period of twelve years (1960-71) are represented in Fig. 1.4-22. As a further example, the monthly data of an arbitrarily chosen year (1963) are listed in Table 1.4-6 for the areas A and B, and the basic data of the same year, calculated for six-day intervals, are also represented in Fig. 1.4-23 for the area A .

184

1 Fundamentals for the investigation of seepage 60 50

40

30

za la

0

Nor Jan. March May July Sept Fig. 1.4-22. Monthly averages of I , and ET, calculated for a 12-year period (1960-71) from the data of area A

There is a special condition characterizing the area investigated: just before the beginning of the recorded period a young forest was planted there, thus the influence of the growing trees on the hydrological cycle can be well observed. Since practically there is no surface run-off from the area because of the very flat and sandy surfwe, and since the change of moisture content Table 1.4-6. Monthly values of the components of vertical water exchange for 1963, determined for areits A and B Month

Monthly preoipiestion

Positive monthly amretion 1,

forarea A

1962

November December

[-I

I

forarea B

-

-

0.76

2.24

72.2 66.2 46.7 36.6 39.9 44.8 30.7 76.3 80.8 26.2

16.10 17.08 76.97 37.09 0.74

21.43 24.66 78.68 49.99 3.47

616.6

147.74

86.6 24.0

Negative monthly accretion ETI C m l

for area -4

13.67 7.43

I-KEZT 2.80 2.46

1963

JmUery Februery March April

W Y June JdY Ausust Ekptember October hnUd

EUm

2.60

182.94

-

0.86

1.26

-

-

-

14.94 66.77 76.03 61.27 33.63 22.19

8.99 67.37 69.19 32.81 20.67 10.12

287.18

195.26


Fig. 1.4-23. Annual tluctuation 01 I g ana

~

2 CaiCuIaTda ' ~

185

xor six-aaymmrvmu (ivoo;

area A )

over a year-long period is negligible, it can be supposed that the difference of annual precipitation and the yearly sum of infiltration to the water table is equal to the amount of water stored temporarily either by interception, or in the soil-moisture zone, and evaporated from this position. Thus the annual actual evapotranspiration can be calculated as the sum of this difference and the vertical withdrawal from the ground-waterspace. These parameters are listed in Table 1.4-7 together with their averages calculated for periods of six years. The averages clearly indicate the increasing trend of actual evapotranspiration, corresponding to the growth of the forest.

186


Table 1.4-7. Annual values of precipitation and actual evapotranspirationfor a period of 18 years A

m

water level in the central obl3ervation well below the wvface

Year

€ I

I4 Ann.

1964 1966 1966 1967 1968 1969

683 614 664 622 491 626

1960 1961 1962 1963 1964 1966

662 421

1966 1967 1968 1969 1970 1971

844 618 384 693 690 429

778 616 669 662

2 g

146 184 208 178 146

2 2

83

2

4

132 188 88

148 66 131 261 163 3 137 191 69

2

2

2 $:

436 430 346 444 346 442 432 233 390 468 603 621 683 466 381 466 499 370

2

3

3

2

$!

189 160 240 187 2 34 162 219 319 303 287 242 199 264 346 384 366 330 411

2

2

2

'?

$$

626 690 686 631 679 604

I

Av.

392 382 362 369 378 409

4

*

846 720

422 399 431 429 436 403

847 811 766 821 928 781

362 316 376 390 348 37 1

661 662 693

2 8

2

766

2 3

cI)

Some conclusions which can be drawn from the summarized data of the analyzed example are the following: It is desirable to select a homogeneous area for the investigation of the ground-water balance. Homogeneity should be ensured for each element of the processes affecting the regime of the ground water [structure of the water bearing layer (permeability, transmissibility, specific yield); surface conditions (type of soil, covering vegetation, slopes influencing the development of surface run-off); climatic factors (precipitation, temperature, radiation, wind velocity), etc.]. In areas covered by inhomogeneous vegetation, only the average characteristics of the vertical water exchange can be determined instead of the actual influence of the different plants. At the same time, if these average values can be correlated with the ratio of the areal extension of the various types of vegetation, the data can be used to characterize the effect of a special cover (in the example presented the influence of forest can be investigated). The water balance is influenced not only by the type of the covering plants, but also b y their phase of growth. Thus the development of trees gradually


187

decreases the vertical recharge of the ground water; this is the result of the increase of interception and soil-moisture storage. These two components are readily available for evaporation and transpiration, thus the part of evapotranspiration originating from interception and soil moisture is increased by the growth of trees. At the same time, the amount of water withdrawn from the gravitational ground water to replenish the evaporated and transpired soil moisture is also increased, probably aa a result of the development of the root structure. Hence the increase of the actual evapo15 transpiration is very rapid (in the example it surpaases the value of 10 mm/year). At the beginning of the investigated period the yearly amount of precipitation was equal to the actual evapotranspiration, but (owing to the increase of the latter) at the end of the period the average yearly deficit was greater than 200 mm. I n the investigated area, where the water table practically had no slope at all, and thus there had been no horizontal groundwater flow, the development of the forest lowered the water table and created a depression cone. The deficit i n water balance of the forest is replenished now by the ground-water flow developing along the slopes of the cone, and the surrounding arem recharge the ground water of the forest. A very rapid decrease of ET,.can be observed as the ratio of the forested area decreases within the investigated area. Such very strong effect can only be expected, if there is no ground-water withdrawal at all under the clearings. This supposition is supported by the circumstance described in the previous paragraph: the water table is considerably lowered under the forest, and thus the vertical flow directed upwards can convey only negligible amounts of water from the ground water to the shallow root zone of the clearings. It can be stated, therefore, a~ a general rule, that the depth of the water table (measured from the lower boundary of the active root zone) is one of the most important factors influencing the amount of water withdrawn vertically from the ground water. Infiltration reaching the water table shows an increasing trend with the decrease of the rate of forested areas. The most likely cause of this change is the smaller interception at clearings. At the same time, it is also probable that the moisture content is generally smaller at the end of dry periods under a forest due to greater evapotranspiration by the trees; thus, during a rainy period, a larger soil-moisture storage occurs above the water table in the forest, than outside. It can be concluded that I , is a function of both the infiltration through the surface and the available storage capacity of the soilmoisture zone. A regular seasonal fluctuation of both infiltration and ground-water withdrawal can be observed, which naturally follows the fluctuation of the climatic 1)arameters.Under the conditions of the investigated area, there is practically 110 vertical ground-water recharge during the summer half-year (from May to October). Infiltration to the water table originates mostly from autumn and winter precipitation, and reaches its maximum with a time lag after the melting of the snow. The seasonal fluctuation of vertical drainage has R more regular pattern, than infiltration. The difference between the actual graph in Fig. 1.4-22 and a sine curve used generally to describe the fluc-

-

188


tuation of climatic parameters (which has its maximum in June and its minimum in December) can be.wel1 explained by the fact that before the maximum of potential evapotranspiration the water table has a higher position than its average, therefore, a surplus withdrawal occurs as compared with the sine curve. Similarly, a deficit can be observed at late summer, because of the greater depth of the water table. It has to be emphasized that both the amount and the seasonal pattern of I, and E T , also depend on the type and the phase of growth of the ctovering vegetation. Thus the conclusions (especially quantitative statements) are valid only for the investigated case (a gradually growing forest with a few percent of clearings). For other surface conditions only rough qualitative approximations can be given on the basis of the example analyzed. However, several of the results summarized here may be applicable for the better characterization of hydraulic processes developing in flow fields influenced by accretion.

1.4.4 Interpretation and determination of the charaderistic curve of the ground-water balance

Returning to Eqs (1.4-9) and (1.4-lo), the remaining component to be investigated is the vertical water exchange between the soil moisture and ground-water zones. It wm already shown in the previous section, that its components (infiltration to, and vertical withdrawal from, the ground water) is a function of the depth of the water table below the surface. In addition to this parameter, both I, and E T , depend on some other variables, which can be divided into three main groups: (a) Climatic conditions: amount and annual distribution of precipitation and potential evaporation. The latter includes the effects of temperature, humidity, wind velocity, etc. (b) Characteristics of the surface: angle and direction of slopes, method of cultivation, vegetation covering the surface, etc. The resultant of some of these effects can be measured by surface run-off. (c) Behaviour of the soil within the unsaturated zone: geological character and structure of the upper lying layer, water retention capacity and the instantaneous soil-moisture content of the top soil. Thus the two function, in general form, characterizing the parameters in question are as follows:

Ig=

( m ; climate; surface; soil);

(1.4-42)

E T g = G2 ( m ; climate; surface; soil). The factors describing the influence of climate, surfaee and soil are timeand space-dependent variables. It has been mentioned that instead of the examination of their modification in time the values averaged for a suficiently

189


long period are used. Similarly, equalization of local effects can be assumed in a large area, thus the individual parameters cun also be averaged over the area. Regarding all variables, except the depth of the water table, 88 constants for a given area, and describing these by their average over the period investigated, Eq. ( 1 . 4 4 2 ) can be simplified: 1, = f Am); ET, = fi(m).

(1.4-43) '

These two relationships - their parameters in any case, but perhaps their structure as well - are functions of the natural conditions (climate, surface, soil), which were regarded as constants for a given site. Some general rules describing the character of the functions can be, however, determined. Infiltration through the surface is the difference of precipitation and the sum of evaporation, run-off and retention on the surface. According to Eq. (1.4-2), i t can be divided into two parts, one stored above the water table, and the other part reaching the ground water. The ratio of these two values depends on the storage capacity of the unsaturated zone influenced by the physical parameters of the soil and the instantaneous water content of the upper layer at the time of infiltration. Although the amount of the stored water changes from case to c a e and, therefore, the amount of water stored in a year also varies every year, an average of the yearly retention of water in the soil-moisture zone can be calculated. Thus graphs can be constructed representing the average yearly storage for various soils as a function of the depth of the water table (Fig. 1.4-24).

For a given soil and in an average year a series of parallel curves can be obtained graphically to represent the relationship between infiltrating recharge storage rapacity of the dry layer rm3/m3J 0

0.0J

alo

at5

azo

0.25S(Z)

Fig. 1.4-24. Characterization of the storage capacity of the soil-moisture zone function of depth

190


and the depth of the water table. The parameter of the individual curves is the infiltration through the surface, which value gives the intersection of the inEltration curve with the horizontal axis (Fig. 1.4-25). To make the mathematical treatment of the relationship more convenient, the curve can be approximated by an exponential function, whose mymptote is parallel to the vertical axis; the position of the asymptote depends

m#z Fig. 1.4-25. Relationship between idltration and the depth of the water table

on the infiltration through the surface. I n a special case (indicated in the figure with a dotted line) the asymptote is the vertical axis itself. The water balance of ground water is very complicated when the water table is near the surface, since very small precipitations may also exert an influence. Another reason why the upper few decimeters of the soil should be disregarded in the characterization of the process of infiltration is the undisturbed structure of the cultivated and the active root zone, as mentioned already in Chapter 1.3. It is advisable, therefore, to fix another point of the curve, than its intersection with the surface. For this purpose the average infiltrating yearly recharge ( I , ) has to be determined at a given level (m,). Thus the final mathematical form of the investigated relationship is ( 1.4-44) I g ( m )= ( I , - a ) exp [ - a ( m - m , ) ] + a ; where the parameters ( I , , m,, a , a ) are functions of the natural conditions. The curve representing the relationship between the amount of ground-water withdrawn and evaporated or transpired, and the depth of the water table is very similar to the curve described previously. This also decreases monotonously with depth. It is evident that the amount is approximately equal to the potential evapotranspiration originating below the land surface, when the water table is near the surface (its average depth 1)eing practically zero).


191

Actual evapotranspiration (ET,+ ETg)decreases with increasing depth of water table, and i t becomes constant at a given depth, supposed to be 2-3 m. under temperate conditions. In the case of such a water table, actual evapotranspiration is only 50-70% of the potential value, depending on the type of soil and vegetation. The total actual evapotranspiration must be divided into two parts; one supplied from the soil moisture and the other

/

/ /

I I ETw+ E G

I

I

pP

i

41

I

6

I

-X-

0

++

calculated from /ysimeefer dar3 (Najor) Blaney bare steppe oasis Kovda steppe Altovszkij Konopljancev W. Friedrich Juhisz J. (ralrulated)

1

Fig. 1.4-26. Relationship between the amount withdrawn from ground water to replenish evaporated and transpired soil moisture and the depth of t,he water table

from ground water (Fig. 1.4-26). The latter, being of interest from the mpect of the present investigation, is also a function of the depth of the water table, and there exists a level from where no water can rise to the root zone. The relationship between the depth of ground water and evaporated losses can also be approximated with an exponential function of a structure similar to Eq. (1.4-44): ETg(m)= ETo exp [ - p ( m - m,)]. (1.445) The process of evaporation depends on the capillarity of the soil, the tension between its grains and water, and the influence of vegetation. Thus the parameters of Eq. (1.4-45) (ET,, m,, p) are also functions of the natural conditions. It is necessary to note here that in lysimeters with constant water table a different character of the ET,(m) function was observed (Fig. 1.4-27) (Major, 1972a). In this case, however, the influenceof the fluctuating water table and the relationship between this latter process and the seasonal variation of ET, is left out of consideration. Supposing a regular fluctuation

192


0

20

Fig. 1.4-27. The ETg(m) function determined from date measured in lysimeters with constant water table

in both the position of the water table and evapotranspiration, which can be approximated with a sine function, the expected ETg(m)function was recalculated from the lysimeter data, and is also represented in Fig. 1.4-26. In this investigation it was also considered that the lysimeter was not covered by vegetation, thus the depth had to be measured below the root zone, and the thickness of this zone also increased with the depth of the water table. The agreement between this curve and that calculated from Eq. (1.445)is acceptable. It must be mentioned that the basic idea of the present analysis is only the qualitative characterization of the processes. The principles concluded in this way do not alter, if another equation is used for the description of the relationships existing between the variables. The character of the relationship supposed to exist between the average depth of the water table and the anwunt of ground-water withdrawn vertically is also supported by the water quality data. It is evident that in a ground-water basin the concentration of salts dissolved in the ground water is higher if the water table is near the surface, because the horizontal inflow is balanced by vertical drainage, and the salts transported by the water there accumulate. It is also expected that the salt concentration has to be proportional to the amount of water drained vertically. Indeed, the data collected by Hazen (1892) at Tafilalet in Morocco fit a very similar curve (Fig. 1.4-28) as the one shown in Fig. 1.4-26. Hazen’s data also indicate that the thickness of the layer influenced by evaporation is greater in the arid zone of Morocco, than it is in the temperate climatic zone, although it is also known that this depth depends not only on the climatic conditions, but also on the capillary properties of the soil.


193

concen:rs?ml; grams uer llfer ll?

0

10

( total oissolved sohds) 20 30 40 50

60

Fig. 1.4-28. The amount of total ealt content in the und water aa a function of the depth of the water ta%

According to Eqs (1.4-9) and (1.4-lo), vertical water exchange is characterized by the difference between recharge and water withdrawal. Both variables are given aa a function of the average ground-water depth, thus their difference can also be plotted against the same depth. This is the characteristic curve of the ground-water balance (Fig. 1.4-29). At the level where the curve intersects the vertical axis, recharge is equal to vertical drainage (equilibrium level). The water table can only develop at this level if there is no horizontal flow, or the inflow is q w l to the outflow. Above this level the zone of horizontally recharged ground water can be found. In the case of such water tables vertical drainage is grbater than recharge, and the differenceis balanced by the surplus of horizontal inflow compared with the outflow. Reversed, when the invwtigatd column i s drained by horizontal flow (outflow being greater than inflow), this difference should be balanced by the surplus of recharge from infiltration compared with the amount of ground-water withdrawn vertically; thus the water table can only develop below the equilibrium level. If ground water had no horizontal movement, its table would develop everywhere at the equilibrium level, thus approximately parallel to the 13

194


infiltration [mm] to0 1-

vertical drainage [ m m l

300 400 500 I I I difference of infitration and vertical drainage C mmJ + 5a a -50 -{an -150 -200 -250

50

100.

0

I

200 I

-

Fig. 1.4-29. Characteristic curve representing ground-water balance as a function of the depth of the water table

surface

I

At

Fig. 3.4-30. Development of water table under a sloping surface

195


surface. The terrain, however, is not a horizontal plain, hence the ground water would have a gradient, if its table were parallel to the slope of the surface. A horizontal ground-water flow would then be started by this gradient in accordance with the permeability of the layers involved, rising the water table in the valleys and lowering it under the hilly areas. Regarding this process in a static way, it would be expected that the water table becomes horizontal in every ground-water catchment. Water tables are actually neither parallel t o the surface nor horizontal, but rather follow the slope of the terrain; thus a dynamic balance develops as shown in Fig. 1.4-30. At higher areas the water table sinks below the equilibrium level, thus fotal salt content 0 0

,

~

"

"

"

/ "

"

'

"

'

2 " "

L %J 3

tbe iota1 salt confenfis negltgible in profiles 0 and @ in profi/e I, @ it is pracfica/lly fbe same as in @ %nd Q

0

- i

140-

: :

i

0@ @

CaCO, content in % of the fofal amount of salf 20 30 40 50

I I I

1' p

---.-.-.

e

65 150

490

........... 230 0300 @ --- 300 @

1 3 (8

450 700

Fig. 1.4-31. Relationship between the salt content of the soil and t,he position of the wmter teble

13*

196


ground water is recharged by surplus infiltration. This amount of water will be transported by horizontal ground-water flow towards lower situated areas, and a dynamic balance will develop a t the levels where the recharge and discharge summarized within the catchment are equalized. The development of this dynamic balance is also indicated by the chemicd composition of the soils. Where the water table lies a t a great depth, the soil is leached, the total amount of salts is very low and none of the well soluble sodium is found in the soil profile, whereas above a high water table surplus evaporation causes salt accumulation, as shown not only by the high salt content, but also by the high Na/Ca ratio (Fig. 1.4-31) (Vhrallyai, 1967, 1970).

A further special point of the characteristic curve of ground-water balance i s its maximum, which divides the range of the possible water table into two zones. Utilizing the ground-water resources continuously a new equilibrium

rn depth b l o w the surf&e K m l Fig. 1.4-32. Classification of shallow ground water according to the iduencing effects


197

can only develop in such cases when the original water table was above this level, because at greater depths the difference between natural recharge and vertical drainage decreases with depth. Assuming the m, parameter in Eqs (1.444) and (1.4-45) t o be equal to the depth of the equilibrium level, the depth of the maximum point can also be calculated (Fig. 1.4-32): 1 BET, m=m,+In 8-a aI,-a or (when a = 0 and therefore ET, is equal to I, a t the equilibrium level): ml=m,,+-ln-. 1 B-a

B a

(1.4-46)

The numerical data of the characteristic curve of the ground-water balance can be determined by lysimeters, measuring the recharge infiltrating to various levels and also evaporation from the water tables of M e r e n t depths (Altovsky and Konopljancev, 1954). When using lysimeter data, the fluctuation of the natural water table should also be taken into consideration. Another possible method of measuring the characteristic curve is the construction of closed ground-water columns (Kovhcs, 1966a). I n the absence of measured data, the curve can be approximated by the use of the relationship representing the connection between the depth and the fluctuation of ground-water (Kovhcs, 195%). Comparing the data collected from semi-arid moderate climatic zones (see Fig. 1.4-26), the average depth of the equilibrium level (naturally influenced by the local climatic, soil and surface conditions) was estimated to be around 2 m in Central Europe (Hungary). This value is in good agreement with the observations of the salt content of soils showing that salt accumulation can be expected in this area, if the average depth of the water table is less than 2 m from the surface (Vhrallyai, 1967, 1970; Szabolcs et al., 1969a, 1969b). It has been mentioned that this characteristic depth is probably greater in arid zones (see Fig. 1.4-28), whereas in humid regions the same parameter may be considerably smaller. This latter supposition is proved by Fig. 1.4-33, which represents a characteristic curve of groundwater balance determined for a swampy area in Byelorussia (Lavrov et al., 1972). Ground-water balance was characterized by the yearly averages in the foregoing. Sometimes the analysis of periods shorter than a year may become necessary. I n such cases the monthly or seasonal mean values of the parameters (depth of water table, evaporation, infiltration) must be determined. Assuming that the general character of the relationship does not vary with the specified part of the year investigated, only the numerical values of the parameters do, the same equation can be used applying a factor expressing the ratio of the parameters (infiltration and evaporation) belonging to the investigated interval (e.g. a month) and the yearly average (Fig. 1.4-34). The final point in the present discussion is the explanation of the way of application of the characteristic curve in seepage hydraulics.

198


inflltra fion to grotind water

I-

tflfll

vertical drainage of grotind wd:er EG t m m l

Fig. 1.4-33. Characteristic curve of ground-water balance for a swampy area in Byelorussia

difference between infiltration and evaparation (Is- E 51 5

B 8 lE

b P

1

0

sa

1.0

3

20

9

L

-2

P

k

sQ

3.0

%

4.0

Fig. 1.4-34. Seasonalfluctuationof the characteristiccurve of ground-waterbelance

When facing the task of the investigation of a ground-water flow influenced by accretion, a method must be found for the numerical determination of the E ( X ) function describing the surface effects. It is already evident from the foregoing that both infiltrating recharge and vertical drainage caused by evapotranspiration and, consequently, also their resultant are functions of the depth of the water table below the surface. It is also an obvious statement that the change of the water table


199

produced by the artificial recharging or draining of a system cauyes surplus evaporation and infiltration, respectively, along the influenced flow space, and these surplus surface effects are proportional to the change of groundwater depth. The surplus surface effects can be numerically determined using the characteristic curve of ground-water balance. The latter represents the relationship between accretion and, the depth of ground water. Thus the difference of the values belonging to the original and the modified water tables yields the surplus effects in question. This way of calculation is valid in both cases, if the water table was originally at equilibrium level and, therefore, the surface effect pertaining to the Btarting condition was zero, and if there was already some surface effect present influencing the groundwater space tjefore the changing of its level, and the original ground-water table wazj at tin elevation different from that of the equilibrium. The detailed description of the method proposed for the hydraulic consideration of accretion will be discussed in Section 4.1.3,where the practical hydraulic application of the characteristic curve derived now from hydrological data will be presented. References to Chapter 1.4 ALTOVSICY, M. E. and KONOPLANCEV, A. A. (1954): Methodological Handbook for the 1nvestigat)ion of the Ground-water Regime (in Russian). Gosgeoizdat, Moscow. ASTM (1958): Procedures for Testing Soils. American Society for Testing Materkk, Philadelphia. A. (1911): Plasticity of clays (in German). Overeatt fr. evemka UT ATTERBERO, Kungl. Lantbruksakademkm Handlinger och Tidakon, 60. 1911. BEAR, J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR,J., ZASLAVSKY,D. and IRMAY, S. (1968): Physical Principles of Water Percolation and Seepage. Arid Zone Research, UNESCO, Paris. G. (1967): Practical Investigation of Subsurface Waters (in French). CASTAQNY, Dunod, Paris. DE WIEST, J. &I. (1968): Flow through Porous Media Chapter 2. Porosity and Permeability of Natural Materials by S. N. Davis. Academic Prees, New Ymk-London. HAZEN, A. (1892): Experiments upon the Purification of Sewage and Water at Lawrence Experimental Station. Health Publication of Ma.gaachueetta State Board, NO. 1. 1889-1891. JOHNSON, A. I., PRILL, R. C. and MORRIS, D. A. (1963): Specific Yield-Column Drainage and Centrifuge Moisture Content. Geological Survey, Water-mpply Paper, 1662-A. Washington. KAMENSKY, Q. N. (1940): Differential Equations of the Nonsteady Ground-water Flow and their Application to Calculate Backwater Effects (in Russian). Izvestiia of Akademii Nauk, USIS'R, No. 4. KAMENSKY.0. N. f1943): PrinciDles of Ground-water Dvnamics lin Russian). Gosgeoltekljz;lat, Modow. ' ~ Z D I A. , (1972): Soil Mechanics (4-th edition) (in Hungarian). Tankonyvkiad6 Budapest. KI~ZDI, A. (1974): Handbook of Soil Mechanics Vol. I. Soil Physics. Elsevier-Akad6miai Kiad6, Amsterdam, Budapest. KINQ,F. H. (1898): Principles and Conditionsof theMovements of Ground Water. 19th Annual Report of US. QS. 1897-98. Koviics, G. (1959a): Determination of the Flow Rate of Ground-water flow on the Basis of Analizing Water Balance (in Hungarian). V'iziigyi Kozlemdnyek, No. 3. KovAcs, G. (1959b): Ground-water Household. ICID Annual Bulletin, New-Delhi,

200

1 Fundamentals for the inveetigation of seepage

KOVACS,G. (1971): Salt Accumulation in Ground Water and in Soil. I A H S Sympoeium cm Qrcyund-wder Pouution, Moecow. KOVACS,G. (1972a): Ground-water Reeources and their Contact with Surf..-water Resources. EUE Seminar on Methodology for the Compilation of Balancse of Water Resources and Needa, B u d a p t , 1972. KOVACS,G. (1972b Asseamnent of Ground-water Re~ources.Sympoeium on Water Resources Planning, e x k o City, 1972. LAVROV,A. P., FADEJEVA, M. V., SACEOK, 0. I. VAXHOVSKY, A. P. and VASILJEV, V. F. (1972): Problems of Underground Water Regimea in Byelorueeian P o W c . UNESCO-IAHS Sympoeium on the Hydrology of Mamh-ridden A r e a , Minek, 1972. LEBEDIEV,A. V. (1961): Investigation of theRegimeand Balance of Ground Water on the Basis of Continuous Observations (in Run~im).Paper in the Volume: Methods of I n v e a t i g a t k and 0 - k in Engineering Geology and Hydrogeology, Gosgeoltekhizdat, M m w . LEBEDIEV,A. V. (1963): Method6 to Inveetigate the Balance of Ground Water (in Russian). Gosgmltekhkdat, Moscow, 1963. MAJOR, P. (1972a): Determination of Infiltration and Evaporation Considering the Capillary Condition of the Covering Layer (Manuscript in Hungarian). SciSntifiO Conierenm of V I T U K I . 1972. &OR, P. (1972b):'Investi ation of Balance Parametere of Ground Watere in Plain6 (in Hungarian). Scientific Repor&, No. 2162. (Manuscript). PRILL, R. C., JOHNSON,A. I. and MORBIE,D. A. (1966): Specso Yield - Laboratory Experiment8 Showing the Efleot of Time on Column Drainage. Geological Survey, Water-eupplyPaper, 1662-B. Washington. ROWE,P. R. 1966): Differenae Approximations to Partial Derivative for Uneven Spacings in the etwork. T r a m a d o m of AGU, 36. SCEOEI&ER, H. (1962): Ground Watere (in French). Masson, PEL&. Srdrr~,W. 0.(1933): Minimum Capillary Rise on Ideal Uniform Soil. P h y s b , May. SMITE, W. O., FOOTE, P. D., BUSANU,P. F. (1931): Capillary Raise in Sands of Uniform Spherical grain^. P h y s b , July. SZABOLCS, I., DARAB,K. and V~ALLYAI, G. (1969a): T h e Tisza Irrigation Syetema and the Fertility of the Soils in the Hungarian Lowland (in Hungarian). Agrokdmiu 4% Talajtan, No. 2. SZABOLCS, I., DAEAB,K. and V~ALLYAI, G. (196913): Alkalization Process Due to Irrigation on the Hungarian Plain. A p k d m i a 4.8 Talajtun, Supplementum. S ~ L Y F. , (1973): Determination of Ground-water A m t i o n and Seepa e Parameters on the Bask of Ground-water Obeervations (in Hungarian). Hidro&iai Rozl h y , No. 6 . TERZAUHI, I(. (1926): Soil Physical Basis of Mechanics of Earth Structures (in German). F. Deuticke, Wien. TERZAunx, K. (1943): Theoretied Soil Mechanice. John Wdey, New York, London, 1943. V~ALLYAI,G. (1967): Salt Acoumulation P'rocea in Soils in the Danube Valley (in Hungarian). Agrokdmia ke Talajtan, 1967. No. 3. VARALLYAI, G. (1970): Salt Movement, Salt Balance and their Importance on Irrigation S tmm on Lowland6 (in Hungarian). Agr6rtudomdnyi Kodemknyek. WO-,~. (1886): Investigation of Water Retention Capacity of Soils (in G m a n ) . Formhungen auf dem Gebiete der Agrikdtur Phyeik.. ZWNXER, F. (1930): Behaviour of Soils in COMection with Water (in German). Handbook of Soil Sciences, Springer, Berlin. Vol. VI.

li

d,

b

Part 2

Dynamic interpretation and determination of hydraulic conductivity in homogeneous loose clastic sediments

I n every case, where the purpose is to describe a special type of movement and to discover the interrelationships of physical processes for the mechanically correct (so-cdled deterministic) characterization of the motion, the detailed investigation of the acting forces is necessarily the first step. There are accelerating and retarding forces forming a balanced system in the case of steady movement. Distinguishing the various types of movement so that in each group the same forces have a dominating role, the limits of the validity zone can be determined, outside of which the motion is affected by other forces. The equilibrium of the accelerating and retarding forces gives an equation in each zone, on the basis of which the correct form of the movement equation can be constructed and even some numerical factors of the latter can also be calculated. The other important step in the theoretical investigation is to find a physical (geometrical) model of the analyzed system. When selecting this model, two contradictory aspects act against each other. The boundaries and structure of the model should be as simple aa possible, t o ensure the mathematical solution of the relevant movement equations. On the other hand, it is necessary to include in the model the largest possible number of parameters from the actual system, because each parameter may describe a special feature of the prototype. Following the determination of the acting forces and the geometrical model, the combination of results provides a conceptual model, which is suitable for the theoretical analysis of the process under investigation. The basic relationships between the interrelated variables may be determined in this way, and the variables having negligible effects can also be selected. The investigation has always t o be finalized with carefully executed experimental research, which serves a dual purpose: firstly it is necessary for the verification of the theoretical relationships, together with the indication of their validity zones, and, secondly some numerical constants of the formulae have to be determined from the data measured. The theoretically derived equations include some experimentally determined components, and thus these relationships are semi-empirical formulae. They have, however, many advantages compared with the purely empirical methods. They are based on the actual dynamic character of the motion

202

2 Determination of hydmdk conductivity

and their application, therefore, reflects the real physical contacts between the variables under consideration. The use of these formulae prevents particular mistakes which are very often caused by applying empirical equations t o practical situations. An example of such a mistake is where an equation derived for a given character of movement is applied in another zone without considering the basic differences between the physical processes and using only some corrective factors, which, though valid in the vicinity of the measurements used for their calculation, are not so in the whole zone. This possible mistake can well be illustrated by an example taken from seepage hydraulics, i.e. where Darcy’s equation is used to describe all types of seepage. Although it is clearly proved that the linear relationship between velocity and gradient is valid only in the case of laminar movement (when the dominating forces are gravity and friction), the same function is very often applied to describe other types of movement as well, using only a numerical factor to correct the resulting error. Naturally, this factor differs from cam to case and investigations determining a factor from a limited number of measurements cannot, therefore, generally be accepted. The method of investigation previously explained, and proved to be physically correct, will be appliedin this part of the book for the characterization of seepage through an homogeneous porous nzedium, the solid m t r i x of which is composed of evenly distributed grains of loose clastic sediments. Using the continuum approach the main aspects in the determination of the geometrical model were explained in Section 1.1.3. It was followed by the analysis of the physical parameters of soil in loose clastic sediments, on the bmis of which the effective diameter ( D J , the shape coefficient of grains (a) and the effective porosity ( n ) were selected as the most important characteristics to be incorporated into the model as parameters. Comparing the various possible types of models, it was found that a bundle of straight tubes lying parallel to the direction of flow and composed of small sections of two different diameters, is simple enough t o be amenable to mathematical treatment, and at the same time is suitable for the accurate reproduction of the most important hydraulic processes occurring in homogeneous loose clastic sediments. The h a 1 result is summarized in Section 1.2.5 where all the equations, from which the model-parameters can be calculated aa functions of the characteristic physical soil data are listed. After having determined the geometrical model, the next step necessary to complete the conceptual model, is the selection and characterization of the acting forces. It has to be followed by the experimental verification of the model. This dynamic analysis and experimental research is discussed in the following part of the book.

Chapter 2.1 Dynamic analysis of seepage As already stated, the creation and maintenance of movement requires the action of an accelerating force, or of a system of such forces. The retarding forces act against the accelerating ones, and they generally increase with the

203

2.1 Dynamic analysis of seepage

velocity of the motion. Thus, there is a limit where the two groups of forces reach a state of equilibrium and a steady movement develops. When investigating the dynamics of seepage, the fist task is to analyze the acting forces. It is necessary t o determine their physical properties and to find the most suitable way of incorporating them in the conceptual model. There are many cases where the dynamic laws established in theoretical physics are not suitable for direct application, because the model would be too complicated, making the mathematical solution more difficult. Acceplable approximations have to be determined in these cases, which ensure both the necessary simplicity and the accuracy required in practice. One important method of approximation is to dbthguish between the dominant and the negligible forces. The ratio of the considered and neglected forces gives numerical values which indicate the limits of the validity zones of the various approximations applied by neglecting certain factors. Similar dimensionless quotients can be derived as the ratios of the accelerating and retarding forces. These parameters provide the theoretical form of the bwic movement equations, which will later be verified by experiments. The stages in the dynamic investigation laid down in the previous paragraphs will be discussed in detail in this chapter.

2.1.1 Forces influencing the flow between grains In seepage hydraulics the most important accelerating force is gravity and in most cmes i t is sufficient to consider only this force, as indicated in connection with the definition of seepage. Apart from this force the pressure of the covering layer may be a dominant factor in deeper lying aquifers, if the comprewibility of the latter is significant. I n the vicinity of the water table, capillarity, and in the unsaturated zone, adhesion between water and grains may also have an important accelerating role. The latter should also be included in the group of retarding forces in he-grained sediments, where the ratio of the internal surface related to the volume of the sample is high. Among the retarding forces, three generally become dominant: adhesion, friction and inertia. Hence, there are six forces, detailed knowledge of which is required for the dynamic investigation of seepage. Four of them are well known and generally used in other branches of hydraulics and soil mechanics. These are: gravity, the pressure of the overlying layer, inertia, and friction. The other two forces (i.e. capillarity and adhesion, both created by molecular interaction) have important roles, mainly in the investigation of seepage, and their more detailed discussion is, therefore, justified here. Qravity is a maas-force (body-force) having potential and caused by the attraction of the Earth. Its size (a)is equal to the product of the acceleration due to gravity ( 9 )and the moving mass ( M ) and is directed vertically downwards: (2.14) c)L = Mg = Veg; where V is the volume of the investigated body, and

e is its density.

a04

2 Determination of hydraulic conduotivity

A force has potential when there exists a potential-function - a singlevalued scalar function of the space coordinate system - whose gradient is equal to the force in question (the differential quotient of the potential function in any direction is equal to the component of the force vector in the same direction). I n the case of a mass-force, the potential may be related to the unit of the mass, and thus, the gradient of thepotentialfunction supplies the vector of acceleration. In a gravitational field, choosing one axis of the coordinate system as vertical, the vector of acceleration has no component normal to this axis, and the derivate of the potential-function, therefore, i n a vertical direction is equal to the acceleration due to gravity, and the potential can be calculated from the following equation: and for a unit mma

u = Mzg; u1= zg.

(2.1-2)

Consequently, the potential is proportional to the distance (21 . , measured from an arbitrarily chosen horizontal plane used as a datum of reference.

unloaded layer

drained layer

Fig. 2.1-1. Hydimstatic pressure distribution and its modification caused by surface load

m5


When the total weight of the grains in the layer and that of the covering layer, is transferred from grain to grain at the points of contact, the water in the pores is loaded only by the overlying water body, and hence, the vertical distribution of water pressure can be characterized by the hydrostatic pressure-distribution curve (Fig. 2.1-la). I n this case, the sum of potentialand pressure-energy is constant at each point, and water movement can be created, therefore, only by outside action. The volume of pores in the layer decreases when the latter is d e c t e d by a surplus load placed on its surface. If the loaded layer is drained (a given amount of water equivalent to the decrease of pores can be immediately discharged from the layer), the surplus load is transferred through the grains as well, and hence, the pressure distribution in the water is not modified by this outside load (Fig. 2.1-lb). Another extreme case is, where the borders of the aquifer are impervious, the layer is compressible and the total load has to be taken over by the water itself (Fig. 2.1-lc) (KBzdi, 1963, 1974). I n an aquifer within a stratum composed of pervious and impervious layers, the outside load is created by the overlying layers. Assuming the stratum to be completely filled with water (the water table is at the level of the surface), the probable water pressure can be limited by two extreme vdues (i.e. the hydrostatic pressure and the loaded condition), in the form of the following unequality (Fig. 2.1-2):

(t=

plane 0 I,g+b, T~pV=Zg plane I Zlg+hl 7sQ~ = =ZY+~I crs-rv)PV

1

pressure differem between plane 0 and plane !related to the datum level

Fig. 2.1-2. Water pressure in natural layer influenod by the weight of overlying layere

206

2 Determination of hydraulic conductivity

(2.1-3)

where yI, is the specific weight of water and ys is that of the solid grains. When the pressure is higher than the lower limit, there is a potential difference between the upper and lower surfaces of the layer covering the aquifer, which may create movement through the impervious, or semiimpervious covering layer (and drain the water from the aquifer). Thus, the upper limit in Eq. (2.1-3) describes the condition at the point of time when the load WM placed upon the layer, while the lower limit characterizes the total consolidation, the development of which can be theoretically expected after an infinite time. The actual water pressure depends on the time elapsed since the loading, the size of the load and the permeability of the layers (that of both the aquifer in question and the bordering formations). A further uncertainty is caused by the fact that many loadings follow each other before the total development of consolidation caused by the earlier loading. A decrease in the load can also sometimes be observed (erosion, exfoliation), and since the layer is not elastic, this is not always followed by complete expansion. For this reason, the expected value of pressure cannot be calculated, only its possible extreme limits can be estimated (Fig. 2.1-3). When calculating the parameters of seepage, the pressure diflerence between the investigated and the draining space can be taken into consideration as an active force. This value divided by the specific weight of water is expressed a.s the equivalent height of a water-column and thus, i t can be taken into account as a supplementary part of gravity. The presence of this type of force requires special investigation, therefore, only in the care of unsteady movement, when the development of consolidation influences the character of movement as well. Inertia is also a body-force similar to gravity. Consequently, it is the product of mass and acceleration. I n this case, the latter is the acceleration of the motion, which is retarded by this force: (2.1-4)

Internal friction is a retarding force i n a viscous fluid, obstructing the relative displacement of two water particles moving along two neighbouring paths (Fig. 2.1-4). This force is directed against the flow and its size is proportional to the surface ( A ) ,parallel to the flow line and perpendicular to the direction of the velocity gradient. The friction acting on a surface of unit size (specific value) is the shearing stress (z), which is the product of dynamic viscosity (7)and the change in velocity in a direction normal to the area investigated (velocity gradient):

S=Az;

z=q-.

dv dn

(2.1-5)


207

pressure bead in wafer column CmJ

Fig. 2.1-3. Values of water pressure measured at various depths

g&yq

- --

2

Fig. 2.1-4 Sketch for the interpretation of viscosity and shearing stress

The ideal (non-viscous) fluid is a theoretical concept. Its molecules are absolutely movable, and their displacement related to each other is not affected by any resistance. The actual water molecules are dipoles, the positive and negative charges being located asymmetrically within the molecule

208


(Fig. 2.1-5). This structure creates an electrostatic field around the molecules, the find result being the attraction between the molecules and the binding of positive and negative poles (Newtonian fluid). The shearing stress between two layers of molecules acting on a unit area is proportional to the velocity difference of the layers. The factor of proportionality is the dynamic

yofy+ Fig. 2.1-5. Dipole structure of water molecules

viscosity, the technical dimension of which was [kp sec m-2]. I n practice, the physical dimension (poise)was also often used, which is roughly 1/100 of the former: 1 poise = l[dyn sec cm-21 = 0.01019 [kp sec m-2]. I n the S I system the physical and the praotical units are the pascal seconds (Pas) and the Nsm+ (1 Pas = lNsm-'= lop). I n many cams the physical behaviour of the fluid is characterized by viscosity and density. It is reasonable, therefore, to introduce a combined parameter, which is the quotient of dynamic viscosity and density v = q/e;

(2.1-6)

known M kinematic viscosity, and its dimension is [L2T-l]. The orientation of the molecules is not constant. The bonds are disintegrated by Brownian movement and new groups are continually developing. Thus, viscosity is a probability parameter formed by a number of randomly bound molecules, which always represents the equilibrium between attraction and Brownian movement. Thus, viscosity can be regarded aa an example of the application of the continuum approach when an arbitrarily chosen parameter is applied to represent a property of the actual group of molecules and to investigate the continuum of a fluid instead of the real molecular structure. It is evident, therefore, that viscosity decreases with inereusing temperature. The equation generally applied in practice to describe this relationship is as follows: 71 (2.1-7) 1 0.0337 t 0.00022 t2 ' qo

+

+

209


where t is the temperature of the fluid at "C, and q,, is the viscosity of the fluid when t = OOC. The latter is the function of the electrostatic field created by the dipole molecules, and hence, this parameter is a material constant of the fluid, which is T o = 1 . 7 7 10-3 ~ [pa 81

in the case of water at a pressure of 1 atm (100 kPa). The theory that viscosity is a result of molecular attraction, is supported by the fact that the relationship between viscosity and absolute temperature ( T ) can be characterized by an equation having the same structure as that between vapour pressure ( p ) and absolute temperature, and thus, viscosity can be directly related to vapour pressure (Madge, 1934):

[:I

In - = A -

BIT;

Inp = A' - B'fT ;

and thus qpB"B

(2.1-8)

c.

1

It is also quite evident that the increasing pressure acting on the fluid decreases the distance between the molecules, and hence, incremes viscosity (although in the case of water some irregularity can be found, as shown in 1.0016 Fig. 2.1-6). Since the compressibility of water is very small (1.0010 times the increase of viscosity as a result of 1 kp/cm2 increase in pressure), this dependence can be neglected in seepage hydraulics. As with the increase of pressure, all kinds of external eflects which increase the probability of the orientation and binding of dipoles, increase the viscosity

-

0.0'

'I 2' 3'

I

4

I

5

pressure lo^

'

6

I

7

'

8

I

'

9 t O

N/mm 2.3

Fig. 2.1-6. Relationship between temperature, pressure and viscosity of water 14

210


of the fluid. Among these influences, the most important is the electrostatic charge of the solid wall surrounding the fluid. The investigation of this effect leads to the study of the next influencing force, i.e. adhesion. The solid wall has electrostatic charges related to the water, which polarize the water molecules along the wall. The molecules are fully orientated in the first layer. They are bound by their opposite charge t o the wall and their charge, which is the same as that of the wall, is orientated towards the

(-)

(-1 (-1 (->

(-) (-)

(-1 (-)

Fig. 2.1-7. Distribution of polarized water molecules along the walls of grains

interior of the water body. The effect of polarization can also be observed inside this double-layer within the water body. The number of orientated molecules, however, is inversely proportional t o the distance measured from the wall (Fig. 2.1-7) (Erdey-Grliz and Schay, 1954; Khzdi, 1972, 1974). It is not yet absolutely clear, whether the adhesive force is influenced by the mineralogical and chemical character of the grains forming the solid skeleton of the seepage field. On the basis of the explanation given in Section 1.2.3 concerning the interaction between water and grains, it is very likely that this relationship is determined by the Van der Waub force, the effect of which can be characterized by a tension ( p ) inversely proportional t o the sixth power of the distance from the wall (a), independent of the mineralogical and chemical character of the latter [see also Eq. (1.3-15)]:

C ' 6

p=-.

(2.1-9)

There are, however, a few exceptions, when the presence of some minerals having a special structure (Na-montmorillonite, amorphous colloids) modi-


211

fies the normal relationship between the tension in question and the water content of the sample. Another observable phenomenon is that the repulsive force influencing the interaction of colloid grains i n the suspension, isagected by the mineralogical and chemical character of small particles (see Chapter 1.2). Since the repulsive force determines the morphological character of the grains (the ratio of the irreversible bonds to the total amount of colloid particles), the mineralogical and chemical character of the grains indirectly influences the size of the active surface of the sample, and hence, the interaction between the water and grains as well. This influence, however, can be neglected in seepage hydraulics, because the actual active surface is always taken into consideration by the application of the effective diameter. There are some publications which state that adhesion is inversely proportional to the third power of the water content, or sometimes to its second power, when the free charges of the grains are bound by monovalent cations (JuhAsz, 1967; Mattson, 1932; Vageler, 1935). The detailed explanation of the soil-moisture retention curve is given in Chapter 1.3, where the meaning of the two separated sections of this curve is also explained. The validity of Eq. (2.1-9) is also proved for the uppermost section of this curve, where the interaction between the grains and water is influenced only (or more precisely, mostly) by adhesion. The lower power mentioned in the papers, is valid in the open capillary zone, where the water content is jointly determined by capillarity and adhesion. I n this zone, however, the relationship between water content and tension can be described more easily by the use of the probability distribution of capillary head, than by formulae expressed in the form of a direct function. The detailed study of such equations, therefore can be neglected here. The tension is perpendicular to t e solid wall, and hence, it has no component in the direction of flow. It is necessary, therefore, to investigate the way in which the motion of water can be influenced by this action. The accelerating froce displaces polarized water molecules in the electrostatic field of the solid grains, and hence,thepaths of the dipoles cross the forcelines of the electric field. It is well known, that this process consumes energy and can be expressed by the action of a force parallel to the flow line (bimilar to friction). The magnitude of this force naturally decreases with increasing distance from the wall (KovAcs, 1957, 1958). A very similar result can be achieved when the eflect of the adhesive force is indirectly taken into account by considering its influence on the flow as a surplus shearing stress. The application of this method means a deviation from the Newtonian fluid, and requires the investigation of a material having a shearing stress in static condition as well. This kind of medium is called Bingham’s body or Bingham’s plastic (Bingham, 1922; Reiner, 1952). In this cme, the sum of friction and the secondary effect of adhesion acting in the direction of flow (S E ) can be expressed by a common equation using a shearing stress (or yield stress z’) having a member independent of velocity, instead of that expressed by Eq. (2.1-5). The existence of the static shearing stress (zo) is proved by the fact that in cohesive layers (and also in capillary tubesof verysmall diameter), there

+

14*

212


is a limiting value of gradient (I,,, threshold gradient), below which there is no movement at all. Thus, zero velocity can occur with a finite gradient. The measured value of the threshold gradient can, therefore, be used to characterize the static shearing stress (Kar&diand Torok, 1955; Kutilek, 1965). Bondarenko and Nerpin (1966, 1967)andBondarenko (1966)havepublished the results of a series of experiments designed to determine the threshold gradient and the static shearing stress. They measured the corresponding velocity and gradient values of the flow of water and ethyl alcohol, respectively, through glass and quartz capillary tubes of known constant diameter. Figure 2.1-8a shows the results measured using water and Fig. 2.1-8b those using ethyl alcohol. The data represented in a v vs. I coordi. nate system prove that the relationship between the two variables cart be well approximated by a linear equution over almost the entire range of velocity. The straight line representing this relationship does not cross the origin of the coordinate system [as it should if Darcy's equation, Eq. (1.1-9) were

(a) wafer

(b) ethyl alcohol 0.3 a2

0.1

0

0

b

r

0.1

I

I

0.2 0.3

d4

a's

d.6

I a'8

d7

0.1

a2

0.3 0.4

0.5 0.6

0.7

b5

LO 0.5 0

2.5

2.0 1.5 1.0

0.5

0

Fig. 2.1-8. Relationship between velocity and hydraulic gradient in the zone of small velocities (after Bonderenko and Nerpin. 1967)


213

valid], but intersects the horizontal axis at a point I , . The gradient of I , is not equal, however, to the threshold value, because the movement of fluid also begins under the influence of smaller gradients inthezoneof very low velocities, where the linear relationship is not valid, as shown by Fig. 2.1-8c, which is the enlarged form of the lower parts of a velocity vs. gradient graph. The figure is also supplemented by a theoretical sketch (Fig. 2.1-8d) showing the interpretation of the threshold gradient ( I , ) and the I , value characterizing the position of the linear relationship, and fkally, that of a third parameter ( I , , ) , applied by Bondarenko and Nerpin for the evaluation of the measurements. The basis of the evaluation is the Buckingham-Reiner’s equation, or more precisely, its simplified form, in which the static shearing stress is assumed to be constant. This constant can be calculated from the velocity and gradient data: v =KI

I,,

-

[ 3 M 4

3

4 I,, + 1 3 L i

[

(Buckingham-Reiner’s equation);

- K I 1 - - Lo ; O-

:rII]

(simplifiedform);

(2.1-10)

and the I , , constant proportional to the threshold gradient: 1

to=IryI,,; 2

where K is Darcy’s hydraulic conductivity, r is the radius of the capillary tube, y is the specific weight of the fluid and I , , is the characteristic gradient, the interpretation of which is given in Fig. 2.1-8d. An important result of this analysis is that the tovalue calculated in this way was found to be independent of the material of both the tube and the fluid and of the diameter of the tube, and it is a single-valued function of the temperature. The intersection of the straight line characterizing the v vs. I relationship and the horizontal axis (Il),is proportional to the I , , value. It can be expected, therefore, that the r y I , product behaves similarly, and the data determined from the graphs of Figs 2.1-8a and 2.1-8b prove this expectation (Table 2.1-1). Bondarenko (1973) has also proved that the non-Newtonian behaviour of the fluid is cawed mostly by the hydrogen bounds between the molecules. The comparison of the virtual hydraulic conductivity values (the ratio of corresponding data of velocity and gradient) measured with various fluids (Fig. 2.1-9) testifies the fact that the rapid decrease of this parameter was observed in the zone of small gradientsonly,if the fluid applied during the experiments had dipole molecules. At the same time, the change in concentration of the dissolved material in the water (KC1in the experiments) did not modify the calculated t oparameter as shown in Table 2.1-1. This gives the r y I , products recalculated from the shearing stress given in the original publication. This result is a negative proof of the previous statement,

214

2 Determination of hydraulic conductivity Table 2.1-1. The characteristic I , gradient measured I,xlO' parameter if temperatun, is r C ]

Pipe Type of fluid

Distilled water y = l.O[pcm-S]

61 103.6 166 198 266

6.4 -

-

-

4,l 2.6 1.3 1.3 1.0

-

-

-

2.80 1.26 1.76

1.30 0.66 0.26

-

-

0.69

0.16

-

-

0.08 -

-

-

-

-

0.40 0.16

0.00

-

average Ethyl alcohol y = 0.8[pcm'*]

67 109.26 166

-

-

6.4

3.66 2.22

3.3

-

-

-

1.1

-

0.68 0.31

0.75

-

0.10

-

-

-

-

-

average

KC1 solution y = 1.0 [ p ~ r n - ~ ] N N N 1 p = 9.807 X

198 -

1.2 1.3 1.3

-

-

-

-

-

-

average

lo-' N

showing that the dissolved minerals do not influence the behaviour of the fluid. It was also found that the heat value of the H-bonds (a),can be related to the shearing stress in the form of an equation having a similar

0 0.2 0.4 a6 0.6 LO 1.2 64 !.6 hydraulic gradient, I Fig. 2.1-9. Comparison of the virtual h draulic conduotivitiea determined by various fluids (after 6ondarenk0, 1973)


215

in capillary tubes by Bondarenko and Nerpin (1967) 1 . ~ ~ 1 0 ~pcm-'] 4

15

3.26

1

20

1

a5 -

-

2.09 2.69 2.14 2.67 2.66

-

3.26

2.41

-

-

-

__ -

1

30

I

40

I

60

1

55

1

58

0.66 0.67 0.41

0.20

0.04

-

-

-

0.00

-

1.43 1.30 1.24

0.28

-

-

1.66

-

0.40

-

-

-

-

-

1.38

0.64

0.24

0.04

0.00

-

2.46

1.66

1.01

-

-

-

0.33

2.88

-

-

-

-

1.47

1.12

0.41

0.13

-

-

2.88

1.97

1.39

0.71

0.23

-

-

-

remlouleted from z,

structure to those giving this parameter as a function of other properties caused by the dipole character of the molecules. The relationship expressing the value of G as a function of zo is the following: (2.1-1 1) where T is the absolute temperature, A and C are constants and the two subscripts ( 1 and 2) indicate the corresponding values of T and zo. Since the G value can be regarded as a constant within a limited range of temperatures, Eq. (2.1-11) can also be used to determine the general pattern of the relationship between the r y I , product and temperature. Choosing a fixed T, value (T,= 273OC, i.e. the temperature in centigrade is equal to can be neglected in the zone of praczero), the dependence of r y I , on TITz tical importance and the (T,*- T,)difference can be supplemented by the temperature expressed in centigrade. The neglect of the T,Tzproduct can be partially compensated by the appropriate selection of the C parameter. Thus, a linear relationship can be expected between the variables under investigation ( t OC and r y 11)in a semi-logarithmic coordinate system. In Fig. 2.1-10 the data summarized in Table 2.1-1 were plotted, resulting in a straight line. The constants and the corresponding basic values of the temperature and the r y I , products, were found to be as follows: C =

216


- - 5 x 10-5; G/AT,T, = 1/50; T , = 273OC; (ry I & = 7 x p cm-2. It is also shown in the figure that by shifting the line horizontally by &5OC

two enveloping curves can be achieved, which border a stripe incorporating tmhewhole range of scatter of the measurements. The probable zone of the

3 --

2--

8verage Wues of tbe measurements determibedw#h o wfer + efhjl alcohol KCI S d U f h R I

I

I

points measured and the I, vs. t relationship within the 10°C < t < 6OoC range of temperature can be described by the following equations: tOC

*

5oc

50

1

--

[log ( r y I,

+

+5x

+ 3.1251 ;

(2.1-12)

I, = -{exp [- 4.6(t°C 156.3)10-2] - 5 x 10-5). V Equation (2.1-12) characterizes the flow of fluids having dipole molecules through a capillary tube with a contltant radius r . The application of this dynamic principle to describe low velocity seepage through fine grained, loose, claatic sediments will be discussed in Chapter 2.3. The interrelation between the threshold gradient (I,)and the I, parameter will also be explained.


217

It is necessary to analyze here the various opinions regarding the character of the t ostatic shearing stress t o clarify the influence of the non-Newtonian fluid on seepage. Most of the authors dealing with this problem assume that the static shearing stress is a function of the distance ( 6 ) measured from the interface between the solid bordering material and the non-Newtonian fluid (Kovbcs, 1957,1958; J u h h z , 1958; KarM and V. Nagy, 1960; Childs and Tzimas, 1971). On the contrary, Bondarenko (1973) argues that this parameter has t o be a constant throughout the entire interior of the fluid, because he does not take account of any physical influence which may modify the property of the fluid. It appears to be proved that the special behaviour of the fluid is caused by the fact that the chains composed of dipole molecules are longer and more stable in the electrostatic field of the wall than in the interior of the fluid. It is reasonable, therefore, to assume that the physical parameter chosen to characterize this property should be proportional to the attraction exerted by the solid material on the molecules of the fluid. Therefore, the hypothesis of a static shearing stress decreasing with increming distance from the wall [zo(6)] is acceptable. On the other hand, the relationship between the attraction and zo is not necessarily linear and a more gentle decrease than that of the attraction indicated by Eq. (2.1-9), can also be assumed. A suitable approximation can be determined only by experiments (as will be ghown in Chapter 2.3) and it must not be linkedto,or influenced by, the high power usedinEq. (2.1-9) to describe the attraction vs. distance relationship (Thirriot, 1969). On the basis of the dynamic analysis of the non-Newtonian behaviour of water, the total shearing stress can be expressed as the sum of the static value t o , depending on the distance from the wall ( 6 ) and the Newtonian parameter being proportional to viscosity, and the velocity gradient normal to the direction of flow. Applying this total shearing stress, the combined retarding effect of friction and the molecular forces can be described by one equation:

consequently,

(2.1-13)

S

+E =A

[

to(6)

3

+ 7-

.

It is very likely that the second member on the left-hand side of Eq. (2.1-13), depending on viscosity and velocity gradient, is also a function of the distance measured from the wall. As shown, however, in connection with the investigation of friction, the viscosity is doubled when pressure is increased 1000 times. Combining this condition with the relationship between tension and the distance from the wall, it can be proved that the modification of viscosity is insignificant in a very large zone, and hence, this effect can be neglected in seepage hydraulics. The sixth dominant force influencing the flow between grains is capillarity, which occurs on the surface of contact between water and air. The most

218


simple explanation of this force can be given by taking into consideration the attraction between water molecules. In the interior of the water body the attractive forces act equally from every direction, and thus, all molecules are in a balanced condition. A molecule on the water surface is attracted only by molecules below the surface. Thus, in an elementarily thin layer along the contacting surfaces of two different media there is free energy, which has to be balanced by pressure differences existing between the two sides of this layer (within the two contacting media). To characterize the amount of free energy its specific value is used, which is equal to the ratio of the surface energy P(A)and the surface ( A ) .Relating this quotient to an infinitely small surface element the interfacial tension is achieved (aik),which is a material constant for any pair of media, i and k, depending only on absolute temperature (T) : ( 2.1-14)

The special value of the interfacial tension developing on the surface of a fluid covered by its own vapour, is called surface tension (a).According to Eq. (2.1-14), its dimension can be calculated as the ratio of energy (work) over surface, which is equal to force over distance. Thus, surface tension can be interpreted m a force acting within the surface along a line having a length of unity [FL-l]. Eotvos law has established a relationship between surface tension and the other molecular parameters of the medium (molecular volume, molecular weight, latent heat, absolute temperature, vapour pressure). The relation between surface tension and temperature can be expressed from this law as follows: (2.1-15) u v213 = c(To- T ) = ct; where v is the molecular volume of the medium and T o is the critical temperature, where the product of av213 becomes zero. Eotvos (1886) proved that the c factor is constant for fluids having R socalled simply composed molecular structure (Table 2.1-2) : (2.1-16)

Some media (especially water, alcohols, fatty acids) exhibit exceptional behaviour. The c factor of ethyl alcohol increases with temperature and only reaches the constant of Eq. (2.1-16) near the critical temperature. The parameter of acetic acid is considerably lower (0.134) than the general constant, but the expected 0.227 value can be achieved if a double molecule (2C2H40,) is considered. The data for water listed in Table 2.1-2 were also calculated assuming the binding of two molecules. As seen in this case, normal behaviour prevails above l O O O C , but below this limit the c factor decreases with decreasing temperature. This relationship also proves the previously mentioned hypothesis in connection with the probable molecular structure of water: i.e. the number of bound water molecules increases with decreasing


219

Table 2.1-2. Experimental data characterizing the relationship between surfaoe ternion o f various fluids and temperature (after Eotvos, 1886) Type of fluid

Ethyl ether Ethyl ether Ethyl ether Ethylene bromide Ethylene bromide Chloroform Mercuric methyl Carbon tetrachloride Carbon dioxide Carbon bisulphide Sulfuric acid Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Water ( 2 S O ) Water Water Water Acetic acid (C,H,O,) Acetic acid Acetic acid Acetic acid (2 C,H,O,)

6- 62 62-120 170-190 20- 99 99-213 20- 60 20- 99 3- 63 3- 13 22- 78 2- 60 21- 78 78-108 108-138 138-168 168-199 199-236 3- 40 40-100 100-160 160-210 21-107 107-160 160-230 21-160

0.228 0.226 0.221 0.227 0.232 0.230 0.228 0.231 0.228 0.237 0.230 0.104 0.136 0.169 0.183 0.202 0.226 0.169 0.180 0.228 0.227 0.132 0.132 0.138 0.211

temperature. As a final result of this discussion the change in the surface tension of water in relation to temperature is given in the form of a graph (Fig. 2.1-11). Let us investigate the points of contact of three immiscible fluids (or gmeous media). I n a section normal to the three surfaces of contact, the balanced condition of the acting interfacial tensions necessitates the closure of the vector triangle formed by the three tensions (Fig. 2.1-12). From this condition, the angles ( aik)of the contact surfaces measured from an arbitrarily chosen axis can be calculated as a function of the numerical values of the interfacial tensions, or - if the axis is equal to the tangent of one of the contacting surfaces - the so-called contact angles (Oik)can be determined (NBmeth, 1963; Bear, 1972):

+

aI2cos aI2 or

023

cos a23

uI2 + a,, cos O,,

+

+ 013 cos a13 = 0; (2.1-17)

613

cos

013

= 0.

This condition indicates at the same time, whether or not the development of the balanced condition is possible. It is evident that if u12 (u23 yU), there is no possibility of fulf3ling Eq. (2.1-17) and, therefore, equilibrtum cannot be achieved at the contact of these media.

>

+

220


75

5 70

.$ 65

8

S! 60

& 55 $

50 -10

u

10 20 30 40 50 60 70 80 90 i o o m

temperature, t tecl

Fig. 2.1-11. Relationship between the surface tension of water and temperature

Fig. 2.1-12. Balanced condition of surface tensions at the points of contact of three immiscible fluids

Equation (2.1-17) can be simplified if one of the contactingmaterials is solid. The molecular forces do not modify its form, and hence, the interfacial tensions between the solid and the two liquid (or gaseous) media lie along a straight line (Fig. 2.1-13): ~ 2 cos 3

0 + 613 -

= 0;

cos 0 =

- u13

.

(2.1-18)

012

It follows from Eq. (2.1-18) that if a,,

> al3, then

cos 0 is positive and

0 < 5 .Fluid 1 is called the wetting fluid in this caae, while in the opposite 2

case

lz < uI3),cos 0 < 0 and 0 > , and fluid 1 is a non-wetting one.

2


221

Fig. 2.1-13. Angle of contact between the surface of two liquid media and a solid wall

Knowing the parameters characterizing the surface tension of a fluid and its contact with the solid wall ( 0 and O ) ,the pressure difference between the two sides of the surface (capillary pressure p,) and the form of the surface near the wall can be determined. The change of free energy when an elementary part of the surface df = = dZ, * dZ, moves in a direction normal to its position by an infinite dC distance, serves as the basis of the derivation of capillary pressure (Fig. 2.1-14) (Levich, 1958). If R, and R, indicate the radii of curvature in the main directions, the modified area of the investigated df element after dC displacement is: (2.1-19)

The total change of free energy can be calculated by integrating individual elements over the whole surface. At the same time, it is evident that equi-

\

Fig. 2.1-14. Sketch for the derivation of capillary pressure

222


librium can develop only if the free energy is minimal and, therefore, the elemental change is zero:

Equation (2.1-20) h w a solution for any infinite d5 displacement if (2.1-21 )

$

-

-

c of Q gX Y of == C017Sf.C017Sf.- Q gX Y

Fig. 2.1-16. Vertical section of free water surface near a vertical wall

For simple geometrical forms, the position of the free surface of a fluid (capillary surface or meniscus) can be determined by integration. Along this surface, the pressure of the upper medium is the atmospheric value (p, = = p A = const.). Considering this condition, the pressure of fluid in a static condition can easily be calculated in the force field of gravity: p , = const. - e g x .

(2.1-22)

For example in the vicinity of a vertical, solid wall (the radius of the curvature being infinite in one direction, R, =-), the condition of the equilibrium is (Fig. 2.1-15):

d2Y -

Equation (2.1-23) can be solved by double integration. The two boundary conditions necessary t o determine the two integration constants are given dY = tan 0 ; and if y = at the two extremities of the surface if y = 0, = oo,x= 0

I.

[

dx

I n a space surrounded by solid vertical walls, the develop-

ment of the meniscus is influenced from each direction by the presence of


223

a wall. If the distance between the opposite walls is small enough, it can be assumed, that the curvature is constant in the section normal to the walls. Thus, in capillary slits or tubes having regular forms in a horizontal plain, the meniscus can be approximated by a cylinder (in a slit), ellipsoid of rotation (in an elliptical tube) or sphere (in a circular tube) (Fig. 2.1-16). In these cases, the curvature and the capillary pressure can be calculated from the geometrical parameters of the horizontal section of the surrounding space and from the contact angle:

Fig. 1.1-16. Approximative forms of menisci in slits and tubes

224


(a) I n a slit having a width of b (Fig. 2.1-16a):

(b) In an elliptical tube the two main axes of which are a and b (Fig. 2.1-14b): a 2

R 1-

l COY

, R , = - -b

1 2 cos

.

0

o

1 ; p,=2a cos 0 -

a

1 +; b

(c) In a circular tube with a diameter of d (Fig. 2.1-16c): d 1 4acosO R , 1R2= R = ; P,= d 2 cos o

(2.1-25)

(2.1-26)

It can be seen that the sign of capillar pressure is determined by the size n of the contact angle: if 0 < - ; p , 2 opposite case 0

> 0 , and, therefore, p 1 > p 2 ; in the

II >-; p , < 0 , and p , < p,;

2

n and finally, if 0 =-,pc= 2

0,

and p 1 = p,. The consequence of the capillary pressure not being equal to zero is that the position of the meniscus i n the mpillary tube or slit will be diflerent from the level of the contacting water body having a large surface. Choosing this latter level as a datum where the surplus pressure is zero, the average height of the meniscus (mpillary rise) above this level can be calculated from capillary pressure:

h, =

4acos @

P

es

; in a circular tube h, = --

Y

.

(2.1-27)

d

Hence, the meniscus is above the reference level if the contact angle is 76

smaller than - , while the surface is lowered in the capillary tube in the case 2

3

of non-wetting fluid 0 >-

[

.

Investigating the contact between water, air and quartz (as fluid, gaseous and solid media), it was found that the contact angle is near zero, although this value changes, t o agreat extent, with the condition of the solid surface. Substituting cos 0 = 1 end the value of the surface tension belonging t o the most commonly occurring temperature (i.e. 15 20°C) into Eq. (2.1-27) a simple approxjmation for the capillary rise of water in a glass tube of diameter d can, in practice be proposed [see also Eq. (1.3-26)]:

-

h, [cm] =

0.30 [cm2]

d [cml

(2.1-28)

225


Some observations show that the contact angle depends not only on the quality of the contacting media and the condition of the solid surfaces, but that it is also influenced by the movement of the meniscus and by the pressure conditions prevailing in the capillary tube. This phenomenon can easily be recognized when a raindrop moves along the slope of a glrtss plate, the drop having a large contact angle at its front and an angle approaching zero at the back (Fig. 2.1-17a), the phenomenon is, therefore, called, the raindrop

(a)

(Q7

(b)

raindrop

wetting

drainage

Fig. 2.1-17. Development of different angles of contact in capillary tubes

effect. The same change in 0 characterizes the meniscus of a water column moving in a capillary tube. When the tube is first saturated and the balanced is condition is approached by lowering the meniscus, its contact angle (0,) smaller than that belonging to the static state (B0),while the opposite inequality is valid in the cme when the meniscus advances in the tube (0, > 0,)(Fig. 2.1-17b). The change in the contact angle enables a drop in a capillary tube to equalize the pressure difference without creating mvement, this phenomenon is, therefore, a special type of threshold gradient in a capillary tube. Let us suppose that a pressure of p , prevails at one end of a horizontal tube and p , at the other ( p , < p z ) . The two sections under different pressure are divided by a drop bounded by menisci at both ends (Fig. 2.1-17c). The pressure within the fluid, p , can be determined from the external pressures and the capillary pressures acting on the menisci: 40

p = p , - pc1=p,- -COB 0, ; d and 4a

p = p , - p , = p , -cos d

0,;

therefore,

p, 16

-p, =

40

-(cos 0,- cos 0,). d

(2.1-29)

226


Thus, a static balance is achieved if the contact angles can develop so that the difference between their cosine values multiplied by 4a/d is equal to the pressure difference. It follows from the condition where p , < p,, that

0,> 0,.

Considering the same phenomenon in a vertical tube, the conditions can also be determined where a drop, having atmospheric pressure on both sides, does not move downwards under the influence of gravity (Fig. 2.1-17d). The surplus pressure just below the upper meniscus is equal to the capillary pressure belonging to this surface. Moving downwards inside the drop, the pressure increases in proportion to the length measured from the upper surface, [the curvature of the surface is neglected here, because of the very small horizontal area, and the length is measured from the centre of gravity of the meniscus which is approximated as d / 3 above the deepest point of the meniscus (Swartz, 1931)l. The surplus pressures inside the drop can be calculated from two directions (the upper capillary pressure increased by the weight of the water column, or the lower capillary pressure decreased in proportion to the distance from the point), and the two valueb have to be equal. Hence, a relationship can be established between the two capillary pressures: 40

40

pcl = -COB 0,; p,, = - COB 0, ; d d d +pie - l y = p,,; -1y = cos0, - co.0, ; 4a

(2.1-30)

+

where 0, = 0 being the lower limit of the contact angle at the upper meniscus (cos 0, = 1). Movement will not develop if d c o s 8 , ~1 - - 1 y 4a

;

(2.1-3 1)

and

021,mit = arc cos It is necessary to note that this condition indicates only the posbibility of static equilibrium, because the contact angle can develop anywhere in the 8, 021imit aa well, and in this caae the 8, < '8,inequality does range 8 not stop the movement, only decreases its velocity.

>

2.1.2 Dimensionless numbers characterizing the various validity zones of seepage As already mentioned in Chapter 1.1, seepage can be classified according to the accelerating forces and further mb-groups can be determined considering the acting retarding forces. Thus, the following types of movement have to be characterized when seepage through loose claatic sediments is investigated:


227

(a) Gravitational seepage through a saturated medium: - Turbulent seepage (retarding force is inertia); - Transition zone (retarding forces are inertia and friction); - Laminar seepage (retarding force is friction); - Microseepage (retarding forces are friction and adhesion); (b) Seepage through a saturated medium influenced by gravity and the pressure of the overlying layers (further sub-groups are the same as those of the previous group) ; (c) Seepage through the unsaturated zone above the water table (accelerating forces are gravity and adhesion, and capillarity has3 to be taken into account as well; retarding forces are friction and adhesion). Considering the fact that the pressure of the overlying layer can be regarded as a supplementary part of gravity (as already mentioned), and because seepage through an unsaturated medium is influenced by numerous forces and, therefore, i t must be investigated separately, the following combinations of dimensionless numbers can be constructed as quotients of the dominating accelerating and retarding forces (KovQcs, 1966; Spronck, 1932; Mosonyi and KovQcs, 1952): Laminar zone

S -

a

Turbulent zone

dv Arl-

dn - v v -

-MK ;

Veg

(2.1-32)

1%

dv

V e-

T

dt - v2 - Fr ;

a Transition zone

dv

T + - S--

(2.1-33)

Is

Veg

dv

ve-+Arldt dn

a

-

(2.1-34)

Veg

(2.1-35)

Microseepage

1, V E + S -- --

es + A r l -

dv

vv dn - I o +-=KO,. 9 Veg la9 When only one retarding force is considered and the others are neglected, similar dimensionless numbers can be constructed as3 the ratio of the forces neglected to the considered retarding force. It is quite evident that these dimensionless numbers can be used, therefore, to characterize numerically the limits of the validity zones of the various types of seepage. The limit between the turbulent and transition zones is evidently characterized by the fact that friction can be neglected as compwed with inertia, ~

16*

228


while the lower limit of the transition zone (the upper limit of the laminar zone) falls where friction becomes dominant over, inertia. Thus, both limits can be expressed as a ratio of inertia and friction: dv VeT dt lv -Re. * (2.1-36) A q -d v v dn

The lower limit of the laminar zone can similarly be expressed by the given numerical value of the ratio of adhesion to friction:

E - I-o-V e g- I , - =lag -_ Ko S

a---

I,

-

(2.1-37)

dv vv MK Arldn Finally, there is a further possible combination of the forces in question i.e. the ratio of adhesive force to gravity El@ = I , = KO,. This dimensionless number characterizes a static condition below microseepage, where friction is zero. Some of the dimensionless numbers given are well known from general hydraulics as the bases of the various model-laws, such aa the Froudenumber from Eq. (2.1-33), the Reynolds’ number in Eq. (2.1-36) or the Mosonyi-Kovhcs’ number [Eq. (2.1-32)], which were previously derived to determine the model-law valid for the characterization of seepage models, and for calculating the transforming factors between the corresponding measurements of the prototype and the model respectively (Mosonyi and KovQlcs,1952, 1956). The structure of the other dimensionless numbers is the same, or very similar to those given before. The only significant difference occurs in the numerator of KO, and KO,, where the sum of two forces instead of one is evident. For this reason, these numbers cannot be used as model-laws, because dynamic similarity can only be ensured in the cme of two dominating forces, when the identity of the ratio of these two forces can be represented in the prototype and the model. When there are three dominating forces (i.e. two retarding forces in the numerator and the accelerating gravity force in the denominator), the identity of the dimensionless number for the two systems ensures only the same ratio of the retardingandaccelerating effects. The ratio of the two retarding forces may, however, be quite different in the model from that in the prototype. The dimensionless numbers characterizing the various zones of seepage [Eqs (2.1-32) to (2.1-35)] can also be used to determine the general forms of movement equations, valid for each zone. It is necessary to take into account that seepage is not a free movement. Only a part of the potential energy (increased with pressure energy) accelerates the movement. The actuul accelerating force can be characterized numerically by the product of gravity and hydraulic gradient:

G’= IMg

= IVeg.

(2.1-38)


229

The substitution of this value instead of gravity, in the equations in question, is in accordance with the theory of similarity as well. The total mechanical similarity includes both the geometrical similarity and the dynamic one. The former is ensured when the ratio of two corresponding lengths in the two systems is constant. Thus, the gradient determined as a quotient of two lengths (pressure expressed in equivalent water columns and the length of seepage) must be the same in the prototype as in the model. Mechanical similarity in Darcy’s zone necessitates the identity of both Mosonyi-Kovkcs’ number and the hydraulic gradient in the two systems. Thus, the general form of the movement equations can be derived as follows: (2.1-39)

if

K = M K ’ la - .9 V

Similarly, a well-known formula can be derived for the turbulent mne (i.e. Chezy’s equation): (2.1-40)

if

A=-.

2

Pr’

In the transition zone the general form is also the same as that proposed empirically to describe this type of movement (Forchheimer, 1924):

if 1 a = --

K 0; lg

; and b =

Y

Ko; l a g

(2.1-41)

Finally, the general form of the movement equation in the zone of microseepage is:

KO$= I , +l v v ; v = - la (K 9oj;II 1129 V

Io)=C1I-c2I0; (2.1-42)

129

c1 = -Koj; V

la 9 1 and c2 = - = K -. V MK’

The accuracy of such application of dimensionless numbers calculated as quotients of retarding and accelerating forces, is also proved by K a r a ’ s measurements (Kartidi, 1963). The points representing his data were plotted

230


on a coordinate system with the logarithm of Reynolds’ number on the horizontal axis and 2/MK‘ on the vertical one (Fig. 2.1-18). The fact that the points lie along a horizontal line in the laminar zone verifies the constancy of the M K ’ value, and also that of Darcy’s hydraulic conductivity.

Fig. 2.1-

-KOV&CB’

2.1.3 Numerical limits of the zones of seepage As already mentioned, the ratio of retarding forces considered and neglected respectively is a suitable parameter to calculate the limits of the validity zones of each movement equation, 88 this quotient meaaures the magnitude of the two forces in question. It is possible, therefore, to determine such fixed numerical values of these dimensionless numbers that indicate where one retarding force becomes negligible compared to the one under consideration. For this investigation, it is necessary to define the characteristic length and velocity from which the dimensionless number in question should be calculated. The Reynolds’ number is proportional to the ratio of inertia to friction. If this number is smaler than a given limit, inertia is small and friction is high. The validity of Darcy’s law can probably be accepted below this limit. When this parameter is above another limit (which is higher than the previous one), the inertia of the numerator is dominant over friction and the

231


latter is negligible. This is the turbulent zone. Between these two limits, there is the zone of the transition condition, where both inertia and friction have to be considered. In seepage hydraulics, according to general practice, the Reynoldd number is calculated by using the eflective diameter as a characteristic length, and substituting Darcy’s seepage velocity. The latter is the discharge of flow divided by the total cross section, including the area of both pores and solid grains [see Eq. (1.1-9)]. Consequently, the Reynolds’ number generally used in seepage hydraulics can be calculated from the following formula: R e , = - Dh .V

(2.1-43)

-

Experimental results, give the upper limit of Darcy’s zone as 2 5 expressed by the parameter calculated from Eq. ( 2 . 1 4 3 ) (Lindquist, 1933; Koieny, 1953; Kar&di andV. Nagy 1960; Kar&di, 1963). According to these 200 as measured papers, the lower limit of the turbulent zone is 100 by Re,. There are, however, some explanations in these papers, indicating that this value gives only the lower limit of the second transition zone, and the real turlbulent zone starts at a higher limit. Some of the physical parameters of the soil (porosity, shape-coefficient) are excluded from the Reynolds’ number calculated from Eq. ( 2 . 1 4 3 ) .It is more correct t o use the Reynolds’ number determined for the model pipe rather than Re,. I n this parameter (Re,),the characteristic length i s the average pipe diameter, and the mean velocity %nthe pores (or in the model pipes) is substituted as the characteristic velocity. Thus, the relationship between the two Reynolds’ numbers can be determined as well:

-

Re because

,-

dn -

veff -

v

Dh

l - n a

v

Re,.

- --,4

l-n

u

(2.1-44) 4

n Dh ; and u l-n

=

-.n V

Using this parameter, the following numerical limits between the various zones can he given, as will be proved in detail in the following chapter:

Re, < 10 laminar (Darcy’s) zone; < Re, < 100 first transition (Lindquist’s) zone; < Re, < 1000 second transition zone; 1000 < Re, turbulent (Froude’s) zone. 10 100

The purpose of all the other proposals concerning the calculation of Reynolds’ number, is to select, in a similar manner, the characteristic length and velocity, so that further information can be obtained by including physical data of the soil other than the effective diameter. The velocity value generally used is either the seepage velocity (v) (Ward, 1964; Harleman et al., 1963; Perez Franco, 1973; Collins, 1961) or the effec-

2 Determinetion of hydraulic conductivity

232

tive mean value in the pores (vetf = v/n) (Maasarani, 1967; Kovhcs, 1969a, 196913; Chauveteau and Thirriot, 1967; Thirriot, 1969; Thirriot and Habib, 1970). Considering the possible difference between volumetric and areal porosity, the n value waa introduced,sometimes with a power lower than unity [as explained by Thirriot (1969) the power can be between 1 and 2/3 and Zampaglione (1969) used a characteristic velocity of the Reynolds’ number of vetf= ~ n - ~ l ~ ] . There is an even larger variety of parameters applied as characteristic lengths for the calculation of the numerical value of the Reynolds’ number. Several attempts have been made to find a length directly proportional to the resistance of the solid matrix. The most natural method is to use the square root of intrinsic permeability for this purpose (Ward, 1964; Harleman et al., 1963; Perez Franco, 1973; Valentin, 1970; Massarani, 1967). Some authors have related this parameter to the square root of porosity

(V,

[E)

(Collins, 1961). Thirriot (1969) haa derived a characteristic length,

assuming that the resistance of the porous medium is equivalent to that of a capillary tube having this length as diameter. According to his result, this length (d) is equal to the product of d*=

K

:v

- and a COmhmt (2.145)

32-.

Substituting the value of intrinsic permeability, according to the modified Koieny-Carman equation (the validity of which will be proved in the next chapter), it is easy to prove that the d o average pore diameter, used in Eq. ( 2 . 1 4 3 ) m a characteristic length, is similarly proportional to

k=-

5 (1 - n)a

d-4--01-n

51-n

(2.146) a

-.

D h 80 -rc

a

The characteristic length used by Zampaglione (1969) is similarly derived from the total resistance of the sample. The most important deviation from the other method, is the consideration of the difference between areal and volumetric values of porosity. The square of the length in question ( A 2 ) is defined as a factor of proportionality between mean pore velocity and hydraulic gradient v l f f = C I ; C = - A9’ ; V

and because


233

therefore (2.1-47)

A grouping according to the two applied parameters, gives an excellent summary of the various proposals analysed in the foregoing paragraphs (Table 2.1-3). The summarization clearly shows that the proposed dimensionless numbers differ only in the effect due to porosity (in the applied power of this parameter), because the use of various numerical constants Table 2.1-3. Grouping of the various Reynolds’ numbers proposed for the characterization of seepage

Square-mot of intrinaic permeebility

fi

Square-root of the ratio of intrimia permeability and porosity

Harleman et d . Masserani (1967)

Rew = v/v

(1963) Ward (1964) Perez (1973)

ReM = v/v

fi

Collins (1961)

rn

Equivalent tube diameter d* or do Parameter derived from the total resistance of the sample

Thirriot (1969) K o v h (1969)

Rey = f Z v / v

fk/n8

Rep = )I&/v Zampaglione (1969)

Rez = v / v

fw

A Square root of the Kovh product ofintrinsjo (unpublished permeability end result) porosity

fax

does not change the character of the Reynolds’ number. On the basis of the comparison,the transforming factors between the various Reynolds’ numbers can also be determined together with the ratio of these numbers to the classical seepage Reynolds’ number (Re,) (Table 2.1-4).

234

‘

I


Relp

Rec

Rex

=

Rez

I

Rer

1

1

1

After comparing the different Reynolds’ numbers, Valentin (1970) proposed that the use of intrinsic permeability should generally be accepted for the calculation of the characteristic length. The only remaining problem is the method of consideration of porosity and the choice of the applied constant. In the first instant, i t seems that this problem is only a question of decision, because knowing the various transforming factors, the numerical characteristics of the limits between the various types of seepage served by the application of Reynolds’ number can easily be transferred from one system to another (e.g. the limits previously fixed in the form of the given Re, values). However, aiming at the establishment of a general seepage law describing the relationship between seepage velocity and hydraulic gradient without any restriction of the validity zone, the power of porosity has to be chosen so that the points representing the measured resistances in a coordinate system based on the applied form of Reynolds’ number, should lie along a curve. Some guidance concerning the constant parameter to be used, can be gained from the constant of the modified Koieny-Carman equation proposed for the calculation of Darcy’s hydraulic conductivity. Considering these objectives, it was found that the multiplication of intrinsic permeability with porosity, is more suitable than the previous proposals, and the application of a constant of 5 is advisable:

Re, = 5-2) V

.

(2.1-48)

Tables 2.1-3 and 2.1-4 are supplemented with this finally proposed parameter.


235

the various Reynolds’ numbers Re8

(using the modifled Rep

Rq

n = -Rep

m

1

Kokny-Cannan equation for the calculation of permeability)

Reg

1 1

Rec = - -ReC 6 n

Rep =

,.

16 1

ReK

Rec=--

I

n

1

-Res

61-n

a

4

1

1-n

a

Rep=--

ReS

In the z.o)ie of microseepage, where seepage velocity is smaller than that pertaining to the laminar flow, the water movement i s also influenced by two retarding forces i.e. adhesion and friction. This zone can also be regarded aa a transition one, which develops between laminar movement and the static condition. As in the case of the use of the Reynolds’ number, the limit between the laminar zone and microseepage can be indicated by a given numerical value of the dimensionless number, proportional to the ratio of adhesion and friction [KO,in Eq. (2.1-37)]. Because of the lack of sufficient measured data, this limit can only be determined on the basis of theoretical consideration. Both Eq. (2.1-39) and Fig. 2.1-18 indicate the constancy of the M K ’ value in Darcy’s zone. Substituting the average pipe diameter in Eq.(2.1-39) as the characteristic length and hydraulic conductivity in the form of the modified KoSeny-Carman equation (aa already shown and as will be explained in detail in Chapter 2.2), this constant can be calculated: n MK~=--I (2.1-49) 80 The dimensionless number M K ‘ can also be determined from grain diameter instead of from the diameter of the model pipe: 1

MK:= 5

a2

n3 (1 - n)2

(2.1-50)

As in the various forms of Reynolds’ number, the square root of intrinsic permeahility can also be substituted as a characteristic length. In this ca-ae

236


the dimensionless parameter is equal to unity, or using the 5 m v a l u e , is inversely proportional to porosity

or 1 MKk=-. 25 n

(2.1-5 1)

Assuming that porosity, n = 0.40, MK; is approximately for a sample of spheres (a = 6), a value which agrees with Karsdi's data (see Fig. 2.1-18). Combining Eqs (2.1-37) and (2.149),the dimensionless number giving the limit in question can be expressed in Darcy's zone as a product of the ratio of threshold and actual gradient, and a constant depending only on Dorositv: (2.1-52) The same parameter if v k o r 5

v& is substituted for characteristic length is I K o,, = 2 ; I

or (2.1-53) This type of relationship must be valid not only within the laminar zone but also at its lower limit. Thus, the numerical value in question can also be determined by a constant value of the ratio between the threshold gradient and the actual one: I , = 1210; (2.1-54)

is the actual gradient a t the limit between the laminar zone and where microseepage, and the constant of 12 is based on the detailed investigations of microseepage given in Chapter 2.3. Comparing this result with Eq. (2.1-12), it can be seen that this limit is about Ih = 0.003 in sand (Dh > 0.01 cm), while in clay (Dh 2 p ) , the limit gradient is higher, about I , = 0.5, although the effect of adhesion is still very small here - smaller than the probable error in measurement of the gradient. The validity zones of the various types of seepage can be represented graphically as the function of either velocity or hydraulic gradient (Fig. 2.1-19). The first graph shows the laminar zone and the zones having a higher gradient. Here the Reynolds' number is linearly proportional to velocity. Its given numerical values indicate the borders between the subsequent zones. This number being proportional to the ratio of inertia and friction, indicates that the ratio of the two retarding forces can also be

-


--

237 3

3

2s I

0 zm2

a

~~

retarding forces

0

retardipQforces Fig. 2.1-19. Graphical representation of the validity zones of the various types of

-age

determined by this parameter. By definition, Mosonyi-Kov4cs' number is independent of velocity in the laminar zone. Its value decreases gradually with increwing velocity in the transition and turbulent zones. The I(v) function is represented in two ways. The solid line shows the connection between dynamically correct relationships in each zone, while the dotted line assumes the validity of the linear function within the whole range of movement. The difference between the two curves indicates the error which may arise by the unjustified application of Darcy's law. The second graph summarizes the validity zones of the types of seepage having small velocity (including the laminar zone as well). Here the w ( l ) function is similarly represented as in the first part but with the gradient now chosen as an independent variable (measured on the horizontal axis). The linear relationship gives a higher velocity than the actual one in this range of seepage. From the dimensionless parameters the Mosonyi-Kov&cs' number is represented as a function of hydraulic gradient. It is nearly constant in the laminar zone and gradually decreases below the lower limit of the latter. It becomes zero at the threshold gradient.

238


References to Chapter 2.1 BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. E. C. (1922): Fluidity and Plasticity. London. BINOHAM, BONDARENKO, N. F. (1966): Nature of Seepage-anomaly in Fluids (in Russian). Symposium on Seepage and Well Hydraulics, Budapeat, 1966. BONDARENKO, N. F. (1973): Physics of the Movement of Ground-water (in Russian). Gidrometeoizdat, Leningrad. BONDARENKO, N. F. and NERPIN,S. (1966): Influence of Viscous-Plastic Properties of Water on its Equilibrium and Transfer in the Unsaturated Zone. Symposium on Water i n the Unsaturated Zone, Wageningen, 1966. BONDARENKO, N. F. and NERPIN, S. (1967): Sheering Stress of Fluids and its Consideration in the Investigation of Surface Phenomena (in Russian). Publication of the Institute of Physico-Chemistry of the Academy of Sciences, USSR. CHAWETEAU, G. and THIRRIOT,C. (1967): Regime of Flow in Porous Media and the Limit of Darcy’s Law (in French). La Houille Blanche, No. 2. CHILDS, E. C. and TZIMAS,E. (1971): Darcy’s Law at Small Potential Gradients. Soil Science No. 3. COLLINS, R. E. (1961): Flow of Fluids Through Porous Materials. Reinhold, New York. DE WIEST, R. J. M. (1969): Flow through Porous Media. Academic Press, New York, London. EOTVOS,L. (1886): On the Itelationship between Surface Tension and Chemical Properties of Various Fluids (in Hungarian). Matematikai 6 Termbzettwlomanyi grtekezkaek, Vol. N ,p. 34. ERDEY-GR~z, T. and SCHAY,A. (1954): Theoretical Physico-Chemistry (in Hiingarian). Tankonyvkiad6, Budapest. FORCHHEIB~ER, PH. (1901): Watermovement through Soil (in German). ZeitschriJt dea Verbundea der deutechen Ingenieurs. FORCHHEIMER, PH. (1924): Hydraulics (in German). Teubner, Leipzig, Berlin. HARZEMAN, D. R. F., MEHLHORN, P. F. and RUDR, It. R. (1963): Dispersionpermeability Correlation in Porous Media. Proceeding of ASCE, HY. 2. JUJL~SZ, J. (1968): Investigation of Seepage (in Hungarian). Hidroldgiai Kiizlii?ay, No. 1. JWASZ,J. (1966a): Dynamic Characterization of Seepage (in Hungarian). Symposion on seepage and Well Hydraulics Budapeat, 1966. J ~ ~ sJ. z(196613): , Calculation of Static Ground-water Resources Available as a Result of Consolidation (in Hungarian). Symposwn on Seepage and Well Hydraulics, Budapeat, 1966. Joadsz, J. (1967): Hydrogeology (in Hungarian). Tankonyvkiad6, Budapest. -I, G. (1963): Hydraulics of Linear Drainage Systems (in Hungarian). (Thesis; Manuscript), Khartum, Budapest. KARADI,G. and TOROX, L. (1966): Applicability of Seepage Law (in Hungarian). HdroMgiai KOd&y, NO. 6-6. KAR-~DI, G. and V. NAGY,I. (1960): Investig&ion ofivalidity of Darcy’s Law (in Hun mian). Conference on Hydradics, Budapest, 1960. ~ Z D I A. , (1963): Neutral Stress and Pressure of Flow (in Hungarian). v k i i g y i Kodemdnyek, 1. K h D I , A. (1972): Soil Mechanics. 4th edition (in Hungarian). Tankonyvkiad6, Budapest. K ~ z D I ,A. (1974): Handbook of Soil Mechanics (Vol. 1. Soil Physics). ElsevierAkademiai Kiad6, Amsterdam, Budapest. KOV~CS, G. (1957): Theory of Micro-seepage (in Hungarian). HidroMgiai KOdizl&ry, No. 3. KOV~CS, G. (1968): Theoretiad Investigation into Micro-seepage. Acta Technica Acudemiae Scientiarum H u n g a r k e , Tom. XXI. 1-2. Kovkcs, G. (1966): Dynamic Investigation of Seepage by Invariant Numbers. Symposium on Seepage and Well Hydraulics, Budapest, 1966.

2.2 Hydraulic conductivity of saturated layers

239

Kovlics, G. (19694: General Characterization of Different Types of Seepage. I A H R Congress, Kyoto, 1969. KovAcs, G. (1969b): Relationship between Velocity of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th I A H R Congreae, Kyoto, 1969. KO~ENY, J. (1953): Hydraulics (in German). Springer, Wien. KIJTILEK, M. (1965): Influence of Interface on Filtration of Water in Soil (in French). Science du Sol, (Prague), 1. KUTILEK,M. and SUINOEROVA,J. (1966): Flow of Water in clay Minerals as Influenced by Absorbed Quinolimium and Pyridinium. Soil Science, 6. LANDAU, L. D. and LIPSCHZTS,E. M. (1974): Textbook of Theoretical Physics (in German). Akademie Verlag, Berlin. LEVICH, V. G. (1968): Physico-chemical Hydrodynamics (in Hungarian, translated from Russian). Akademiai Kiad6, Budapest. LINDQUIST,E. (1933): On the Flow of Water through Porous Soil. ICOLD let Congreee, Stockholm, 1933. MADQE, E. W. (1934): The Viscosities of Liquids and their Vapour Pressures. PhY8k8, N O . 2. MASSBANI, G. (1967): Critics of the Validity of Darcy's Law in Heterogeneous Porous Media (in French). CRAS, July. MATTSON, S. (1932): Soil Science. Baltimore. MOSONYI,E. and KovAcs, G. (1952): Model Law by Joint Consideration of Gravity and Friction (in Hungarian). Hidroldgiai Kodony, No. 7-8. MOSONYI, E. and KovAcs, G. (1956): Model Law of Filtration (in French). I A H R Congreee, Dijon, 1956. N~METH E., (1963): Hydromechanics (in Hungarian). Tankonyvkiad6, Budapest. PEREZ FRANCO, D. F. (1973):Non-Darcian Flow of Ground Water towards Wells and Trenches. Particularly within the Turbulent Range of Seepage. Technical University of Budapest (Manuscript). REINER,M. (1962): Theory of Rheology (in French). Dunot, Paris. SPRONCK, R. (1932): Hydrodynamic Similarity and the Investigation of Models (in French). Annalea dea Travaux Publiqua de Be2gique. SWAE~TZ, C. A. (1931): The Variation in the Surface Tension of Gas-saturated Petroleum with Pressure of Saturation. P h y e k , vol. l. THIRRIOT, C. (1969): Hydrodynamics of Flow in Porous Media (in French). 13th IAHR Congress, Kyoto, 1969. THIRRIOT,C. and HABIB,J. (1970): Experimental Study of Flow with Low Reynolds' Number in Clays (in French). Sciences Agrmomique, R e n w . VAGELEB, P. (1936): Cations and Water Content of Mineralogical Soils (in German). Berlin. F. (1970): Non-linear Resistance of Porous Media (in German). MitteilunVALENTIN, gen, Inatitut fiir Hydraulik und ae&eeerkzl.n.de, Miinohen, No. 6. WABD,J. C. (1964): Turbulent Flow in Porous Media. PV'OCeedilage of ASCE, H Y . 6. ZAMPAQLIONE, D. (1969): Turbulent Seepage Flow in Filters 13th IAHR Conpeas, Kyoto, 1969.

Chapter 2.2 Hydraulic conductivity of saturated layers The interpretation of Darcy's seepage velocity [Eq. (1.1-9)] and that of hydraulic conductivity [Eq. (1.1-ll)] h w already been discussed. It hw also been stated that the simple v=KI (2.2-1) relationship can be accepted for the characterization of all types of seepage

240


through saturated media, if the concept of hydraulic conductivity instead of Darcy’s original parameter (KO), is used in a more generalized form. The Darcy coeflcient is composed of two factors: one characterizing the properties of the solid matrix (intrinsic permeability) and the other describing the behaviour of the transported fluid. These two components have to be supplemented by a third one, depending on the condition of the flow. The third factor should be the function of either the seepage velocity or the hydraulic gradient, because the apparently linear relationship between v and I in Eq. (2.1-1) should be modified in such a way and for each zone of seepage that a suitable mathematical expression may be achieved connecting the two basic variables. Investigating seepage through loose clastic sediments, the intrinsic permeability is influenced by two properties of the solid matrix: (a) The size and form of grains from which the layer is built up; (b) The packing of the grains, which determines the size of the pores between the grains.

Thus,four components have to be investigated and determined for the complete characterization of the hydraulic conductivity of saturated, loose claatic sediments: (a) Geometrical parameters of the grains in the layer (size, shape and distribution); (b) Porosity of the layer; (c) Physical parameters of the propagating fluid (the ratio of specific weight and dynamic viscosity, or that of acceleration due to gravity and kinematic viscosity) ; (d) Flow condition of the seepage investigated (considering the dominant forces among those listed in the previous chapter). In Chapter 1.2, a geometrical model waa proposed to characterize the water transporting channels between the grains of the layer. Combining this simplified structure with the movement equations based on the consideration of the acting forces, a conceptual model can be established. According t o the explanation given in the introduction of this part of the book, the next phase is the verification of this model and the determination of its numerical constants on the basis of experimental data to ensure the practical application of the theoretical results. The h a 1 verification, which is the object of this chapter, will be based partly on data from publications, and partly on the author’s own measurements.

2.2.1 Determination of Darcy’s hydraulic conductivity Accepting the hypothesis whereby there are only two main forces to be taken into account in the zone of laminar movement (i.e. gravity and friction), their equilibrium can be expressed in a mathematical form for a model pipe with a diameter of do. The well-known Poiseuille’s equation can be


241

derived in this way. The equilibrium of a cylinder concentric about the axis of the pipe having a radius r and a length bf I , gives the following equation (Fig. 2.2-1):

I y r Z n l + 2 r n l q -d=v O . dr

(2.2-2)

Fig. 2.2-1. Symbole used for deriving Poiseuille's equation

After solving this differential equation with a boundary condition, where the velocity at the wall of the pipe is zero (w = 0 ; where r = ro),the velocity at a point at a distance of r from the axis can be determined: (2.2-3)

Integrating the product of the velocity and an elementary area (U) along the total surface of the cross sect,ion, the flow-rate through one pipe with a radius of ro can be achieved r.

This value, divided by the total area, is the mean velocity:

v = -Qo- - r zI. Y (2.2-5) A 8rl The number of pipes in the model system crossing the unit area of the sample [see Eq. (1.2-20)] is known, and thus, the total discharge and the virtual seepage velocity can also be calculated:

'

vs -- - =

As

Q o N = - *I ;Y

3211 where A , is the total cross-sectional area of the sample. 16

(2.2-6)

242

2 Determination of hydmulic conductivity

Substituting the effective grain diameter of the sample instead of the pipe diameter [see Eq. (1.2-19)], the following relationship can be determined: vS --_1 y n3 [?)'I. (2.2-7) 2 7 (l-nn), The hydraulic conductivity of the model pipes with constant diameter cdculated from Eq. (2.2-7) is 2.5 times greater than the actual value deter-

4

Fig. 2.2-2. Comparison of flow rate in pipes having constant or changing diameter

mined by reliable experiments as mentioned in Chapter 1.2 in connection with the detailed discussion of the establishment of the model system. It was also explained that this difference can be eliminated if the pipe is constructed from short sections with diameters of d1 and d, so that the volume of the pipe remains the same as that of the pipe with a diameter of do, and the 1 : 3.5 [Eq. (1.2-22)]. The aboveratio of df and df is equal to 1 : 3 mentioned increase in flow-resistance or the decrease in discharge can also be proved mathematically (Fig. 2.2-2):

-

Qi = Qz

m(I1

= QI-2 = a&,;

+ I,) = I , 1;

(2.2-8) Q1+

= p I l d : = BIzda = 2 P I ,


243

d, =1 ; 3.5 and d o 1.5 then a

N

0.4

On the basis of this explanation, the determination of the virtual seepage velocity can be based on the proposed geometrical model and can be calculated from Eq. (2.2-7) multiplying its right-hand side by 0.4. Comparing this result to Darcy’s equation, the theoretical value of hydraulic conductivity can be determined, which agrees with the dynamic analysis and includes all the effects of the influencing factors: (2.2-9)

The variables influencing this parameter can be divided into three groups:

”] [I

(a) The parameter expressing the behaviour of the fluid -, or using the

, the equivalent ratio is - ;

kinematic viscosity

V

(b) The characteristics of the grains in the layer (Dh/a) expressing the size, distribution and shape of the grains; (c) The effect of porosity

ns

(1

).

- n)2

To verify the proposed formula, i t is of value to investigate the ways in which these groups of variables were considered in the previously developed equations. If the relationship between the hydraulic conductivity and one of the groups is similarly described in a formula aa in Eq. (2.2-9) all measurements and observations used to determine the formula, also prove the validity of the relationship. The first investigations aimed a t exploring the relationship between permeability and the behaviour of fluids, have expressed their results in a form which illustrates the dependence of the K D parameter on temperature. Thus, Hagen (1869) observed a modification of 3 yoper degree centigrade between 12.5”Cand 23.4OC, while Havrez measured an increase of six times from 0°C to 100°C. This relationship waa later expressed in the form of mathematical equations (Seelheim, 1880; Hazen, 1895; Forchheimer, 1924). Among these formulae, Hazen’s equation is the best-known:

K D = llSD?,(0.7 16*

+ 0.03t).

(2.2-10)

244


The observations listed do not contradict Eq. (2.2-9). Temperature is only an indirect parameter to characterize the property of fluids and the correct method of expressing this relationship is by viscosity, which is inversely proportional to hydraulic conductivity. All the observations listed and formulae proposed, supply practically the same numerical ratio between two given values of temperature, as does the use of viscosity for the same temperatures. This statement is supported by the fact that in recent investigations, every research worker used the ratio of specific weight to dynamic viscosity (or the equivalent of acceleration due to gravity to kinematic viscosity) to express the dependence of hydraulic conductivity on the behaviour of the transported fluid (Forchheimer, 1924; Lindquist, 1933; Terzaghi, 1926, 1943; Koieny, 1953; Carman 1956; Chardabellas, 1964). On the basis of the above, the relationship between permeability and the parameters characterizing the properties of the fluid, can be regarded as an accurate and reliable one in the form expressed by Eq. (2.2-9). The variables used to describe the effect of porosity in the same form as they are applied here were first found in Koieny’s equation (Koieny, 1953): n3 e3 (2.2-11) Pk = (1-n)2 lfe Most of the recently published formulae use the same expression (Leibenson, 1947; Zamarin, 1928; Carman, 1956; Zauberei, 1932). There are, however, several proposals which differ from the above. From a theoretical point of view, the most important is Schlichter’s equation. This is the first attempt to combine all variables, independent of viscosity and grain diameter, into one factor, which is therefore named Schlichter’s number (Schlichter, 1899). Recalculating the effect of porosity from his formula, the relationship in question can be expressed by a parameter approximately proportional to

Ps=-.n2

(2.2-12)

1-n

The same formula expressing the influence of porosity can also be achieved theoretically by using the drag force model (Goldstein, 1938; Ward, 1964; De Wiest, 1969). More recent publications generally accept Koieny’s formula theoretically and suggest other equations only from a practical point of view to make the calculation easier. These proposals are mainly based on experimental measurements and can be summarized in the following general form:

P = ea;

(2.2-13)

where P is the parameter expressing the effect of porosity; e is the void ratio; and a is a constant determined from various measurements (a = 2 Terzaghi, 1926, 1943; a = 3 Chardabellas, 1964). JuhBsz (1967) published two different formulae to calculate this parameter. One is a member of the group


245

described by Eq. (2.2-13) with a power of a = 2.8 and the other has the following form:

Pi= n4.

(2.2-14)

The various proposals cannot be directly compared by using their absolute values, because the latter also include some constants, which may differ in each case. Good results can be achieved, however, by investigating the

porosiiy, n Fig. 2.2-3. Comparison of the results of the various methods proposed to calculabe the relationship between porosity and hydraulic conductivity

relative change of parameter P with porosity. In Fig. 2.2-3, the ratio of PIPs8 (i.e. the P parameter at a given porosity divided by that belonging to the most probable porosity n = 0.38) was plotted against porosity. On the basis of the graphs representing the proposals of several authors, it can be stated that the relationship determined by Eq. (2.2-11) and also used in Eq. (2.2-9), occupies a central position among the different formulae. This form has been accepted by most research workers and the other equations proposed do not give an appreciable difference for the probable values of porosity. Thus, the accuracy of Koieny's parameter in Eq. (2.2-9) can also be accepted. The investigation of the various methods published in the literature to calculate the influence of size, distribution and shape of the grains on hydraulic conductivity, is the most complicated study because of the numerous publications and the wide variety of basic ideas. There is, however, one point on which all the authors are in general agreement, i.e. hydraulic conductivity is proportional to the square of a characteristic diameter. This statement is proved by the formulae listed in Table 2.2-1 and by the graphs in Fig. 2.2-4. The latter combines the results given by Romer (see De Wiest, 1969) and Bear (1972). In both publications, intrinsic permeability is related t o the mean diameter of nearly homogeneous samples. The memurements used

Table 2.2-1. Comparison of formulae proposed for the calculation of permeability coefficient Author

I

Form

--

of the formula

KrBber

Temperature considered,,i the formula or m e d in the

[“a

KD=41 Di0

10 20

Seelheim

KD = 37 Di0

12

Hagen

KD KD

10

Hezen

Jhky KarAdi

DZo = 116(0.7

= 36

+ 0.03t)Df0

10 20

KD = 100 D:o

10

KD = 100 D2 (W balls) KD = (90 140) D:, (heterodisperse gravel) KD = 200 eS D:o

10

N

Terzaghi

2 2600 v

Dz (lead balls)

20 10 20

The given or aswmed validity =oneof the c0eta-t of miformity

-

--

1.2

m e multiplying factor valid in the c88e of the Itse of effectivediameter

43

m e recalculated shape wetacient assuming n 0.38 aa porosity

-

7.16

The

Q,

we Of diameter [cml 0.064-0.210

8.16

1.6

40

7.68

0.016-0.068

1.6

39

7.63

0.028

1.6-2.6

46 60

6.92 6.92

2.0

39

7.63

1.0

100

6.10*

2.0

40-62

1.6

46

1.0

10

6.31-7.86

%

7.00 8.00

Y

61

6.00

G

69 32 43 60 63

6.10 8.36 7.16 6.66 6.60 6.90-10.8

r

; e.

Lindquist

KD=

Chardabelles

KD = 266 e3 D2 (lead balls) KD = 140 ea De K D = 186 ea D 2 KD = 216 e3 D2 KD = 230 e3 Dz direct measurements

10 10 10 10 10

Zeuberei

330 N 3 6 h a KD = 0 (1 - n)*

18

47-60

Veroneae

K D = ~-g D 2 2300 v

10

34

7.96

Romer

K, = 6 . 6 4 ~1 0 - 4 s Dz

10

1.0

49

6.24

0.01-

Bear

K,= 6.17 x 10-4 S DZ

10

1.0

46

6.80

0.008- 0.1 J

Nola: n = 0.49

2

N

-

N

0.016-0.026 0.026-0.060 0.06O-O.100 0.123 0.022-0.081 ,

-

7.37-7.62

0.66

8

P a

x u

2.2 Hydraulic conductivity of satuxated layers

o

247

Krumbeh and Monk (lQ43)

particle size, D rcml

particle size, 17 Ccnl

Fig. 2.2-4. Relationship between intrinsic permeability and grain diameter (after Romer me De Wiest, 1969 and Bear 1972)

by Romer are listed in Table 2.2-2 while Bear has based his relationship on data measured by Krumbein and Monk and by the Institute Franpis Du Petrol. The opinions related to the interpretation of the characteristic diameter in the cttse of a heterodisperse sample, are more diverse. Assuming that friction is proportional to the surface of the grains and gravity to the volume of water, which can also be substituted by the volume of grains using porosity values, a result can be achieved according to which Kogeny's effective diameter is the most reliable parameter of a heterodisperse sample. This is because it is based on the identity of the specific surface (the ratio of the surface to the volume of grains) in the prototype and the model system. The same result was previously expressed by Koieny (1953) and Carman (1956). The proposal of Zamarin (1928) for the calculation of another characteristic diameter will first be compared to Kogeny's effective diameter. According to his formula, every fraction of the grain-size distribution curve has to be weighed by different factors when calculating the characteristic diameter. The reciprocal value of the latter is equal to the sum of the products composed of the weight of the grains belonging to a fraction and the total weight of the sample (A S f ) ,the reciprocal value of the mean diameter (D f ) and the previously mentioned factor of the same fraction (ai): (2.2-15)

248


Table 2.2-2. Experimental reaults between intrinsic permeability and diameter for nearly homogeneous porous media (after Romer) Diameter of mean particle siee D

t-4

u-ty we5cient

poroefty

IntrlnsiC penneability

U

n

k [em']

Investigator

Spherical shape 0.63 0.30 0.20 0.20 0.20 0.14 0.092 0.0646 0.0468 0.039 0.0383 0.0322 0.0273

0.646 0.14 0.092 0.046 0.0116

1.0 1.0 1.0

0.40 0.37 0.38 0.38 0.37

1.0 1.0 1.16 1.08 1.o 1.0 1.13 1.0 1.0 1.0

-

0.36

-

0.33 0.39 0.40 0.39 0.33

1.16 1.08 1.12 1.0

16.1 x 64.6 x 34.6 x 22.0 x 10-0 24.6 x 16.7 x 10-0

Brownell et al. (1960) Sunada (1966)

Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Ward (1964) Ward (1964) Harleman et al. (1963) Ward (1964) Ward (1964) Ward (1964)

6.7 x 26.2 x 10-7 18.7 x 10-7 10.3 x 10-7 11.6 x 10-7 9.0~10-7 5.77 x 10-7

2 . 6 6 ~lo-' 9.46 x 10-@ 4.83~ 1.10 x lo-" 8.10~

Banka et al. (1962) Harleman et al. (1963) Harleman el al (1963) Harleman el al. (1963) Raimondi et al. (1969)

The detailed analysis of this equation shows that this parameter is very similar to the ratio of the effective diameter to the average ghape coefficient. = 10 mm, ai= 1; The factor ai,increasing with decreasing diameter (Di and Di= 0.001 mm, ai= 1.8), expresses practically the same effect, which is represented by the increasing shape coefficient in the zone of fine grains. To prove this statement, the shape coefficients were recalculated from Eq. (2.2-15) using the constant in Zamarin's formula, and are listed in Table 2.2-3 as a function of the diameter. The data, compared to those repreTable 2.2-3. Probable shape coefficients belonging to various diameters recalculated from Z~MI&S formula Shape cOe5OiEnt Qmln diameter

[a1 1.0 0.1 0.01

0.001 0.0001

reoelculated from Zemarin's formula

8.0 9.2 10.6 12.1 14.4

average value in Fig. 1.2-5

8-1 1 9-1 2 10-13 12-16 16-18


249

sented in Fig. 1.2-5, verify the statement that Zamarin’s formula gives practically the same result as the theoretically derived equation [Eq. (2.2-9)] thus all measurements which served as the basis of Zamarin’s method, simultaneously, testify the validity of the latter as well. Although Paladin (1964) uses D5, as the characteristic diameter, his investigation also proves the accuracy of the use of the effective diameter. He applies a corrective factor depending on the coefficient of uniformity, to supplement D50and to express the eflect of grain-size distribution. This factor is proportional to the ratio of D50 to Dh, as shown in Chapter 1.2 (see Fig. 1.2-6). This fact can be accepted as proof of the accuracy of using effective diameter. Further comparisons are hindered by the fact that the various authors use different characteristic diameters (mostly D5, or D,,,). The relationship between these values and the effective diameter given in Chapter 1.2 (Fig. 1.2-6 and Table 1.2-2) make it possible, however, to recalculate the probable value of the shape coefficient included in the constant factor of the various formulae proposed. If the parameter remains within a realistic range of values, the investigated empirical equations and all the measurements used for their development prove the validity of Eq. (2.2-9). This recalculation wm easy in those cases where all the other variables were included in the formula in question, or the validity zone of the neglected variables was given. In other cmes these values were taken into consideration on their probable average i.e. porosity at n = 0.38 and temperature at t = 10°C and t = 20”C, the latter including the possible range of temperature. The result of this comparison is shown in Table 2.2-1. Because of the large number of formulae, it is not possible to list all the methods presented in the literature. The equations shown in Table 2.2-1 represent examples over a long period of time and from as many different countries m possible, ensuring the reliability of the presently proposed method by the investigation of a very large number of observations. As shown by the result of the comparison, a shape coefficient of a = 6 can be recalculated from equations, the bmic data of which were measured by using a sample of spheres, while the samples composed of natural sand or gravel generally resulted in a shape coefficient between 7 and 8, which agrees well with the expected theoretical value. The analysis of Beyer’s data also yields very reliable verification of the theory developed. Beyer has collected numerous data and presented them in a table which gives the expected coefficient of permeability as a function of D,, and D6,. The data in this table can be expressed in the same form as the general equation proposed by Hazen KD = CG,;

(2.2-1 6)

where the coefficient of C can be calculated from Beyer’s data atj a function of the coefficient of uniformity (U= D6dDl0),since both the corresponding Db0 and D,, values are given in the table. The best method of comparing Beyer’s data with Eq. (2.2-9) is to represent the recalculated C factor as a function of U (Fig. 2.2-5). The figure was supplemented by further values:

250


25, the fine grains started to move readily under the influence of relatively small gradients (Cistin, 1965).

3.2 Motion of grains in cohesionless soils

353

The same basic idea is reflected by the proposals for the design of filters, according to which, to ensure the stability of the flter, the uniformity coefficient of the filter material (Uf) should be lower than a given value:

Uf< 6 (Moscow-Volgostroj construction in 1935); U f < 16 20 (Creager et d.,1945); Uf< 20 ( U . S . Army Corps of Engineers, 1955).

-

(3.2-4)

The results of Lubotchkov's investigations (1965) are in slight contradiction to the previous statements. He has found that the possibility of movement depends to a great extent on the shape of the grain-size distribution curve, and hence not all materials having uniformity coefficients higher than 20 are liable to suffusion. He has pointed out at the same time that the condition satisfying Eq. (3.2-1) does not completely exclude the possibility of movement. If the layer is composed of the skeleton of coarse grains and the pores are fUed with fine particles, the latter can move through the voids of the skeleton, although the smallest grain is larger than the pore diameter calculated from the distribution curve of the mixed sample of coarse and h e grains. For this reason, Lubotchkov has proposed the investigation of the shape of the distribution curve and has prepared standard curves as a second limitation of the geometrical condition (Fig. 3.2-1). On the basis of the same investigation, Lubochkov has elaborated an analytic method as well. Its theoretical basis is the hypothesis that the Zuyer

am

0.1

4 relative diameter 47l5X

Fig. 3.2-1. Matching curves of relmtive grain-size distribution cheracterizingthe upper mnd lower boundmry of samples not sueceptible to suff'usion (mfter Lubochkov, 1962) 23

354

3 Permeability of natural layers

is not susceptible to su&&m when the slope of the distribution curve is eqml to, or smaller than a given limit in each diameter-interval. A simplified mathematical form of this condition can be written as follows (Fig. 3.2-2):

AS,IA& < l ; if Dn-1 - Dn 4.0 D, D"+l

AsJAs, < I ; 2.6

Fig. 3.2-2

if%=--

DrI

- 10

(tolerance safety factor 1.0);

Dn - 5 (tolerance safety factor 1.5);

Dn+1

of grains

355


. ASJ AS1 1.7

Dn< 1; if -Dn

D

1

= 2.5 (tolerance safety factor 2.3).

Dn+l

(3.2-5) Here and in Fig. 3.2-2, the symbols used are as follows: D, is an arbitrary diameter on the distribution curve; Dn-l and Dn+l can be determined from D, by multiplying or dividing it by 10; 5 or 2.5 according to the safety desired; AS1 and AS, are differences between the percentage in weight be0.05 01 az

0.5

D77L7J

Fig. 3.2-3. Application of KBzdi’s method to determine the self-filtering ability of a sample 23.

356


longing to the particular diameters:

AS1 = S,-I - S,; AS, = S , - Sn+l.

(3.2-6)

In his method, K6zdi (1969a) hus investigated the whole distribution i n a similar way. After dividing the sample at an arbitrary point of the distribution curve (D,) into two parts, the coarse skeleton can be ragarded as the filter of the remaining fine particles (Fig. 3.2-3). Applying Terzaghi’s filter law (according to which DidDS, < 4 < D{S/DS, where the symbols f and s refer to the filter and the soil respectively, while the subscript indicates the percentage by weight of particles of a certain diameter, it is possible to determine, whether the h e particles are able to pass through the pores of the skeleton. The 15 percentage by weight of the skeleton (Si;)and also the 15 and 85 percentages of the h e particles (Si5and S’&) can be expressed as a function of prcentage belonging to the selected D, diameter on the original distribution curve (S,): sF5= 0.858, 0.15; Sf5= 0.158,; (3.2-7) si5= 0.858,. Thus, the material forms a self filtering system (the geometrical condition excludes the possibility of movement), if the following inequality is valid at every point of the distribution curve:

+

‘D(OSSS) > D(OsSn+0.15) > 4 D t ~ . 1 5 S n ) *

(3.2-8)

It should be noted here that the lower grain size limit of the filter in Terzaghi’s equation, ensures the highest possible permeability, and the condition of stability i s expressed by the upper limit only. All three members of the inequality can be represented in the form of a graph as a function of the D, diameter chosen arbitrarily [i.e. the

4D(0.s5Sn)= 4@5 VE4*D n ; the D ( o . ~ S , , + o * U= ) DL VS. Dn and the = 4 0 4 , ) vs. D , curves respectively].

4D(0.1sS,)

-

The last curve gives the requiredlower limit of the representative diameter of the skeleton ( D k ) ,but this limit has no practical importance in this investigation as already mentioned. The critical particle size (Dcr)can be indicated where the upper limit (40$ vs. D, curve) intersects the Di; vs. D , curve. The grains being smaller than this size (in the range where D;, vs. D, curve runs above the upper limit) are able to move through the large pores of the skeleton. If the two curves have no point of intersection and the Di; VS. D, curve is everywhere below the upper limit, no suffusion is expected. There is a further limitation to be investigated. Uding the same division of the sample as before, the porosity of the remaining skeleton (n.) can also be determined depending on the original porosity ( n ) : V V - V,(1 - 8,) nu= =n S, = n S,(l - n ) ; (3.2-9)

V

+

V

+


357

where V is the total volume of the sample, V , is the volume of the solid particles and two simplifying msumptions are made. Firstly, the specific weight of the grains does not vary with grain size, and secondly the skeleton is uniformly distributed in the total volume of the layer after the fine particles are removed. As in Eq. (3.2-7), the diameters of the skeleton belonging to the 10 and 60 percentages by weight can be determined and their ratio yields the uniformity coefficient of the remaining coarse grains:

because and

+ 0.1; Sio = 0.45, + 0.6. &,

(3.2-10)

= 0.9s,

According to a previous investigation, the highest porosity (that'belonging to the loose condition of the sample) is a function of the uniformity coefficient and the shape coefficient, and it is independent of the diameter, as long as the latter is greater than 0.2 mm. Assuming an average shape coefficient of a = 10, the highest possible porosity of the skeleton is:

nhax= 0.3 + 0.15 exp

[

-

~

(3.2-11)

Comparing Eqs (3.2-9) and (3.2-11), it is possible to determine whether the grains of the remaining skeleton contact each other after removing particles smaller than the chosen D, diameter, or the new porosity (nu)is greater than that belonging to the loosest possible condition (n;,,). I n the latter case, part of the weight of the overlying layer is carried by the fine grains and the forces at their contact points also act against suffusion. Considering the geometrical condition of grain movement, the upper limit of unhindered suffusion has to be determined first, calculating the diameter Do below which the expected porosity [Eq. (3.2-9)] is lower than the possible maximum [Eq. (3.2-ll)] (Fig. 3 . 2 4 ) . The possibility of suffusion can be investigated only along the lower part of the distribution curve, where

D,

< Do.

(3.2-12)

Any critical value (calculated on the basis of methods given by Lulmtchkov, KBzdi, or by the method to be subsequently presented) above this limit, does not imply high susceptibility to suffusion, because the forces at the points of contact of the fine grains hinder their movement and, therefore, a relatively high gradient is necessary to initiate movement. It is worthwhile to mention here, however, that in this case a high velocity will cause the subsidence of the layer, displacing the fine grains, relative to each other, and resulting in a loss of stability. 'IB To ensure the uniformity of the mathematical treatment, the self-filtering bility of the layer can also be expressed using the model system of cupillary

3523


0.05 0.1 0.2

a5

I

2

5

lo

0.7

0.6 0.5 0.4

q 6 20 el; fa

u'

.-Y* @

10

0

005 0l L?Z 05 1.0 20 50 M O 20.0 D,,cmm_7 Fig. 3.2-4. Consideration of the maximum possible porosity in the determination of the self-filteringcapacity of a sample

tubes aa in the derivation of Eq. (3.2-2). Dividing the sample at a given diof the skeleton can also be calculated. ameter D,, the effective diameter (0;) Considering now the movement of a group of fine grains it is not necessary to compare the smallest particle to the pore diameter because the development of arching effect may be expected. According to the literature on filter design, the diameter belonging t'o 85% of the moving mms measured by weight appears to be a representative parameter (Cedergren, 1968). For safety, it may be compared to the average diameter of the pipes of the

3.2 Motion of grain8 in cohesionless soih

359

capillary tube model (see Eq. (3.2-2)]. Substituting n = 0.3 and a = 10 values as average parameters, the following relationship is obtained

n D;

4.0 -- < @s

l-n

a

:

where 4.0 n - 0.17 ; a l-n

(3.2-13)

thus

-

Di: < 5.0 D(0SesS").

Accepting the D,/D;, 1.25 ratio on the baciis of the analysis of the distribution curve (see Fig. 1.2-6) it is easy to see that this theory corresponds well with Terzaghi's and KBzdi's methods, not only formally, but also numerically. From the above, it is concluded that the first step in investigations concerning cluffusion is to check whether the geometrical conditions permit movement in the layer or not. Three conditions must be considered in this invest,igation: (a)The smallest grain is capable of movement through the pores of the layer [Eq. (3.2-2)]; (b) A group of fine grains below a given limit (D,) can move through the pores of the remaining skeleton [Lubochkov, Eq. (3.2-6); KBzdi, Eq. (3.2-8); or using the capillary tube model, Eq. (3.2-13)]; (c) Whether or not aftar removing fine grains smaller than the critical diameter determined according to the second condition (Dcr= D,) the etabili t y of the skeleton is e m r d [for comparison see Eqs (3.2-9) and (3.2-11).

Tf the second condition indicates the possibility of movement, but D,,

>

> D , according to the third condition, the subsidence of the layer can be

expected, rather than the development of suffusion. After checking the geometrical possibility of suffusion in the layer, it must be determined, whether or not the expected seepage-velocity is high enough to move the fine grains. The purpose of this investigation is the determination of the hydraulic condition of the motion of fine grains. The emieet way to express this hydraulic condition of movement is to establish the balance between the acting forces. In the case of a grain at reat on a horizontal surface, this Condition can be calculated by considering the weight of the grain (a)less the upholding force (af=a-U)and the drag force created by the flowing water (8):

af = V(y,- y o ) ;

s = Ivy,;

(3.2-14)

where V is the volume of the grain; yr and yo are the specific weights of the grain and the water respectively; and I is the hydraulic conductivity maintaining the movement of the water.

360


In the critical condition the angle formed by the resultant vector of the two forces with a horizontal direction, is equal to the angle of friction (KCzdi, 1969a): - = tan&, = tan@ = ___. I c r ~a (3.2-1 5 ) at Yt- Yo Accepting the validity of this relationship aa the first rough eatimation for characterizing the movement of small grains through the pores of a layer, and assuming the validity of Darcy’s law, and knowing the permeability of the layer, the critical value of both the seepage velocity and the effective velocity can be calculated

s

Yt - Y v ., vU,-ct= K tan @ -

and

K tan @ Yt - Y o . =n Yv

(3.2-16)

It is quite evident, that the velocity calculated in this way is considerably smaller than the actual critical value, because in the layer the grain is compelled to move in a complicated conduit including rising sections and not on a plain surface. The only consequence which can be drawn from Eq. (3.2-16), is that the critical seepage velocity is a function of permeability and friction, assuming the specific weights of both the grain and the water are constant. Another simplified model is the investigation of a flow directed u,pwards i n a vertical pipe (Cistin, 1966).The balanced condition is expressed in this case by the equality of the settling velocity ( w ) and the effective flow velocity. It is well known, however, that the velocity of settling is decreased by the wall of the container, if the diameter of the latter is small. The collision of grains with each other and with the wall has the same effect. Cistin has estimated that the critical effective velocity is smaller than half the settling velocity because both negative effects in the pores are of considerable magnitude: veffer 0 . 5 ~ and ; vsWcr Q.5nw. (3.2-17)

7.75; and at the limit b, = -.

(3.4-45)

Cr

The probable number of the openings having length is

R

width of

bh

within a unit

or neglecting once again the members of the series having higher c parameters and considering only the last one 8,'-

sa, exp (- 7.75) = 4 . 3 ~

sa,.

(3.4-47)

It is necessary to note, that the a parameter and thus, the probable specific number of the openings having a width of b h aa well, is the function of the Ab interval chosen arbitrarely. It is reaaonable, therefore, to deter, to the limiting mine a design value for the interval [ ( A b ) h ]proportional width, and transform all the results calculated by using any Ab interval to this system. The determination of such a design value is an arbitrary step once again, and i t can be proposed that the b, width should be the mean value of the 5-th d b interval. Considering the listed conditions, the probable specific number of openings having the limiting width and also those within a length of L can be calculated 1

( A b ) , = -b, = 0.22 b, ; 4.5

(3.4-48)

30*

468


If a characteristic length is defined in such a way aa to be the shortest size of a field regarded as a continuous seepage field, the requirement is that the probable number of an opening having a width of (1 f O.ll)bh has to be 1 within this length s h = 1; consequently (4.4-49) ~ b 1 0 4 ~b c, 104 L=--=---* ab, a, s a, 7.75 The data calculated in this way are also listed in Table 3.4-1 1 , and they agree well with the observations around pumped wells (see Fig. 3.4-22), and also with Kir4ly’s hypothesis, where the spacing of large water transporting 4 km. formations has to be about 2 The final step still to be solved is the combination of the general movement equation [Eq. (3.4-32)] and the model. Unfortunately, the equation gives the hydraulic gradient as a function of mean velocity, while in the model the velocity haa to be substituted in explicit form. For expressing the velocity from the general equation in an easily amenable form,some terms have to be expanded into series, which require the application of some new restriction of the validity zone. Thus, the equation loses its general character, but it remains suEcient for the combined characterization of the laminar and transition zones. For this derivation Eq. (3.4-32) has to be modified

-

where

and

I

1;

4B

0 means i d t r a t i o n to the ground water while E < 0 indicates its vertical drainage. Accretion related to the unit area of the phreutic surface ( E , ) can be calculated from the parameter given for a horizontal surface ( E ) and the /3 angle between the water table and the horizontal plane: En = E cos p, (4.2-1 4) although the difference between E, and E is generally neglected, because usually the slope of the water table is very small. The amount of accretion is equal to the flow rate through a unit area (e.g. m3sec-1 over m2), thus its

+&

-&

Y

-u-

X

Fig. 4.2-6. Development of stream lines in seepage field influenced 33

accretion

514

4 Kinematic characterization of seepage

dimension is equal to that of velocity [LT-11. When the purpose is to determine its influence on the boundary conditions, the flow rate crossing the water table (either recharging or draining the seepage space) has to be considered first. The flow rate or flux is equal to the seepage velocity perpendicular to the phreatic surface, thus it can also be expressed depending on the component of gradient vector in the same direction: 8,

=V, =K

as, I , = - -. an

(4.2-1 5)

The angle between the internal normal of the water table and the vertical axis is B (equal to the declination of the water table), while the angle closed by the internal normal and the horizontal direction is n/2 - B. The I , component of the gradient can be expressed, therefore, as the function of the I , and I , projections of the gradient: I , = I , sin p I , cos B . (4.2-16) Combining the last two equations, and considering that accretion is positive in the case of infiltration, and thus its direction is opposite to that of the y axis, the following equation may be written:

+

sin B I ,

+ COB?!,

I + ;I I,

- =0.

(4.2-17)

The left-hand side of this equation is a scalar product of two vectors: the first is the unit vector perpendicular to the phreatic surface, the coinponent of the other are I , (in z direction) and I ,

+ - (in y &

K

direction). The

scalar product being zero indicates that the two vectors are perpendicular

to one another, the second vector has to be tangential, therefore, to the phreatic surface. It follows from this relationship that the scalar product of the pressure gradient vector and this new vectorial parameter are also zero on the surfaces with constant pressure, where the pressure gradient is normal to the surface: (4.2-18) I , -8P + I -=o.

ax

1, + -A:

The surface where Eq. (4.2-18) is valid can be either the water table ( p = 0 ) if capillarity can be neglected, or the capillary surface, because the pressure is also constant along the latter ( p , = - yh,). From this relationship the boundary conditions can be expressed in the form of a function between the components of the hydraulic gradient or the velocity vector on a surface having constant preaure and influenced by accretion, similar to Eq. (4.2-9):

or

(4.2-1 9)

4.2 Boundary and initial conditions

515

The boundary condition along such a border can also be expressed by using the stream-function. It is evident that the change of the flow rate between point A and B is equal to the sum of recharge or drainage along this stretch. T h e digerence of the stream-functions belonging to the two points in question has to be equal, therefore, to the product of the accretion function and the elementary horizontal length integrated from A to B (Polubarinova-Kochina, 1952, 1962): B Y B -YA

=

(4.2-20) A

Until now, when investigating the boundary conditions along the free surface of the seepage field, i t was always supposed that the seepage is in steady state, and that the kinematic parameters, and thus the position of the investigated surface as well, are time-invariant. In the case of unsteady flow the conditions are basically different, because the surface i s raised or lowered in time. When the surface propagates, dry pores of the layer become completely or partly saturated, though when the surface is withdrawn the water stored in the pores above the new position of the water table is drained into the seepage field. The amount of water stored during the first process or released from storage in the second case, is calculated generally as the product of the change of the water table and specific yield, although the latter parameter is neither exact, nor completely characteristic. This is because of the prolongated process of drainage and thus the time lag occurring between the lowering of the water table and the complete release of the stored water (see Section 1.4.2).It is for this reason that proposals can be found in the literature to use the instantaneotks dewatering weflcient ( m ) instead of specific yield, which is the amount of water released from the pores just after the lowering of the water table, and which is, therefore, always smaller than the specific yield (Bear et al., 1968). The most important difference in the boundary conditions along a timevariant free surface and those for steady seepage [see Eqs (4.2-6) and (4-2-7)], is the fact that the surface is not a stream line, thus the constancy of the stream-function is not valid. The instantaneous stream lines intersect this border at angles different from n/2 the border is not therefore, a potential line either. Two examples are shown in Fig. 4.2-7 to represent the development of the flow net in the case of a falling water table. The first is the generally quoted case of an earth dam neglecting the effect of capillarity above the water table. The second example is constructed on the basis of experimental data (Vauclin e f al., 1975) and clearly indicates the influence of the soil-moisture zone. Investigating the boundary conditions along the water table and assuming the negligible effect of capillarity and accretion, it can be stated that the potential is proportional to the elevation of the investigated point fitted to the water table, because the pressure is zero at this surface. When the total differential value of the potential has to be determined, i t is necessary to consider that the potential changes also with time, aa the phreatic surface 33.

516

4 Kinematic characterization of seepege

wafer level at the sfarflng time point of the process

wiibouf capillary effect

impervious lager with capillary effect

/no

0

0

-1 1

~

'

~

'

1

'

1

cm 300

2011 '

1

'

'

1

1

capillery urpce

0

1

'

1

1

1

~

1

~

~

'

'

1

~

'

~

water table

200

400

cm 300

velocity vector Instantaneous flow net in an intermediate time point of the draining process of earth dams cm

Fig. 4.2-7.

'

T=I.O h

-I

moves upwards or downwards. Eq. (4.2-9) has to be supplemented with a term, expressing the change of the potential with time:

or

(4.2-2 1)

v:+v;4,+Kvy=ns-.av at


517

Equation (4.2-21) i s valid also for the characterization of the cupillary surface but capillary suction has to be considered as a component of the total potential at the water surface. Similarly the influence of accretion can be considered also in the characterization of boundary conditions along the phreatic surface of unsteady seepage: + E+E n as, I ; + I; + I , K -. = 3- ; K K K 2 at

vi

+ v$ + v,(K + + K E= n, aP, . E)

(4.2-22)

at

impervious layer

1 exit &-e Fig. 4.2-8. Development of seepage and boundary conditions in the surroundings of the free exit faoe

Finally the forth main type of surface bordering the seepage field is the free exit surface (CJ, where the percolating water leaves the porous medium and enters into a space of free air characterized by atmospheric pressure (ptot= p o ; p = 0 ) . The position of the free exit face is geometrically fixed, time-invariant and known a priori, as that of the exit face covered with water. There is one basic difference between the two types of exit faces caused by the different materials contacting the seepage field i.e. the pressure along the free exit face is constant ( p = 0). Consequently this contour is not a potential line, but the velocity potential changes similarly to the water table (the potential is proportional to the elevation of the point in question above the reference level) (Fig. 4.2-8):

v = KYk.

(4.2-23)

The free exit surface is not a stream line either because water leaves the field through this surface, thus the stream lines cross it. Another boundary condition can be derived, however, from the fact, that the pressure is constant along the contour, thus the pressure gradient has to be perpendicular to it at every point. From this condition, and considering also the relationship between velocity potential and pressure gradient, an equation can be determined, which gives the boundary condition in the form of interrelated horizontal and vertical components of the hydraulic gradient vector (Bear et al., 1968): (4.2-24)

518

4 Kinemah characterization of seepage

and because

(4.2-25)

consequently

I,

--

-

sin

-Iy-l. cosp

’

or

(4.2-26)

I, =

-

(1

+ I , cotan p) ;

where fi is the angle between the tangent of the investigated point at the free exit face, and the positive direction of the horizontal x axis. There is one further important aspect, which has t o be mentioned concerning the influence of the free exit surface on the development of seepage within the flow field. Along stream lines crossing this stretch of the contour, the total potential i s smaller than that between the head and tail water, and its value changes depending on the height of the intersection of these stream lines and the free exit face (KovBcs, 1965). As already shown, the total potential of the system of stream tubes reaching the exit face below the level of the tail water is constant and equal t o the product of hydraulic conductivity and the elevation of the head water above the tail water [AT = K d H ; see Eq. (4.2-4)]. In the case of stream tubes having higher position, the total potential has t o be calculated aij the difference of the constant entry potential and the changing value of the exit potential, e.g. choosing the level of the tail water as datum, ;t8 i t is shown in Fig. 4.2-8:

I n connection with the kinematic classification of seepage as previously mentioned, the seepage field covered by a semi-permeable layer can be regarded as a transition form between the c o n h e d and unconfined systems. The same behaviour can be observed in the characterization of the boundary conditions as well. I n the case of steady seepage the upper contour of the field (the lower boundary of the covering layer) i s a stream line, the boundary conditions being described as those along a n impervious boundary. If the seepage is unsteady, this border does not remain a stream line, because the water originating from the change of storage crosses this contour. Its positioii does not change, however, in time. I t s role is similar, therefore, to the free surface with accretion, but the latter changes in time. Another difference is that the pressure is not constant either in time or along the border between the seepage and the covering semi-pervious layer.


519

4.2.2 The investigation of the layered seepage field As already mentioned, in the determination of the hydraulic parameters of seepage the knowledge of the boundary conditions around the contour of the field, the initial condition for the whole field in the case of unsteady flow, and the flow conditions of the field (hydraulic conductivity or intrinsic permeability and porosity) is necessary. If the flow conditions diger considerably within the field, so that they can be regarded as constant in a region, while in other regions the parameters are different (but similarly constant within the separated regions), the seepage field has to be divided into so many parts (layers) as is necessary to have homogeneous (or almost homogeneous) regions. When investigating the seepage through such layered fields, the special conditions along the internal boundaries have to be considered a.s well. The layered structure is called regular, if the bordering surfaces between the different layers are parallel planes, and thus the thicknesses of the layers are constant. This type of structure is quite common in nature, especially in the water transporting systems of the large sedimentary basins, where the aquifers may be well approximated with such regular structures. I n regularly layered flow fields two basic cases of flow can be emphasized: seepage parallel to the bordering surfaces of the layers, or flow perpendicular to these surfuces. If the flow is parallel to the layers, the total resistivity of the formation against seepage can be calculated on the basis of an analogy between this process and electrical resistances connected in parallel (Fig. 4.2-9) :

(4.2--28)

where K , is the hydraulic conductivity of the i-th layer, mi is its thickness and K , , is the resultant hydraulic conductivity parallel to the dividing planes of the formation composed of n layers. In the c a e of seepage perpendicular to the surfaces bordering the layers, the combined hydraulic conductivity can be determined similarly, consider ing now its analogy with the electrical resistivity of a system connected in series:

i.

li

K1= -q1 . 9

(4.2-29)

2-

Kf

where li indicates the thickness of the i-th layer, which is measured in this case in the direction of flow (Kamensky et al., 1935; Kamensky, 1943; Shea and Whitsett, 1958). It is evident from Eqs (4.2-28) and (4.2-29), that even a thin layer having very high hydraulic conductivity determines basically the total

520


t

L

L

t

i

1

TT

--7

CL

inpervlbus boundary

..

x

.. . . . . . _ .- . .., .

..... .

impervious boundary impervious bmdrlry Fig. 4.2-9. Characterization of resistivity parallel or perpendicular to the borders of the layers in layered seepage field

permeability of the formation, if the seepage is parallel to the layers. Similarly the total formation can be regarded as impervious in the caae of normal flow, if there is one (even a relatively thin) very impervious layer in the formation (Bear et al., 1968). When solving practical problems numerically, the seepage field may be characterized by its original geometrical data. In this caae the flow condition is described with one of the resultant hydraulic conductivities calculated from Eqs (4.2-28) and (4.2-29) respectively, depending on the direction of flow. Another possible way, which may he applied, is to choose an arbitrary , hence, determine the hydraulically value of hydraulic conductivity ( K O )and equivalent length of the jield (lflctive)belonging to the chosen permeability in the case of perpendicular seepage, or the hydraulimlly equivalent thickness of if the flow is parallel to the layers (see Fig. 4.2-9): the formation (mflctlve)

Seepage perpendicular to the layers:

(4.2-30)

(4.2-31)

4.2 Boundmy and initial conditions

521

When the investigated seepage is normal to the surfaces dividing the of layers, the ratio of the thickness (li) and the hydraulic conductivity (Ki) the layers can be applied to simplify the numerical characterization (Galli, 1959).This parameter has a dimension equal to that of time [TI,and it is called seepage resistance. The total resistance of a, formation expressed with these parameters can be written in the following form:

li

n

etot = 2 ei = 2 - . 1

1

Ki

(4.2-32)

If the direction of seepage closes an angle with the surfaces bordering the layers, different from 0 and nI2, the formation can be characterized as a layer having transverse anisotropy. In large sedimentary basins the development of this type of anisotropy is generally characteristic, because the pervious formations are composed of layers of only slightly differing permeability, with nearly horizontal bedding planes. Thus hydraulic conductivity in each horizontal direction is the parameter representing the flow parallel to the layers (KN= KII)while the vertical permeability is described by the parameter calculated for perpendicular flow ( K , = K l ) . The coefficient of anisotropy can be calculated aa the ratio of parallel and perpendicular hydraulic conductivities (or intrinsic permeabilities) considering that the thickness of the layers is identical in Eqs (4.2-28) and (4.2-29), although they are there indicated by different symbols (mi = l j ) :

(4.2-33)

There are, however, some restrictions to be considered, when the layered formation is characterized as an anisotropic field: (a)The variation of permeability of the layers should not be high, because extreme permeability can basically influence the behaviour of the formation as already mentioned. In practice it is a generally accepted limit that the ratio of permeability of any single layer should not be higher or lower than the average value multiplied or divided by a constant from 3 to 5. (b)The layers should be relatively thin, because in opposite cases the approximation with an anisotropic field is inaccurate, and, therefore, the investigation of the actual layered system is preferable. I n solving practical problems layers having a thickness more than one metre are generally considered separately, but the limit depends also on the total size of the seepage field, and also on the relative permeability of the neighbouring layers. (c) When the formation is composed of groups of layers, the field has to be divided into more than one anisotropic units (an example is shown in Fig. 4.2-10, where a double layered anisotropic field is considered). The requirement concerning the ratio of permeability should be satisfied in each unit but the difference is larger between them.

522


3' B

;s"

8%

$ Lsg

Fig. 4.2-10. Characterization of layered flow fiold with anisotropy

Because of the limitations mentioned in connection with the application of anisotropy as the simplest characterization of layered seepage fields, there are many cases, when the consideration of the layered structure and thus that of the internal boundary conditions along the surfaces dividing two layers becomes necessary. These internal boundary conditions are very similar to the law of light refraction i.e. if a stream line enters a porous medium having smaller permeability than the preceding layer, and it crosses the bordering surface at an angle different from 4 2 the line breaks so, that its angle t o the normal is greater than that in the more permeable layer. Going in the opposite direction (from the less permeable medium into a more permeable one) the ratio of the angles with the normal is also opposite, the entrant angle is larger than that of the emergent one. The basis of the mathematical characterization of the internal boundary condition is fhe continuity of both pressure and flow at the intersection of the stream line with the bordering surface. T h e pressure and the potential energy (elevation above the reference level) have to be the same on both sides of a point on the surface. It follows from this condition that the ratio of the velocity potentials in the two contacting layers at the same point on the surface is equal to the ratio of the permeabilities or hydraulic conductivities (Fig. 4.2-1 1 ) :

K" z" = z'; p' = p"; consequently rp" = -rp' . K'

(4.2-34)

If the two velocity potentials are proportional to one another along the dividing surface, their differential quotients in this direction have t o follow

4.2 Boundary ark1 initial conditions

523

Fig. 4.2-11. Break of st,ream lines at the border between two layers

this pro1)orti oiiality : 1

W f-

1 alp'

(4.2-35)

K" as K' as On the basis of the definitions of velocity potential and hydraulic gradient. i t is quite evident that Eq. (4.2-35) determines a condition for the hydraulic gradients in the two contacting layers a t the intersection of the stream h i e with the bordering surface; viz. their components parallel t o the surface have to be equal: I," = I;. (4.2-36) Tlie other basic condition is the continuity of flow,which requires that the compoitenls of secpugc velocities pcrpendiculur to the surface should also be epcal : v; = VA. (4.2-37) Accepting the vdidity of Darcy's law the ratio of the normal components of the hydrnulic gradients can also be determined from Eq. (4.2-37), i.e.

K"I; = KfIL.

(4.2-38)

I n mi isotropic, homogeneous medium the gradient vector is tangential t o the path (in the case of steady flow t o the stream line as well). Therefore, the angles closed by the path (or stream line) and the normal of the bordering ?' can be calculated in both layers from the ratio of the surface (i.e. ,iand coml)onents of the gradient vectors:

r)

I' = tan

I:,

p'

I," I;

; -- - tan

r.

(4.2-39)

Combining Eqs (4.2-36), (4.2-38) and (4.2-39) the h a 1 result can be achieved: the rutio of the tangents of the two angles closed by the path and the n o r m 1 of the dividing surface i s equal to the ratio of the hydraulic conductivities of ihe contacting layers: tan K" -=(4.2-40) tan p' K'

524


It follows from Eq. (4.2-40) that if the resistance of the second layer is very high compared to the previous one, the path (or stream line) does not enter i t (more precisely it hardly enters) and, therefore, the second laver can be regarded as impervious. This interpretation gives a further e s l h i a tion why the approximation of the layered field with an anisotropic medium is not acceptable if there is a very impervious or very pervious member i n the series of layers. Here the ratio of hydraulic conductivities of two neighbouring layers is too high. However, there is no complete refraction a t the borders of the layers (as it can be observed in optics), but according to the correct characterization, the stream lines hardly enter the layer having low hydraulic conductivity. It is the reason why a sedimentary layer is never absolutely impervious. This is only a relative term, the acceptance of which depends on the character of flow, the size and form of the seepage field, and also on the proportion of the total flow rate, which can be regarded as negligible. To prove the last statement, let us consider as an example the seepage field below the horizontal impervious foundation of a dam (Fig. 4.2-12). The difference between the elevations of the head and tail water maintains a steady seepage under the dam. The first supposition should be that the depth of the pervious layer is infinite. The flow net can be mathematically determined in this very simple case, and the vertical distribution of seepage velocity along the symmetrical axis of the flow field can be calculated. If the field is not homogeneous, but composed of two layers, the original flow net is distorted. The modification of the vertical distribution of seepage permeable layer of infinite depfh

practica/& impervious lower layer

2b

w8ter transporf through the lower layer is noi'n~l@ible 2b

Fig. 4.2-12. Interpretation of the depth of seepage field, in layered Hystem

525


velocity depends evidently on the depth of the second layer, and on the ratio of hydraulic conductivities. The lower lying layer conveys a portion of the total flow in any case (even if i t has low conductivity and its surface is in a deep position). It is necessary, therefore, to decide first, what ratio of the two flow rate is regarded as negligible. After determining the total flow rate (e. g. by integrating the graph of vertical velocity distribution in both layers) and comparing the water conveyance of the lower layer to it the relative imperviousness of the lower layer may be judged. When this limit is set e.g. at 10% of the total flow rate, the lower layer is called impervious, when the upper layer transports more than 90% of the total flow rate. It follows that a layer can be regarded as impervious in a deep position, while the same layer conveys more than the limiting rate if its surface is in a higher position. If the criterion of imperviousness is determined by the percentage of total flow rate, even the lower part of a homogeneous layer can be neglected from the point of view of water t,ransport, as i t would be impervious. It can also be seen in Fig. 4.2-12 that the critical depth of a given layer (where its water transport can be neglected) cannot be determined with absolute values of depth given in metres, because i t depends not only on the ratio of hydraulic conductivities, but also on the size and form of the seepage field. On the basis of the similarity of flownets the maximum depth of a given stream line is influenced by the geometrical data of the system aa well (in the example by the length of the foundation i.e. 2b). This fact draws attention to insufficient investigations where an attempt is made to determine the active depth of a pervious layer, below which neither natural effects nor human activity on the surface can initiate seepage. Such approximation can be accepted only, if they consider the whole structure (form and

0.0

0.50

m, m

LOO

0.0

0.5

m, m

-

l.0

Fig. 4.2-13. Ratio of total water conveyance of a double layered seepage field and the fictive flow rate of a similar homogeneous field

526


size) of the seepage field and its contact with the actions creating and maintaining the flow. For numerical characterization of the relationships explained only qualitatively in the previous paragraphs some of the results of an experiment are shown in Figs 4.2-13 and 4.2-14. The object of the experiment was to determine the influence of the layered flow field under different foundations composed of horizontal and vertical elements (concrete blocks and sheet piles) (ojfaludy, 1974). I n Fig. 4.2-13 the measured points and the constructed curves are shown, which characterize a double layered system of two dimensional seepage when sheet piles are not applied, and the foundation is a horizontal plane placed on the surface of the layer. The measured total flow rates (9) are related to a fictive value calculated by supposing only one layer below the foundation, the thickness of which is equal t.0 the total thickness of the two different layers ( m = m1+m2) and its hydraulic conductivity is equal to the parameter of the more pervious original layer [qol if the upper layer is more permeable ( K , > K,) and qo2in the opposite case ( K , < K , ) ] . These fictive values are calculated from Eq. (5.3-67) hy substituting m aa the thickness and K1 or K2 as hydraulic conductivities. The first graph of Fig. 4.2-13 represents the case K ,

>K ,

and the second one shows the q/qo2 value as the function of

,-' ] I : [

[

q/qoL=

m

m

f

if K ,

< K,.

The curves approximating the measured points can be espressed in the form of mathematical equations as well:

---

l0-

--g 05-

00

0.5

LO

2 m

Fig. 4.2-14. Ratio of water conveyance of the upper layer and the total flow rate in double layered system 88 a function of the rate of the hydraulic conductivities


527

> K,;

if K ,

q=-

K*AH x

where

1.5-

and 901=

if K ,

KIAH -

(4.241)

7c

K , (Fig. 4.2-14):

(q) can also be approximated mathematically and, if

(4.242)

Supposing that a is the negligible fraction of flow rate (e.g. if the limit, mentioned previously is lo%, then a = O . l ) , the condition of the imperviousness of the lower lying layer can be expressed also in mathematical form from Eq. (4.2-42):

52 m

[ :[ 1.67 -sh

(1 - a)ar sh

528


Applying one or more sheet piles the situation becomes more complicated e.g. if the thickness of the upper lying pervious layer is smaller than the depth of the sheet piles, all stream lines, therefore, have to penetrate into the second layer, and the water transport is determined by the water conveying capacity of the less pervious material. Although Eqs (4.2-41), (4.2-42) and (4.2-43) are valid only in the case of a very simple form of the seepage field (as shown in the figures), they are suitable to visualize the problems arising in connection with the use of the term impermeable. They can be applied as rough estimations in many practical cases to characterize the expected development of flow through a layered system. Another special type of layered seepage fields is the case of a vertical (or almost vertical) position of the contacting surfaces, if a nearly horizontal unconfined flow develops through the system. The special aspects to be considered in this cme can be well demonstrated in the section of a multilayered earth dam (Fig. 4.2-15), which is one of the most frequently occurring practical examples of this type of seepage field. Where the water enters a less pervious layer from a pervious one, the less pervious layer causes a backpressure effect in the preceding part of the seepage field.’ At the contacting surfaces, where the change of hydraulic conductivities is the opposite (the first layer is less and the second more permeable) a free exit surface has to develop in a similar way to the contact of the seepage field with the tail water (Kov&cs 1968). It is necessary to emphasize here that a stretch of the seepage field must never be investigated separately, always the whole field has to be considered from the entrant potential surface to the final exit face. The latter is also a potential surface below the level of the tail water. Also resistances to flow within the entire section have to be determined. It is for this reason, in the example given in the figure that the dynamic equilibrium can be described within the body of the dam, including not only the free exit face where the water enters into the atmosphere, but also the internal free exit face (or the internal free exit faces, if there is more than one impervious layere, from which the water flows into more permeable layers). The equations characterizing this dynamic equilibrium have to be solved simultaneously, satisfying the given boundary conditions.

Fig. 4.2-16. Development of phreatic surface in a vertically layered earth dam

529

4.2 Boundary rand initial Conditions

4.2.3 Application of hodograph image and other special transformations for the characterization of boundary conditions Among the various kinematic formations, the path (or the stream line in the case of steady movement) can be used for the characterization of seepage velocity. It gives information, however, only on the direction of the velocity vector (the latter being tangential to the path). For characterizing the size of the vector the hodograph curve i s frequently applied on the basis of Kirchhoffs proposal. This curve runs through the end points of velocity vectors represented in a coordinate system independent of the field of the actual movement. All the vectors start from the origin of this new coordinate system (hodograph space), and the curve follows the order of the points along the real path. When investigating a two-dimensional movement, the hodograph (which is generally a three-dimensional curve) becomes also two-dimensional, and it can be represented on a so-called, hodograph plane (Fig. 4.2-16). In seepage hydraulics the use of hodographs assists us in representing the phreatic surface, the position of which is not known a priori, but considering the boundary conditions along this surface its hodograph image can be easily constructed. It is for this reason that the application of this method is very wide in this subject. It is necessary to note also that in homogeneous and isotropic seepage fields the velocity vector is proportional to the vector of the hydraulic gradient and the factor of proportionality is the hydraulic conductivity (accepting the validity of Darcfs law). The velocity hodograph can be supplemented, therefore, with gradient hodograph, which differs from the former only in the unit size measured on the axes of the coordinate system. The advantage of the application of the gradient hodograph is that in this case dimensionless quantities are represented on the axes (the components of the gradient vectors). Tf a hodograph is constructed for each point of the seepage space, the end points of the velocity or gradient vectors will completely fill a limited space,

t"

hodograph plane

"X

Fig. 4.2-16. Interpretation of hodograph curve 34

530


bordered by the end points of those vectors which belong to the contour surface of the actual seepage space. Limiting the discussion to the investigation of two dimensional flow, the end points of the vectors, belonging to the contour line of the seepage field, determine a closed curve on the hodograph plane, which is called hodograph contour. The construction of the limited hodograph field surrounded by the hodograph contour is a special type of mapping of the actual seepage field, because contact can be established between the relevant points of the two fields. If this contact can also be determined analytically and expressed in the form of mathematical formulae, the equations can be used for the calculation of the hydraulic parameters, as will be described in Chapter 5.1 and 5.2. In many cmes there are possibilities only for the graphical construction of the hodograph contour, and not for the analytical description of the relationship between the flow plane and the hodograph plane. This method, however, is also very useful in solving practical problems, because valuable information can be gained in this way on the velocity conditions along surfaces, the positions of which are not known on the actual seepage plane (as was already mentioned it gives the hodograph image of the phreatic surface, and even the velocities at special points of this surface can be determined). For this reason, the literature of seepage hydraulics deals generally in great detail with the construction of hodograph contours and with their practical application (Hamel, 1934; Vedernykov, 1934; Polubarinova-Kochina, 1952; 1962; Aravin and Numerov. 1953; Harr, 1962; Bear et al., 1968; Bear, 1972). The reader interested in special problems of the construction of hodographs is referred to the listed publications. Here only the main rules of the method will be summarized and some examples will be given to demonstrate its application. Along a potential line (or if three dimensional flow is investigated, along an equipotential surface) the gradient is always perpendicular to this line. The hodograph images of the points of a potential line are fitted. therefore. to straight lines,starting from the origin of the hodograph plane and perpendicular to the potential line at the point in question. The sections of entry a n d exit faces contacted by surface-water bodies are always potential lines. If these contours are straight lines, the relevant hodograph points form a line crossing the origin and normal to the contour in question (Fig. 4.2-17; lines a' and a"). If the entry or exit face is not a plane (in the case of two dimensional seepage its contour is not a straight line) the position and the form of the hodograph contour cannot be determined, because only the direction of the velocity (or gradient) vector is known at every point, but not its length. In the case of an earth dam, shown as an example, it is evident that the toes of both the upstream and downstream slopes are singular points with zero velocity (saddle points). Their hodograph image8 are fitted, therefore, to the origin of the hodograph plane. Along a stream line of a flow net the direction of the velocity vector is similarly known, the vector being everywhere tangential to the stream line. On the basis of this condition the hodograph images of impervious contours can be constructed, these borders being stream lines. In general cases these hodograph contours are also undetermined curves, the position and form of


531

if the influence of the caoillary fringe has to be mnsiderd Fig. 4.2-17. Construction of a hodograph field characterizing the seepage field of an earth dam

which is not known. But if the bordering impervious surface of the seepage space is a plane (the contour of the flow field is a straight line) its hodograph image is also a straight line crossing the origin and parallel to the contour in question. In the example the lower impervious boundary of the earth dam is such a contour (Fig. 4.2-17: line b). Because this border of the seepage field is horizontal, and crosses the toes of the slopes (where the velocity is equal to zero) the relevant hodograph contour has to be horizontal as well, and i t has to start and end at the origin of the hodograph plane. The ho34*

532


dograph line moves from the origin along the horizontal axis of the hodograph plane ( I , or v, axis). Velocity or gradient reaches its highest value at point 8 (the position of which is not known either in the seepage field or in the hodograph plane), from where the hodograph contour turns back to the origin following the same path. If the role of the capillary zone has to be considered, as is indicated in the figure, the hodograph images of the capillary exposed contours (faces) have also to be determined. They arealso stream lines, the geometrical position of which is fixed and known. In the exemple they are straight lines, the relevant hodograph contours are, therefore, also straight lines crossing the origin of the hodograph plane (Fig. 4.2-17 lines c' and c"). Both ends of the hodograph lines can also be determined, but we shall return to this problem after characterization of the hodograph image of the phreatio surface. There is one further section of the of seepage field contour, the position of which is geonietrically determined and independent of the kinematical character of the movement; i.e. the free exit face. The boundary condition along this contour is described by Eq. (4.2-24). Considering this equation, it can easily be proved, that the end points of the gradient vectors, starting from the origin of the hodograph plane have to be fitted on lines crossing the vertical axis at a point Iy = -1 and being perpendicular t'o the free exit contour at the relevant points. I n the general caae the position of this hodograph contour is also undetermined, but if the free exit surface is a plane, the lines transforming the various points of this stretch onto the hodograph plane become identical. Thus, only one straight line (crossing the point I , = -1; I , = 0 and perpendicular to the freeexit face) characterizes the position of the hodograph contour (Fig. 4.2-17; line d ) . Knowing the velocity or gradient values at the two bordering points of the free exit surface the validity zone of the hodograph line can also be indicated. There is only one further part of the contour of the seepage field which remains to be investigated, i.e. the phreatic surface. It is necessary once again to make distinctions according to the conditions along this surface. There are three aspects which have to be taken into account: whether the capillary eflect has to be considered or can be neglected, whether the flow i s steady or unsteady and whether accretion influences the seepage field through this surface or not. The most simple case is the steady seepage without capillary influence and accretion. The boundary conditions are given in this caae by Eq. (4.2-9). Considering the form of the equation giving contact between the components of the hydraulic gradient vector, i t can easily be seen, that the image of the phreatic surface o n the gradient hodograph plane is a circle with a radius of R = 112, the centre of which is at a point fitted to the vertical axis C(0:-112) (Fig. 4.2-17; line e). The position of the circle indicates that the gradient along the phreatic surface can never have a vertical component directed upwards, if the seepage is steady, capillarity is negligible and there is no accretion. As already proved, Eq. (4.2-9) characterizes the upper surface of the capillary zone and not the phreatic surface if such a zone develops above the water table. This is because in this case the surface with constant pressure

533


( p = 0 ) is not a stream line, while the capillary surface satisfies the two conditions given by Eqs (4.2-6) and (4.2-10): i.e. i t is the stream line with constant pressure ( p = - yh,). It can be stated therefore, that if the capillary effect is not negligible the hodograph circle [R = 1/2: C(O:-l/2)] describes the hydraulic gradient along the capillary surface (Fig. 4.2-17; line e l ) , while the image of the water table is a curve with undetermined form and position within the hodograph field. I n the caae of unsteady seepage without capillary eflect and accretion the hodograph image of the phreatic surface i s similaTly a circle, the centre of which is also at the point C(0:-1/2), but its radius changes from point t o point and from time t o time (Morel-Seytoux, 1961). The numerical value of the radius can be determined from Eq. (4.2-21) (4.2-44)

It follows from Eq. (4.2-44), that the radius is greater than 112, if-89, >0, at

and the upper stretch of the circle is above the horizontal axis of the hodograph plane, indicating that t,here is a stretch where the movement is directed upwards along the water table. I n the opposite case, if-aQ1< 0 , at the circle is smaller than that representing steady conditions and, therefore, the water particles composing the phreatic surface can move only downwards (Fig. 4.2-18). Posifive accretion (recbaue)

neg8itve accrefion(draimje)

unsteady seepale

+f t

Fig. 4.2-18. Modification of the hodograph circle characterizing the water table unde, the influence of accretion and the unsteady state of seepage

534


Changing both the radius of the hodograph circle and the position of its centre, the hodograph curve of a water table influenced by accretion can also be determined. I n this cam both parameters depend on the ratio of accretion and hydraulic conductivity. On the basis of Eq. (4.2-19) their numerical characterization can be given by the following formulae:

consequently the hodograph circle intersects the vertical axis of the coordinate system a t a point I , = -1: I , = 0. If the seepage field is vertically recharged (positive accretion E > 0), the radius of circle is smaller and its centre is a t a deeper position, compared t o the uninfluenced condition. I n the case of vertical drainage (negative accretion E < 0), the circle having a larger radius and higher centre rises above the horizontal axis indicating the possibility of vertical upward movement of the water particles situated along the water table (Fig. 4.2-18). Knowing the position of the various stretches of the hodograph contour, the images of the corner points of the seepage field can also be determined. It has already been shown that the two toe-points of the slopes are transformed t o the origin of the hodograph plane (singular points with zero velocity) and point 8 (maximum velocity along the horizontal impervious boundary) is fitted to the horizontal axis, but its position along this line is unlmown. The transformation of these points is not influenced by the conditions prevailing along the water table, their images may be fixed, therefore, a priori. The further form of the hodograph contour depends, however, o n the character of the phreatic surface. Two different conditions of steady seepage will be analyzed further on aa examples, one without capillary influence and the other considering the effect of the capillary zone, but both without accretion. The hodograph has the most simple form, if the steady seepage i s not influenced either by capillarity or accretion (Fig. 4.2-17c). Starting from point 1, the image of which is the origin, the hodograph contour goes along line a’. Point 2 is the intersection of the entry face and the phreatic surface, its image has t o be fitted, therefore, t o bath line a‘ and line e (hodograph circle). Thus, the gradient (or velocity) is completely determined at this point. The hodograph contour from here follows line e upwards having smaller and smaller gradient, uiitil reaching the inflection of the water table (point 9), the position of which is not known either in the seepage field or in the hodograph l’lane. It turns back from this point and goes once again along line e because this contour of the seepage field is still a phreatic surface. The next corner point is that, where the water table meets the free exit surface (point 5). Its image haa t o be, therefore, the intersection of line e and line d. The gradient along the water table is equal t o the slope of the latter, the position of point 5 determines, therefore, not only the size of the gradient here but also the apgle of the slope. It follows, however, from Thales’ law that the line is perpendicular to line d . Knowing that the latter is perpendicular to the


535

exit face, i t can be stated, that the free exit face is tangential to the water table at their intersection. After point 5 the hodograph contour has to follow line d , but it is also clear, that below the level of the tail water i t is represented by line a", so that point 7 should be fitted to the origin once again. Lines d and a" being parallel to one another, the hodograph contour can be continuous and closed only if the image of point 6 is at an infinite distance in the direction of lines a" and d. After point 7 the hodograph contour follows the horizontal axis, reaches the point of maximum velocity along the impervious boundary (point a), the position of which is undetermined, turns back to the origin andcloses the hodograph field, when reaching point 1 once again. When the capillary zone has to be considered as a part of the seepage field (Fig. 4.2-17d) the hodograph contour starts similarly from the origin (point 1 ) and moves along line a'. Point 2 is, however, a singular point in this caae with i n h i t e velocity (external corner point). Its hodograph image is, therefore, at an i n h i t e distance, characterized by a section of circle with infinite radius, extending from the direction of line a' to that of line c' (because point 2 has to be fitted to both lines). The hodograph contour comes from infinity along line G' and reaches the origin, because point 3 is a saddle or internal corner point (a singular point with zero velocity). Point 3 being the intersection of the capillary exposed entry face and the capillary surface, the hodograph contour has to be continued along the hodograph circle which is the image of the capillary surface (line e'). This condition states that the capillary surface always joins horizontally to the capillary exposed face. Because point 4 is the meeting point of the capillary surface and the capillary exposed exit face, its image is the intersection of lines e' and c". But the same point is also the intersection of lines c" and d on the hodograph plane, thus the images of points 4 and 5 cover one another. It follows from this condition that the capillary exposed exit face is also a tangent to the capillary surface, as is the free exit face to the water table in the previous example. The stretch of the border of the seepage field is a stream line, where the pressure changee from -p, to zero, thus the potential at a point here is 9 = K(Y - PI; where

(4.246)

Pc > P > 0. This contour being a straight line in the example, and if the linear change of pressure p can also be assumed, the derivate of the potential along this line is constant, and, therefore, both velocity and hydraulic gradient are also constant. Thus, the hodograph image of this stretch is reduced to one point (point 4 identical with point 5 ) . The continuation of the hodograph contour is the same as it was in the previous caae. It goes to infinity along line d (point 6) turns back to the origin along line a" (point 7) and after following the image of the horizontal impervious boundary through point 8, it is closed at the origin (point 1).

536


There are some other special functions, which can be applied in a similar way to the hodograph. Mapping the original physical seepage field into a new plane according to these functions, the position of the images of some undetermined contours of the flow field can easily be constructed in the transformed system and also some hydraulic parameters along the contours can be calculated. The well-known, and widely applied mapping method among these functions, i s the Zhukovsky’s function (Zhukovsky, 1923). In the case of steady, two-dimensional seepage the product of the pressure and a constant value can be expressed as a linear function of velocity potential [see Eq. (4.1-12):]

(;1

because p = K - + y . (4.2-47) Y The 8,function is harmonic in x and y, because both g, and y are harmonics. The conjugate of 0,can also be determined:

01 --K-=g,-Ky; P

0 , = y + Kx.

(4.2-48)

The complex number composed of the two variables is called Zhukovsky’s function, which can be expressed also as a function of the position of the investigated point on the actual phjsical plane (described by the complex form of the position vector: z=z+iy) and the potential- and stream-functions belonging to this point, (these may be included in a summarized form into one function called complex potential, t,he detailed definition of which will be discussed in Section 5.1.1 w = g, iy):

+ 0 = 0,+ i0,= w + iKz.

(4.2-49)

Considering the Zhukovsky’s function as a complex number and representing i t in a coordinate system the horizontal axis of which is 0,and the vertical a, the boundaries of the flow field can be determined on the new plane, if both the position and the boundary conditions at each point of the contour are known (the position determines the z value, while the boundary condition gives g, and y from which w can be composed). The images of special boundaries can be constructed on the Zhukovsky’s plane, even if some of the basic data are undetermined. Thus, the water table (independent from the influence of the capillary fringe) and the free exit face are mapped to the vertical axis of the new plane, because the pressure is zero along those lines, and the velocity potential is proportional, therefore, to the elevation above the reference level: 0 , = 0 ; i f p = O ; andg,=Ky.

(4.2-50)

Along a stream line the 0, value is a unique function of the position of the point in question, because the stream-function is constant here:

0,= C, + Kx ; if y = C1= const.

(4.2-51)

Equation (4.2-51) is valid for the water table if the influence of capillarity is negligible, and accretion does not affect the flow. The same relationslip


537

describes the image of the impervious boundaries of the field, (although the numerical value of the constant is different if the lower rather than the upper contour is investigated). Thus the Zhukovsky’s function €or the phreatic surface of a steady two-dimensional seepage without capillary effect und accretion is: (4.2-52) 0 = i(C, K x ) .

+

As in equation (4.2-51), the O1value can be expressed a8 a unique funct,ion of the position of the investigated point along potential lines (e.g. entry and exit faces), where the potential-function is constant:

0,= C, - K y ; if

v = C, = const.

(4.2-53)

T h e image of the capillary surface is parallel to that of the water table, because the pressure is constant along this contour as well, although it is not zero [see Eq. ( 4 . 2 - l o ) ] , and thus Zhukovsky’s function can also be used to describe this contour, the capillary surface being always a stream line: 0 1

= -Kh,;

if and

(4.2-54)

To characterize the capillary exposed exit face it may be supposed once again that the pressure changes linearly between - p c and zero [see the explanation given in connection with Eq. (4.2-46)]. Accepting this approximation the image of the contour in question on Zhukovsky’s plane can be described with the following equation:

because and

q=K

[i

y1--

3 ; +-y

, O J .

9, = C , + K z ; [see Eq. ( 4 . 2 - 5 1 ) ] . where

and

(4.2-55)

538

4 Kinematic characterkation of seepage

Special hydraulic interpretation can also be given to Zhukovsky’s function if Eq. (4.2-49) is differentiated according to the z complex variable: d@ - d(w +iKz)dw --dz dz dz

+ i K = - V* + i

~ ;

(4.2-56)

[because the derivate of w complex potential according to z gives the negative conjugate vector of velocity (-v*) as will be proved in Section 5.1.11. Thus, knowing the interrelation between Zhukovsky’s function and velocity along a contour (e.g. by constructing the images of the contour for Zhukovsky’s and the hodograph planes) the position of the contour on the real physical field can also be determined. z=

J -v* d@+ iK-.

(4.2-57 )

Any other functions differing from Eq. (4.2-49) only by multiplying it with a constant factor can be similarly applied to characterize special boundary conditions, and they are known generally as Zhukovsky’s functions (Polubnrinova-Kochina, 1952, 1962). Thus, the complex pressure where

-+ ip’ ; Yo,;

P =p p’=

(4.2-58)

K

is also a type of Zhukovsky’s functions. Similarly the following generally applied forms @=i(w iKz) = Kz - iw ;

+

or

. U’ G=z-~-;

(4.2-59)

K

can also be applied instead of Eq. (4.2-49) (Aravin and Numerov, 1953, 1965).

Examples showing the application of Zhukovsky’s function can be found in the books of Polubarinova-Kochina (1952, 1962); Aravin and Numerov (1953, 1965); Harr (1962) andBear (1972). The role of Hamel’s mapping function (4.2-60)

is the same aa that of the previous methods (i.e. to characterize the geometrically uncompletely defined contours of the flow field) (Hamel, 1934). The special properties of this function are described and examples of its application are given by Muskat (1937) and Bear (1972). Readers interested in studying further details in connection with this method are referred t,o these publications.


4.2.4 Consideration of initial conditions The definition of the initial or starting conditions has already been given in the introduction of this chapter. It waa explained that the kinematic equations can describe only the change of the hydraulic parameters in the m e of unsteady movement. Therefore, the calculated data have to be related always to an initial situation, when the parameters (velocity, pressure, flow rate) are known for the entire seepage field, because the time-variable values of these quantities can only be obtained by adding the changes to the initial values. The most simple solution is to choose the static equilibrium as an initial cmditwn, supposing that at the beginning of the investigated period the total potentials were equal along the entry and the exit faces of the seepage field. Thus, the initial condition is characterized by zero velocity, zero potential difference and zero flow rate at every point of the field. The pressure may be determined from the position of the head and tail water which are supposed to be a t the same elevation at the start of the process. If the seepage is created and maintained by continuously changing potential diflerence, a time point can be determined in most caaes, which can be regarded aa the start of the movement, when the static equilibrium can be applied as for the initial condition. The method can be used even in those cases, when the period in which the seepage is actually investigated, does not include this starting time. Knowing the time variant boundary condition, they can be extrapolated either backwards or forwards in time outside the investigated period to determine the time point, when the static state can be accepted, as the most simple initial condition. There are cases, when the mvement has an initial transition period before developing that state of seepage, in which the changes of boundary conditions can be characterized (or at least approximated) with monotonous mathematical functions. An example of this type of seepage is the unsteady movement around a well, the detailed analysis of which is given in connection with Fig. 3.1-11. If a mathematically easily amenable relationship is determined for that period of the process when i t can be considered to be regular, a starting point can be calculated by extrapolation, which really does not fit the curve describing the actual variation of the parameters in time. This fictive initial time point can beaccepted, however, if the characterization of the transition period can be excluded from the investigation, because starting at the fictive time point from the supposed initial condition (e. g. static state), and following the estimated change of boundary conditions, the hydraulic parameters can be well approximated at every moment after the elapse of the critical transition period. It happens many times that such approximate hypotheses have to be applied, which give infinite value for one or more parameters at the beginning of the movement, e.g. if an unsteady seepage maintained by a given and constant draw down of the contacting surface-water body has to be investigated. Choosing the static state aa the initial condition, i t has to be supposed that the draw down was created instantaneously at the beginning of the in-

540


vestigated period. It then incrertsed from zero to a definite value during an infhitesimally small period, which hypothesis gives infinite velocity and flow rate at the first instant. Although this discrepancy of the method could cause considerable error, if the purpose were the characterization of the process from the beginning of the movement, good approximation of the later stage of seepage can be achieved in this way, if the calculation of the parameters in the first short period can be neglected. Investigating seepage with periodically fluctuating time-variant parameters, i t is very probable that after starting from static equilibrium the first few periods are not yet regular, the parameters show a continuously changing trend apart from the fluctuating variation. It is necessary to have some time elapsed after the beginning of the process to achieve a state of movement, when the characteristics have the same numerical values at the corresponding time points of two subsequent periods. The calculation has to be continued, therefore, until a considerable difference occurs between the periods, and the values describing the hydraulic parameters in the last period can be accepted as the solution of the given problem. The static state has been analyzed up to now as an initial condition and various approximate hypotheses have been mentioned to facilitate the application of this simplest characterization. If the actual movement cannot be approximated, however, in this way other initial conditions can be chosen. A general state of the unsteady movement at a moment chosen arbitrarily is not suitable for this purpose, because both the determination and the mathematical description of this condition are hindered by numerous difiiculties. The other way is, therefore, to accept the steady state seepage as an initial condition, supposing that before the boundary conditions started to change in time, they had been constant, and the movement had developed as the result of those time-invariant conditions. Naturally, the situation is very similar to the application of the static state, a8 initial condition. The steady seepage (which is fully described for the entire seepage field by knowing velocity, pressure or potential, and flow rate at every point) may be characteristic at one time point of the continuous series of changing stages, or i t may be a hypothetic stage at a fictive time point providing a good approximation of the actual parameters in the investigated period. The time-variant period of movement can start from the steady state with continuous. smooth transition, or i t may be necessary to assume an instantaneous change. The latter case results in infinite parameters at the rapid change of the process, the period surrounding this time point should be excluded, therefore, from the investigation. Finally periodically fluctuating movement can be superimposed over a steady seepage as well. I n this crtse the calculation of more periods is necessary to check whether the difference occurring between two subsequent periods is negligible or not, and to determine the parameters of the stabilized waves, a8 is done when the investigation starts from static equilibrium. As a final consequence, i t can be stated that in practice either the static state, or the steady movement is applied as an initial condition, because the complete characterization of the seepage field cannot be expected in other cases. Attempts have to be made always to determine how the actual time-

References

541

variant movement could have originated from one of the possible initial condtions. Generally the methods and approximations listed in the previous paragraphs are applicable, but universal rules and laws cannot be stated. References to Chapter 4.2 ARAVIN,V. I. and NUMEROV,S. N. (1953): Theory of Movement of Fluids and Gaseous Material through Non-deformable Porous Medium (in Russian). Gostekhizdat, Moscow. S. N. (1966): Theory of Motion of Liquids and Gases in ARAMN,V. I. and NUMEROV, Undeformable Porous Media. I P S T , Jerusalem. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR, J. and DAOAN,G. (1962): The Use of Hodograph Method for Groundwater Investigation. Dept. of Civil, Eng. TECHNION, Haifa. S. (1968): Physical Principles of Water BEAR, J., ZASLAVSKY, D. and IRMAY, Percolation and Seepage. UNESCO, Paris. CHILDS,E. C. (1959): A Treatment of the Capillary Fringe in the Theory of Drainage. Journal of S o i l Science. GALLI,L. (1959): Approximating Method for the Calculation of Seepage under Hydraulic Structures through Layered Formations (in Hungarian). Viziigyi KozZemdnyek, No. 3. HAMEL, G. (1934): On Ground-water Flow (in German). Zeitschrift jiir Angewandte Maternatik und Mechanik, No. 3. HARR,M. E. (1962): Groundwater and Seepage. McGraw-Hill, New York. KAMENSKY, G. N. (1943): Principles of Ground-water Dynamics, (in Russian). Gosgeoltekhizdat , Moscow. G. N., KORCHEBOKOV, N. and RAZIN,K.J. (1935):Flow of Ground Water KAMENSKY, in Heterogeneous Strata (in Russian). Gosizdat, Moscow. KovAcs, G. (1962): Consideration of the Covering Layer Situated before the Dam when Calculatin: the Parameters of Seepage under the Dam (in Hungarian). Viziigyi Kozlemdnyek. No. 3. Kovbcs, G. (19ti5): Influence of Development of Free Exit Face on Flow Rate Percolating through an Earth Dam with Vertical Faces (in Hungarian). HidroZdgiai Koz16ny, 9. KovAcs, G. (1966): Hydraulics. (in Hungarian). VITUKI, Budapest, Vol. 111. KovAcs, G. (1968): Seepage to Ground Water Created by Hydraulic Structures. Actu Techriaco dradernirce Scientinrum Hungaricae, Tom. 60. No. 3-4. LUTHIN,J. N. (1966): Some Observations on Flow in the Capillary Fringe. IASH Symposium wk JVrzter in Unsaturated Zone, Wageningen. MOREL-SEYTOUX, H. J. (1961): Effect of Boundary Shape on Channel Seepage. Stanford Unia. Dep. C. E . Technical Report, No. 7 . MUSKAT, M. (1937): The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York. NEMETH,E. ( 1 963): Hydromechanics (in Hungarian). Tankonyvkiad6, Budapest. POLUBARINOVA-KOOF~INA, P. Ya. (1952): Theory of Ground-water Movement (in Russian). Gostekhizdat, Moacow. POLUBARINOVA-KOCHINA, P. Ya. (1962):Theory of Ground-water Movement. Princeton University Press, Princeton. PRANDT, L. and TIENTJENS,0. G. (1934): Fundamentals of Hydro- and Aerodynamics. McGraw-Hill, New York. H. E. (1958): Predicting Seepage under Dams on SHEA, P. H. and WHITSETT, Multilayered Foundations. Proceeding8 of ASCE, Vol. 84. p. 1. TERZAQHI, K. (1943): Theoretical Soil Mechanics. John Wiley, New York. ~JFALUDY, L. (1974): Investigation of the Length of the Floors of Weirs and the Depth of Sheet Piles from the Aspects of Seepage (in Hungarian). V I T U K I Report, No. 2639. Buclaprst.

542


VAUCLIN, M., VACHAUD, G. and KHANJI,J. (1975): Two Dimensional Numerical Analysis of Transient Water Transfer in Saturated-unsaturated Soils. Modelling and Simulation of Water Reeourcee Systeme. North-Holland PublishingCompany, New York. VEDERNYIKOV, V. V. (1934): Infiltration from Canals (in Russian). Gostroizdat, Moscow. Yomas, E. G. (1966): Horizontal Seepage through Unconfined Aquifers Taking into Account Flow in the Capillary Fringe. IASH Sympoeium on Water i n Umaturated Zone, Wageningen. ZHWOVSKY, N. E. (1923): Seepage through Dams (in Russian). Collected Work9 Vol. 7. Gostekhizdat, Moscow.

Chapter 4.3 Kinematic characterization of non-laminar seepage Substituting the relationships describing the resistance of porous medium to laminar flow into the two basic differential equations of hj-dromechanics expressing the conservation of both mass and energy (i.e. equation of continuity and Navier-Stokes’ equation), Laplace’s differential equation is achieved as the final mathematical form of the kinematic charwcterizution of seepage. Investigating this relationship in detail, it was proved that any flow satisfying the Laplace’s equation has velocity potential and fulfils also the Cauchy-Riemann conditions, thus i t is irrotational. A further result of the analysis is the justification of the fact, that the Laplace’s equation is valid for the characterization of all laminar seepage having a linear relationship between seepage velocity and hydraulic gradient within the whole flow field except at the singular points. The further kinematic terms (stream-, and potential-functions, flownet, etc.) derived on the basis of tlie Laplace’s equation can, therefore, be directly applied when investigating this type of seepage. In most practical cases the assumption that the linear relationship between velocity and gradient is valid, is acceptable. Thus, the solution of the Laplace’s equation satisfying at the same time the boundary and initial conditions describing the influences along the border of the seepage field (as well as the starting state inside the latter if the flow is unsteady) provides the hydraulic parameters of seepage (flow rate, pressure and velocity) with adequate accuracy. This is the reason, why handbooks dealing with the flow through porous media discuss generally the analytical, numerical or analogue solution of these mathematical systems, as is done in the nest part of this book. There are, however, few cases when the application of the hypothesis previously mentioned leads to larger error, than the generally acceptable one. However, its application may neglect some of the important characteristics of seepage. Perhaps the most common example of this situation is the investigation of the unsaturated flow, which was analyzed in Chapter 2.3. It may happen sometimes, however, that the use of Laplace’s eqmtion hccs to be excluded even f r m the study of saturated seepage as well. It is necessary, therefore, to find relationships suitable to substitute Laplitce’s equa-

543

4.3 Non-laminar seepage

tion, when other than laminar flow in a saturated field is analyzed, and to determine the differences between the corresponding parameters calculated on the basis of different hypotheses. The objective of this chapter is to summarize the methods applicable to the characterization of non-laminar seepage through saturated porous media.

4.3.1 Differential equations equivalent to Laplace’sequations for the various zones of seepage

When the seepage belonging to the validity zone of Darcy’s law was investigated, Laplace’s equation was derived by simplifying the Navier-Stokes’ equation. The dynamic principles characterizing laminar seepage [i.e. there is only one accelerating body force (gravity), and one retarding force (internal friction), thus the resistance can be expressed by Darcy’s equation]. The general kinematic relationships substituting Laplace’s equation could be derived similarly, if the forces found to be dominant in the validity zones of the various types of seepage were taken into account in the Navier-Stokes’ equation. The necessary dynamic analysis was, however, already executed in Chapter 2.1, and the results were summarized in the form of relationships between seepage velocity and hydraulic gradient, valid in the different zones of seepage. The combination of these movement equations with the equation of continuity gives, therefore, the same result, and ensures at the same time an easier method of investigation. The movement equations in question are analyzed and summarized in Chapters 2.1 and. 2.2. Expressing the differential change of head (i.e. hydraulic gradient) as the function of seepagevelocity, the following formulae can he repeated here: dh For turbulent flow --- bv2 ; ds In the transition zones

-

On the basis of Darcy’s law

-

For laminar flow

dh - - av ; ds dh__ -av+c; ds

(considering threshold gradient) For microseepage _ -dh = c1 00.9 ds

(4.3-1)

+ cp.

These equations have to be combined with the equation of continuity valid for steady seepage [using the two-dimensional form here to ensure more simple representation, see Eq. (4.1-20)] to get the relationships equiva-

544


lent with Laplace’s equation and to characterize the different types of seepage. Before performing this step, i t is necessary, however, to analyze, whether some elements of the flownet determined by assuming the potential character of flow remain applicable in the case of non-laminar seepage, or not. For the interpretation of the stream-functions and the derivation of mathematical formulae describing the position of stream lines, there was only one hypothesis applied: i.e. the velocity vector i s tangent to the stream lines at every point of the field, except at the singular points. This supposition is fixed by Eqs (4.1-16) and (4.1-25) in mathematical form and this interpretation of the stream lines is acceptable for any type of seepage. Only one restriction has to be mentioned, viz, in the case of turbulent flow when at a given point the direction of the velocity vector varies in time even in the case of steady seepage, the instantaneous position of the actual fluctuating vector will not be considered, but its average value, which determines the main direction of flow, is regarded as the tangent of the stream line. The next principle used for the derivation of the stream-function was the statement, according to which Eq. (4.1-25) has a solution, if the velocity i s continuous and derivable at every point of the seepage field, the solution results in the stream-function [Eq. (4.1-26)]. Excluding the singular points from the investigated field, as, when the potential seepage waa analyzed, the continuous and derivable character of seepage can also be assumed in all the zones of seepage. Once again the restriction mentioned before is taken account of, i.e. that in the case of turbulent flow the fluctuating velocity is characterized by its mean value. The result of this explanation is that the concept of stream-function can be generalized. On the basis of the relationship between its total and partial differential quotients the components of the velocity vector can be expressed depending on the stream-function: (4.3-2)

The fulfilment of this equation ensures at the same time, that the equation of continuity is also satisfied. In constrast to the stream-functions, the concept of potential-functions cannot be applied, if the flow i s not laminar, because the basis of their derivation was the existence of seepage potential, the prerequisit of which is the linear relationship between velocity and gradient. The equipotential lines, however, can be substituted by curves interconnecting the points of the seepage field, where the hydraulic head (the sum of elevation above a reference level and pressure head) has the same numerical value. The development of hydraulic head within the field is determined independently of the type of seepage and it is governed by the same principles as the development of potentials in the case of laminar flow. Along the entry and exit faces (except the free stretch of the exit face) the value of hydraulic head is constant and determined by the boundary conditions. The integration of the change of head along a stream line results in the total available head (the difference between the levels of the head and tail water). Movement

4.3 Non-laminar seepage

h

AH

a9

0.8

0.7 860.5 0.4 0.3

DIZ

- - -- - -- - -

545

0.1

calculated from L@?lace’squation calculafed porn O h ’ s equafion observed curve I

I

Fig. 4.3-1. Sketch of the flow field, investigated by Oka (1969)

being possible between two points only, if the hydraulic heads prevailing there are different, the velocity vector cannot have components parallel to the curves interconnecting the points having the same head. It follows from this condition that the two system of curves (i.e. stream lines and lines of equal heads) make an orthogonal network (the two types of curves intersect each other everywhere at right angles). This system, however, is not a net of orthogonal squares because the equality of the elementary changes of the potential- and the stream-function is ensured only in the caae of laminar movement (see. Fig. 4.3-1). Substituting the concept of velocity potential with hydraulic head in this way, relationships have to be looked for between this parameter and the components of the velocity vector instead of Eq. (4.1-12), which gives the 35

546


same relationship between potential and velocity. It has already been proved, that in a homogeneous field the stream lines are normal to the lines of equal heads. It follows from this condition, that the gradient vector of the head is parallel to the velocity vector at every point of the field. Considering this fact the relationships in question can be determined from Eq. (4.3-1):

For turbulent flow

ah --=

8X In the transition zones

i3h blvl v,; - - -- b 8Y 1v

--vx+b

(4.3-3)

1%; (4.3-4)

On the basis of Darcy’s - ah ah = av, ; - - = avy ; law ax aY For laminar flow C (considering threshold - av, -v, ; gradient) I V I

(4.3-5)

+

C

=avy+-vy;

(4.3-6)

I V I

For microseepage

+v, ; Iv I C

0.1%

(4.3-7)

Recalling the results of the dynamic analysis of the various types of seepage, i t can be stated that Eq. (4.3-5) describing the potential seepage and derived on the basis of Darcy’s law does not provide an accurate relatiow ship for any zone, because theoretically the existence of the threshold gradieiLt ought to be considered in the laminar zone as well, and therefore. Eq. (4.3-6) would be the exact solution. It was, however, also proved that the difference between velocity values calculated either from Darcy’s equation or by using the correct relationship [Eq. (2.2-56)] is negligible over almost the entire range where practical problems occur. The potential theory and Laplace’s equation derived from the linear relationship is acceptable in practice for the characterization of laminar seepage. Equation (4.3-7) relates f o microseepage, the validity zone of which is determined with the ratio of the actual and threshold gradients ( I o I < 121,,). The error caused by the use of Darcy’s law in this zone is not too high, except in the close vicinity of the threshold gradient. It has to be considered also that the flux transported through the mass of porous media composed of very h e grains (e.g. clay, the threshold gradient of which may be considerable) is relatively small. At the same time these materials have

H h, t i e drainage of the soil column starts. The tube is contacted with free surface water at its top (Fig. 5.1-ld):

+

if t

--f

00

;H

t

+ h, + x f ( t )-+

+

z f ( t ); I ( t ) + 1 ; and &(t) + R A .

In this case only the wetting process (infiltration) can develop, because both gravity and capillarity act in the same direction. Uncertainties are caused even in the characterization of this very simple form of unsteady seepage by the fact that the saturated zone and the dry part still undisturbed by seepage can never be divided by a plane with a well determined position. There are channels between the pores, in which water moves faster and, therefore, the wetting front is always a partly saturated zone of finite length with hydraulic conductivity smaller than that belonging to the saturated condition, and changing from point to point in both directions. The capillary suction is also a random variable within the cross section, which can be only roughly approximated with its average value. For a seepage field different from the most simple form investigated in the previous paragraphs, there always exists a single valued continuous solution of the differential equation, if the boundary and initial conditions are known. The only problem is, how to find this solution. The graphical solution (Leliavsky, 1955) has already been discussed in Chapter 4.1 [see Fig. 4.1-5 and Eqs (4.1-37), (4.1-38) and (4.140)]. In the introduction to this part of the book models and numerical methods were mentioned as auxiliary means, and the analytical methods will be discussed here in detail.

6.1 Two-dimensional potentional seepage

563

The possible analytical solution can be divided into two parts. The application of diflerent mupping methods to simplify the flownet until the orthogonal system of straight stream and potential lines is achieved, is the first one (Muskat, 1937; Milne-Thomson, 1955; Polubarinova-Kochina, 1952, 1962; Bear et al., 1968; Bear, 1972). The second part of the solution is the approximation of the actual seepage with a one dimensional flow (Dupuit, 1863; Boussinesq, 1904; Forchheimer, 1924). I n this case the validity limits of the approximations and the probable errors of the methods have to be determined. The theoretical bases of the first group of analytical methods will be explained in Chapter 5.1 while its practical application and some other methods will be dealt with in the subsequent chapters.

5.1.1 Complex potential. Conjugate velocity The mapping methods are based on the mathematical analysis of complex numbers and functions of complex variables. Here the basic concepts of complex and conjugate complex numbers, the symbols and terms applied generally, as well as the use of algebraic operations with complex numbers are assumed t o be known. Consequently only a short summary of the application of functions of complex variables is given, as an introduction to the conformal mapping. If z is a complex variable (which can have any z = x + iy value on the z complex plane), the result of special algebraic operations executed with z and some constants is a new complex variable which is the algebraic function of z : 2u = f ( z ) ;

w=cp+iy;

z=x+iy;

(5.1-6)

Each value of w complex number indicates similarly a point on a complex plane (now on plane w having a real axis and an imaginary axis y). Thus the function w = f ( z ) creates a contact between the points of the two planes, which can be either single- or multi-valued, depending on the algebraic operations applied within the function. After having determined the contacts between the real and imaginary parts of the variables (by dividing either their real and imaginary parts or their absolute values and the angles of the complex vectors), the relationship between the coordinates of the points on the two interrelated planes can be characterized. To assist this operation, the real and imaginary parts as well as the absolute values and the angles of the complex vectors determined by some algebraic funct,ions of complex variables are listed in Table 5.1-1. A curve on the z plane can be regarded as a line of points. Each point has its corresponding pair on the w plane. Thus, the i m a g e of a two-dimensional curve y = y ( s ) can be determined on the w plane, and also its equation tp = y(v) can be calculated. It can be stated similarly, if the y = y(x) curve 36 *

564

5 Movement equations describing seepage

Table 5.1-1. Real and imaginary parts as well as the absolute values and the angles of vectors determined by some functions of complex variables Function w =f(z t i Y )

w z $. iY)

+ iy)

sin (z COB

(zI iy)

tan

(3:

1

Real part

Imaginary part

1 y*) 2 sin z ch y - ln(z*

+

*arc tan

X

+cos z sh y

cos z ch y

$.sinsshy

sh(z i iy)

sin 22 cos 22 ch 22 sh z cos y

sh 2y 22 ch 22 k c h z siny

ch(z I iy)

c h x cosy

+shz s i n y

th(z

t iy)

iy)

Absolute value

Rew

+

sh 22 ch 2z COB 2y

+

COB

+

+ sh*y ~ccos*z+ sh*y )/sin2z

+ sinzy )/sh?z + cos*y )/&z

1

angle arg w

*arc tan (cotan x t h y ) *am tan (tan z t h y )

+arc tan (cth a tan u ) +arc tan (th z ten y) " I

sin 2y ch2z cos2y

+

is a closed one surrounding a single contacted continuous field of the x plane, the image of this field can also be constructed on the w plane. When determining the borders of the interrelated fields it i s necessary to divide and select those parts of the planes, inside which the mapping function in question creates a single-valued reationship between the corresponding points. Thus, the singular points (where the mapping function is undetermined) have to be excluded from the field, or if the applied function is multi-valued (e.g. square root), one solution has to be indicated as a main value and the image determined by this value has to be regarded as the transformed form of the field investigated originally. Limit value, continuity and differential quotient have the same formal interpretation for functions of complex variables, as given in the case of functions with real variables. If the w = f ( z ) function is single valued, continuous and differentiable at each internal point of a continuous field, this function is analytic (or regular) in this field. The necessary and sufficient condition for a function to be analytic is, that it has to satisfy CauchyRiemann's condition:

w =p

+ i y = f(x) = f(x + iy) ;

p = f d x , y ) ; y = f2(% Y ) ;

(5.1-7)

aw . -&L&.%L _ ax a y ' ay aY This idea leads to the hydrodynamic interpretation of functions of complex variables. It was proved earlier [see Eq. (4.1-33)] that velocity potential and stream-function are interrelated functions of the x and y coordinates of the seepage field, and their relationship is characterized by the fact, that their differential quotients satisfy in all cases the Cauchy-Riemann's condition repeated here as Eq. 5.1-7.

565

6.1 Two-dimensional potential seepage

It can also easily be proved that the fulfilment of Cauchy-Riemann’s condition ensures a t the same time the conformal character of mapping executed by an analytic function (the mapping is proportionate concerning both distance and angles; Nbmeth, 1963). The total differentials of z and w variables are as follows: (5.1-8) dz = dx idy ;

+

d w = dg,

89 + idy = -dx + ax

Using this relationship the differential quotient of w according t o z can be written in the following form:

dz

dx

dx

+ idy (5.1-9)

+ idy

A function is differentiable within a field if its differelitid quotient dw ix single-valued a t every point of the investigated field. The - ratio, dz however, would be infinitely multi-valued depending on the choice of the pairs of dx and dy, except in only one case, when the numerator choosing any pair of dx and d?j - can be divided by (dx + idy). This condition is satisfied if

9 + i - aw = aw - m a p , 2-

;

(5.1-10)

ax

ax ay ay i .e. if Cauchy-Riemann’s condition is fulfilled. It follows from Eqs (5.1-9) and (5.1-10) that if the w function can be dw differentiated with respect t o z within the investigated field, the -quotient

dz

is constant in the vicinity of a point and independent of the investigated direction (dx and dy elementary lengths can be chosen arbitrarily).This fact proves that the ratio of two interrelated elementary lines (dz and dw) is also independent of the direction of the investigated lines, and thus the mapping executed by an analytic function ensures the proportionality between the original field and the image. At the same time the proportionality of the elementary lengths is a SU& cient condition of the mapping to be proportionate also concerning the angles, viz. an arbitrarily chosen curvilinear triangle and its image have t o be similar t o one another, if the lengths of the sides are elementarily small, because dii) the - ratio can be regarded as constant, independent of the position of dz

566


the point within the surrounding area. The similarity of triangles ensures the identity of the corresponding angles at the same time. Further hydrodynamic consequences can be drawn from the fact that the mapping executed by analytic function is conformal. Let us construct an orthogonal network of squares on the w plane composed of straight lines characterized by v1,rp2, . . . p,,, constant values having a difference of d = q i - 'pi- ; and another set of straight lines being normal to the former, having the same distance of d and described by yl,y2, . . . yk constant ( A = - yi - yi-l relationship is also valid). Applying the inverse function of w = f ( z ) (which is also analytic, if the original function is regular), an orthogonal network of two sets of curves is reproduced on the z plane, the network being composed of curvilinear squares. A part of the z plane bordered by the curves corresponding to straight lines on the uf plane determined by the constant values of vA,v B and yA, pB respectively, can be regarded as the flow net covering a given seepage field. This is because a constant q value belongs to each of its potential lines, and each stream line is characterized with constant y value. At the same time these two functions (p, and y) satisfy Cauchy-Riemann's condition, and, therefore, the kinematic conditions explained earlier are also necessarily fulfilled. For this reaaon, using the analogy of the name of p,(x, y) potential-function, the y ( x , y) streamfunction is frequently called conjugate potential, and the name of the uy = f ( z ) function is complex potential. It follows from the hydrodynamic interpretation of potential- and streamfunctions that the di#erential quotient of complex potential according to the z variable can be used to determine the vector of seepage velocity. If only dz the inverse function of the complex potential [ z = f(w)] is known, the dw differential quotient gives the same result, providing us with the reciprocal value of the conjugate of the velocity vector. Combining Eqs (5.1-9), (5.1-10) and (4.1-33) the result is as follows:

dw

---

dz

ap, ay +i-=--

ax

ay

ax

. & = - (v, - i v y ) .

8-

ay

(5.1-11)

ay

The velocity vector can also be regarded as a complex number

v = v,

+ ivy = 1 v I eie.

(5.1-1 2)

I v I e-'",

(5.1-1 3)

Its conjugate vector is

v* = v,

- ivY --

the negative,value of which is identical with the differential quotient of the complex potential. Differentiating the negative inverse function of the complex potential according to the latter, the reciprocal value of the conjugate velocity is achieved as previously mentioned:

567

6.1 Two-dimensionel potential seepage

Thus, the angle of this vector is identical with that of the velocity vector itself, while its absolute value is equal to the reciprocal value of the absolute value of the velocity vector. Consequently, it can be proved that by differentiating the complex potential, or its inverse function, the conjugate velocity vector or its reciprocal value can be determined at each point of the seepage field. The conjugate velocity vector provides us with the absolute value and the angle of the velocity vector at the same time. It can be stated, therefore, that the velocity can be calculated at each point of the field, if the analytic mapping function of complex variables is known, which transforms the flow net into an orthogonal network of squares composed of straight lines on the plane of complex potential. The problem can be similarly solved, if the inverse of this mapping function is given.

5.1.2. Solution of seepage problems by applying mapping The main rules of mapping by using analytic functions of complex variables were summarized in the previous section. It was also shown that a hydrodynamic interpretation can be given for the mapping function which is called complex potential. Considering the relationship between the complex potential and the hydraulic parameters, the seepage velocity can be culculated from the mapping function at euch point of the field of a two dimensional seepage. The problem can be regarded tt8 completely solved, if the other parameters (flow rate, pressure and its distribution) can also be determined at any point or at least along special lines and at given points. When discussing the kinematic characteristics of seepage, it was previously proved that hydraulic gradient and velocity are closely interrelated, thus the knowledge of velocity simultaneously determines the gradient as well, and from these two parameters all the others can be calculated. The most direct application of conformal mapping is, therefore, the determination of the complex potential (or its inverse function), and after differentiating it, the vector of seepage velocity can be calculated for every point of the field by using Eq. (5.1-11) [or Eq. (5.1-14)]. By integrating the productof seepage velocity and an elementary length normal to the former along a potential line the discharge transported through a porous medium of unit width by the two dimensional seepage is achieved (Fig. 5.1-2):

pv dn = q.

(5.1-15)

A

Similarly the integration of the product of the velocity and an elementary length can be used to determine the potential OT pressure mlue at a given point P.I n this case, however, the elementary length is parallel to the velocity vector and the integration is executed along a stream line:

(5.1-16)

568


impervious boundwy Fig. 6.1-2. Velocity distribution along a potential line and a stream line, respectively within a two-dimensional seepage field

This method is frequently used if the only purpose is the determination of seepage velocity at a few points, because knowing the mapping function, this parameter can be directly calculated. It would be, however, very laborious to determine velocity and gradient for many points of the field- to facilitate the calculation of pressure and flow rate in this way. To solve these types of problems some other application of the mapping method is preferable. The use of the mapping function to construct the flow net by determining the relevant lines on the x plane belonging to constant values of q ~ a n d y respectively, also requires very long calculation, although the hydraulic parameters can be easily calculated after the flow net is determined [see Eqs (4.1-37), (4.1-38) and (4.140)]. Cases are very rare, therefore, when this method is followed to determine the hydraulic parameters. A t the same time the essential part of this method is the graphical representation of curves given in analytic form. Consequently, this method dots not need any further detailed investigation. There are problems - especially when an unconfined system ie investigated and thus the position of the upper boundary of the field is undetermined - when the hodograph mapping discussed in the previous part of'the book can be used to determine the complex potential (Hamel, 1936; Vedernikov, 1934). Considering the boundary conditions, the hodogralh contour can be constructed even in the case of unconhed flow. Knowing the hodograph image representing the velocity vector, the contour of the field on the plane of conjugate velocity (v*)or on that of its reciprocal value also be determined. If an analytic function 1 v* = fl(w) ; or - = f,(w) V*

(5.1-17)


569

can be found, which contacts the corresponding points and contours of the new plane and the w plane of complex potential, the problem of mapping [the determination of w = f ( z ) function] is solved. It is known from the foregoing that dw 1 dz v*=-= fl(z) ; and - = - - = f2( w )9 (5.1-1 8 ) dz V* dw respectively. From one of these relationships, after separating the variables, the w = f ( z ) function (or its inverse) can be calculated by integration. The solution is achieved also, if a n intermediate auxiliary complex plane ([ = E ill) can be found instead of the direct relationship between the complex potential and the hodograph of the conjugate vector of the seepage velocity. I n connection with this intermediate plane, there is a requirement that, after mapping by analytic functions both the complex potential and

+

I

the v* hodograph or the hodograph of - onto this new [plane, the images V* ll of the corresponding points either transformed from the w plane or from the hodograph should be identical. After determining the mapping functions establishing the required con-

I

tact from the w to the [ plane and from the v* or - to the [ plane respecv7 tively, the basis of the further analytical investigation is once again the relationship between the conjugate velocity vector and the differential quotient of the complex potential:

or using the inverse function of the complex potential as the basis of t,he investigation

(5.1-20)

After separating the variables and integrating the equation in question the z complex variable can be determined as a function of the 5 complex number describing the position of the points on the intermediate complex plane. The images of the contours bordering the seepage field are known on the plane. Thus, the previously undetermined position of the phreutic surface can be recalculated using z = z ( [ ) function by transforming its image to the

570


actual seepage field. The knowledge of the relationship between w and complex numbers was also a prerequisite for the application of this method [see Eqs (5.1-19) and (5.1-20)]. If the two equations [i.e. z(l) and w ( Q ] can be combined the direct mapping function between the actual seepage field and the plane of complex potential can also be determined. Here only the main concept of the application of hodograph mapping was summarized. Examples to show the practical use of this method will be given in Chapter 5.2.The application of Zhukovsky’s function will also be demonstrated there. The theoretical basis of this latter method is very similar to that of hodograph mapping. Similarly an analytic function (or a, series of such functions) has to be found, which transforms the contour of the field determined on the Zhukovsky’s plane onto the plane of complex potential (where the contours are also determined by the boundary conditions). This short summary has demonstrated, however, the complexity of the problem and the large amount of calculation, which has to be executed, if the application of one of these methods is intended to be used to determine the hydraulic parameters at every point of the field. This method leads back practically to the same type of work, which ought to be performed in the case of a geometrically fully determined seepage field, if all data are to be calculated on the basis of the complex potential. As will be shown, in the latter case an easier way can be followed by transforming the flow net into a plane, where the stream lines become parallel to each other and, therefore, Eq. (5.1-1)can be applied - at least as a good approximation - to calculate the necessary parameters. When investigating unconfined flow the hodograph mapping or Zhukovsky’s transformation can be wed only to determine the position of the water table, and afterwards the fixed contours can 1963,1964). be mapped similarly, as those of a confined field (KOV~CS, The easiest application of mapping for the determination of the hydraulic parameters of steady two-dimensional seepage is the reduction of the problem to the investigation of one dimensional seepage characterized by parallel straight stream lines. It was also mentioned in the introductory part of this chapter that this purpose can be achieved in many cases by using exact mathematical methods if the original /low net can be transformed into the system of orthogonal straight lines. Applying this method, the mapping is regarded as only a pure geometrical process. Its result is a rectangular flow field on the new plane which is characterized in this cam by u real and v imaginary axes [w(u,v ) ] to emphasize that the hydrodynamic interpretation of the complex potential (and that of potential and y stream-functions) i s not considered. In such a system the Laplace’s equation can be directly solved, and the result of integration is the set of characteristics looked for. Naturally in this case the parameters are given as the functions of the geometrical data of the field determined by mapping. The mapping functions express, however, the relationships between the geometrical parameters of the actual seepage field and those of the transformed system. These functions can be used, therefore, to introduce the original data into the hydraulic formulae, and h a l l y the interesting kinematic parameters can be related to the sizes of the actual field, which means the practical solution of the problem.


571

To demonstrate the application of the method, a very simple seepage field is chosen as an example, the flow net of which can be easily transformed into the required form, using only one mapping function (Weaver, 1932). The purpose of the investigation should be the calculation of the discharge below the horizontal foundation of a dam having a width of 2b, and the determination of the pressure distribution along the contour of the foundation, which is placed on the horizontal surface of a permeable layer. Further hypotheses are: (a) Both the entry and exit faces are horizontal planes; (b) The field is homogeneous, isotropic and bordered by an impervious bed; (c) The vertical section of the lower contour of the pervious layer is an ellipse, the half axes of which are interrelated to the half width of the foundation by the following equations (Fig. 5.1-3) :

N2 - M2 = b2; N =nb;

(5.1-21)

M=bvn2-l.

It is also supposed that the difference between the levels of head and tail water ( A H total pressure head) as well as the hydraulic conductivity of the

head water

Fig. 6.1-3. Mapping of elliptic seepage field

572


permeable layer ( K ) are known, thus the total potential digerenee inducing and maintaining the seepage can be calculated:

dv = pi-

~ 1= 2

K ( H 1 - H,) = K d H .

(5.1-22)

This very simple seepage field can be mapped onto a w plane, (where the contour surrounds a rectangular field) by the following analytic function : w =u

+ iv=

arc sin-

. x+iy

z

b

= arc sin-.

(5.1-23)

b

To simplify the mathematical contact, the analysis of the inverse function is preferable : z =x

+ i y = b sin w = b sin (u + iv).

(5.1-24)

The first step of calculation is toprove, that the mapping function or its inverse satisfies Cauchy-Riemnn’s condition. It is necessary, therefore, to determine thc relationships between the coordinates of z and w by selecting the real and imaginary parts:

x = b sin u ch v; y =b

COY

(5.1-25)

u sh 2).

The derivatives of x and y with respect t o u and v respectively, have to he determined to check the correctness of Eq. (4.1-33):

8X = b cos

ZL

8ZL

ch v = - = b cos u ch 8V

V;

(5.1-26) 8X -b

sin

ZL

sh

= - @= b sin u sh

v;

aU

aV

:~nd thus Cauchy-Riemann’s condition is satisfied. Consequently, the map-

z 7E ping function is analytic and single-valued within the field - - < u : +2 2 z rxcept the singular points w - - . 0 j ; w2[+;. 011.

[ ,[

The next step of the investigation is the determination of thc coordinates of the corner points of the field on the w plane by using the relationships given in Eq. (5.1-25):

P,(O, 0); P,(-b, 0);

W,(O, 0);

573


~ ~ (arshVn2 0 ,

- 1);

or considering that sh2y = n2 - 1 ; n2 = sh2y

+ 1 = ch2y;

W5 (0 ,arch n).

(5.1-27)

The seepage field is mapped in this way onto an orthogonal quadrangle bordered by parallel straight lines. The images of the entry and exit faces are parallel t o the imaginary axis, and their distance (the length of the field) can be calculated either from the positions of the W , and W , points or from the W, and W 4positions:

AU = u,

-

u1 = u

-

u, = z.

(5.1-28)

The stream lines indicating the upper and lower impervious boundaries are also straight lines parallel t o the real axis and, consequently, they are normal to the Imtential lines determined previously. T h e width of the field between these stream lines is:

LIV

=

v3 - vl = v5 - v o = v4 - v2 = arch n .

(5.1-29)

0 1 1 the w plane the internal stream and potential lines are straight. Identical lines are parallel while the two different groups of lines are perpendicular to each other. The investigation is reduced, therefore, in this way t o the analysis of the most simple flow pattern i.e. the one dimensional seepage tlirough a rectangular flow field. Applying the boundary and flow conditions according t o the data of the original system, the problem can be solved by using Eq. (5.1-1). The necessary data are as follows (Fig. 5.1-4):

(a) Hydraulic conductivity K (original flow condition); (b) Total pressure head AH (original boundary condition); (c) Area of the cross section in the transformed system (considering that a unit width of the two-dimensional seepage is investigated)

A = Av = arch n ;

(5.1-30)

(d) Total length of the transformed field AU = X

;

(5.1-3 1)

574


head warer

pressure bead

I

impervious boundary

'

potentiat surnce Fig. 5.1-4. Flow model of on0 dimensional straight stream tube of unit width

(e) Distance of a point fitted to the upper impervious boundary of the field in the transformed system from the entry section 7d

u =2

+ arc sin-.5b

(4.1-32)

The parameters t o be determined can be directly calculated as the functions of the geometrical data of the new simplified system:

Specific discharge (flow rate through a section of unit width of the Dermeable laver): AH AH 4: AK= AvK; (5.1-33) Au Au Pressure distribution along the foundation: -P= H,+ h ;

Y

(5.1-34)

Equations (5.1-33) and (5.1-34) are valid for any stream tube having a cross section of A = 1. Av and a lengthof Au if the flow condition is characterized with homogeneous, isotropic hydraulic conductivity of K , and the boundary conditions are given in the form of a constant pressure head of A H . These relationships can be used, therefore, in every case after transforming the actual flow field of two-dimensional seepage into the new system

575


characterized by straight, one dimensional flow. The contact between the geometrical parameters of the field achieved by mapping (i.e. A u ; Av; 2 4 ) and those of the original system will d$er from case to case. I n this example the width of the foundation is 2b; the position of the impervious bed is charact-erized by the M and N parameters; and the x parameter describes the position of the investigated point along the foundation. This contact, however, is always determined by the mapping function (or by the series of such functions, if the transformation is executed in more than one step), and Au, Av and u variables can be substituted by considering the relationships determined from the mapping functions [in the present caae these relationships are listed in Eqs (5.1-30); (5.1-31) and (5.1-32)]. Thus, the final solution of the investigated example can be given in the following form: (a) Specific discharge ar ch n . n

p = K A H ~,

(5.1-35)

(b) Pressure distribution along the foundation

=AH

2

n

5.1.3 Basic mapping functions applied most frequently

It can be stated, from the results of the previous section that the easiest and most generally applicable method of determination of the hydraulic parameters of steady two-dimensional seepage is to transform the original contour of the field into a rectangle by using mapping purely geometrically without considering its hydrodynamic role. T h e hydraulic problem i s simplified in this way to a geometrical one: to find an analytic function of complex variables (or a series of such functions) which execute the expected transformation of the original contour. It is quite evident that the actual borders of the flow field (bedding planes, river beds, etc.), which are generally irregular in nature, have to be approximated at first with straight lines or other mathematically amenable curves. For the application of this method i t is necessary t o know the basic mapping functions and the results achieved by each of them (the two formations on the original and new planes interrelated by a given analytic function). Various combinations of these basic functions enable us to map even a very complicated flow net into a rectangular field (Muskat, 1937; Polubarinova-Kochina, 1952, 1962; Nkmeth, 1947, 1963; Gruber and Blah6, 1973; Harr, 1962; Rear et al., 1968; Bear, 1972).

Table 6.1-28

rnappgng functions and

relationships between the coordinates of the interrelated plane

the orfhogoml tt?ijectories interconnected by mapping functions

unambiguously interrelated fields

fie two typesoffucfionsareinversefinc(ionq they contacf berefore, the sane ortnogona/tmjecforfes, but the role of the planes is changed . .

&-+-&!!q-+ ..

.!

Y

c;

G

II

II

u

ISS.

t


v

N

Y

578


Some of the simple mapping functions are listed in Table 5.1-2, where the interconnected orthogonal trajectories on z and w planes are also shown. If the mapping function is multi-valued, the transformed image of a plane covers many times the other plane which describes a multi-fold Riemann’s surface. I n this caae - as already mentioned - one of the results of the multi-valued function has to be selected at3 a main value, and thus the fields of the planes can be indicated within which the mapping function ensures a single-valued relationship between the corresponding points. The interconnected fields are also shown in Table 5.1-2. The explanation of the result of mapping by the various functions is summarized in the following paragraphs. B) depends on the character of The effect of linear function (w = Az the constants A and B respectively. They may be real, imaginary or complex numbers. In special cases A may be unity or B equal to zero, when they do not influence the mapping, because the position of the field does not change if the position vector is multiplied by unity or if zero is added to it. The effect of the B additive member is the shifting of the points a definite distance parallel to a given direction, corresponding to the vector described by the B complex number. If the B is a real number the shifting occurs in the direction of the real axis and similarly, the points move along the imaginary axis if an imaginary B is added to the x complex variable. Multiplication with a real A value causes the linear extension or shrinking of the plane [the result is extension if 1 A I > 1, while I A I < 1 indicates shrinking (negative extension)]. If A multiplying factor is an imaginary number its effect is a similar extension and the plane is also turned anticlockwise by an angle of n/2.The factor may be generally a complex number ( A = a + ib). In this cam its effect is a combination of extension and turning. The size of the components can be calculated from the following equations: The angle of turning b B = arc tan ;

+

a

The rate of extension m = vu2 + b2

I 3

.

(5.1-37)

The fractional function w = - can be characterized by two reflection steps. The points of the original plane are reflected at first to the unit circle (its centre is the origin and its radius is unity), and the result is reflected once again to the real axis. The image of the origin is a circle in the infinite distance, while the infinite points are crowded in the origin of the image plane. There are two points the position of which do not change (i.e. +1 and -1 on the real axis). If a point is described with a complex number of Reie on the original plane, its image is determined by another complex number (re‘*), the angle of which is the negative value of the former, while


579

its absolute value is equal to the reciprocal value of that of the original vector: z = Rei@; and w = reie ; where (5.1-38) .B=-@andr=-.

1

R

The z = 0 point is a singular point. The mapping method creates contact between the u = const. lines (parallel to the imaginary axis of the w plane) and the circles tangent to the z axis in the originof the z plane (their centre being fitted to the y axis). Straight lines on the w plane parallel to the real axis and determined by v = const. values are also interrelated to circles crossing the originofthe z plane, but now they are tangent to the y axis and their centres line-up along the z axis. The flownet composed of these two sets of curves is called dipole or dublett in the literature (Fig. 5.1-5). az b The linear-fractional function w = - can be regarded as a combi-

+

I

cz+d

)

nation of three other mapping functions: i.e. t = cz + d (linear function): a bc-ad 1 s (linear function), and, theres = - (fractional function): w = t c C fore, the effect of this mapping can also be characterized as the superposition of the result of the three components. The point characterized by the cz - d = 0 value is a singular point. The simple fractional function being

+

Fig. 5.1-5. Flow nets of simple and turned dipolee 37

5 Movement equetions describing seepage

580

one of the components, the flownet transformed into an orthogonal system of straight lines on the w plane by this mapping is also a dipole. Its centre is shifted and its main axes are turned according to the constants of the linear mapping. The power function (w= zn) is generally multivalued. A sector of the z 2n plane having a central angle a = - (the field closed by the real axis and n

2nl

a straight line crossing the origin and having a slope a = - is mapped n onto the total w plane cut along the positive real axis. That part of the field of the z plane, which is closed by the real axis and the line having an angle a n = - = - , is interrelated, therefore, to the positive imaginary half of the

2 n w plane. To calculate the corresponding complex number in the w system, the absolute value of the original vector ( z ) has to be raised to the n-th power, while its angle should be increased n times. Special cases of this mapping function are achieved by substituting determined even numbers instead of n. For example the function of w = z2 maps the flow net in a right-angledcorner onto the upper half of the w plane. Using any value of n the function is irregular at the point z = 0 (singular point). *-

The root function (w= v z = z'/") is also a special case of the powerfunction (with a fractional number as a power). Its effect is, therefore, identical with that explained in the previous paragraph. From the point of view of the investigation of seepage the square-root ( n = 2 ) is important, which maps the positive imaginary half of the z plane into a quadrate of the w plane bordered either by the + u and + v or by the +u and -v axes. (The square root being double-valued the total z plane is mapped either onto the upper half or onto the lower half of the w plane, and, therefore, i t is always necessary to indicate, whether the positive or the negative result of the square root is regarded as the main value in the calculation.) This mapping transforms the seepage field having the positive real axis (+x) as an impervious boundary and drained along the negative real axis (-x) into the field of one dimensional flow. This mapping being a type of power-function, the z = 0 point is a singular point. The logarithmic function (w= f l n z ) transforms the total z plane onto a horizontal stripe of the w plane extending in both directions into the infinite distance and having a height of 2n. Depending on whether the multifolded z plane is cut along either the positive or the negative real axis. The primary field on the w plane lies between the real axis and the straight line parallel to that characterized by the u = +2n value, or i t is bordered with the lines u = -n and u = +n. The further possible stripes interconnected with the other folds of the z plane join one of the stripes mentioned previously (depending on the place of the contact of the further folds on 2: plane) shifted by a distance of 2kn along the imaginary axis (where k is an arbitrarily chosen positive or negative integer), according to the z = = r exp [i(@ 2kn)l = r exp (i0relationship. )

+

581


By such mapping the = const. lines of the image are interconnected with the radii of the z plane running into the i n h i t e from the origin, if the sign of the function is positive (source), while the negative function is represented by radii coming from the infinite and submerging at the origin as flow lines into a sink. The mapping function is called, therefore, the transformation of a source or a sink, which has a singular point at the origin. The exponential function (w = eZ)is the inverse of the logarithmic function discussed previously. Its effect can, therefore, easily be understood, if the roles of the prototype and of the image are exchanged. The trigonometric functions i.e. the sine function (w = sin z ) and its inverse (w = arc sin z ) were already analyzed in the example given in the previous section (see Fig. 5.1-3). As waa shown there, the arcus sine function maps the half plane into a stripe parallel t o the imaginary axis, having a width of two units bordered by the real axis at one side and extending into infinity in the other direction. The cosine mapping (w = cos z ) can be derived from the former. According to the relationship

COB

L 1

z = sin - f z

,this transn

formation is a combination of the sine function and a shifting of - along 2

the real axis. The practical application of further trigonometric functions (tangent or cotangent mapping), as well aa that of hyperbolic functions is generally of less importance, and they can be constructed aa the superposisin z ez - e-z tion of other functions e.g. uj = tan z = -; or w = s h z = cos z 2 Their detailed explanation is, therefore, neglected here. The linear-fractional function can be regarded aa the overall result of three mapping exercises executed after each other, aa waa proved previously. By the superposition of the eoects of other simple mapping functions in a similar way very complicated seepage fields can be transformed into straight flow tubes of constant area by the different series of functions. The number of possible variations is increaaed by the fact t h a t the potential and stream lines are conjugated trajectories, thus their role can be exchanged. This operation can also be expressed mathematically as mapping characterized by multiplication with the imaginary unit. An example of this transformation is ahown in Fig. 5.1-6, where two sets of flow Lines are compared i.e. that encircling a foundation of 2b width, and the other characterizing a seepage recharged in infinity and discharged along a linear drain also having a width of 2b. The other principle providing baais for the superposition of various mapping methods is the additive character of the complex potential. If there are two analytic functions of complex variables and the relationships are determined between their components

i

w1 = f l ( z ); and w, = f 2 ( z ) ; where u1 = ul(x. y) ; and v1 = q ( z , y) ;

1.

582


zi plane

Fig. 6.1-6. Mapping of elliptic and hyperbolic stream lines

and similarly

(5.1-39) = u ~ ( xY) , ; and ~2 = v ~ ( xy) , ; the same relationships can also be calculated for the new function formed 88 the sum of the previous two functions: ~2

where and

29.

= w1

+

w2

u = Ul(X,

v = Vl(G

= fl(4

+ f&) ;

+ u2(x, Y) ; Y) + Y) -

(5.140)

Y)

V2(G

After h d i n g the derivatives of the individual members of the equations, it can be proved that the sum of the two functions satisfies Cauchy-Riemann’s condition provided the original functions also fulfilled the condition: -+-=8% 8% 8% I 8% ; ax ax ay ay and au, au, av1 av2 . ----+-= aY aY ax ax ’ if (5.1-41) 8% - av1 -8% - av1 . ax ay ’ a y ax ’ and au, av2 . au, 8% -=---ax ay ay ax

. 9

583


It can be stated, therefore, that the sum of analytic functions (even if they are multiplied by different constants) is also always regular. It is quite evident, that innumerable variations of mapping functions can be determined in this way. Only a few of them are listed in Table 5.1-2 and those explained in the following paragraphs have the greatest practical importance. Superposition of a source and a sink having identical discharge (strength) + a = ln(z a ) - ln(z - a ) . The flow can be formulated by to = In z2-a net of the system can be graphically constructed locating the source at + a point and the sink at -a point of the real axis (Fig. 5.1-7). At these two points the mapping function is irregular (singular points). The stream lines are circles crossing both the source and the sink, their centres are fitted to the imaginary axis. The potential lines are also circles, the centres of which are on the real axis outside f a points going gradually to infinity ;t8 the size of the radii of the circles increases. (The imaginary axis is also a potential line being a circle with infinite radius.) The upper half of the z plane is interrelated to an infinite strip on the w plane parallel to the real axis and having a width of n (between the lines u = 0 and u = n). Since any potential or stream line can be regarded as an impervious boundary, the image of the field recharged along a potential line and drained at the sink (or along a potential line surrounding the close

I

w plane

1

+

irnaje of the fird

quarier of the z plane

impervious bounda,-q

Fig. 5.1-7. Flow net characterizing the superposition of a source and a sink having aame strength

584


vicinity of the sink) can be determined also by this mapping function. It is quite evident, that one need to consider only that stretch of the infkib strip which lies between the images of the two bordering potential lines. The practical example of the application of this mapping function is the horizontal flow net of a single well along a river, when the bank of the river is substituted by a straight line (the imaginary axis of the original plane) and the wall of the well by a small circle around the sink. Naturally, a recharging well in the vicinity of the river bank can be similarly characterized, considering the other half of the z plane (that between the imaginary axis and the source). Infinite series of sources having identical discharge and located at equal

distances along the imaginary axis. The mapping function transforming this very complex flow net into one dimensional flow can ah0 be determined aa the sum of simple analytic functions. The number of the sources being infinite, the mapping series is composed also of an infinite number of functions, the limiting value of which can, however, be determined thus: w=

+ [lnz + ln(z + ih) + ln(z - ih) + ln(z + 2ih) +

+ ln(z - 2ih) + . . .] = +In s h yZhZ .

(5.142)

There are two groups of singular points. The sources ( z = 0; f i h ; h 2 i h ; . . .) are cavitational points with infinite velocity, while the points dividing the h distances of the sources into two halve z = &i - ; -f 3i - . . . are stagna2 ( 2 tion points with zero velocity. The field of the z plane bordered by the lines h h y = - y = - - and x = 0,being open in the fourth direction extending 2’ 2 into infinity and having a width of h , is transformed on the w plane into a strip, infinite in both directions, parallel to the real axis and bordered n n by u = -and u = - - lines (its width is, therefore, equal to n). This 2 2 mapping function is applied in practice, when the task is the characterization of the horizontal flownet of a series of wells located along a river. Naturally, in this case the series of sinks is investigated instead of sources, but i t does not change the character of the mapping function, the more detailed analysis of which will be given in Chapter 5.2 where the practical application of the composed mapping functions will be discussed. It has also to be noted here, that infinite series can be composed of the superimposed flownets of a source and sink, or those of sources and sinks located alternately along an axis. The number of variations can be further increased by multiplying the functions which are the members of these series by different constants. This means in practice that the yields of the sources and the discharges of the sinks are different.

h l

+

+

5.1 Two-dimension'al potential seepage

585

+

The sink in a uniform flow (w= -az b In z ) is also a flownet having an infinite number of variations depending on the ratio of the two constants a and b. The position of the line dividing the part of the field recharging the sink from that, where the flow is continued in the main direction, is determined also by this ratio (Fig. 5.1-8). The stream-function is zero along this

Fig. 6.1-8. Sink in uniform flow

water divide, this fact providing us with the basis for the calculation of the position of this line, and also of its intersection with the x axis (this latter point and the sink itself are singular points of the field): a

x = y cotan - y ; b

(5.143)

b xo=-. a The half width of the recharging part of the field (the distance of the asympt,oteof the water divide line from the real axis) can also be calculated from this relationship by substituting an infinite value instead of 2:

h, = n-

b a

= n x, .

(5.1-44)

586


5.1.4 Application of Schwartz-Christoffel’s

mapping

When two-dimensional seepage is investigated, the section of the seepage space and the flow plane is a two-dimensional field, the contour of which is composed of straight lines, or the stretches of the contour can be well approximated by such lines. For this reason, Schwartz-Christoffel’s mapping is frequently used in seepage hydraulics. This method transforms a polygon composed of arbitrarily chosen straight lines into one line without any corner point (Schwartz, 1869; Christoffel, 1867). Schwartz-Christoffel’s transformation gives the inverse function of the mapping (i.e. the z complex number is expressed as a function of the w vector representing the position of a point on the image plane) in the form of an integral equation (Fig. 5.1-9): z =5 z =AS-

+ i y = P(w)= P(u + iv) ; dw

(w- u p n (w- U Z ) +

. . . (1c - u n ) a +

(5.145)

+B.

As shown in the figure, this mapping transforms the polygon given on the z plane into the real axis of the w plane. The denominator of the expres-

tY

w plme

Fig. 6.1-9. Representation of Schwartz-Chhtoffel’s mapping

587


sioii to be integrated has as many factors as the number of corner points of the polygon. The basic value of the factors is the difference of the independent variable w and the real coordinate of the image of the relevant corner point ( u l ,u 2 .. . u,). The power of this basic value can be calculated m the angle of the polygon at the corner in question related to n. The angle is inemured between the elongated line of the previous side and the following stretch of the polygon, if the direction followed along the contour is determined by the increasing order of the coordinates of the corners on the w plane (ul< u2 < . . . < un).The sign of the angle is negative if the line of the previous side covers the following one, when the former is turned clock-wise. In the other case a is positive. This interpretation gives the possible limits of the angles and also that of the powers of the factors: (5.146) The A and B constants in Eq. (5.1-45) can be any complex number. Their value is determined by choosing the positions of three uicoordinates arbitrarily considering only the possible easy execution of the integration. Fixing the posit,ion of fewer corner points than three, the problem is undetermined, while i t becomes too determined if the coordinates of more than three points are known. As a first example of Schwartz-Christoffel mapping a polygon composed of three sides will be transformed into the real axis of the w plane. The polygon is the contour of a stripe parallel to the y axis, having a width of 2b symmetrically located on both sides of the y axis (bordered by the lines x = + h and x = --b respectively) closed below by the real axis x and extending into infinity in the other direction (Fig. 5.1-10a). It haa two corner and P J , where the angles are equal to one another points (PI The position of the two corner points on the 10 plane can be arbitrarily chosen : WA-1; 0 ) ; P,(--b; 0 ) ;

Pz(+b; 0 ) ;

W,(+l; 0). Because of the symmetry, it is evident, that the two origins are corresponding points: Po(0; 0 ) ; Wo(0; 0). The coordinates of the two corner points are consequently u1= -1 and u2= + l . Thus, the equation to be integrated is as follows: z=AJ

(w

+

A dw 1)1/2 (w - 1)1/2 + B =

dw

7J v1 -

___ f w2

A B = Tart sin w 8

-+ B.

(5.1-47)

The B constant is determined by the condition given for the origins ( z = 0 and w = 0 are interrelated points), and it is equal to zero ( B = 0). The other

588


la)

z plane

4+y

Fig. 5.1-10. Application of Schwartz-Chrktoffel’smapping

F

pair of interconnected points is w = 1 and z = b. From this condition the multiplying factor can be calculated: A n - A 2b b = - A a r c s.i n l = - - , -=-. (5.148) i i 2 i n Thus, the final form of the mapping function is 2b . z = - arc sin w . (5.1-49) 7c

This mapping is the inverse function of that applied previously [see Eq. (5.1-23) and Fig. 5.1-31. Comparing the results of the two transformations, i t can easily be seen, that the only difference is the exchange of roles of the z and w planes respectively. The second example is the straightening of the polygon forming a step on the z plane (Fig. 5.1-lob). The coordinates of the corner points on the z and w planes, the angles at the corners and the relevant powers of the factors can be listed as follows: Symbols of points

1

2 0 0

Coordinates on the plane Coordinates on the w plane

X

v

0 t a -1 0

Angles

a

_ _II

f,

Powers

a/n

-112

+1/2

z

Y U

2

+1 0

n

589


Consideringthese values the mapping function can be given in the following form:

A i

= -(arc

sin w -

V m )+ B .

(5.1-50)

The constants have to be calculated from the coordinates of the interrelated points: a.

A=-; x

and B = - - .

a 2i

(5.1-5 1)

Combining Eqs (5.1-50) and (5.1-51) the final form of the mapping function is achieved: 2 a

1 n

- = -(arc

1 sin w - v 1 - w2) - -. 2

(5.1-52)

More complicated contours can also be mapped by Schwartz-Christoffel’s method, as shown in the third example, where a horizontal foundation having a width of 2b supplemented with a sheet pile located symmetrically and having a depth of 1 is investigated. It is assumed that the foundation is laid on the horizontal surface of a permeable layer with i n h i t e depth (Fig. 5.1-1Oc). The original contour has five characteristic points, among them only three are corners, because the direction of the contour does not change at points 1 and 5, and, therefore, a, = a5= 0. Thus, it is not necessary to fix the position of these points on the w plane, but its coordinates will be calculated from the mapping function. The a priori determined data required for the derivation of the mapping function are as follows: Symbols of points

Coordinates on the z plane Coordinates on the w plane Angles Powers

1 2

Y

-b 0

2 0

0

21

+1

2)

0

3

4

0

0 0

-1 0 0

a

0

+n/2

-n

ah

0

+1/2

-1

6

+b 0

-1

0 fR/2

+112

0 0.

The integral equation and the solution of the mapping function can be written in the following form:

z=Af

w dw (w - 1)1/2(w+ 1)’/2

-+ B = i A v 1 - w2 + B . (5.1-53)

590


The next step is once again the determination of the constants in the equation. Now the interconnected coordinates of points 3 and 4 can be used for this purpose. The result is

A=-1;

and B=O.

(5.1-54)

Thus the mapping function transforming the given contour into a straight line is determined by the following equation: 2 = - ill/l-

(5.1-55)

w2;

from which the coordinates of points 1 and 5 on the w plane can be calculated:

Wl(u1; w1); W5(u5; us);

u.1

= 71@1'+1;

VI+]+

w,=o; (5.1-56)

2

u5 =

--

1 ; w5 = 0.

If the z plane is a seepage field, the entry face of which is the x < --b stretch of the horizontal real axis, and the water percolates around the foundation and the sheet pile until it reaches the exit face, which extends along the x > b stretch of the real axis, the transformation analyzed above does not result i n one dimensional flow, only simplifies the original flow net. On the new plane the seepage equivalent to the original system can be characterized by horizontal entry and exit faces located along the u real axis (the entry face is the u

r[!)'

>+

extends from the point of u = -

+ 1 stretch of the axis, while the exit face

[+I' +

1 to negative infinity). Between

the faces, a horizontal impervious boundary with a width of 2

VIY+

can be found and the seepage develops above this boundary. The image on the w plane is evidently double-valued, because both the negative and the positive value of w provides the same result. By considering negative w as the main value the seepage field from the z plane can be mapped onto the lower half of the w plane. Thus, i t can be stated, that a seepage under a foundation composed of a horizontal block and a vertical sheet pile is simplified to the investigation of a horizontal foundation without a sheet pile (in both systems an infinite depth of the permeable layer is assumed). As already shown, however, the stream lines are confocal ellipses having their foci at point W ,and W,in the simplified system which can be mapped as straight lines by applying the sine transformation [see Eq. (5.1-24)]. Combining the two steps, considering the recently achieved form only as an intermediate plane (using in Eq. (5.1-55) the 5 variable instead of w), and mapping the newly formed flownet into straight by using Eq.

5.1 Two-dimensional potentional seepage

591

(5.1-24) (where the 5 plane is used instead of the original z plane) the complete solution of the transformation is aa follows (Fig. 5.1-11): z=-iil~1-~2=-iil

5 = Bsin w ; where (5.1-57)

After the determination of the geometrical parameters of the seepage field on the w plane and substituting them into Eqs (5.1-33) and (5.1-34), the discharge and the pressure distribution along the foundation can be calculated. Some difficulty will arise when applying this relationship for solving practical problems, because the infinite depth of the permeable layer was presupposed in the example. The cross-sectional area of the field is also infinite, which evidently also results in infinite discharge.

5

I

Fig. 5.1-1 1. Mapping of seepage field with infinite depth below a foundation composed of horizont,al block and vertical sheet pile into the system of one-dimensional flow

592


When investigating the seepage under a foundation i t is necessary, therefore, to find a transformation which also considers the position of the lower impervious boundary of the field. This requirement makes the mapping function more complicated. The integral equation can be derived similarly on the baais of Schwartz-Christoffel’s theory but there are many caaes when the integral cannot be solved in closed form, or it leads to functions, which cannot be handled in analytic form (elliptic functions, integral-sine, integrallogarithm, exponential-integral, etc.). In these cMes the numerical solutions can be used either by having expanded the expression to be integrated into series, or by using the pretabulated values of the special functions. The investigation of the seepage under a foundation i n a pervious layer of finite depth, provides a good example to demonstrate the problems occurring in connection with the application of Schwartz-Christoffel’s mapping. For the characterization of such a seepage field Pavlovsky’s mapping methods are generally applied (Pavlovsky, 1922, 1956; KovAcs, 1960; Nkmeth, 1947, 1963; Leliavsky, 1955). The simplest form will be discussed here that is when the foundation contacts the surface of the permeable layer along a horizontal plane, and the mapping function will be derived for this cwe (Fig. 5.1-12). The seepage field is an infinite strip bordered b y two parallel lines. The upper is the real axis, which is divided into three parts: i.e. entry face (z -b), upper impervious boundary (--b < 2 + b ) . The lower boundary is the surface of the impervious bed, the depth of which

- . In the

opposite case the model describing the seepage into a draining trench can be used. Accepting this approximation, the task is the determination of the thickness of u virtual layer ( m > M > d ) in which a fully penetrating well yields the same flow rate as the investigated field (KovAcs, 1966). The first step in mapping is the application of the sine transformation to obtain the image of the flow field on the auxiliary plane 2 C = sin n;

m

therefore 5 Y 5 = sin-nnh-n;

m

m

(5.2-18)

and

It' is necessary afterwards to shift the origin to the centre of the image of the exit face

C' = 5 - 6;

606


z plane

w plane

VA

+

'

.

L

Fig. 6.2-6. Transformation of flow field betweeavertical entry face and partially penetrating drain

therefore

E'

and

= 5 - 6;

(5.2-19)

q' = 7;

where

6=-

7

2

dl

l+cosn-.

m

The field can easily be retransformed into a s e m i - i n f i b strip between two impervious boundaries which is fully penetrated by the vertical exit face. During this mapping i t can be ensured, that the length of the new field should be equal to the original distance between the vertical exit face and the drain (L):

L arc Bin . C' ., w =A

1-6

therefore

E' = 51.11 A u Av -ch - ; 1-6

L

L

6.2 Combined applicetion of various mapping functions

607

and

Av ; '' - cos-A u sh -L

1-6

(5.2-20)

L

where

1-6

As shown by the figure, the entry face is not absolutely vertical after mapping, but the approximation is acceptable, if the-

L

ratio is greater

m

L a than unity; this limit can even be decreased to - > 0.5, if ->0.7. Among m

m

the hydraulic parameters the specific flow rate can be calculated as the function of the geometrical data of either the new or the original field

M L

q= KAH-;

where

M = -nL. ,

(5.2-2 1)

A

:1

while within the validity zone mentioned previously - >-

the pressure 1 ) distribution along the upper boundary may be approximated 'with a linear relationship X

h= AH-.

L

(5.2-22)

The exit velocity is always constant in the transformed system, and its original local value can be calculated by considering the ratio of the corresponing stretches of the exit face in the two systems [see Eq. (5.2-14)].

Vertical entry face; draining trench (Fig. 6.2-7)

Similar to the investigation of flow fields with horizontal entry face8 the actual confour line of the trench is approximated in this case also with the retransformed image of a straight line on the w plane characterized by u = u,,= const. value (Kovhcs, 1975). At first the flow field (which is a semiinfinite strip) should be mapped onto the positive imaginary half of an auxiliary plane ( t plane): .

e

t = sin -x ;

m

608


4

3

i4

5

Fig. 6.2-7. Transformationof flow field between vertical entry face and draining trench

therefore X Y r = s i n --rich-n; m m

(5.2-2 3)

and X Y s = c0s-nnh-n. m m

The seepage field in this plane is identical to that on the t plane, when the flow between a horizontal entry face and a draining trench is mapped (see Fig. 5.2-4). After shifting the origin into the centre of the exit section t' = t

- 6,;

5.2 Combined application of various mapping functions

609

therefore r’ = r - 6,; and s’ = 5;

where 1

(5.2-24)

the steps of mapping are the same as those described previously [Eqs (5.2-6), (5.2-8) and (5.2-9)]. Thus the relationship between the corresponding geometrical paramehm of the original field and its image on the w plane can be determined: 61 ch -

m

and (5.2-25)

or

On the basis of the combined transformation functions [Eq. (5.2-25)] the hydraulic parameters can be directly calculated from Eqs (5.2-10) and (5.2-14).

The influence of both the position of the closing faces and the main geometrical parameters on the hydraulic characteristics can be well demonstrated by cumparing the results of the various models. Thus, Fig. 5.2-8 shows the flow rate as the function of the ratio of the two main geometrical parameters (Llm) determined by considering different positions of the closing faces. Supposing that the difference between the results of the models is negligible, if it is smaller than 5 % of the flow rate, the following conclusions can be drawn from the figure: (a) In the caae of partially penetrating drains the hydraulic parameters can be calculated by the more simple models characterizing the fully penetrating drains, if the ratio of penetration dlm > 0.8; 39

610


-.......... ...-...

--- -+-

R50J 08 10

3.0 4.05080 80 IOU L/m Fig. 5.2-8. Comparison of the results of models established for t.he de1,ermination of hydraulic parameters of seepage between surface-water and drain 20

(b) I n the range of Llm > 1 the difference between the results of models for horizontal and vertical entry faces is generally smaller than loo/,, thus the actual form can always be substituted with one of these approximations (which is nearer the actual form) without causing a larger error than 5%; (c) In the zone of Llm > 4 5 the digerewe of /low rates determined by horizontal or vertical entry face is smaller than 5%, the more simple and accurate models can be applied; (d) If Llm > 8 10, the largest dioerence between the results of the various models becomes less than 5 % and, therefore, the flow rate above this limit can be calculated from the simplest Dupuit's equation [Eq. (5.2-li')]. The differences caused by the form and position of the closing faces are more considerable if the pressure head distribution is investigated instead of the flow rate. In this case a suitable model has to be selected more carefully, especially if the piezometric head has to be determined at special points (along the entry face, in the vicinity of the draining structure, on the protected side of the ground-water space, etc.).

-

N

5.2.2 Combination of mapping functions applied on two different planes of the flow space Yield of riparian wells

The flownet of three-dimensional seepage cannot be transformed into one dimensional flow by the simple application of mapping functions. There are, however, special cmes, when the stream lines are fitted to cylindrical surfaces (see Fig. 4.1-8). A section investigated perpendicular to these flow surfaces,

611


reveals a two-dimensional orthogonal net composed of the curves representing the intersections of the plane of section with both the flow surfaces and the equipotential surfaces. The conformal mapping has at first to be applied on this plane, thus obtaining a system of parallel flow planes. As a second step the contour of the seepage space on the new flow plane has to be mapped onto the plane of complex potentials to achieve the required one dimensional seepage equivalent to the original system. It is necessary to note, that after executing the first mapping in the plane perpendicular to the flow surfaces, the geometrical parameters of the new system have dimensionless values, and they have to be combined with the flow nets on the flow surfaces. These nets however, are distorted because the lengths in one direction parallel to the intersection with the plane of mapping, were transformed and are characterized by dimensionless quantities, while in the direction normal to the former the original measurements remained unchanged having a dimension of length. Before starting the second mapping this contradiction has to be eliminated. It is necessary to determine, therefore, the most characteristic length in the plane of the first mapping, and to multiply the result of the first transformation so that the image of this length should be equal to the original size. The multiplying factor also has a dimension of length, thus the geometrical parameters of the mapped system will be homogeneous having the same dimension in the plane of the transformation aa in the normal direction. The best example to demonstrate the application of two conformal mappings executed along two surfaces normal to one another, is the determination of the yield of riparian wells. If the wells fully penetrate the aquifer and the entry face is vertical, the stream lines are really fitted to vertical cylindrical surfaces, and thus the method of double conformal mapping gives the accurate solution. In the case of a non-vertical entry face the difference is negligible, while larger discrepancy is caused if the well is a partially penetrating one, because the stream lines coming from the direction of the river bed may cross the axis of the well in the lower part of the aquifer and turning back they approach the screen from behind. Thus the model can be applied as an approximation for any position of the entry face, and it is even acceptable for the calculation of the yield of partially penetrating riparian wells, if the ratio of penetration is high enough ( d / m > 0.5 0.6). The transformation of the flow field between the bank and a single well, aa well as the application of the method for the characterization of one member of a series of wells running parallel to the bank, will be demonstrated here as practical examples.

-

Single well near the bank of a river (Fig. 5.2-9)

There is more than one method of solving this problem (e.g. the method of images), but here the use of double conformal mapping will be shown. The horizontal sections of the flow surfaces are identical to the stream lines of the well-known flow pattern characterizing the superposition of a source and a sink having equal flow rates. With conformal mapping (see Fig. 5.1-7) 39*

612


Fig. 5.2-9. Transformation of the cylindrical flow surfaces of a single riparian well into parallel flow planes, applying mapping in a horizontal plane

this flow field is transformed into a semi-infinite stripe of 2n width (the image of the centre of the well is at infinity). The length of the transformed active field is determined by the distance between two parallel straight lines, which are the images of the bank and the horizontal section of the well screen. The multiplying factor, which has to be applied after the mapping in the horizontal plane, can be determined where the distance between the images of the bank and the well screen (the length of the flow field) is equal to the distance of the centre of the well from the bank intheoriginal system. Thus, the width of the new seepage space (the length of drain equivalent to that of the well) can be calculated from the following equation:

B-L-

2n 2L’

(5.2-26)

In r0

Considering the explained relationships, the yield of the fully penetrating well can be calculated as the flow rate to the drain of B width and being of L distance from the bank. In the case of unconfined flow this value is determined by Eq. (4.1-44). After substituting the geometrical parameter on the basis of the transformations, the yield of a riparian well is:

H2, - H L n K H : - H ; . Q=Bq=KB

2L

2L

2111-

7

(5.2-27


613

where H , is the height of the water level in the river above the lower impervious boundary of the aquifer, and H , is the elevation of the water level in the well above the same datum. If the seepage field is confined having a thickness of m Eq. (5.2-17) can be used for the determination of the yield:

& = Bq=KBm

- H Z = Km 2 4 H ,

L

- H,) 2 L

(5.2-28)

In r0

In the case of partially penetrating wells (when the method can be applied as an approximation) the flow net of the vertical flow plane obtained by the first mapping has to be transformed to get a one dimensional flow. This second m p p i n g depends on the position of the entry face and o n the rate of penetration. The steps of the necessary transformation have already been discussed in the previous section for each profile of practical importance. Thus, the task is the combination of the mapping executed in the horizontal section with one of those models. O n e well from a series of riparian wells (Fig. 5.2-10)

Supposing that both the spacing ( B )of the wells and the distance between the wells and the bank ( L )are constant, the horizontal section of the vertical flow surfaces shows a pattern identical with the flow lines of the infinite series of sources or sinks. The mapping function derived for the character-

z plane

Fig. 5.2-10. Transformation of the cylindrical flow surfaces of a member of a series of riparian wells into parallel flow planes applying mapping in a horizontal plane

614


ization of this flow net [seeEq. ( 5 . 1 - 4 2 ) ]can be used, therefore, for the determination of the geometrical parameters of the flow field belonging to a drain hydraulically equivalent to the seepage space occupied by the flow from the river bed to the well in question: tu = [In z

+ ln(z + iB)+ ln(z - iB)+ ln(z + i 2 B ) + ln(z

-

i2B)

+ . . .] =

ZZ

= lnsh-;

B

therefore

zc = -In2

and

’[

2

X

ch 2 n - - cos

B

tan

(5.2-29)

j.;)

v = arc tan

Applying this mapping method, the geometrical parameters of the new system having parallel flow planes can be determined. The basis of the determination of the necessary multiplying factor is that the width of the hydraulically equivalent drain should be equal to the spacing of the wells (KovQcs,1 9 6 2 ) : B, = B ;

(5.2-30)

n

2

where the last relationship gives the distance between the bank and the image of the corner point (the intersection of the axis of the wells and the line normal to the bank and half the distance between the wells) on the transformed plane. Knowing the geometrical parameters of the flow space after the first mapping method executed in the horizontal plane, a system of parallel flow planes is achieved. The vertical dimensions of the seepage field in this flow plane and the position of the entry and exit faces are determined by the original data, and the horizontal by mapping. Considering the form of the new seepage field, the model suitable to characterize the flow pattern in question can be selected from those analyzed in the previous section. After combining i t with the mapping explained above the hydraulic parameters characterizing the seepage from the river to one member of a series of wells along its bank can be calculated.


615

5.2.3 Combined application of hodograph and conformal mappings Infiltration from canals

The Dupuit's hypothesis can be used for the approximate determination of the hydraulic parameters of unconfined seepage only i n the m e of horizontal or nearly horizontal flow, when the supposed constancy of the velocity along a vertical section is acceptable. Investigating the other types of unconfined seepage and even in the case of nearly horizontal flow if more accurate analysis is required, the position of the seepage line (the intersection of the water table and the flow plane) has to be determined first. This analysis can be followed by conformal mapping, aiming at the determination of the field of the one dimensional seepage equivalent with the original system. As already mentioned, one of the possible methods of determining the seepage line is hodograph mapping. In this section an example will be discussed to demonstrate the combination of this special mapping and the conformal transformation. The practical example chosen for this purpose is the infiltration from irrigation canals. Before going into details of the application of hodograph mapping, it is necessary to investigate the general character of this type of seepage varying with time. On the basis of this analysis the validity zones of the various methods applicable to the description of the different flow conditions can be clearly indicated (KovAcs, 1963b). The process of infiltration from irrigation canals starts with the filling up of the canals. The first stage is the saturation of layers around the canal, when the infiltrating water has no contact with the continuous ground water, but the wetted front moves slowly downwards until i t reaches the water table, or more correctly, the upper surface of the closed capillary zone. During this propagation the water entering from the canal is used for the saturation of the wetted zone, where the pores were previously empty. After reaching the water table only a part of the infiltrating water is stored within the extending saturated zone, the other recharges the ground-water space. A unified seepage field develops in this way and the horizontal water conveyance of the groundwater space aflects the process of infiltration. On the basis of the explanation given in the previous paragraph, the process of infiltration can be separated into two main phases, i.e. the free seepage, and that influenced by the ground water. Both main types can be further divided by distinguishing unsteady and steady conditions. The process starts with the saturation of the layers around the canal which is an zcnsteady free seepage (Fig. 5.2-lla). Theoretically, the steady state of the free infiltration can develop only if the wetted front can propagate until reaching an infinite depth without the influence of the water table (Fig. 5.2-11b). In practice there is a point in time when the front just contacts the lower saturated zone and the process changes from the free stage into the influenced one. This t'ransition phase can be regarded as the quasi-steady state of the free infiltration (Fig. 5.2-11c). If there is a relatively very permeable layer below the water table, which is able to ensure the horizontal transport of the

616

6 Movement equetions describing seepage

(a) free unsteady infiltration

7 (c)quasi-steady free infiltmfion

h e depth ofthe pervious lager! is infinite

(d)influenced

unste8dy infl'ltration

(influenced steady infiltratiou)

Fig. 6.2-11. Verioua flow conditione characterizing infiltration from irrigation canal


617

infiltrating water without considerable rise in the gradient and the elevation of the water table below the canal, the quasi steady state of the free movement can become stable. The amount of infiltrating water is small because of the relatively low hydraulic conductivity of the layer between the canal and the water table. In other cases the water conveying capacity of the ground-water space in a horizontal direction hinders the further development of the infiltration. The seepage line indicating the continuous upper surface (contacting the points having pressure equal to atmospheric) of the combined flow field of the infiltration, extends gradually sideways. Part of the infiltrating water is stored in the pores becoming saturated as a result of the extension of the ground-water mound, and the remainder is drained by the horizontal flow (Fig. 5.2-11d). Thus this phase is also unsteady, but the movement is of the influenced type. The sideways propagation of the slope stops when the drainage becomes equal to the gradually decreasing amount of infiltration and a dynamic balance develops within the seepage field (steady influenced infiltration, Fig. 5.2-lle). It is necessary to note here, that drainage may be caused not only by the nearby canals having a lower water level, but also by the negative accretion resulting from the rise of the water table. The analysis of the various phase of infiltration clearly indicates that in the case of influenced seepage the largest part of the seepage field is characterized by nearly horizontal flow and, therefore Dupuit’s approximation is acceptable, though it may have to be supplemented by the determination of the local resistance caused by the strong curvature of the stream lines in the vicinity of the canal. Thus, the free infiltration (its unsteady and steady state) is the form of seepage which will be used as an example to demonstrate the combined application of hodograph and conformal mapping. The computation of the unsteady free infiltration has been dealt with by few authors. Perhaps the most frequently quoted relationship is given by Averjanov (1950). He has expressed the time-dependent flow rate of infiltration as a product of the water loss belonging to the free steady state ( t = 00) and a factor greater than unity and determined the latter partly on the basis of some theoretical consideration and partly from experimental measurements. In contrast to the previous type of movement steady free infiltration has been investigated by many research workers and a great number of theoretically well established results are known (Koieny, 1931; Vedernikov, 1934; Pavlovsky, 1936a, 1936b; Riesenkampf, 1940). The listed investigations are equally based on the application of both hodograph and conformal mapping. The differences between them are caused, in general, by the diflerent profiles of the canal considered aa boundary condition. They can differ further, according to whether the capillary eflect is taken into account or not. Among the various methods, which all characterize the steady free infiltration, Verigin’s derivation (1949) is presented here. The advantage of this method is not only the relatively simple method of studying the capillary water transport and the rapid convergence of the series into which some expressions are expanded, but also the possibility of further developing the method

618


for the description of the unsteady state of free infiltration on the basis of the results achieved in the analysis of the steady state (Kov&cs,1963a). The basis of Verigin’s derivation is a diagram specifying the bondary conditions around the seepage field of the free steady infiltration (Fig. 5.2-12). These conditions along the various stretches of the boundary are as follows:

1

1-

potential line

1-

Fig. 5.2-12. Seepage field characterizing free steady infiltration from irrigation canal

h

(a) Stream line bldl: stream-function y = 9 4 2 ; pressure head p l y = - -h, = const.; (b) Stream line b,d,: stream-function y = -qd2; pressure head p l y = - - h, = const.; (c) Special boundary blcl: stream-function ly = 942; pressure head varies from p l y = 0 at point c1 to p l y = -h, (at point bl); (d) Special boundary b,c,: stream-function ly = -942; pressure head is similar to that of the previous stretch p l y = 0 at point c, and p l y = -h, at point b,; n (e) Potential line c1c3c2: the potential is constant or choosing the water h

h

level as datum it is equal to zero cp = K (The symbols are defined as follows: h, capillary height, go specific discharge infltrating from one metre length of the canal; velocity potential.) The seepage field of the two dimensional flow is represented on the complex potential plane (w = rp i y ) aa well aa on the hodograph plane of the

+

reciprocal vector of the conjugate velocity stripe parallel to the real axis (Fig. 5.2-13). between these two planes, an auxiliary plane (( = E

.

) by an infinite

the relationship

+ i q ) has to be intro-

619


Fig. 5.2-13. Conformal and hodograph mapping to determine the hydraulic parameters of free steady infiltration

duced. Both stripes can be easily mapped onto the positive imaginary half of the 5 plane by applying Schwartz-Christoffel's transformation:

In 6

2

(5.2-3 1 )

dz dz dw 1 Considering the relationships between - , - - and -[see d5 dw ' dC V* (5.1-20)] the following equation can be given dz dC

-

h, -1n--, 1 1+C. nln6 C 1-C

Eq.

(5.2-32)

620

6 Movement equa.tions describing seepage

the solution of which (after expanding the expression t o be integrated into series) yields z=

[ c + ~ c ~1 + - ~1 ~ + - : ' + . .(5.2-33) .

1 -2 2h

nln 6

52

72

Substituting the corresponding parameters characterizing the width of the water in the canal ( z = S/2 and 5 = 6 ) a formula is achieved t o calculate the unknown 6 parameter as a function of the ratio of the width and the capillary height: SJ2=?

1 1 6+-63+-65f-67+ nln 6 9 25

1

...

49

(5.2-34)

The other important geometrical parameter is the fictive width of the water level ( S othe distance of the two extreme stream lines at the elevation of the water level between points b, and b,), which characterizes the influence of the capillary water transport. Its size can also be determined from Eq. (5.2-33), by substituting the 5 values belonging to point b, or b,: s o / 2 = - z

n2h ln6

[

1 1 1+-+-+-+ 9 25

1

...

49

(5.2-35)

Using Eqs (5.2-34) and (5.2-35) the 6 parameter and the So/S ratio can be presented either in tabulated form (Table 5.2-1) or by graphs (Fig. 5.2-14). Table 5.2-1. Determination of effective water table width. Relationship between actual water table width and capillary suction pertaining to saturat,ion Ratio of mtual water Ratio of effective and actual water table width saturation

d

0.00 0.01 0.02 0.04 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.92 0.94 0.96 0.98 1.00

f%lS

0 0.0028 0.0065 0.0158 0.0272 0.0404 0.0564 0.159 0.318 0.565 0.948 1.56 2.66 5.25 12.3 16.6 22.5 34.4 71.5

00

123.4 61.7 30.8 20.6 16.4 12.3 6.1 4.1 3.0 2.4 1.96 1.66 1.40 1.20 1.16 1.12 1.09 1.06 1.00


621

After determining all the auxiliary variables, substituting the [ values belonging to points b, and b, into Eq. (5.2-31) and calculating the difference between t,he w(b,) and the w(b,) vectors, the specific discharge can be calculated. By using Eq. (5.2-35) this hydraulic parameter can also be expressed as the function of the virtual width ( S o ) :

nKh,

qo=-- 2KS0. In 6

(5.2-36)

There is an important conclusion which can be drawn from this formula. At infinite depth, where the gradient is equal to unity, the area of the cross section having unit size perpendicularly to the flow plane has to be equal to 25,. Namely, t,heflow field is twice aa large at infinite depth, aa at the elevation of the water level in the canal. One point (i.e. b,) and the vertical asymptote of the bordering stream line is fixed in this way. Considering these two fixed conditions the border of the flow field can be approximated by curves, which can be easily mapped aa parallel straight lines to get a one dimensional flow pattern (Fig. 5.2-15). The first stage of mapping changes the distance between the two aaymptotes from 25, to n:

therefore (5.2-37)

622


Fig. 5.2-15. Conformal mapping to determine the hydraulicparameters of free unsteady filtration

and

This dimensionless plane is further mapped t o transform the t w o asymptotes onto the imaginary axis of the t plane. This result can be achieved by applying the logarithmic functions z’=lnt=lne+it9; therefore z‘= In e; and (5.2-38) y’ = 6 ; where

--

e = 1 / 1 2 + s2;

8

and 6 = arc tan - . r

The basic hypothesis of the derivation is that the contour of the seepage field can be approximated by a curve, the image of which on the t plane is a straight line parallel to the vertical axis and running through the images of the points b, and b,. The ordinates of these two points in the polar system


623

. After shifting the origin to the intersection of the real axis and the vertical image of the contour,

t‘ = t therefore

Ic

- cos - ;

4

r’ = r - 0.7071;

(5.2-39)

and s’=s;

a second supposition is made. The actual contour of the canal is approximated by a curve, the image of which i s a semicircle, passing through the images of points c1 and c, and having its centre fixed to the origin. It should be noted here, that the original seepage field assumed by Verigin is enlarged to simplify further investigation.Two curvilinear triangles (with 4-11-14 and 5-12-15 as corner points) are attached to the field. This enlargement is negligibly small and does not cause appreciable differences in the results. Verigin has already pointed out the uncertainties occurring in the boundary conditions along the lines 4-11 and 5-12. It is evident that the water rises in the soil above the water level due to capillarity. It can be assumed, therefore, that the actual conditions are better characterized by the enlarged field than by the original one. Applying one more mapping function the contours of the flow field become parallel straight lines: w = In t ’ ; therefore u = lne’; and v = 6’; where e’ = = I / ( r - 0.7071)2 s2 ; (5.240) and

v

v

9’

+

S

6‘ = arc tan - = arc tan r’ r - 0.7071 Some further approximations have to be accepted using the mapped field to determine the hydraulic parameters of the unsteady free infiltration. It has to be supposed, that the form of the wetted zonedoesnot differ considerably from that of the potential line intersecting the y axis at the same point aa the wetted front at the moment of the investigation. A further assumption is that the change in the length from the canal to the wetted front as well as the decline i n the actual flow direction from the vertical are negligible. On the basis of these two hypotheses the piston flow in the mapped system can be investigated instead of the actual unsteady free seepage. Comparing the forms of experimentally determined wetted fronts to the calculated potential lines and considering the uncertainties occurring in the vicinity of the front (random character of capillary suction, uneven velocity i n the pores having different


624

sizes, etc.) the suppositions can be accepted. Thus, the hydraulic parameters can be calculated aa the functions of the y depth reached by the wetted front at a time point t :

v = K Y+hC.

Seepage velocity

9

uy-

Velocity of propagation

K Yfhc. vefl = n uY-uA q=nK

Specific flow rate

uA

Y uy

(5.2-41)

9

+ hc - uA

The geometrical parameters of the mapped field depending on measurements of the original system are determined by the transformation functions: uA= In eX = In

1.5- 1.4142cos

u B = In & = In 0.7071 = -0.3466; uy = h e ;

[

KO

= I n exp - - -0.7071

(5.242)

1.

The relationship between the position of the wetted front and the time elapsed since the start of the process can be determined if the velocity of propagation is taken into consideration. Thus, all the hydraulic parameters can be expressed as time-dependent variables:

therefore

where 2 yo = h',-ln x

+ 0.70711;

[exp (uA)

(5.2-43)

the last parameter being the virtual depth of the canal. The second expression to be integrated can be solved directly, while the first one leads to an integral exponential function. Therefore, tabulated values, numerical solutions or the expansion into series have to be applied.


625

5.2.4 Characterization of unconfined field by the application of Zhukovsky’s function Development of the water table beside a draining trench

As already explained in Section 4.2.3 the position of the seepage line of an unconfined field can be determined not only by the hodograph mapping but also by applying other special mapping functions. The application of Zhukovsky’s mapping function in the following example demonstrates the determination of the hydraulic parameters of seepage around a trench draining the ground water. Recalling Eq. (4.249) it can be seen that the ground-water table drawn down by the trench, the free exit face and the bottom of the trench (it is supposed that the water depth is equal to zero) is mapped onto the vertical axis of the Zhukovsky’s plane, because the pressure along these contours is equal to zero at each point (Fig. 5.2-16). The hodograph plane of velocity, or that of its conjugate value, can also be easily constructed (v = v, + ivy or v* = v, - ivy plane) to determine the vertical ordinates of points 3 and 5 on 0 plane, while the parameters of points 2, 4 and 6 are known from Eq. (4.2-48)

Fig. 5.2-16. Application of Zhukovsky’s mapping to determine the hydraulic parameters around draining trenches 40

626


(5.2-44)

P =2

+ -2b -

Applying the 1

t= V*

- iK

(5.2-45)

relationship the mapped image of the hodograph field becomes a triangle being slit at its lower peak. This form can be mapped onto the lower half iq) by applying the Schwartz-Christoffel’s of an auxiliary plane ( 5 = 5 formula:

+

After solving the integral numerically for a given a angle of the slope, the ilparameter can be calculated. Considering the identity of the 0 and the 5 planes and the corresponding ordinates of points 2 and 6 on the two planes, the e2ordinates of point 3 and 5 can be also determined: @=im[;

where (5.2-47)

and, therefore,

Combining the relationships derived here and the equations establishing contacts between the z vector of the original system, the conjugate velocity and the Zhukovsky’s potential [seeEqs (4.2-56) and (4.2-57)], the hydraulic parameters can be calculated, as analyzed by Polubarinova-Kochina (1952, 1962) for special values of the angle of the slope. References to Chapter 5.2 AVEBJANOV, S. V. (1960): Seepage Losees from Irrigation Canals (in Russian.) Wrotechnica i M d b a c i a , No. 9-10. KOVLCS,G. (1960a): Calculation of the Flow Rate of Seepage under Dam (in Hungarian). Viziigyi EozZem&nnyek, No. 2. KOVAW,G. (1960b): Design of Draining Trenches along SurfaceWaters. Part I (in Hungarian). Hidroldgiai Kodony, No. 6.

5.3 Horizontal unconfined steady seepage

627

Kovilcs, G. (1961a): Design of Draining Trenches alongsurface Waters. Part I1 and I11 (in Hungarian). Hidroldgiai Kozlony, No. 1. KovAcs, G. (1961b): Yield of Riparian Drains (in Hungarian). Hidroldgiai Kozlony, No. 4. KOVACS,G. (1962): Yield and Influence on Pressure Head of the Series of Riparian Wells (in Hungarian). Hidroldgiai Kozlony, No. 2. Kovlics, G. (1963a): Free Seepage from Irrigation Canals. V I I I . Concegno di Zdraulica. Pisa. KovAcs, G. (196313): Characterization of the Steady Influenced State of Seepage from Recharging Irrigation Canals (in Hungarian). Hidroldgiai Kozlony, No. 1. K O V ~ C G. S , (1966): Yield of Partially Penetrated Wells (in English). Symposium on Seepage and Well Hydraulics, Budapest 1966. Kovilcs, G. (1976): Interaction between Rivers and Ground-water (in English). I A H R Symposium o n Groundwater, Rapperswil, 1975. KOZENY,J. (1931): Infiltration from Rivera and Canals (in German). Wusserkruft und Wasserwirtschajt, No. 3. PAVLOVSKY, N. N. (1922): Theory of Ground-water Movement around Hydraulic Structures (in Russian). Leningrad. PAVLOVSKY, N. N. (1936a): Basis of the Solution of See age Problems Concerning Free Infiltration from Canals (in Russian). Izwestia V N I I 6 No. 19. PAVLOVSKY, N. N. (193613): Free Infiltration from a Circular Shaped Canal until I n h i t e Distance (in Russian). Izvatiu VNIIQ, No. 19. POLUBARINOVA-KOC, P. YA. (1962): Theory of Ground-water Movement (in Russian). Gostekhizdat, Moscow. P. YA. (1962): Tbeory of Ground-water Movement. POLWARINOVA-KOCHINA, Princeton University Press, Princeton. RIESENKAMPF.B. K. (1940): Ground-water Hvdraulics (in Russian). Proceedinm, - . University of Sbatov, Vol. XV. No. 6. VEDERNIKOV. V. V. (1934): Infiltration from Canals (in German). Wmserkraft und Waseerwirtschajt, NO. ii-12: VERIQIN,N. N. (1949): Infiltration of Water from the Canals of Irrigation Systems (in Russian). Dokl. Akad. Nauk SSSR, No. 4.

Chapter 5.3 Horizontal unconfined steady seepage (Dupuit’s equations and the limits of their application) In the previous chapters the application of those mapping methods, which can be used to transform the flownets of two and three dimensional seepage space into a new system, where the stream lines are straight and parallel to each other waa discussed. These methods are necessary to determine the hydraulic parameters of seepage, because the one dimensional confined flow is the only form of seepage, for which Laplace’s equation can be directly integrated in closed form, as shown in Chapter 4.2 [see Eq. (4.1-41)]. It ww also explained there and repeated once again in the introduction of Part 5 that there are several cmes in practice, when the actual seepage field (or at least a part of it) is approximated by the simple flow pattern of horizontal confined flow. Also the hydraulic parameters calculated in this way are accepted as good approximations of the real values, if the actual conditions are within the validity zones of the hypotheses applied for the derivation of the simplified relationships. 40*

628


The use of Dupuit’s equations to describe the unconfined steady seepage was mentioned ;t9 an example of this approximation [see Eq. (4.1-44)]. Since the publication of this method (Dupuit, 1863) based on the laminar character of seepage (Darcy’s law) and on the supposition of the constancy of seepage velocity along a vertical section of the seepage space being normal to the flow direction (Dupuit’s hypothesis), this equation is indeed the most frequently applied relationship in seepage hydraulics. Its field of application wm enlarged by deriving further relationships suitable to characterize axial symmetrical flow (Thiem, 1870) and to consider the accretion of the seepage field (Hantush, 1964). It was proved recently that the hypotheses seeming at first to be very rough approximations, can be justified theoreti1973). cally as well (Charnyi, 1951; KOVBCS, Further extension of the examples investigated by using Dupuit’s equations, is ensured by dividing the seepage field into separate stretches. There are many practical problems, when the unconfined horizontal seepage is characteristic in the largest part of the field but special conditions in the vicinity of the entry and exit faces (e.g. strong curvature of the stream lines) exclude the application of Dupuit’s hypothesis for the whole system. Separating these disturbed stretches and calculating the local resistances byconsidering the actual conditions, the seepage through the remaining part of the field can be described by Dupuit’s equations. The combination of the results determined for the various stretches provides us with the required hydraulic parameters by considering the superposition of the head losses and the continuity of the flow rate. This important practical application justifies the detailed discussion of Dupuit’s method. The derivation of the basic equations, the validity zones of the applied hypotheses, the special boundary conditions to be considered and the determination of local resistance (in some cases of divided seepage fields), will be summarized in this chapter.

5.3.1 Derivation of Dupuit’s equations The basic hypotheses used for the derivation of Dupuit’s formulae are as follows: (a) The validity of Darcy’s law in the homogeneous flow field; (b) The fixed geometry of the contour lines of the two dimensional seepage field with known parameters, [horizontal lower boundary; vertical entry and exit faces, the distance between them being L; constant elevation of both the head water ( H , ) and the tail water (H,) above the impervious lower boundary; and in the case of a confined system the horizontal impervious boundary above the flow field having constant thickness ( m ) ] (Fig. 5.3-l), (c) The steady state of flow as a result of the constant boundary conditicns (Hl and H2); (d) The constant velocity along a vertical section which can be expressed in mathematical form as the product of hydraulic conductivity and the deri-


629

Fig. 5.3-1. Symbols used for the derivation of Dupuit’s equations in the case of twodimensional seepage

vative of the seepage line [yo(x)]with respect to the flow direction: V(X)

= - K-,dY0

.

(5.3-1)

dx

where the seepage line (yo)indicates either the depth of the flow field (unconfined seepage) or the elevation of the piezometric line above the lower impervious boundary. Combining Eq. (5.3-1) with the equation of continuity, a differential equation for the determination of the percolating flow rate is obtained: For confined and semi-confined systems: q(x) = mw(x) = const. ; q = - K mdYO ; ax For unconfined systems q(x) = yo(x)w(x) = const.; q = - Kyo-. dY0

ax

(5.3-2)

The differential equation can easily be integrated in the case of a confined (or semi-confined)aquifer, because the boundary conditions are well determined (if x = 0; yo = H , and if x = L; yo = H 2 ) .The resulting integration is

9=

and

mK L

-w,

- H,) ;

(5.3-3)

630

5 Movement equations describing Reepage

Originally, the differential equation describing the unconfined flow waa similarly solved, using the same boundary conditions. Since the existence of the free exit face and its influence on the flow developing in an unconfined field became known, the accuracy of the boundary conditions had to be revised. It can easily be proved by hodograph mapping that the intersection of the exit face and the tail water level is theoretically a singular point. The same conclusion can be drawn from the simple physical study of the boundary conditions. Below the tail water the stream lines are normal to the exit face, the latter being a potential surface. The section of the phreatic surface with the flow plane is also a stream line having a slope determined by the boundary condition which characterizes the stream lines of constant pressure in general. If this seepage line joined the lower water level, this point would be the intersection of two stream lines and stream lines intersect each other only at singular points with infinite velocity. Therefore, the exit point (intersection of the seepage line and the exit face) cannot reach the level of the tail water, because the hydraulic gradient on the surface always haa a finite value (the slope of the water table cannot be infinite), while the intersection of the seepage line and the tail water level would require an infinite value. In a negative way this analysis proves the necessary development of the free exit face between the exit point and the tail water (Muskat, 1937). The influence of the special boundary conditions along the free exit face on the whole seepage field can be well demonstrated by comparing the potential differences along the stream lines, supposing at first the lack of the free exit face and its development in further cams (Fig. 5.3-2). The Dupuit’s hypothesis supposes not only the constant velocity along a vertical section, but involves at the same time the constant potential diflerence along euch stream line, which corresponds to the first version of the lower boundary conditions (Fig. 5.3-2a). This approximation is not acceptable because of the development of the free exit face and, therefore, the consideration of x = L,yo = H z boundary condition does not give the correct solution of the basic differential equation. If there is no tail water ( H z = 0), the potential distribution along the exit face ( y k = Kyk) is linear and the potential difference can be represented with a trapezium shaped distribution (Fig. 5.3-2b). The condition is considerably modified by the existence of the tailwater, because below its level, the potential is constant (fpk = KHz if yk < H2), while above the tail water the potential remains unchanged (fpk = K y , if yk > H z ) . The graph representing the distribution of the p t e n tial d i f l e r e m is divided, therefore, into two parts: i.e. along the exit surface contacted by the surface water the diflerence is w m t a n t , while along the free exit face the relationship between the potential diflerence and the height of the investigated point is linear (Fig. 5.3-2c). A method to calculate the specific flow rate by studying the lower boundary conditions was derived by Charnyi (1951). The symbols used in the derivation are shown in Fig. 5.3-1. The flow rate is determined by integrating the product of the horizontal seepage velocity

(

2)

vx = -- and the


potential difference along flowlines related fo the entry

pofential didribufion ~Iongthe entry surf8ce

seepage field

631

poiential distri- poteniial dfference butiffn along the along flowlines exit suM8ce reiaied to ibe exit points of the latter

(cl With fail wafer and free exit surface Fig. 6.3-2. Distribution of the potential difference measured along the stream lines and represented along the entry and exit faces

thickness of an elementary horizontal layer (dy) between the impervious lower boundary and the water table: (5.34) y=o

Y-0

The velocity potential along the surface [cp = Kyo(x)]is known.Thus, the previous integral can be transformed by using the relationship between the total and the partial derivatives of defhite integrals:

0

thus

0

(5.3-5)

632


Integrating Eq. (5.3-5) with respect to x the constant can be determined by considering the boundary condition along the entry face (5= 0; y o = H,; rpB = K H , = const.). The result is aa follows: (5.3-6) 0

The h a 1 form of the equation derived to calculate the specific flow rate can be achieved by substituting the lower boundary condition (x = L; y o = H 3 ) , where the integral of the potential haa to be divided into two parts expressing the difference of the potential distribution below and above the tail water level (see Fig. 5.3-2c):

r 7 . r 0

~ ( ydy) = KHZ dy 0

K (Hi - Hi) . +H,Ky dy = 2

(5.3-7)

Substituting this relationship into Eq. (5.3-6), the surprising result is that the theoretically correct relationship to calculate the flow rate is identical to Dupuit’s original equation determined by the uncorrect consideration of the lower boundary condition :

K q =-(H, 2L

- H,).

(5.3-8)

This virtual contradiction can be explained by the fact that the two approximations applied in Dupuit’s derivation (the supposition of the constancy of the potential difference along each stream line and the substitution of the depth of the tail water instead of the height of the exit point) compensate one another. It is quite evident that the position of the seepage line cannot be determined by using this incorrect substitution of the lower boundary condition, because the water table has to intersect the exit face at the height of the exit point, and not at the level of the tail water. Considering only the geometrical conditions (i.e. the water table has to be fitted to the x = 0, y o = H , and x = L, yo = H 3 points) and neglecting the hydraulic interpretation of the boundary conditions, the differential equation given for unconfhed systems [Eq. (5.3-2)] can be solved by expressing y o as the function of the horizontal distance from the entry face yo =

vm).

(5.3-9)

Dupuit’s parabola is, however, only a very rough estimation of the actual position of the water table. Analyzing the unconfined seepage field between vertical entry and exit faces and the hodograph fields of both the gradient and the reciprocal value of its conjugate, as well as the image on the complex


633

potential plane (Fig. 5.3-3), i t can be seen that the parabola does not satisfy the boundary conditions at the entry and the exit faces (it is not normal to the entry face and i t does not join tangentially the exit face at the exit point). There is, however, an internal part of the field (indicated by hatching) exluding the vicinity of the closing faces, where the proportionality 1

between the fields on the complex potential and the - planes may be

I*

assumed aa an approximation: w-KHj

Q

K

K .

V*

v;x

- ---

’ (5.3-10)

and considering Eq. (5.1-14) 1 H j l w - - - -- -dz _--dw v* P v;x K9

After substituting the corresponding values belonging to the border of the stretches supposed to be proportional, the solution of this differential equation gives a relationship similar to Dupuit’s curve. This result indicates that the p = ky, boundary condition is approximately satisfied along the internal &retch of the parabola.

( w plane)

( ~ p l m ?) Fig. 6.3-3. The unconfined seepage field between vertical entry and exit, faces and its images on the hodograph and complex potential planes

634


Using Hamal’s mapping function an exact mathematical relationship can be derived to determine the position of the water table (Muskat, 1937).This method is, however, very complicated and, therefore, the use of Dupuit’s equation by substituting H , as the lower boundary condition [Eq. (5.3-9)] is generally accepted in practice. It is necessary, however, to remember the uncertainties of this approximation, and to apply more exact methods for the determination of the seepage line, if its position is critical from the point of view of the investigation. The same uncertainties justify the use of other simple geometrical approximations [e.g. the substitution of the seepage line by an ellipse passing through the x = 0, yo = H , and 2 = L, yo = H , points and having a vertical minor axis ( b = H 1 - H,) fitted to the x = 0 coordinates axis, also a horizontal major axis ( a = L ) at the elevation of the exit point, whose curve satisfies the required conditions at the two closing faces]. The acceptability of the very rough approximation of the seepage line can be justified by considering the relatively small differences, which may occur between two smooth curves passing through two fixed points (i.e. x = 0, yo = H , and x = L, yo = H,). A further factor supporting this hypothesis is that the position of the water table has practical importance only in the vicinity of the exit face (here i t influences the stability of the solid matrix) and, therefore, the height of the exit point is the parameter, which has to be determined very carefully. There are two theoretically based methods published in the literature for the calculation of the exit height. Polubarinova-Kochina (1952, 1962) gives mathematically correct derivation, in which the acceptance of Dupuit’s flow rate is a basic hypothesis, while the Hungarian method (Kov&cs,1973) uses some approximations, but provides a general relationship between the flow rate and the exit height using the fixed geometrical parameters of the field as independent variables. The comparison of the two methods proves their satisfactory accuracy and also gives important information about the character of seepage. head water ,,; .................................. ................................... . . . . ...... . . . ..... ..: . .:.... . . . .;.:.:. .: ......: T I . .. . :..:..,.; ....:. .

\ . .‘.:.I:

exit joint 7 -~ -

Fig. 6.3-4. Mapping of the flow field by wing Polubarinova-Koohma’s method

635


Poluburinova-Kochina’s derivation is based on the analytic theory of linear differential equations. This highly sophisticated mathematical method is used t o map the curvilinear polygon surrounding the seepage field into the upper half of an auxiliary plane (Fig. 5.3-4). It has, therefore, many similarities to Schwartz-Christoffel’s mapping, which solves the same transformation in the case of polygons with straight sides. The angles formed by the stretches of the contours of the seepage field a t the corner points are an essential feature. It is necessary, therefore, to analyze the hydraulic and geometrical conditions determining the internal angles of the field, as in the case of conformal mapping. Because the relationships are more complicated in the case of a curvilinear polygon than those achieved by mapping tt field bordered by straight lines, hypergeometrical functions have to be applied instead of the elliptical integrals characteristic to the solution of Schwartz-Christoffel’s mapping. The general form of the hypergeometrical functions is a8 follows:

P( a, b, c, 2) = 1

+ ab + a(a-1.2+- 1)c(cb(b++1) 1) --2 C

22

+

. ..

(5.3-11)

By using various auxiliary functions, going through a very complicated chain of mathematical relationships and applying the results of hodograph mapping, the geometrical parameters and the specific flow rate through the various sections of the seepage field can be expressed by a series of integral equations: b 1

H2

H,

I

=A b

- H2 = A

i -

CLdC

5 CV(5 - 1) ( 5 - a ) ( 5 - b ) C- 1 (1 -

1-C

=AJ,;

d5

5) Y(a - 5 ) ( b - 5 )

-=A J4 ;

(5.3-1 2)

636


To solve thifi system, the consideration of a further hydraulic condition is necessary. As dready mentioned, the validity of Dupuit's flow rate [Eq. (5.3-8)] is assunzed as a supplementary hypothsis. The relationship can also be expressed by the integral equations listed earlier: 2 JIJ, = Jt - 4 . (5.3-13) corresponding symbols in PafubariflovaKochina's orgina/ publicafton and in fnis texf

:

10

3 Hf

015

0

-

4

Fig. 5.3-5. Graphical representation of Polubarinova-Kochina's resultascharacterizing the height of t,he exit point


637

From Eqs (5.3-12) and (5.3-13) the general solution can be achieved by using numerical methods after expanding the relationships to be integrated into series. The final results are given in the form of graphs (Fig. 5.3-5). To ensure the unified discussion of the various problems the symbols and the quantities here differ from those applied by Polubarinova-Kochina in the original derivation. Figure 5.3-5 represents, therefore, the height of the exit point as the function of the geometrical parameters of the seepage field in both systems, and also lists the corresponding symbols. The original description emphasized that, if the seepage field is not influenced by tail water ( H , = 0 ) , the reliability of the relationships is higher than that of the other curves characterizing the hydraulic parameters of seepage under the influence of tail water of various depths. This is because in the latter case the calculation requires the use of successive iteration. Apart from this difference there are special caaes, when substituting the geometrical parameters of the flow field simplifies the equations: e.g. in the case of a very large relative length (if L } H , tends to infinity) both the height of the free exit face ( H , - H,) and the specific flow rate percolating through this section [(J(~,-~,)],can be expressed as functions of the same constant. Thus, a relationship can be established between the two variables: and (5.3-14)

consequently q(H,-H,)

= 1.3469 K(H3 - H,) ;

where 1.8319 = 2G and G is Catalan's constant. If the entire height of the exit face is in contact with the air (there is no tail water, H , = 0 ) Eq. (5.3-14) gives a direct relationship between the total specific flow rate and the height of the exit point. Thus, the latter can also be expressed as a function of known geometrical parameters:

consequently

[2l0

(5.3-1 5 )

H = 0.371 -2; L

where the subscript 0 indicates, the parameter H , = 0. As already mentioned the baaic hypothesis of this derivation is the aasumption that Dupuit's flow rate according to Eq. (5.3-8) is valid. A further approximation applied in determining Eqs (5.3-14) and (5.3-15) is the supposition that the seepage field is of infinite length. Polubarinova-Kochina has found that the hyperbolic relationship between the relative height of the exit point ( H 3 / H 1 )and the relative length of the field ( L / H , ) is valid, if the latter is greater than 1.5. Comparing the results with those of the other theoretical method and to measured data the validity zone can even be

638


extended and Eq. (5.3-15) can be accepted as a good approximtion in the zone LIH, 2 1. The mathematical basis of the other method is conformal m p p i n g of the seepage field assuming the position of the water table to be approximated by Dupuit’s parabola [Eq. (5.3-9)]. After a suitable shifting of the origin of the coordinate system represented in Fig. 5.3-1, the new vertical axis intersects the focus of Dupuit’s parabola. The equation of the seepage line in the new system is (Fig. 5.3-6): yo=A Bz; where

+

B=

- H2

a= 2p;

(5.3-1 6)

Fig. 5.3-6. Conformal mapping of the flow field of unconfined horizontal seepage

639


and p=--.

H:

-

Hi

2L Using an auxiliary pkane c(5, q), whose contact with the origiiial system is expressed by a well known analytic mapping function,

consequently (5.3-17)

and and the image of the water table becomes a straight line parallel to the re a axis. Its distance from the latter is:

;i!V! - H 2 = va.

qo = const. =

(5.3-18) 4L A further approximation in this method is the hypothesis, that the series of curve8 intersecting the parabola at its maximum point ( P o )and covering the whole field between the water table and the impervious lower boundary, can be regarded as a system of stream lines independently of the position of the entry and exit faces. These lines are not perpendicular to the vertical closing faces, although the whole entry face and the exit face below the tailwater level are potential surfaces, which have, theoretically to he intersected by the stream lines at right angles. The images of the curves, supposed to be the approximative stream lines compose a series of almost parallel curves from infinity on the 5( 5 , q) plane. They are bent upwards near the imaginary axis and intersect it at the point Zo(O, qo). This system is identical to a strip of the flownet characterizing the infinite series of sources (or sinks) located along the imaginary axis at points Zol(O, qo);Z,(O, -q,,); Z,,(O, 3q0); Zo,(O, - 3 q 0 ) ; . . at a distance of 2n, from each other. This svstem of stream lines can be transformed into parallel straight lines by appliing the following mapping functions [see also Eq.(5.2-29)]:

.

."

w =u

+ i v = lnsh-([x

consequently and

- iqo);

2rlO

u=-ln-[ 2

2

-E

ch

) +cos [

x- 7 ~ O q ~ . ) ] ;

(5.3-19) x

tan -(rl - 7 0 ) v = arctan

270

9

~

JC

t h -5 2rlO when the image of point P o is transferred to infinity.

640


By this form of mapping the original seepage field is transformed into a series of elementary lamellae having a thickness of Av. The length of these lamellae depends on their distance from the horizontal axis of the w(u, v) plane : Au(v) = u ~ ( v) uK(v). (5.3-20) Knowing the potential diflerence along each lamellae (which depends on the position of the corresponding stream line in the original field, im explained in Fig. 5.3-2) it can also be expressed as the function of the imaginary ordinate of the I I ~plane and

Avl = K ( H , - H,) = const. ; if 0 5 Y k dv2 = K ( H l - Y k ) = & ( V ) ; if H ,

H2 ;

(5.3-2 1)

5 Y k 2 H3.

The water conveyance through a lamella can be calculated from the listed parameters in both zones and summarizing these quantities, the total specific flow rate is achieved:

and dq, = K

- Y k ( v ) d v ; if H , 5 yk

H, ;

(5.3-22)

fwv) consequently

where the limits vl, v, and v3 can be determined from the mapping functions, substituting yk = 0, yk = H , and Y k = H , values respectively: 7c

vl=

- -;

2

v2

= f ( H 2 ); and

w3 = 0.

( 5.3-23)

If the seepage field is not influenced by tail water ( H , = 0 ) Eq. (5.3-22) can be simplified: 0

(5.3-24)

When solving the integrals in Eqs (5.3-22) and (5.3-24), i t has t o be considered that both Y k and Au have t o be substituted as the functions of the v imaginary variable. The result gives the specific flow rate depending on four

641


geometrical parameters or, using dimensionless quantities, i t can be expressed as function of three independent ratios of the four parameters: or

(5.3-25)

Among the four types of geometrical data (i.e. length, depth of the head and tail water, and height of the exit point) only three are known a priori ( H l , H2,L). Eg.(5.3.-25) does not give, therefore, a single-valued solution but a series of interrelated q and H , values. Thus rating curves can be constructed

El

showing the relationship between the relative exit height - and the flow rate. It is advisable that this latter value should be related to Dupuit’s flow rate calculated from Eq. (5.3-8), to get a dimensionless quantity independent of hydraulic conductivity. If a seepage field which is not influenced by tail water is investigated and the relative length of the field is used as a parameter, one set of rating curves can be determined (Fig. 5.3-7).

qL7 Fig. 6.3-7. Relationship between the relative height of the exit point and the dimensionless charaoteristicsof flow rate if the field is not intluenced by tail water 41

642


In general, there are two variables to be considered (LIH, and H,/H,) apart from the interrelated values of H $ H , and q / q D . The rating curves belonging to different given values of LIH, can be represented using H,IH, aa the parameter (Fig. 5.3-8). The result of this method can be compared with Polubarinova-Kochina’s derivation only if the same hydraulic hypothesis is used i.e. the water conveyance is equal to Dupuit’s flow rate. Applying this supposition the exit height belonging to various LIH, and H J H , parameters can be determined from the graphs in Figs 5.3-7 and 5.3-8 aa the intersections of the curves and the vertical line belonging to the q / q D = 1 abscissa. These data are plotted in Fig. 5.3-5 and show good agreement between the two methods, and thus the reliability of both derivations. A further important conclusion can be drawn. Although Churnyi’s theoretical analysis has proved that the correct specific flow rate can be calculated from Eq.(5.3-8), the rating curves show that the field could transport higher amounts of water. The possible increase in flow rate is relatively small.

‘:r %? 0.4

0.2

0

fl!

h? 0.8

0.6 0.4 0.2

0 Fig. 6.3-8. Relationship between the relative height of the exit point and the dimensionleea characteristics of flow rate if the field is influenced by tail water

643


The difference between the absolute maximum belonging to the infinite relative length (resulting H d H , = 0.5 relative exit height) and Dupuit’s value is only 12.5%, but the least increase in the discharge causes considerable rise of the exit point, the rating curve being very flat. Considering the principle of “lex minimi”(which is a basic law of theoretical physics) i t is expected that the movement oould reach a steady condition only in a system having a given energy content (represented by the difference between the levels of head and tail water), if the maximum possible flow rate were transported. Combining this principle with the rating curves, the probable flow rate and the exit height can be determined &B the functions of LIH, and H,IH, by determining the position of the maximum points of the rating curves.

Figure 5.3-9 shows the points representing both the maximum possible flow rate and the exit height belonging to this value, aa the functions of the relative length of the field, supposing that the field is not influenced by tail water ( H , = 0). The equations presented in the figure, mathematically describe the curves through the points in the range LIH, > 1. A third curve H also constructed in the figure shows the interrelation between -2and L / H , Hl on the baaia of Polubarinova-Kochina’s investigation [Eq. (5.3-15)] compared to that of the points determined at the intersection of the rating curves and the q/q, = 1 ordinate in Fig. 5.3-7 (a dotted line passes through these points). The first conclusion drawn from the figure is the accurate correlation between the results of the two methods, as shown in Fig. 5.3-5.

0

2.5

50

z5 % 8

Fig. 5.3-9. Representation of the maximum possible flow rate and the exit height belonging to this value and to Dupuit’s flow rate, if the seepage is not influenced by tail water 41*

644


It can also be seen, that the ratio of qma&D does not differ considerably from unity, while the position of the exit point belonging to the maximum water transport is much higher than that determined by taking Dupuit’s flow rate into account. There is an interesting question to be answered in connection with this analysis: i.e. why the flow rate determined by Charnyi’s theoretically correct derivation is smaller than that calculated on the basis of the principle of lex minimi”. The approximations applied to construct the rating curves (especially the use of Dupuit’s parabola) can cause some discrepancy but it may be neglected since the results achieved in this way are similar to those derived from the hypergeometrical functions. There is another possible reason: a part of the available energy i s consumed by resistances not considered in the investigation and, therefore, the water transport haa to be lower than the maximum water conveying capacity of the field. Some interesting aspects in connection with this problem can be made clearer by analyzing the development of the exit velocity. The hydraulic model described in Eq. (5.3-22) can be used to calculate the exit velocity by dividing the elementary flow rate by the thickness of the corresponding lamella measured on the original seepage field: 66

and

(5.3-26)

where v k is the exit velocity (the subscript is used to make a distinction between velocity and the imaginary ordinate of the w plane). The symbol du(yk) is used to indicate that the du value has now to be expressed aa the function of the height of the investigated point along the exit face of the dv original seepage field and- is the differential quotient of the imaginary du part of z calculated on the &is of the mapping functions. The velocity distribution calculated by using Eq. (5.3-26) can be represented in the form of graphs. An example is shown in Fig. 5.3-10. In this case the relative length of the field is LIH, = 2 , each graph representing the exit velocity belonging to a given value of H31H, using H d H , as a parameter. Hence, i t is evident that the highest velocity develops when seepage is not influenced by tail water. The distribution above the surface of the tail bater does not change, as in the non-influenced caae, while below this level the velocity is decreased. The numerical values do not indicate the theoretically expected increase i n velocity in the vicinity of the singular point. The velocity distribution can also be characterized by the maximum, mean, and minimum velocity values. The mean can be calculated as the ratio of the flow rate and the exit height:

645


Thus, the dimensionless parameter of the mean exit velocity

[

]

6: L H, ‘ H be expressed depending on the - , -, -and3- ratios. The curves in Fig. q

qD

Hl

Hl

Hl

5.3-11 represent this relationship if H , = 0 and are calculated from the corresponding values determined by the rating curves in Fig. 5.3-7. Two special

cams are emphasized in the figure, namely, those belonging t o Dupuit’s flow rate and the mean exit velocities belonging t o the pomible maximum

=O K

Fig. 6.3-10. Velocity distribution along the exit face calculated from Eq. (6.3-26)

646

6 Movement equations describing seepago QQQQQb

eKl.+jw-
1. This result can also be derived mathematically from Eq. (5.3-15):

Another possible summarization of the calculated velocity data, is the construction of graphs representing the minimum (developing at the exit point) and the maximum (at the level of the lower impervious boundary if H2 = 0) velocity values as the functions of the relative height of the exit

13

point - and using LIH,

&B

a parameter (Fig. 5.3-12).

The curves characterizing the minimum exit velocity have a vertical enveloping line, which indicates that in the probable zone of the exit point, this parameter is constant, independent of any geometrical parameter

-

v k mln (5.3-29) 0.55. K On the curves constructed to show the maximum exit velocity, the points belonging to Polubarinova-Kochina’s exit height (Eq. 5.3-15) can be

determined. These points indicate a practically constant maximum exit velocity. If the same exit height is characterized by the intersections of the rating curve8 with the q / q D = 1 ordinate in Fig. 5.3-7 and the maximum


647

Fig. 6.3-12. Maximum and minimum exit velocity as the functions of relative exit height and field length assuming H z = 0

exit velocity is determined as the function of this height, the numerical values of the maximum velocity do not differ considerably from the previous data, the points being scattered along the vertical ordinate determined by Polubarinova-Kochina’s method. Thus, this parameter can also be characterized by a constant, independent of the geometry of the field: v k max

--2.5.

K

(5.3-30)

The interesting result of this analysis is that in the case of seepage not influenced by tail water all parameters (mean, maximum and minimum values) of the ezit velocity are constant. They do not depend on the relative length of the field, or on the absolute depth of the head water, if i t is assumed that Dupuit’s flow rate is the correct water transporting capacity of the system, and the exit height is determined by Eq. (5.3-15). The contradiction between this result and the potential theory, can be explained by the development of,?tonlaminarflow.In the vicinity of points which ought to be singular points according to the potential theory, the velocity would be high, approaching an infinite value and in any case, higher than indicating the upper limit of the validity of Darcy’s law. The laminar character of flow is a basic principle of potential theory and, therefore, i t can be supposed that in this zone of the field, seepage is not a potential movement and the resistance is higher than that calculated from the equations generally applied. The seepage field can be regarded as a selfregulating system, in which the increase in velocity raises the resistance, and this process hinders the further rise in velocity. There exists, therefore, an upper limit of velocity, and the height of the exit point develops at the elevation to which this limit value belongs, as a maximum exit velocity. This explains why part of the energy

648


has to be consumed to overcome the resistance occurring above that calculated a8 the retarding effect against potential seepage. It is reasonable, therefore, that the flow rate is smaller than the maximum possible water transporting capacity of the field. Thus, Dupuit’s discharge is not only a flow rate, the validity of which can be proved by potential theory, but the exit point hawing this value also has a special role, viz. the maximum velocity developing i n this m e is equal to the upper limit, the existence of which was explained previously.

5.3.2 The influence of the capillary water conveyance The investigation of the hodograph of an unconiined.seepage field, clearly indicates that the development of the capillary fringe above the gravitational field considerably changes the boundary d i t i o n s of the latter (see Fig. 4.2-17), and thereby influences seepage i n the whole field. Eq. (5.3-8) gives only the flow rate percolating through the gravitational part of the system. The water transported in the capillary zone has to be added to this value to get the total flow rate. The two amounts (i.e. the gravitational and the capillary discharge) are not idependent of one another, because, at the entry and exit faces, the total flow rate crosses the sections of the gravitational field. Water cannot enter into the porous medium above the level of the head water, or exit into free air from the capillary zone having lower pressure than atmospheric. Thus the capillary fringe is recharged and drained in the form of water exchange between the two zones of the seepage field (Fig. 5.3-13). As an approximation, it can be supposed that the regular error caused by considering the capillary wafer tramport independently, is negligible (KOV&CS, 1973). The capillary fringe can be regarded, therefore, as a closed, nearly horizontal tube, and its water conveyance can be calculated as the product of the cross-sectional area and seepage velocity. This is equal to the capillary height since a unit width of the flow space of the two-dimensional seepage is investigated (A=h,) multiplied by the seepage velocity. It can also be supposed that the total potential is proportional to the difference in eleva-

Fig. 5.3-13. Water transport through the capillary zone

649


tion of the lowest points of the capillary exposed faces Thus, the capillary flow rate is

['p

= K(H, -H,)]. (5.3-31)

Accepting the supposition of independent capillary and gravitational water conveyance, the total amount transported through the field is equal to the sum of flow rates calculated from Eqs 5.3-8 and 5.3-31:

H2 = 0

or, if

(5.3-32)

Hyl+--' I:]-

((&=K--

1--

2L

The possible range of the factor by which the parameter is multiplied can easily be limited. Its highest value occurs if there is no tail water and in this case 1--< H3 1;

(5.3-33)

HI while in general the following inequality has to be considered: HI > H 2 , and, therefore, if H2 H 3 H I ---f

1--

>H3 >

--t

a 3

HI

+ 0.5.

(5.3-34)

Because the water cannot leave the capillary zone through the capillary exposed exit face, it has to percolate downwarrds into the gravitational field. Thus, the exit velocity would be increased by the capillary water transport if the exit height remained unchanged. On the basis of the explanation given in the previous section in connection with the existence of an upper limit of the exit velocity [Eqs (5.3-28), (5.3-29) and (5.3-30)] it is expected that the exit point would have to be raised by the capillary water conveyance to keep the exit velocity below this limit. Experiments executed to measure the height of the exit point (Muskat, 1937; Polubarinova-Kochina, 1952, 1962; Maione and Franzetti, 1969; Ching-Ton Kuo et al.,1969; Kov&cs,1973) have proved that the exit points are always at higher levels than those calculated from Eq. (5.3-15) in the case of the absence of tail water (but below

650


the maximum possible height) and the difference becomes greater aa the rate of the capillary water transport increases (Fig. 5.3-14). The measured values are suitable to check the reliability of the theories of both the determination of the rating curves showing the flow rate vs. exit height relationship (Figs 5.3-7 and 5.3-8) and the calculation of the capillary flow rate [Eq. (5.3-31)].

Fig. 5.3-14. Comparison of the measured and calculated values of the exit height

All the characteristics of 60 measurements executed without and 43 with tail water influence were carefully determined ( H I ,H,, H,, L, h, and 9).

The frequency distributions for both the 919, and the

'

-C ' values

were cal-

qD

culated separately for both sets (Fig. 5.3-15). The mean of the original data,

I:([

uninfluenced by tail water shows some difference from unity

I

= 1.041

[r 2 I

o mean = while after correction a normal distribution around unity

ot i a n

1

= 1.004 indicates only the influence of errors in measure-

ment. If the exit face is partly covered by tail water the original data have no larger discrepancy from unity

3

= 1.007 than the corrected

651


number of dafa

distributfuff of fbe measured dafa .

-

--

... ...

-----

without fail water with &il wafer complete set of dafa

distribution of the corrected dafa .............. witbuuf faif water wifb tail water complete set of dafa

-----

600

b 02

106

108

1/0

q and q - %. values Fig. 5.3-15. Frequency distribution of qD

qD

l , [i

(I - QC

= 0.993]. The capillary reduction, however, mean values decreases the variance in the measured data and thus the more accurate correlation justifies the correction. The accordance of the mean calculated

- Qc by combining the two corrected sets with unity, is excellent Q=o.gg~]. qD Imean

[(

Checking the theory by measurements can be continued by comparing the actual exit height with the calculated quantities determined a s the ordinates of the relezwat rating curve belonging to the measured qfqDvalues. This control

652


proves the reliability of the rating curves [ q / q D = f(H$H1; H,IH,; LIH,)] determined by using the mapping method. The AH$Aq=

H3

- (H3)D

Q ~- Qn ratios calculated from the measured data were practically identical with those indicated by Fig. 5.3-7 and 5.3-8 [the symbol ( H J Dmeans the exit height belonging to Dupuit’s flow rate]. The conclusions drawn from this analysis are as follows:

(a) Dupuit’s flow rate is acceptable aa an accurate value characterizing the water transport through the gravitational seepage field ; (b) The total discharge can be calculated aa the sum of the gravitational and capillary water conveyance, and the latter may be determined assuming the complete independence of the capillary zone of the gravitational field [Eq. (5.3-32)]; (c) The influence of capillarity is relatively smaller in the presence of tail water, because the rise in the exit point decreases the gradient in the capillary zone; (d) The actual height of the exit point has to be calculated by considering the total flow rate and using the rating curves in Fig. 5.3-7 and 5.3-8. The relationships given previously ensure the complete characterization of horizontal unconfined seepage through a seepage field with vertical faces, even in the case of considerable capillary influence. I n practice however, Eq. (5.3-32) cannot be applied directly because the H 3 variable is not a n a priori known basic parameter, but i t haa to be determined as a function of the flow rate. Successive approximation would be necessary for the simultaneous calculation of the two unknown values ( H 3 and 9). To avoide this laborious process, a further approximation haa to be applied, which results in a method convenient for the computation of both the capillary water transport and the position of the exit point modified by this surplus flow rate, if the seepage is not influenced by tail water. Substituting the relative height of the exit point from Eq. (5.3-15) into Eq. (5.3-32) (supposing that the influence of capillarity on the position of the exit point is negligible) the first approximative value of the capillary flow rate can be calculated:

(Ej0=2( --‘ 1 - - H3 -

L

H,)

This equation indicates a linear relationship between the capillary discharge and the relative length of the field. The actual height of the exit point is greater than that determined from Eq. (5.3-15)and the capillary water transport is, therefore, smaller than this calculated value. It can be presumed, however, that the linear relationship remains valid in the following general form :

L

L

(5.3-36)

653


In the case of the first approximation, both the tangent to this line and its intersection with the vertical axis of the coordinate system are known values calculated theoretically ( m = 1 and b = -0.371). I n general, these parameters have to be determined from the measured data, after representPC

-

L

ing the experimental results in a system having - -and -as ordinates. qD

2hc

The graphical representation of the measurements proves that the linear relationship is an acceptable approximation for each series (Fig. 5.3-16). The scattering of the b additive constant is relatively small and irregular, while the tangent to the lines shows a definite contact with the parameter of 2nc -,

L

as indicated in the figure. The rn and b parameters were calculated by applying regression analysis (Table 5.3-1). The coeEcient of correlation ( r > 0.999) indicates that the linear relationship is almost a functional contact. The intersections of the straight lines with the vertical and the horizontal axes ( b and blm respectively) show a scatter from -0.221 to -0.376 and

Fig. 6.3-16. Relationship between the5!! qD

L L and - parameters Hl

654


Table 6.3-1. Results of the regremion analysis concerning the relationship between capillary height and the position of the exit point Symbol of the

serieaof measurements

1

m

0.873 0.834 0.818 0.910 0.898 0.838 0.960 0.916

1

b

-0.286 -0.239 -0.298 -0.221 -0.207 -0.369 -0.376 -0.320

1

0.9992 0.9995 0.9995 0.9997 0.9999 0.9982 0.9999 0.9999

0.010 0.012 0.016 0.023 0.017 0.026 0.035 0.021

0.328 0.275 0.364 0.243 0.231 0.428 0.392 0.360

0.0196 0.0218 0.0330 0.0100 0.0111 0.0240 0.0047 0.0062

from $0.231 to $0.428. Although this variance is relatively high, its influence does not appreciably affect the position of the line, because the

L > 1.0 and the HI

application of the whole method is limited to the range-

horizontal discrepancy caused by the variance of b is negligible. This fact as well as the irregular scattering independent of the capillary height and probably caused by errors of measurement, support the hypothesis that the intersection of the line and the horizontal axis can be considered as a constant equal to the theoretical value blm = 0.371. For the calculation of the m parameter, the following empirical relation(5.3-37)

Substituting these values into Eq. (5.3-36), a relationship can be determined for the calculation of the capillary flow rate. The equation contains only a priori known independent variables [i.e. geometrical ( L ,H I ) and physical ( K ,hc) parameters] and therefore, it can be directly applied to design :

consequently

(5.3-38) 0.05

L Combining Eqs (5.3-32) and (5.3-38), a direct relationship can be determined between the relative length of the field and the height of the exit point, considering the capillary influence as well:


\

Fig. 6.3-17. Comparison of measured and calculated exit heights if the field is not influenced by tail water, and the capillary water transport is not negligible

655

656


Fig. 5.3-18 . Measured exit heights aa the functions of the relative length of the field in the case of seepage influenced by tail water

This equation is the modified form of Polubarinova-Kochina's hyperbola, supplemented with the egect of capillary water transport. Its reliability can be proved by comparing the measured and calculated values, which exhibit conformity in the range LIH, > 1.0 (Fig. 5.3-17). Analyzing the influence of capillary water transport on the exit height in seepage fields influenced by tail water, the measured data can be com-

5.3 Horizontal unconfined steady seep8ge

c

i

4

k

8

in

P

14

657

L I H ~10

pared with the curves calculated from Eq.(5.3-39)(Fig. 5.3-18). As already mentioned, the egect of capillarity is not so prominent if a part of the exit face is covered by water, and, therefore, the establishment of a separate method for the characterization of this condition is unnecessary. Instead of further investigation, the division of the total height of the exit point into t w o parts is proposed: (a) The height calculated from Eq.(5.3-39) neglecting the effect of the tail water; (b) The surplus height caused by the presence of the tail water. The second part (5.340)

Hl 42

Hl

6 Movement equatione dmoribing eeepage

Fig. 6.3-19. Empirical approximation of A -Ha values HI

659


H l and 2 H by using an empirical forcan be approximated aa a function of -

L L mula determined from measured data (Fig. 5.3-19). It is quite evident that H$Hl has to tend to unity, when H l / L -tends to H,IL. To unify the data i t H

is reasonable, therefore, to relate the d3-

H,

value to the difference between

values belonging to the Hl/L = H,/L variable,

Hl

and express this ratio as the function of the difference of

1% $1 -

. After

representing the calculated points in the form of graphs, it was found that - as paramthe points queue along curves characterized by different values eters, while if the independent variable is divided by

r2,

the points

indicate only one curve with a negligible scattering. The jinal form of the empirical formula determined in this way is aa follows:

y =a

+ exp[(2.7 + b ) x ] ;

where

Y=

2-[2Io

., x =

-[(2IolH,-

L

and ’

H l- -H ,

L

L .

H.

L

a = f0.03; b = 50.4.

(5.342)

This relationship can be controlled theoretically by comparing the curves calculated from Eq. (5.3-42) with Polubarinova-Kochina’s graphs (see Fig. 5.3-5), or with data determined by using the rating curves in Fig. 5.3-8. The capillary influence has to be neglected in this comparison, because it is not considered by Polubarinova-Kochina. Hence, this comparison serves only to show the reliability of the determination of the second separated part of the exit height. Figure 5.3-20 shows the final result of the comparison, indicating that the empirical formula agrees not only with the measured data but with the theoretical methods as well. It can also be seen in the figure that the mean 42*

660


of the strip covered by the measured points does not describe the relationship best fitted to the theoretical results, but a curve running a little bit higher is more suitable for general characterization. The final form of the equation proposed to calculate the exit height influenced by both tail water and capillarity is, therefore, as follows: 2.5 L

L

wHb ihe pmumed "...... 1 . Lcalculated . ewation

0

+

L

values ra/cufated fbeorefim,@ using Pufubarihow -KOfcbha's neibod, and by conformal mapping

Fiy. 6.3-20. Comparison of empirical and theoretical values of the exit height,assuming the seepage field is influenced by tail water

661


5.3.3 Characterization of horizontal unconfined seepage infI uenced by accretion

As already explained in connection with the kinematic classification of seepage, a dynamic equilibrium can develop in a semi-infinite unconfined field only, if the water crossing the starting section of the field (recharging the field through the entry face, or drained by the surface water at the exit face) is balanced by the accretion of the ground water along the flow space (Section 4.1.3). This type of seepage field (generally termed a leaking aquifer) is the characteristic case, when the width of the influenced zone has a physical explanation: i.e. there is a limit to which the influence of the surface-water (either recharge or drainage) may extend to achieve a balanced state between the horizontal flow and the sum of the vertical accretion. The equations describing the relationships between the hydraulic parameters of horizontal unconfined seepage have to be further developed by supplementing them with the influence of accretion. Dupuit’s modified and extended formulae are suitable to calculate the time invariant flow rate as the function of the distance meamred from the starting section [q(z)] and to determine the width of the influenced zone (L).The flow equation baaed on Dupuit’s hypothesis remains unchanged [Eq. (5.3-l)]. I n this case, the equation of continuity states that the change of flow rate along an elementary horizontal length (dz)is equal to the accretion [ ~ ( z )prevailing ] within the same dx distance [see Eq. (4.2-20) and Figs 4.1-10 and 4.2-6)]:

dgo = E ( 2 ) ; dz

consequently X

Q(4 = Qo - J I 4 4 1 dx = A 4 4 = y o ( 4 4 4 ;

(5.3-44)

0

because in the cam of two-dimensional unconfined field the area of the cross section ( A ) is equal to the y o depth of the ground-water flow. The equation can also be applied to characterize serni-wnfined seepage. In this special case the area has to be substituted by the thickness of the pervious layer, which may be approximated with a constant value ( m ) ,or its change along the field m ( z ) , may also be considered: X

0 4 = 90 - J 1 “(4I dz = m ( 4 4 4 .

(5.3-45)

0

Knowing the numerical value of accretion as a function of the distance measured from the starting point [ ~ ( x ) ]the , hydraulic parameters can be determined. When solving Eqs (5.3-44) and (5.3-45) the boundary conditions to be considered are as follows: (a) In the case of recharging the ground water from the surface water: at the section x = 0 the depth yo = H I ;

the accretion ~ ( zis) negative;

662


at the end of the influenced zone ( x = L ) the depth is equal to the original ground water depth yo = H,; (b) Investigating the drainage of ground water by surface water: yo = H,, at the point x = 0, assuming that the influence of the free exit face is negligible; E ( Z ) is positive; and the original depth prevailing at the end of the influenced zone is yo = H I if x = L. The width of the influenced zone ( L ) can be determined from the equations knowing that the total interaction between the surface water and the seepage field is balanced by accretion within this distance, and, therefore, the horizontal flow rate is equal to zero. It is necessary to analyze the practical determination of the numerical value of accretion. As already discussed in Section 4.1.3,the accretion is generally proportional to the vertical change in the position of the water table (or the piezometric surface). The earlier proposals, giving accretion aa a constant value depending for example only on the yearly precipitation and the physical soil parameters of the covering layer, as in Kriiger’s table (NBmeth 1942),cannot be accepted. It would be logical to determine the vertical @ e& ? as the function of the depth of water table below the surface, or m r e precisely as that of the change in this depth [ ~ ( d y ) ] Using . this concept the numerical value of accretion can be determined by the application of the charactmistic curve of the ground-water balance (see Fig. 4.1-ll), the interpretation and construction of which is discussed in Section 1.4.4.The basic differential equation can easily be established in this case as well:

d2y dx2

yo-+

dY0

1 K

- +--(dy)=O;

(dx)

(5.3-46)

but solving of this relationship raises considerable problems even in caaes, ) when the most simple functions are used to approximate the ~ ( d yrelationship. LBczfalvy (1958)solved Eq.(5.3-46) by using linear relationship, and Kovhcs (1962)by applying an exponential function, but their results cannot be proposed for practical use, because they are very complicated, and their use would not be in keeping with the low accuracy of the numerical factors in the equations. Because of the difficulties emerging in connection with the integration of Eq. (5.3-46),it is generally accepted that the accretion be approximated by a function describing its horizontal distribution. This function should be a steadily decreasing one with its maximum at the starting point ( E ~ ) and , it should have zero value at the limit of the influenced zone (if z = L; E = 0). In the first publication, in which the vertical effects were described in this way (JuhQsz, 1953),a linear relationship and/or a parabola of second order were used. A broader generalization can be achieved, however, by applying a parabola of the n-th order (Kovhcs, 196313):

I z)”

e=&E01--.

(5.3-47)


663

Substituting this relationship into Eq. (5.3-44), the latter can be integrated in closed form. The solution is shown in Fig. 5.3-21 in the investigation of a horizontally drained seepage field. The parameters to be determined are aa follows: - L the width of the influenced zone; - go the drained specific discharge; - q(s) the steady flow rate changing along the seepage field; - y o ( z )the curve describing the position of the water table or the piezo-

metric surface. The solution is given separately for semi-conhed and unconfined fields. In the m e of a semi-confined system:

::Y

L=

-(n+1)(n+2)(H,-H2);

(5.3-48)

unconfined system Fig. 6.3-21. The development of the water table (or the piezometric surface) of a horizontslly drained ground water, considering the iduenoe of accretion

664


For an unconfined system (assuming that the development of the free exit face can be neglected, thus H , = H a ) :

The n parameter of these equations has to be determined considering the local conditions. The same relationships can also be applied for the charac-

unconfined system Fig. 5.3-22. The development of the water table (or the piezometric surface) of a horizontally recharged ground water considering the influence of accretion


665

terization of a horizontally recharged flow field. It waa found, however, that in this case a more rapidly decreasing horizontal distribution better approximates the actual conditions. A proposal was made, therefore, to use another type of E ( Z ) function instead of Eq. (5.3-47) (Fig. 5.3-22) (KovBcs, 1963a): E(Z)=-E0

[

1-

where

li-ia] 1-

1--

=-EO[l-l/l-Z"]; (5.3-50)

Using this approximation the integral can be solved only after expanding Eq. (5.3-50) into series. Applying the following symbols:

y(n;z ) = [I

-

v l - z"] ;

1 a(n;z ) = 1B(n;z ) dz ;

B(n;z ) = y(n;z ) dz ;

t ( n ) = y(n;z = 1 ) = 1 ;

o(n)= B(n;z

= 1) ;

(5.3-51)

e(n)= a(n;z = 1) ;

the results of the integration can be summarized in the form of the following equations: For a semi-confined system:

(5.3-52)

For a n unconfined system:

(5.3-53)


666

As a numerical example, the parameters summarized in Eq. (5.3-51) are given in Table 5.3-2 where n = 12 in Eq. (5.3-50) (which was found to be reliable to characterize the vertical surface effect, i.e. the increase of evapotranspiration aa a consequence of the rising water table). It can be seen from the table of values, that at a distance longer than one third of the length of the influenced zone from the entry point both the change in the position of the water table and the accretion caused by this change, can be practically neglected, if the indicated horizontal distribution of the vertical drainage is used. Table 6.3-2. Numerical values of the parameters listed in Eq. (6.3-51) using n = 12 power to the Eq. (6.3-10) e (12) = 3 . 0 2 ~ (I (12) = 4.76 x lo-' T (12) = 1.00 z -

a(e; 12)

L

e (12)

0.00 0.01 0.02 0.06 0.10 0.20 0.30 1.00

1.00

0.99 0.98 0.96 0.90 0.80 0.70 0.00

1.ooo 0.868 0.733 0.464 0.213 0.041 0.007 0.000

B 0;12) 0

Y(G 1%

(12)

1.000 0.839 0.714 0.462 0.214 0.046 0.008 0.000

1.000 0.668 0.632 0.322 0.163 0.036 0.007 0.000

When investigating the development of the water table and ground-water flow between two canals, the ratio of the distance between the canals to the depth of the flow field is generally greater than 10, mentioned previously as the limit, above which the non horizontal character of seepage around the recharging and draining structure can generally be neglected, especially if the influence of the curved stream lines on the flow rate is investigated (see Fig. 5.2-8). The same condition (the relatively large length of the field) decreases the horizontally transported amount of water and increases the total accretion along the field if these two values are compared to one another. For this reaaon, accretion has to be considered in most cases, in the investigation of the development of ground-water flow between two canals. If seepage is influenced by accretion, the interaction of the two canals within the enclosed seepage field cannot be characterized by the superposition of the separate influences of the canals, because the accretion is itself the function of the new position of the water table. The original differential equation [Eq. (5.3-44)] has to be solved considering simultaneously the boundary conditions at both ends of the flow field as well aa the horizontal distribution of accretion along the field (Kovhs, 1963c, 1964a). If the interaction of a recharging and draining canal is investigated, the new water table crosses the horizontal surface of the ground water, which develops under static condition (Fig. 5.3-23). The dynamic equilibrium can be


r

667

lb

Fig. 5.3-23. Interaction of recharging and draining canals if the seepage field is influenced by accretion

characterized by considering that the inflowing recharge is equal to the sum of the negative accretion accumulated from the starting section until the cross section, where the new water table crosses the original horizontal ground-water surface, and the horizontal flow rate through this vertical section. The same relationship can be determined for the other part of the field, namely, that the amount discharged is equal to the sum of the integrated positive accretion and the horizontal flow rate at the section, which divides the two parts of the field having been influenced by opposite accretions: and

(5.3-54)

where the definition of a(n)is similar to that in Eq. (5.3-51). Its numerical value depends on the approximation of the horizontal distribution of accretion and may be different in the two parts of Eq. (5.3-54). Taking this condition into account and accepting a given e(%) function the differential equation can be solved and the position of the influenced water table determined. Between two canals having the 5ame character, (two recharging canals or two drains) two different conditions can be distinguished. I n the first case the role of the canals does not change in the combined system (Fig. 5.3-24). In the other case the action of one of the two canals is modified under the influence of the other canal (Fig. 5.3-25). If i t wasoriginally a drain, it will recharge the ground-water space between the two canals and, inversely, the original recharging canal will drain it. This case occurs when the water level in one of the canals is relatively close to the original water table. The character of the canals remains unchanged, as indicated in Fig. 5.3-24, if the accretion between the canals can balance the action of the two canals.

668

5 Movement equations describing seepmge

Fig. 5.3-24. The development of the water table between two canals having the same character, Considering in addition the influence of accretion

This condition is fulfilled if the water table, calculated by considering the single effect of one of the canals, intersects the vertical section of the other canal, between the water level of the latter and the original horizontal water table. Calculating the balance of the flow rates, it is necessary to take into account the fact that accretion is not zero at that point of the seepage field, where there is no horizontal flow, because the position of the water table may here be changed Q1= + d n ) A&]; and (5.3-55)

+

Y2

=

&I&,+)

+ 4.

When the previous condition is not fulfilled (when the water level in one of the canuls is between the original water table and the water surface calculated by assuming that the seepage space is influenced only b y the other canal) the role of the canal (the level of which is a greater distance from the original water table) remains unchanged and its effect is balanced partly by accretion and partly by the other canal. Following the sketch given in Fig. 5.3-25, the balance of the field in this case can be expressed by the following equations: r/1= l e 1 4 4

and

+ q2;

(5.3-56 )

where the interpretation of the p(z, n ) function is similar to that in Eq. (5.3-51). Naturally the action of the second canal (the water level of which lies near the original ground-water surface) changes only in the direction of the common flow field, while outside i t maintains its original character.

669

6.3 Horhontal unconfined steady seepage

wafer iable, which would devehpunder fhe influence ofwnal f. only

----

h

I

-

A

I

e

I-

-

_

1

,

.

,

_

,

__

L

~

,

l

.I

I

__

-

.

Fig. 5.3--25. The development of the water table between two similar canals if the role of one canal is modified within the common field

Equations (5.3-54), (5.3-55) and (5.3-56) do not give the final explicit form of the mathematical models; they are [together with Eq. (5.3-44)] only the starting points for the derivation of the models. Many different formulae can be determined in this way according to the horizontal distribution of accretion considered in the investigation. The correct approximation depends, however, on the local conditions of the leaking aquifers. It is advisable, therefore, t o develop the models by always taking into account the most probable local influences.

670


5.3.4 Local resistance occurring in the vicinity of the entry and exit faces An important hypothesis for the derivation of Dupuit's equations is the assumption that the lower boundary o€ the seepage field is horizontal and the vertical entry and exit faces cross the pervious layer at full depth. If the latter condition is not satisfied (the closing faces are not vertical or they do not penetrate until the lower impervious boundary) the seepage cannot be approximated as a horizontal flow because high resistance may occur within the curved zone of the flownet, consuming a considerable part of the total available energy. This decreases the amount of water transported through the field and influences the position of the developing water table. Investigating the flow between two surface-water bodies in Section 5.2.1, it was found, that the position and the depth of the closing faces do not influence the water transport of the field, if the ratio of length and depth of the latter is greater than 8 to 10 ( W m > 8-10 see Fig. 5.2-8). A similar result can be achieved by comparing the yield of fully and partially penetrating wells. The equation for the transformation of the vertical flow plane of a partially penetrating well was derived in Section 5.2.1 [see Eqs from (5.3-18) to (5.3-21), and Fig. 5.2-61 (Kov&ca,1966). Using those relationships, the yield of a well, penetrating to a depth d below the upper boundary into the aquifer having a thickness m and recharged horizontally around the well at a distance R , can be calculated and compared to other values determined by using more simple approximations. Such approximations may be the substitution of the actual aquifer by a pervious layer having an impervious boundary at the bottom of the well, or to neglect the partial penetration of the well and to calculate the yield by supposing that the well is screened along the whole depth of the aquifer (Fig. 5.3-26). The yields to be compared can be calculated from the following equations: (a)By using the mapping functions, the result of which can be considered in the form of a modified thickness of the aquifer (Eq. 5.2-21):

R

&=2nKAH

Rn

13

1

R

=KAHR--

ch - - 8 lnarch ro 1-8

nm

m 2n

R

R

("n3

ch - - - 6

ln-

arch

where

1--6

(5.3-57)

(b) Assuming fully penetrating wells:

m m 2n =K A H R - - . R R R In In -

(&)DL = 2n K A H --

(5.3-58)


671

Fig. 6.3-26. Comparison of the yields of partially penetrating wells determined by various approximations

(c) Finally, if the partially penetrating character is considered in the form of an aquifer closed by an horizontal lower boundary at the level of the bottom of the well d d 2n (&)02= 272

KdH -= K A H R - - . R R In TO

R

(5.3-59)

In TO

The figure clearly indicates that the second approximation (the limitation of the aquifer at the lower level of the well) is not acceptable,if the rate of pen-

d etration - < 0.8 while in the other cme its application is unnecessary, m because the first approximation provides acceptable results. Thus the use of the (Q)mvalue can be a priori excluded.

672


Another interesting result is, that in the case of axial symmetrical flow, the

R depth of the field has to be compared to In- and not to the length of the field. r0 If the same comparison were executed in connection with partially penetrating drains recharged from both directions, the result would be the same aa that achieved in Section 5.2.1 [i.e. if Llm > 8 10, the curvature of the

-

impervious layer Fig. 6.3-27. Flow nets in the vicinity of recharging canals

flownet is negligible if the flow rate is calculated]. I n the caae of axial symmetrical flow

R

In --r 0 > 8 - 1 0 m 2n

(5.3-60)

is the condition indicating the limit, above which the local resistance around the well is negligible compared to the total resistivity of the field. This result shows that the yield is more influenced by the rate of penetration in

R

the case of axial symmetrical flow if the - ratio is smaller than 550. r0 If the ratio of the length and depth of the seepage field does not allow the simple, direct application of Dupuit’s equations, or the task is the more accurate determination of the water table (or the piezometric surface), which requires more detailed investigations, a frequently applied method is to divide the field into stretches. Then within the main part, the flow may be apprixomated as horizontal seepage, while the local resistances characterizing the remaining stretches can be determined separately using other suitable formulae. One good example to demonstrate the application of this method is the investigation of the seepage i n the vicinity of recharging canals (Kovhcs, 1964b). The flow nets determined in sand box models (Fig. 5.3-27) clearly shows that the supposition of vertical potential lines is acceptable in almost the whole field except close t o the canal. To determine the hydraulic parameters of the seepage the position of the dividing section has first t o be fixed (Fig. 5.3-28). The first stretch having a length of L, between the axis of the canal and the dividing section has to be characterized by local resistances, while the second stretch ( L J extends t o

673


second sfrpcb of seepage Fig. 5.3-28. Interpretation of the separation of the various stretches of the flow field

the draining structure, which is now supposed to be a fully penetrating canal with vertical faces, to simplify the lower boundary condition. The aeepage in the secondstretch can bedescribed by Dupuit’s equation (using either the most simple form or that which considers accretion, depending on the upper horizontal boundary condition of the field). The dividing section, substituting the entry face of the second stretch, is called, therefore, the starting section. As already emphasized, in every cme the total field has to be investigated from the actual entry face to the exit face, where the water leaves the porous medium. This requirement determines two conditions, which have to be considered in the investigation. These are: the sum of the head losses along the

the various stretches

[5Aht) has to be equal to the total available pressure head i- 1

( A H ) ,and the continuity has to be expressed in the form of equal rates i n each stretch (ql = q2 = . . . = q., - . . . = qn), if there is neither recharge nor drainage along the field. In the field investigated here which have only two stretches, these conditions are expressed by the following equations: (5.3-61)

Considering that the width of the wetted zone below a recharging canal is a well determined parameter, if the seepage is in a free steady state, i t can be proposed, that the starting section should be identical with the vertical asymptote of the wetted field. Thus the length of the first stretch should be equal to the virtual width of the canal [see Eq. (5.2-35)]: (5.3-62) 43

674


In this way the position of the starting section also depends on the capillarity of the material composing the field. It is evident, however, from S Fig. 5.2-14, that the influence of capillarity is negligible, if - > 25. In this hC case the length of the first stretch may be independent of the physical soil parameters of the field S

L , = S ; if - > 2 5 . hC

(5.3-63)

It will be demonstrated in connection with the analysis of the local resistance in the vicinity of the canal, that Ah, decreases at first if the length of the first stretch (L,)related to its depth (H) increases. The Ah, = f LI

l.Ul ,

4 =4 value, and relationship has a local minimum somewhere around the H 3 L, parameter indicating, that a increases afterwards with the increasing H

considerable length is also included in the first stretch, where the flow could be characterized by horizontal flow. It has to be noted that below large beds if 35 > 4H and, therefore, - >- the flow net becomes horizontal within H L1

4 31

the part of the field covered by the bed. Therefore, the starting section can be located at the edge of the water surface (or that of the virtual width if capillarity is not negligible). In this case the water table joins the level of the surface water and thus the head loss within the first stretch is negligible:

L; = S d 2 ; and Ah, = 0 if 35,

> 4H;

S and -hC

< 25 ;

or

(5.3-64)

Li = 512; and Ah, = 0; if

35

S > 4H; and >2 5 .

hC The four possible variations in the determination of the starting section are represented in Fig. 5.3-29. It is quite evident that the total pressure head is consumed along the stretch characterized by horizontal flow if 3 5 > 4H (or 35, > 4 H ) , and, therefore, Dupuit’s equation can be directly applied using the vertical section at the edge of the surface water as the starting section, without studying the local resistances. In the other cme, the remaining problem is to determine the head loss between the perimeter of the canal and the starting section. Acceptable approximation is obtained in the investigation of the field lying along the contour, which is symmetrical about the axis of the canal and bounded by the impervious lower boundary, the starting section and the


(5)

(b)

starting section

675

starting section

/

L’-

=I? V $1 -4 & 313 V

WlCu

B

K =s

imprviaus ager k~ -4

impervioos ayer

$. -.J

~ ~ 1 - s ~

Fig. 5.3--29. Possible variations in the determination of the starting section

water table. By comparing the stream lines of this field to those developing in the surrounding of a rectangular internal corner of a large continuous field the similarity is obvious (Fig. 5.3-30). The mapping function transforming this latter flow net into two orthogonal sets of stra.ight lines can be used, therefore, to calculate the local resistance near the recharging canal:

w =u

+ iv =

22

= (x

+ iyy;

therefore u = 5 2 - y2;

(5.3-65)

and 2,

contour of the canal

impervious layer

= 2xy.

t-

border ofthe rectangular mrner

(b) Fig. 6.3--30. Comparison of stream lines in the vicinity of a recharging canal and in the surrounding of a rectangular internal corner of a continuous field 43*

676


Applying this mapping method and approximating the images of some uncertain contours with straight lines, a field is achieved, the cross-sectional area ( A )of which is a linear function u of the real ordinate of the w plane (Fig. 5.3-31). Accepting this relatively very rough approximation the specific flow rate can be expressed depending on the change of pressure head along contour o f

Fig. 6.3-31.

the u axis, and after integrating the elementary pressure head according to the u variable between two vertical sections. Thus, relationship is achieved between the local head 108s and the flow rate: dh du

q= KA(u)-;

Ah,

=

(5.3-66)

7

dh -du= du

du. -[H'-&)']

-[H'-er]

Substituting the linear relationship previously mentioned between A(u) and u the result of the integration is: 3

H'+7LI

Ah1 =

s

1 Q 1 du =--[ln(b-au')]o K(b - au') K a

Q

0

where

.'=.+H2-($)

2;

and 2 -Ah1 -1

Ll A(u') = L1H - -

H

3

Ha+-Lf

u' = b - au';

;

677


consequently 2--1 Ah1

[yqq

H LH H In2 1---, i ) = K H L , 4 H

Ah,

[

3+

.

(5.3-6 7)

The numerical determination of the three dimensionless quantities

r+

;

among which only Ll is an a priori known geometrical parameter,

H

H ’ KH

has to b e executed by using successive iteration together with the characterization of the second stretch to ensure that Ah, and q satisfy the conditions given in Eq. (5.3-61). To demonstrate that the results are acceptable in practice, although very rough approximations are applied, the calculated local resistance values are compared in Fig. 5.3-32 to those of the water tables observed in the three sections of a recharging canal. The conclusions drawn from the figure are aa follows:

I

0 -

---__-.

\\

/I.

1 3

@ @

.

.-.

‘Cy

I

s

Go

I

/-;4 /

f!ucfuaflon of the water !eve! in the investigated strefch

R:

intersectionof the starting section and tbe theoretical water hble calculated by considering the O l C31 losses

8 8 ,observation well

SI

@ QR

static water fable before fiM?Jg the canal Fig. 5.3-32. Comparison of calculated local resistances of natural water tables observed in the vicinity of a recharging canal

678


(a)There are many cases, when the local resistances cannot be neglected, as they considerably influence both the water transport and the position of the water table; (b) Within the stretch characterized by almost horizontal seepage the fiuctuation is very similar to that of the natural water table, due to the seasonal fluctuation of accretion (infiltration and evapotranspiration), but in the vicinity of the canal, where local resistances occur, these eflects can be neglected because the length of this stretch is very short, and thus the influence of accretion is not considerable; (c) Although the accuracy of the method described here is not high, it can be accepted in practice to give rough information on the development of flow conditions near the recharging canals.

Another example frequently used to show the application of the separation of the various parts of the seepage field to simplify the hydraulic characterization of the flow, is the seepage under horizontal impervious foundations (Fig. 5.3-33). The mapping method suitable for mathematically correct determination of the hydraulic parameters of this seepage field was derived by Pavlovsky (1922) and discussed in Section 5.1.4. The results, on the basis of which the specific flow rate and the pressure distribution along the horizontal foundation can be calculated were summarized in Eqs (5.1-66)and

2.5

5.0

z5

a m

MD

Fig. 5.3-33. Comparison of the various approximations proposed for the calculation of the specific flow rate under horizontal foundation


679

(5.1-67). In Section 5.2.1 an example of the practical application of this method was also shown, because the flow field represented in Fig. 5.2-2 is half of that analyzed below, and considering the symmetry of the field the equations applied for the characterization of confined seepage between the horizontal entry and vertical exit face can also be applied directly here. Only the geometrical parameters have to be considered precisely. Thus, the original relationship for the calculation of the specific flow rate composed of the total and supplementary total elliptical integrals can be substituted in practice, by a more simple formula (Kovbcs, 1960) [see also Eq. (5.2-2)]: q=-

KAH 32

m

arsh 1.5 -. b

(5.3-68)

Before the derivation of Pavlovsky’s method it was generally accepted that the actual water transporting layer could be substituted by an elliptical seepage field, the depth of which is equal to that of the pervious layer and its length determined so that the edges of the foundation should be the foci of the ellipse. The equation for the calculation of the water conveying capacity of this field waa also previously derived as well [see Eqs (5.1-30) and (5.1-33)]. Using the geometrical parameters of the actual seepage field the specific flow rate is in this case approximated by

KAH m qL-4 = -arsh b

32

.

(5.3-69)

Finally, the most simple approximation is the supposition that the flow is horizontal below the foundation between two vertical faces: m qo = K A H - .

2b

(5.3-70)

The comparison represented in Fig. 5.3-33 shows that the use of Eq. (5.3-69) is unreasonable because i t causes a regular error generally higher than 10 yo and its structure is no simpler than that of Eq. (5.3-68). As in the previous results, the difference in the flow rate between the mathematically correct calculation and the simple use of Dupuit’s approximation, is not higher than 5 yoif the ratio of the length anddepth of the field is higher than 8-10

if-> 1 . 8-10

In this special caae even the pressure distribution can be approximated by simple relationships. In Fig. 5.3-34 the results of the various methods used for the determination of the pressure distribution below the horizontal foundation are compared. Dupuit’s approximation results in a linear relationship, while the distribution can be expressed by a trigonometric function, if an elliptical (or infinite) seepage field is assumed [see Eqs (5.1-32)

680


Fig. 5.3-34. Comparison of the various approximations of the pressure distribution below horizontal foundation

and (5.1-34)]. These two curves envelop the set of distribution curves bem

longing to various -ratios and are Calculated from the theoretically correct b equation [Eq. (5.1-67)]. The discrepancy, however, between the accurate and approximative values, is relatively small. It can be proposed, therefore, m 1 that the linear relationship is applicable if - < - ;while Eq. (5.1-34)gives b 2

;1 3 .

a better approximation in the opposite case - >-

These aspects were

also considered when the confined seepage from surface water having a horizontal bed to a fully penetrating drain was analyzed [see Eq. (5.2-2)]. The summary of the theoretical and practical investigation of seepage under a horizontal foundation proves that the practical methods bawd on the theoretically correct conformal mapping provide the designers with relationships suitable for the accurate determination of the hydraulic parameters. Nevertheless, i t is worthwhile to analyze the previously generally applied methods, to attempt to approximate the correct parameters by dividing the field into separate stretches. Using these assumptions is still the easiest way to calculate the /low rate and prassure distribution i n special layered fild8. As mentioned in the previous discussion (Fig. 5.3-33), the difference between the correct flow rate and that calculated by the simple use of Dupuit’s 2b equation is larger than 5% if - < 8-10. To eliminate this error i t was m proposed earlier that the field should be divided into three stretches, assuming a vertical stretch to have the same cross-sectional area as the horizontal section before and after the horizontal foundation. Its length being equal to four tenths (Dachler, 1936)or half (Kamensky, 1943)of the thickness of


o

r

2

3

4

5

1

681

_ rn

Fig. 6.3-35. Comparison of the flow rates calculated by using Dechler’sand Kemensky‘s approximation to the theoretically correct value

the pervious layer (Fig. 5.3-35). Considering these hypotheses the flow rate can be calculated from the following equations q1= K A H

m

2b

+ 0.8m (Dachler) ; (5.3-71 )

or q2 = K A H

rn (Kamensky) . 2b + m

The comparison of the results of these equations with the theoretically correct flow rate (calculated from Eq.5.3-68) proves the accuracy of these methods (especially that of Dachler’s approximation). Although in the case of a homogeneous field there are similarly simple and theoretically better baaed models, the basic idea of this approximation can be usefully applied, if the field i s covered by a semi-pervious layer. In the case of layered fields, where they are composed of a series of semipervious lamellm covering the aquifer (which can also be built up from several layers differing slightly in permeability) the whole field has first to be substituted with a double layered system, in which a covering layer of lower permeability lies above the water transporting formation I n Fig. 5.3-36 the first sketch shows the original system and the second one represents the

682


(b) ~

X

Q

$1 4-

eomin.o............. . .lauer . . . .I:.I .I I. ..................... ................ ............ ...........:.. .. :::........................ ..... .

.

parameters characterizing the members of tbe covering forma fion :t,I K,, i t2I h2 .,,.. ; tn, hn; parameters characterking the members ofthe pervious formation :n/I KO, j mz I Ko2 I ..... j mn ,KO,,

Fig. 5.3-36. Transformation of the layered water transporting system to an homogeneous field

result of the transformation. In the next step the field has to be homogenized by enlarging the thickness of the covering layer by multiplying the original size by the ratio of the average permeabilities of the two layers (Fig. 5.3-36c):

t , = t - K" . (5.3-72) Kt Since the covering layer, having relatively high resistance, is substituted by a long vertical stretch before and behind the horizontal foundation, the vertical flow within the water transporting layer can be neglected, compared to the resistivity of the upper lying layer. Thus, the equation proposed for the calculation of the specific flow mte is aa follows (Kamensky, 1943; Galli 1959): (a) If the covering layer is different before ( t l ;K , ) and behind (t2; K,) the horizontal foundation:


683

AHmK,

(b) Having thesamecovering layer ( t ;K,) on both sides of the foundation:

AHmK,

!7=

2b

+2

vk

(5.3-73)

K,p

The same aspects applied in subsequent steps can be followed if the water transporting field below the foundation is composed of more pervious and semi-pervious layers (Shea and Whisett, 1958). In this case horizontal flow is always assumed in the pervious layers and vertical flow is assumed in the semipervious ones. There is one rough approximation in Dachler’s method: i.e. the supposition, that the active width of the vertical flow is equal to the thickness of the horizontal water transporting layer. This hypothesis may cause considerable error in the case of a layered field. To eliminate this source of error Juh&sz (1968) has developed a method combining the basic aspects of Dachler’s derivation (separation of vertical and horizontal stretches of flow) with the use of finite elements (Fig. 5.3-37). Let us first consider the water conveyance through a horizontal stripe of thickness dz, at a depth z below the lower boundary of the covering layer. The vertical stretches joining the stripe have a thickness of a dz,where the a

separated sfreches of the seepage field

:

(1) vertical tbrough tbe covering layer before the dam ( 2 ) vertical in tbe pervious layer before the dam

(3) borizontal in the pervious layer ( 4 ) vertical in tbe pervious layer bebind the &m (5) vertical tbmugb tbe cwering layer oebind the dam

Fig. 5.3-37. Application of Dachler’s method for the cheracterization of finite stripes of the seepage field

684


factor is provisionally unknown. This water transporting element of the field can be divided into five stretches: (a) Vertical jlow through the covering layer before the foundation area A = a,dz; length 1 = t,; hydraulic conductivity K,; (b) Vertical flow in the water transporting layer before the foundation

A = a&; 1 = z; K O ; (c) Horizontal flow

A = dz; 1 = I,

+ a,z + a,z;

KO;

(d) Vertical flow in the water transporting layer behind the foundation

A = a,&; 1 = z; KO; (e) Vertical flow through the covering layer behind the foundation

A = a&; 1 = t,; K,. Considering continuity and the requirement that the sum of the head losses along a stream line should be equal to the total available pressure head [see Eq. (5.3-61)] a relationship can be determined between the dq and dz elementary variables:

where

B = lo+ K O

[- tl

+

a1K1

1-

t2 ; a2K2

and 1

C =a1

1 + -+

a,

a2

+ a,.

(5.3-74)

By integrating this relationship between the z = 0 and z = m limits, the specific jlow rate can be determined as the function of geometrical and physical soil parameters, but i t also depends on the undetermined a, and a, values, the effects of which are included in the B and C factors: rn

--JB+Gz=Cln 9 dz AHK, 0

1

B+Cm B

(5.3-75)

685


If the thickness and the material of the covering layer on the two sides of the foundation do not differ, the a1 value is equal to a, and the two factors can be simplified if

t , = t, = t ; K , = K , = K , ; B , = 1, 2-,t K O . aK+

a, = a2= a;

+

(5.3-76)

and

The a value has to be determined by trial and error method considering the principle of “lex minimi”, by which the highest possible water transport has to develop as the result of a given energy content. Thus the -pstrameter AHK,

‘ [

has to be calculated as a function of a, and the highest value aa well as the a parameter belonging to -

can be accepted aa the final result. If the

cA:K.,m.d

similarity of the covering layers in front of and behind the foundation is an acceptable assumption [see Eq. (5.3-76)] this trial and error method only re-

‘

quires the calculation of one -vs. a curve to determine its local maxi-

AHK, mum. If different t , and t2,K , and K2,as well as a, and

a2 values

‘

have to be

considered, u, and a2 combination which gives the highest -value, has

AHK,

to be used. Even the anisotropy of the water transporting layer can be taken into account in this way. If the horizontal hydraulic conductivity is indicated by the K O rr

symbol ( K H = KO), and ;z = A-isH

K”

the coefficient of anisotropy, the hydraulic

conductivity of the vertical sketches within the water transporting layer haa to be considered by using

K K v = 2 parameter. Thus, the C factor in I

Eqs (5.3-74) and (5.3-75) has to be modified: (5.3-77)

or the simplified form, if the same layer covers both sides of the aquifer (5.3-78)

Suitable a values have to be determined by a trial and error method in this case as well, as previously explained.

686

5 Movement equat,ions describing seepage

References to Chapter 5.3 CHARNYI, J. A. (1961): The Proof of the Correctness of Du uit's Formula in the Case of Unconfined Seepage (in Russian). Dokl. Akad. Nauk. &SR No. 6. CHENQ and Y A N O - SSHA ~ (1969): An Experimental CHINO-TON KuO, M O W - S O ~ O Study on the Boundary Conditions for the Flow t h o u g h Porous Media. 13th Congress of I A H R , Kyoto, 1969. DACHLER,R. (1936): Ground-water Flow (in German). Springer, Wien. DUPUIT, J. (1863): Theoretical and Practical Studies on Water Movement in Open Channels and through Permeable Layers. (2nd edition) (in French). Dunod, Paris. G m , L. (1969): Approximative Method to CalculatetheParameters of See age through a Layered Field below Hydraulic Structures (in Hungarian). Viziigyi &zlem h y e k , No. 3. HANTUSH, M. S. (1964): Hydraulics of wells. (From Advances in Hydrosciences edited by V. T. Chow). Academic Press, New York. J ~ s z J. , (1963): Data about the Ground Water in Plains Especially Regarding Aspects of the Backwater Effects Caused by Barrages (in Hungarian). V'iziigyi Kozlemdnyek, No. 2. JWASZ, J. (1968): Investigation of Seepage Developing under Dikes in the Case of a Thick Water Transporting Layer (in Hungarian). HidroMgiai Kozlony. No. 8. KAMENSKY, G. N. (1943): Bases of the Dynamics of Ground Water (in Russian). Gosgeoltekhizdat, Moscow. KOVLCS,G. (1960): Calculation of the Flow Rate of Seepage under Dams (in Hungarian). Viziigyi Kozlemdnyek, 2 . KOVLCS,G. (1962): Design of Drains Discharging the Ground Water (in Hungarian). $pit&- ks Kozlekedkstudomanyi Kozlemknyek, No. 2. KOVACS,G. (1963a): Characterization of the Steady Influenced State of Seepage from Recharging Irrigation Canals (in Hungarian). Hidroldgiai Kozlony, No. 1. KOVACS,G. (196313): Practical Method for Hydraulic Design of Drains Discharging the Ground Water (in Hungarian). &pitks- 6s Kozlekeddstudomanyi Kozlemknyek. NO. 1-2. KovAcs, G. (1963~): The Development of the Water Table in the Vicinity of Canals with Nearly Constant Water Level. 5th Congress of ICID, Tokio, 1963, Vol. 17. KOVACS,G. (1964a): The Development of Ground Water under the Influence of More Canals (in Hungarian). HidroMgiai Kodony, No. 4. KOVLCS,G. (1964b): Local Seepage Resistances in the Vicinity of Canals Recharging and Draining the Ground Water (in Hungarian). Hidroldgiai Kozlony, 10. KOVACS,G. (1966): Yield of Partially Penetrating Wells. Symposium on Seepage and Well Hydraulics, Budapest. KOVACS,G. (1973): Characterization of Steady Seepage through Homogeneous Earth Dams With Vertical Faces. V I T U K I Publications in Foreign Languages. Budapest, 6. LkOma-, S. (1968): Determination of the Yield of Drains (in Hungarian). HCdroldgiai Ko.z.?ony, 1. -ONE, U.and FRANZETTI, S. (1969):Unconfhed Flow Downstream of an Homogeneous Earth Dam with Impervious Sheetpiles. 13th Congress of I A H R , Kyoto, 1969. MUSEAT,M. (1937): The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York. Nh?E!t%, E. (1942): Water Problems of Modern Agriculture (in Hungarian). Budapest. PAVLOVSKY, N. N. (1922): Theory of Ground-water Movement around Hydraulic Stmatures (in Russian). Leningrad. POLUB~LI~INOVA-KOCHINA, P. YA. (1962): Theory of Ground-water Movement (in Russian). Gostekhizdat, Mosaow. POLWARINOVA-KOCHINA,P. YA. (1962): Theory of Ground-water Movement. Princeton University Press, Princeton. S ~ AP., H. and WEISETT, H. E. (1968): Predicting Seepage under Dams on Multilayered Foundations. Proceedings of ASCE, Vol. 84. ! ~ I E M , A. (1870): The Yield of Artesian Boreholes, Shaft-wells and Filter-galleries (in German). Journal fur Gasbeleuchtung und Wasservermrgu~g,Vol. 14.

6.4 Horizontal unsteady seepage

687

Chapter 5.4 Investigation of horizontal unsteady seepage In the foregoing chapters, when the conditions of development of steady seepage and those of the application of Dupuit’s equations for the description of this state of flow were discussed, the known position of both the entry and exit faces was mentioned rn a basic requirement. Analyzing the kinematic classification of seepage, i t was emphasized that in a field having i n h i t e length, steady seepage can develop only in a confined water transporting layer, where the actions caused by the changes of the boundary conditions propagate with very high velocity. In such systems the time-variant movement can be approximated with the series of parameters calculated by supposing a steady state. If the field is unconfined or it is covered by semi-pervious materials, and thus the change of pressure modifies the stored amount of water the velocity of propagation is restricted by the velocity of water transport. Steady seepage can develop in these systems only if the horizontal flow is balanced by the accretion of the ground water. In other cases the actions created at the starting section of the field (i.e. the entry or the exit face, the position of which is known) propagate slowly until infinity, and this state of seepage can only be characterized by methods describing the unsteady flow. Analyzing the methods generally applied in engineering practice to solve seepage problems the errors most frequently occurring and the largest differences between the calculated and actual values, are caused by the fact that the equations derived by assuming steady state conditions were also applied to determine the hydraulic parameters of unsteady seepage. The main ambition of engineers always was and is to find simple, practical solutions. For this reason, Dupuit’s equations are generally applied to characterize even unsteady flow and the obstacles caused by the unknown position of one of the closing faces are overcome by using arbitrarily chosen limits of the influenced zone without any physical meaning or explanation. Thus, most of the practical problems are solved by approximating the movement as a steady flow, although the natural character of seepage in most cases is unsteady. It is necessary to develop a system of mathemtically easily applicable methods having a firm theoretical basis so that the equations composing this system ensure at the same time the uniform treatment of the largest part of the various types of unsteady seepage frequently arising in practice. Since any method considering at least the unsteady character of flow, may bridge the gap existing between theory and practice, the hypotheses applied to simplify the solution of the differential equations can be based on relatively rough approximations, because the errors caused in this way are probably smaller than those made by using steady state equations. Considering the above, the objective of this chapter is that after the derivation of the differential equations describing the horizontal (or nearly horizontal) unsteady seepage, their solution should be given in a unified form. The application of this method should be demonstrated by practical examples as well.

688


5.4.1 Derivation of Boussinesq‘s equations and the problems in connection with their linearization Let us first investigate the continuity of flow within an elementary prism having a nearly horizontal impervious lower boundary (Bear et al., 1968). In a coordinate system having x,y and z orthogonal axes the position of the lower boundary is described by the functions [ = [ (x,y). The height and form of the water table can be similarly characterized by Z = Z (x,y) function. The depth of the flow field (h) can be calculated a t each vertical line in this way (Fig. 5.4-1):

h=Z-c.

(5.4-1)

The change of the depth between two points is

az (dh), = U -d[ = -& ax or where

(dh), = - I , d s

az dy - -& ac +aY

ax

ac

- -dy ;

8Y

+ ix& + i y d y ;

- IYdy

(5.4-2)

Fig. 6.4-1. Interpretation of the symbols used to derive Bouseinesq’s equations


689

indicate the slopes of the two surfaces in the x and y direction respectively. The flow being unsteady, the depth changes not only according to location, but i n time as well. This change is equal to the modification of the position of the water table in time, because the height of the lower impervious boundary is a fixed geometrical parameter. Thus, the partial differential value of the depth according t o time is

(5.4-3) The determination of continuity requires the analysis of the water balance in the prism. Considering that the water occupies only the pores of the column, the total amount of water stored within the investigated space can be calculated for the time points o f t and t At:

+

V,(t) = n ( t ) Ax d y h ( t ) ; and

V,(t

+ At) = n(t + At) Ax A y h(t + At);

(5.4-4)

where n(t ) is the time-dependent value of porosity. Neglecting the compressibility of water, the difference of the two volumes has to be equal to the difference of the water flowing in or transported out of the column during the investigated At time-period. Further simplification can also be made by supposing a non-deformable solid matrix and substituting, therefore, the n, specific yield as a time invariant parameter characterizing porosity. The final form of the balance equation is

where F is the accretion having a dimension [LT-l], positive if the ground water is recharged vertically, and negative in the opposite case. A further assumption is the validity of Darcy’s law and the vertical position of the potential surfaces (Dupuit’s hypothesis). Thus relationships can be determined between the components of seepage velocity and the slopes of the water table

v , = K I , = - K K - ;az ax

and

vY= K IY 44

az

- - K--.

-

aY

(5.4-6)

690

6 Movement equatiom describing seepage

By substituting these values into Eq. (5.4-5) the general form of Boussinesq’s equation can be achieved:

(5.4-7)

The final hypothesis applied to simplify the basic differential equation is the aasumption, that the lower impervious boundary i s horizontal, its plane can, therefore, be chosen as a reference level, and thus the height of the water table is equal to the depth of the field (2 = h). The new form of Eq. (5.4-7) is a2h2 a2h2 -+72 ---(5.4-8) ;)* ax2 ay -

(x:

If the investigation is limited to the characterization of one flow plane (twodimensional investigation, but the flow being horizontal the movement is really one dimensional) the form of Boussinesq’s equation already quoted in Chapter 4.1 is attained: 1a2h2 n, a h E - __- - -

2 8x2

K at

K

(5.4-9)

There are only two slight differences between this equation and Eq. ( 4 . 1 4 5 ) . Here the influence of accretion is also considered, which was pre-

viously neglected. The other is a difference in the symbols applied. The depth of the field was previously indicated by y, while here h is used, because one of the axes of the three-dimensional coordinate system is indicated by y. After restricting the analysis to a flow plane, the y symbol will again be applied as flow depth. A further frequently used version of the differential equation of unsteady seepage is the form of Eq. (5.4-8) expressed in polar coordinates, which is more suitable for the characterization of axial symmetrical flow: (5.4-10)

Before discussing the approximations necessary for the analytical solution of these differential equations it is worthwhile summarizing the hypotheses used so far and in this way survey the reliability of the basic equations: (a) Both the water and the solid skeleton of the flow space are incompressible. (b)The change in the position of the water table is immediately followed by the change in storage and, therefore, the storage mpacity is time-invariant constant, equal to the specific yield ( n J : (c)The impervious lower boundary is horizontal, and (d)Both Darcy’s law and Dupuit’s hypothesis are valid.

691


In spite of the detailed approximations there are only a few very special forms of seepage when Boussinesq’s equation can be solved directly by analytical methods. Such special solution may be achieved by the separation of the x and t variables by expressing the h ( x , t ) function as a product of two functions one depending only on x and the other on t. This method was applied by Boussinesq (1904),to determine the parameters of two-dimensional flow, when the field is drained by an infinite series of sinks located along the lower horizontal impervious boundary. It was also his proposal to combine the x and t variables into one a parameter, within which the interrelated role of x and t was determined a priori. This method was used by Polubarinova-Kochina (1952, 1962) in the form of h ( z , t ) = f(4;

where (5.4-1 1)

to determine the specific flow rate crossing the vertical starting section (where the surface water and the seepage field contact one another), if the level of the surface water was instantaneously raised or lowered from an elevation of HI to H , (Fig. 5.4-2). The result of this derivation gives the flow rate as a function of the geometrical and physical soil parameters depending also on time: n q= i H , H , K A (5.4-12) 2t‘

li

Following a similar derivation the propagation of a ground-water wave can also be determined, if the original level of the surface water was below

.or@inal

impervfous boundary

A

Fig. 5.4-2. Interpretation of the symbols used in the examples demonstrating the direct solution of Boussinesq’s equations 44*

692


the lower impervious boundary of the field and, therefore, the wave pene$rates into dry soil after suddenly raising the level of the surface water. The length of propagation in this cwe is (Fig. 5.4-2c):

s o ( t )= 1.62

V T -.

(5.4-13)

A special type of propagation of the ground-water wave is also described by Irmay’s method (1961). The basis of this derivation is similarly the separation of the x and t variables. The h ( x , y ) function is not composed, in this case as the product of the two new functions but as their sum. The final relationship is suitable for the characterization of the action of surface water, the level of which rises gradually and is described by a linear function of time (Fig. 5.4-2d). It can be stated, however, that both the direct methods listed here and those described in the various publications (Polubarinova-Kochina, 1952; 1962; Aravin and Numerov, 1953, 1965; Bear et al., 1968; and Bear, 1972) apply special hypotheses t o find the possible solution of the relevant differential equation and, therefore, they always satisfy those boundary conditions which correspond to the assumed approximations. Thus, their use in practice is limited to the cases characterized by the special conditions studied. To obtain more generally applicable solutions, various approximations are proposed to linearize the basic differential equations. Several linearization techniques are described in the literature (apart from those publications listed in the previous paragraph Jacob and Lohman, 1952; Charnyi, 1951; Bochever and Verigin, 1961; and Kar&di, 1963). The method most frequently used, assumes that the change in the depth of the ground-water flow along the field and in time is negligible compared t o the average depth of flow ( h 9 H I - H,; where h is the average depth, H , is the natural depth and H , is the highest point or smallest depth along the field and within the investigated time period; Fig. 5.4-3). Accepting this approximation the area of cross section is supposed to be constant and thus, this value multiplied by the slope of the water table and hydraulic conductivity gives the specific discharge according to Dupuit’s hypothesis. This product differentiated according t o the horizontal distance, has to be equal t o the change of depth in time multiplied by the specific yield. Thus, the

impervious boundary Fig. 5.4-3. Interpretation of symbols used for the linearization of Ronssinesy’s equation

693


simplified form of the Boussinesq’s equation valid for one dimensional flow after neglecting the effect of accretion is (5.4-14)

There are also proposals as t o how the average depth of the field should be calculated. Verigin (1949; 1952a) for example gives twice the weight t o oiezometrich e

pf(-FJu layer

h

,,\,

I

‘X

/

I

imperiwus bourdqf

Fig. 5.4-4. Derivation of Roussinesq’s equation for semi-confinedfield

the origiilal depth ( H , ) and consider the minimum depth ( H , ) as having only a weighting factor equal to unity: (5.4-1 5)

Another possible method of linearization could be to raise the power of h in the case where i t is differentiated by the time and substitute here the average depth of flow: (5.4-16)

Finally, special functional mapping can also be used to simplify the differential equations. The application of such solutions depends, however, on the boundary conditions t o be satisfied. Summarizing the above, i t can be stated that the general solution of Boussinesq’s equation requires its linearization, the most frequently applied basis of which is the supposition of a constant, time-invariant cross-sectional area along the field. This hypothesis is equivalent to the application of the models determined for semi-confined aquifers [see Eqs (4.1-16) and (4.1-17)J For this system, Boussinesq’s equation can be directly derived, resulting in a linear differential equation. Using the symbols represented in Fig. 5.4-4, where the height of the piezometric line above the lower impervious boundary is indicated by y to distinguish between the semi-confined and unconfined systems, the combination of the equation of continuity and Dupuit’s hypotheses provides the new form of the differential equation. I n the most

694


simple caae, if there is no accretion along the field or its influence isnegligible the equation is as follows:

and

therefore

m K -a2Y - n s-- .

a22

aY at

(5.4-17)

Considering that the basis of linearization is the supposition of a constant average depth which is equivalent to the thickness of the layer in the new model ( h = m) and the role of the water table is taken over by the piezometric surface ( h = y ) , it is evident that Eqs (5.4-16) and (5.4-17) are identical. Thus the use of this made1 provides the same results and the same accuracy, as the various methods of linearization (Kov&cs,1966). The application of this model will be demonstrated by using a practical problem as an example, the solution of which is given by the equation most frequently applied among those derived when assuming unsteady flow: i.e. TheisJacob’s equation for the characterization of unsteady seepage around a pumped well (Jacob and Lohman, 1952) (Fig. 5.4-5). The suppositions applied in the derivation are aa follows: (a)the horizontal water transporting layer extends to infinity is each direction and the boundary condition does not change there; (b) the storage capacity characterized by the specific yield (n,) is time-invariant and does not depend on the position of the investigated point either.

i impervious boundary Fig. 6.4-6. Axial symmetrical unsteady flow around a pumped well

695


Thus, the equation of continuity for an annular area of the aquifer around the well and Dupuit’s hypothesis can be combined in the following form: - &(r

and

+4

1 At

= n, 2r iz

[Y@) - Y(t

+41;

& ( r ) = 2 r n m K - aY ; ar

thus

a& = ns2rn-;aY at

ar

and (5.4-18) which is equal to Eq. (5.4-10). if it is linearized by aasuming the constant depth of the field, and the influence of accretion is neglected. Considering that the piezometric level was at H, above the impervious lower boundary before pumping and supposing that the pumped yield is the solution of the differential equation results in the vertical constant (Q0), coordinate of the draw-down surface above the reference level as the function of r and t independent variables:

-

t ) = Hi -

4n mK U

where u=-

r2 n,

(5.4-19)

4mKt

is a new independent variable combining r and t . The solutions contains the exponential integral function, which can be expanded into series:

-

j ~du=-o.5772--ln

u+u+

u2 2.2I

u3 u4 +- -+ . .. . 3.31 4.41

-

(5.4-20)

0

To solve Eq. (5.4-19) Theis (1935) proposed the application of a graphical method, while Jacob has proved that i t can be considerably simplified, if u < 0.01, because in this case the square and the higher powers of u can be neglected. Consequently, there is a time point to each value of r above which the draw down (the difference between the original horizontal piezometric level and its value changing at a certain time and location) can be expressed in a relatively simple form: Q0 1.5(mKt/ns)1/2. 8 ( r , t) = H , - y ( r , t) = ’ 2nmK r if n 1 t>r2---S--. ( 5.4-2 1) mK 0.04.


696

The numerical application of this method is demonstrated in Chapter 3.1 where the determination of hydraulic conductivity by the evaluation of the unsteady state pumping tests is discussed (Fig. 3.1-11).

5.4.2 Application of the differential equation of unsteady flow to the characterization of seepage in an infinite field

For the general application of the simplified model, based on the supposition of a semi-confined water transporting layer (which directly provides a linear form of the basic differential equation) in an infinite seepage field Eq. (5.4-17) has to be integrated between two time points ( t l and t z ) along the entire length of a semi-infinite field (0 < 5 < 00).The draw-down [s(z, t ) = = H l - y ( z , t ) ] is substituted instead of the depth of the ground-water flow, which is naturally negative, if the level of the surface water is raised at the starting section. The resulting integration is expressed by the following relationship: tl

n , [ s ( z ;t l ) - s(s; tz)]dz. I,

(5.4-22)

0

It is worth-while to note the physical interpretation of Eq. (5.4-22). It states that the product of the fiow rate through the starting section (the area multiplied by hydraulic conductivity and hydraulic gradient) and the elementary time unit summarized between two time points, is equal to the change in the stored volume. The latter can be calculated as the difference between the two water tables corresponding to the time points f , and f , multiplied by storage capacity. The total change in the volume of the seepage space can be characterized by the area enveloped by the two seepage lines in the c u e of two dimensional flow, when a unit width of the space normal to the flow plane is investigated. The solution of Eq. (5.4-22) is complicated by the fact, that unsteady seepage may be created by different influences and, therefore, the boundary conditions (the change in the level of the surface water at the starting section) have always to be considered i n the solutions of practical problems. To achieve a generally applicable solution (or at least one which satisfies the boundary conditions occurring most frequently in practice), a combination of the approximations mentioned in connection with the direct solution of the original differential equation is proposed. That is the separation of the variables, and the use of an a priori determined relationship between time and distance (KovAcs, 1966):

45,t ) = f l ( t ) q l ( Z , t ) ;

(5.4-23)

where f l ( t ) is t ~ known e boundary condition [ f l ( t ) = s ( 0 ; t ) ] , influencing

697


the investigated unsteady flow. Thus, for a semi-infinite flow field the q~ function has t o satisfy the following conditions: ifz=O; if

2

v ( x , t ) = l ; (O 0. I

1

I

Fig. 5.4-7. Hydraulic parameters of unsteady seepage created by constant discharge or recharge

Considering Eqs (5.4-23) and (5.4-25) this flow rate can be expressed as the ratio of f l ( t ) and fz(t)functions introduced in this method: (5.4-32)

Selecting one pair of the possible corresponding f l ( t ) and fz(t) functions, the following solution can be proposed

fz(t) = v

e

t ;

n S

fl(t)= and

SVF

(5.4-33)

700


Thus, the final result in the form of the hydraulic parameters to be determined is aa follows:

and

Gradual change in the level of the surface water (Fig. 5.4-8)

The boundary condition cannot be changed by a finite value within an infinitely small time interval1 in practice. Dracos (1963) haa found that the exit point cannot follow the rapid lowering of the surface water, even if the rate of change is finite. According t o his investigation the highest possible velocity of the lowering of the exit point is

K

-sin2 B ;

(5.4-35)

n where B is the slope of the surface dividing the seepage space from the free water body. It is advisable, therefore, t o approximate the boundary condition with a smoothly changing function t o avoid the occurrence of an infinite flow rate a t the time point t = 0. With a suitably determined function i t can also be proved, that the rate of lowering should be less, than the limit value determined by Eq. (5.4-35). One of the possible approximations may be the use of the following function: t (5.4-36) sn(t)= f l ( t ) = S , t h 2 - ; tn max

*

I

.~ -_L-

e:X=O)

I

I

I

t

t

.-L

3 -

l

l

s,x=71t

pig. 5.4-8. Hydraulic parameters of flow created by the gradual change in the water level


701

where t o is the time point after the beginning of the operation, when the required s, draw-down has to be practically achieved. The substitution of this relationship into Eq. (5.4-26) provides the other time-variant function:

(5.4-37) to Considering the f l ( t ) and f 2 ( t ) functions determined in this way the following formulae give the final solution:

Prom Eg. (5.4-38) the expected maximum flow rate can also be calculated, which is an important parameter, for example in the operation design of (5.4-39)

-

The development of this flow rate is expected to occur around the time 0.5 to. point t If the limiting velocity of lowering has to be considered e.g. in the form of the Dracos' equation, the corresponding s, and t o values have t o be chosen by taking into account the fact that the derivative of the 42, t ) function, according t o time a t time point t = O and at section x = 0, should not be higher than a given value 1

-s ,

2

-

K . <sin /3.

(5.4-40)

702


This condition determines the t o value depending on physical ( K , n) and geometrical (/?) parameters, when the a priori given s, draw-down can be achieved. The combination of constant flow rate and constant draw down (Fig. 5.4-9)

In practice the lowering of the water table generally starts with a pumping of the constant flow rate, and after achieving the required depth, the water level is maintained in the well with decreasing pumping. The parameters. belonging to this method of operation can be determined by combinc L

Fig. 5.4-9 Combination of constant discharge and constant drawn-down

ing the two boundary conditions previously analyzed and described in Eqs (5.4-27) and (5.4-31). In this derivation it has to be considered that after a t o period of pumping at constant flow rate, a water table develops in the seepage field, which is identical with that created by the maintenance of a constant lowered level at the starting section after a period of t o / 2 having elapsed, measured from the fictive starting point of the rapid change of the water level. Thus, at this time point the two processes can be combined to one another without discontinuity, and, therefore, both the boundarv conditions and the results have to be determined separately for the periods t < t o and t > t o respectively: for t

< to m

Kn,

for t > t o so(t) = a m ; -~

qo(t)= smv> f 2t - t o (5.4-41)

(5.4-41)

703

6.4 Horizontal unsteady seepage The influence of surface water with fluctuating level (Fig. 5.4-10)

The unsteady ground-water flow created by the periodically changing level of surface water (flood waves on a river) can be characterized by this type of mathematical model as well, if the hydrograph of the river (its level determining the boundary condition and changing in time) can be well approximated with a periodic mathematical function (e.g. with some combination of trigonometric functions), (KovBcs,1962). Such time variant boundary condition may be (5.442)

where 2h is the amplitude of the wave and 2t0 is the wave length. Let us suppose that the change of the water table at each time point can also be well approximated with an exponential function. The function which expresses the draw down depending on the 5 and t variables, has to be divided into two parts aa before but one part should now be a single valued function of 5,while the other remains time- and space-dependent. Applying the boundary condition given in Eq. ( 5 . 4 4 2 ) , and accepting the approxima-

Fig. 5.4-10. Hydraulic parameters of flow created by periodically fluctuating boundary condition

704


tion mentioned before, the position of the water table depending on time and distance can be calculated:

h s(x,t)=-exp(-Az) where

vz

(5.4-43)

There are two important parameters characterizing this type of unsteady ground-water flow, firstly the curves enveloping the possible extreme positions of the water table as the function of the distance measured from the starting and secondly the time lag between the culminations of the section [smax(x)]; wave at the starting section and at a distance of x from i t ( A t ) : z (s ,),

= -f he-AX;

(5.4-44)

and

t At = 3!A x . n

5.4.3 Unsteady seepage in a horizontally limited field In the previous section a unified and simplified method was discussed for the determination of the unsteady seepage developing in a semi-infinite field considering different boundary conditions at the starting section, the position of which is geometrically determined and known. Apart from the hypotheses generally applied in seepage hydraulics two further basic suppositions were accepted: i.e. the field is covered by a semi-permeable layer and the vertical section of the draw down surface can be approximated with an exponential curve. The first hypothesis is the physical interpretation of the approximation usually applied for the linearization of Boussinesq's equation, and thus i t can be used for the characterization of limited seepage fields ~ t 8 well. By using an exponential function to describe the piezometric line (or the water table) the actual propagation of actions originating from the change in the boundary condition is neglected and it is supposed that the influence of the change can be observed in the first moment along the entire field, (naturally at a very rapidly decreasing rate as i t corresponds to the exponential relationship). This second supposition is not acceptable, therefore, when a horizontally limited seepage field is investigated and another approximation has to be sought, which satisfies the special boundary wnditions occurring at the external vertical border of the field. At the end of the seepage field opposite to the starting section two extreme boundary conditions can develop (Fig. 5.4-11). The field may be bordered either by an impervious layer or contacted by surface-water where there is no resistance against the water exchange between the layer and the surface


705

seepagefield' having free ror~tac?'w7b fbe suVace waters a f both sides (a)consiaanf mter level at one side (s,,,~=0) (b) different influems at tbe sides (smff s , ~ (c) s,mmetrical& infiuenced fieid (sml =s;nz) seepagefield bavng impervious border &one side (d)

~I

impervious boundary

(Cl ..

fb) \-I

impervious boundary

mpervious bounddry

Fig.6.4-1 1. Possible boundary conditions along the external vertical border of a limited seepage field

water (entry or exit face). Between these two basic conditions a continuous series of transition forms may developif water exchange is partially restricted by semi-pervious material covering the contacting surface. Investigating only the absolutely closed field and that undergoing water exchange without the development of any resistance at the contacting surface, further cases have to be distinguished within the second group. The level of the surface-water recharging or draining the field at the external face may be constant (steady boundary condition), or it m y change in time (unsteady boundary condition). The change of the potential along the contacting surface may be the same aa that at the starting section or may differ from it. Although there are many possible variations, the hydraulic investigation of one type is sufficient, (the field drained or recharged by the external contacting surface water having constant level) because the other conditions can be derived from that as indicated in Fig. 5.4-11. If the water level changes not only at the starting section, but the external boundary condition is also 46

706


unsteady, the influences of the two actions have to be determined separately and the results can be superimposed. Section 1 is regarded at h t as a starting section with given boundary condition, assuming constant water level at Section 2. I n the second step the actual change of the water level at Section 2 is the action.creating the unsteady flow in the field, which has constant water level at Section 1. If the changes of the two boundary conditions are identical, the draw-down curve is symmetrical. It is evident that the flow pattern in a field with an impervious vertical boundary is equal to half of the flownet developing a symmetrically influenced layer having double length compared to the original field.

impervious boundary Fig. 6.4-12. Development of unetedy eeepage in horizontally limited seepage

field

On the basis of the explanation given in the previous paragraph the task is the determination of a method suitable for the approximation of the hydraulic parameters of the unsteady i b w in a finite, two-dimensional seepage @ld having a length of L,bordered below by a horizontal impervious bed, covered by a horizontal semi-pervious layer and having in this way constant thickness of m (Fig. 5.4-12). The boundary condition at the starting section is given as an a priori known parameter of the flow [ H , ( t ) ] ,while i t is supposed that the field is recharged or drained without resistance at the other and by a surface water having constant level (HZ = const.). The basic equation [i.e. Eq. (5.4-17)] remains unchanged, while its integrated form [Eq. (5.4-22)] has to be slightly modified: 11

n,[s(s;t l ) - s(z; t z ) ] d s . tl

(5.445)

0

In this case the draw-down curve is approximated by the product of two functions the h t depending only on time and being equal fo the given boundary condition, and the second combining the influence of both variables s and t (Kovhcs, 1975). Equation (5.4-23) can be used, therefore, without any change from the previous investigation. The initial and boundary conditions determining the form of the possible approximation have to be revised and modified, according to the actual data of the field investi-


707

gated. Thus instead of Eq. (5.4-24) the following conditions have to be satisfied : if x = 0; rp(x, t ) = 1; ( 0 < t < 00); if x = L; rp(z,t)= 0 ; (0 < t if t = 0; p ( z , t ) = 0;

< (0 < z < L ) . 00);

(5.4-46)

There is a further special condition, which has to be considered in a finite field. In the cases discussed in the previous section the flow remains unsteady for an infinite time, and, therefore, the validity zone of the equations derived for the characterization of that type of flow is 0 < t < 00. Thus, the V(Z,t ) = = 1 if t = 00 requirement, has to be taken into account. The validity zone in time is, however, limited if the seepage field is not infinite, because the steudy state of flow m y develop at a finite time point between the closing sections, if the levelis constant at one side of the field, while at the other end it is drained or recharged to a constant level or by a constant flow rate. Indicating the time point of the development of the steady state by t,, the validity of the equations describing the unsteady seepage is 0 0.

(5.4-5 1)

It follows from this condition, that f;(t) = 0, except at the time point t = 0. Substituting these values into Eq. (5.4-50), a relationship can be determined in implicit form: (5.4-52)

11 - f2(t)I exp [f2(t)l = exP 0

0.1

0.2

0.3

0.4

0.5

firt)

Fig. 6.4-13. The change in the piezometric level in time in limited flow field caused by rapid draw-down

6.4 Horizontal unsetady seepage

The corresponding values of thef2(t)and

709

mK t parameters are represented in Ln,

Fig. 5.4-14. From the graph, the upper time limit of the validity zone of the unsteady state can also be determined

mK

if f2(t) = 0.5; -t, = 0.193; L2n, consequently L2ns t, = 0.193 -.

mK

(5.4-53)

After determining f2(t),all the hydraulic parameters (the position of the piezometric surface, the flow rate depending on time and distance, as well as the flow rate through the starting section), can be calculated:

go@)= 8 ,

"[

- -- 11.

L

f2W

I

Substituting the f 2 ( t ) = 0.5 value, which belongs to the upper limit of the validity zone ( t J , the parameters become equal to those describing the steady

mk

Fig. 6 . k 1 4 . Relationship between the f&) and -t parameters in the cam of n, unsteady seepage through finite field caused by the rapid change in the surface-water level

710


seepage created by a pressure difference s, = HI - H,:

and

because (5.4-55) and

[-&

- 23 =O; if fi(t) = 0.5.

Unsteady flow created by constant discharge or recharge (Fig. 6.4-15)

I n this case the boundary condition at the starting section is given in the following form: qo(t) = 0; if t = 0; and

mK L

= qm = const.; if t

POU) = -ffl(t)

----

0.0905

-........... 0.179

0.290

10

0

smax8

...--... 0.414

y

- 0.5

> 0.

(5.4-56)

R.009 0.039 0.I18 0.282 0.5

Fig. 6.4-16. The change in the piezometric level in time in limited flow field caused constant discharge


711

From this condition the following relationships between the two timedependent functions [ f l ( t ) and f 2 ( t ) ] can be derived:

qrnL fA4 = -

mK 1

f2(t)

.

-f2(t) '

(5.4-57)

Substituting these values into Eq. (5.4-50), expressions are achieved for the calculation of the f l ( t ) and f 2 ( t ) functions depending on the time elapsed since the start of the process. The stages of this derivation are as follows: (5.4-58)

consequently

mK 1 -t = z + -+c; L2n, 2 where z = 1 - f2(t) and the C constant of integration can be determined from the condition, where, at the time point t = 0 the power in Eq. ( 5 . 4 4 8 ) has to be i n h i t e to achieve the p (2,t ) = 0 condition independent of z. Thus the final forms of the two time-dependent functions are:

(5.4-59)

where smaXis the maximum possible draw down defined by Eq. (5.4-63). Further hydraulic parameters can be calculated by substituting f l ( t ) and f2(t) into the corresponding basic relationship:

(5.4-60)

712


As in the previous caae the t, upper limit of the validity zone of t,he unsteady seepage can be determined from the condition, where the f 2 ( t ) function has to be equal to 0.5 at this time pont:

2L2n,

mK t, - -t

2~2n,

2~2n,

2 -

(5.4-61)

2

1 L2ns

t,=TmK. When the unsteady movement reaches this upper limit, the hydraulic parameters become equal to these characterizing the steady state q&)

= qm = const. = so, TaK.,

L s,(5,

t ) =,s

L-x. ~

L

,

because (5.4-62)

It also follows from the relationships derived here that there is a maximum poasible lowering of the water level belonging to a given discharge, because the development of the steady state ensures the continuous recharge at the other end of the field. This maximum draw down is ,,,s

qmL . = so, = TaK

(5.4-63)

The discharge to be pumped has in practice to be determined, therefore by considering the required lowering of the water table.

References to Chapter 5.4 ARAVIN, V. J. and NUMEROV, S. N. (1963):Theory of Motion of Liquids and Gases in Undeformable Porous Media (in Russian). Moscow. S. N. (1966):Theory of Motion of Liquids and Gaaes in AVARIN,V. J. and NUMEROV, Undeformable Porous Media. Jerusalem. BEAR,J. (1972):Dynamics of Fluids in Porous Media. Elsevier New York, London, Ameterdam. BEAR,J., ZAELAVSKY, D. and IRMAY, S. (1968): Physical Principles of water Percolation and Seepage. UNESCO, Arid Zone Rmearch, Park. BOOHEVER, F. M. and VERIQIN,N. N. (1961): Methodological Guide to Calculate Ground-water Resources for Water Supply (in Russian). Moscow. BOUSSINESQ, J. (1904):Theoretical Research on the Flow Rate of the Ground Water percolating in Soil, and on the Yield of Sources (in French). Journal Mathbmatique pure et Appliqde, Vol. 10.

5.5 Model laws for sand box models

713

CHARNYI,J. A. (1951): Methods of Linearization of Non-linearDifferential Equations of Heat-transport Type (in Russian). Id. A d . Nauk USSR, 6. DRACOS, T . (1963): Two-Dimensional Unsteady Unconfined Ground-water Flow (in German). VAWE-Mitteilung, No. 67. IRMAY, S. (1961): Unsteady Flow Through Porous Materials. 9th Cmgress of IAHR, Dubrovnik, 1961. JACOB, G . E. and LO-, S. W. (1962 : Unsteady Flow to a Well of Constant Draw-Down in an Extensive Aquifer. T A U, 33. -1, G. (1963): Hydr8dics of Linear Draining Structures. (Doctoral Thesis, manuscript in Hungarian). Kbartum, Budapest. KOV~CS, G. (1962): Dimensioning Flood-control Levees for Underseepage. A d o Technica Acaokmiae Scientkmm Hungaricae. KOVACS, G. (1966):Physical Int retation of Linearization of Differential Equations characterizing Unsteady Seepage~ympos~um m Seepage and WeU Hydraulice, Budapest, 1966. KOV~CS, G. (1967): Practical Characterization of Unsteady Seepage in the Vicinity of Drains (in Hungarian). Hidroldgiai KOd6my. Kovdos, G. (1976): Interaction between Rivers and Ground Water (in English) IAHR Symposium, Rapper&, 1975. POLWARINOVA-KOCI, P. YA. (1962): Prop ation of the Ground-water Wave in the Case of Infiltration from Canals (in Russian)%oU. Akad. Nauk SSSR. 6. THEIS,C. V. (1936): T h e Relation between the Lowering of the Piezometric Surfam and the Rate and Duration of Discharge of a Well UsingGround Water Storage. TABU. VERIQIN,N. N. (1949): Unsteady Ground-water Flow in the Vicinity of Reservoirs (in Russian). Dokl. Akad. Nauk USSR, 6. VERIQIN,N. N. (1962a): Ground-water Movement along Reservoirs (in Russian). Bidrotechnicheskcye Stroitdstvo. 4. VERIQIN, N. N. (1962b): The Regime of Ground Water under the Influence of Raising and Lowering the Water Levels in Reservoirs (in Russian). Gidrotechnicheskcye Stroiteletvo. 11.

B

Chapter 5.5 Model laws for sand box models

As already mentioned in the introduction of Part 5 models are used in many cases to measure the hydraulic characteristics of the investigated seepage, or at least to determine some parameters of the relationships describing the flow, instead of the direct solution of the differential equations. The models may be a small scale form of the actual physical process (hydraulic, OT sand box models). Sometimes the transport of quantities other than water or even other phenomena producing the network of orthogonal trajectories within a given field are used for this purpose (analogue models). Reference was made to some recent English publications (Bear et al. 1968; Bear, 1972)) where the detailed description of the application of both hydraulic and analogue models can be found. It is well known that constructing a small scale model of the seepage field to be investigated, flling i t with porous material and water, and h a l l y applying similar boundary conditions along the entry and exit faces of the model to those prevailing in the prototype some of the hydraulic parameters of seepage through the model can be measured, or visualized. The stream lines are generally determined by dyeing the water at given points, while

714


the pressure inside the sand box is measured and the potential lines are constructed from the observations (Fig. 5.5-1). Originally, this method waa only used to determine the flownets of complicated seepage fields, and the hydraulic parameters were calculated from the geometry of the net [e.g. by using Eqs (4.1-37), (4.1-38) and 4.1-40)].

Fig. 5.5-1. Observation of pressure in sand box models

The discharge percolating through the model can also be measured. This value aa well aa the observed pressures, can be recalculated for the prototype, if the model laws valid in the case of hydraulic seepage models, are known. There were, however, very few publications on this topic. It waa felt necessary, therefore, that a short summary of the model law should be given in this chapter, while the reader interested in other problems of modelling is referred to the books quoted already.

5.5.1 General derivation of model laws for hydraulic models The hydraulic model is a scale (generally decreased) copy of the flow space (and its boundaries), where the hydromechanical process to be investigated actually develops. There are special caaes, when the proportionality is not required for all the sizes of the prototype, and some measurements are deliberately distorted and modelled in a predetermined rate unlike those in the other models (distorted models). Filling the model with fluid and applying the same boundary conditions aa those prevailing in the prototype a hydrodynamics1 process develops in the small scale system, the hydraulic parameters of which can be directly measured. The fluid and the other materials transported in the model (e.g. bed load in the caae of open channel models) may be the same aa those moving in the original system, or different materials may also be used. In the latter cwe the differences between the physical properties of the transported material have to be considered when evaluating the data measured in the model.

715


The hydraulic parameters of the original system (which are not meaaurable, because of the large size of the prototype or because i t is only a designed condition and not existing in reality) can be calculated from the data of the model, if the similarity between the two hydrodynamic processes is ensured. This similarity is expressed by the model laws, which give a relationship between the interrelated data of the two systems. The first requirement of the total hydromechanical similarity is the constant rate of the corresponding geometrical parameters of the prototype and the model (geometrical similarity). If this condition is satisfied, any size of the original system (1) divided by the corresponding size in the model (1’) gives a constant value called the transforming factor of length ( A ):

1 - = A = const. I’

(5.5-1)

The next condition is the kinematic similarity, which ensures the constancy of the rate of the interrelated kinematic parameters. This condition requires that the form of paths of two moving particles (one in the prototype and one in the model) should be similar, and the time ( t ) needed for a particle to move along an 1 stretch of its path in the prototype related to the model 1 time ( t ’ )necessary for a particle in the model to paas an I’ = -stretch of the 3,

corresponding path should be conatant, independently of the place and the time point of the investigation: t t’

- = z = const. ;

(5.5-2)

where t is the transforming time factor. Finally, the hydromechanical similarity is regarded as t o t d , if the ratio of all the external and internal forces acting at the interrelated points of the prototype (P)and the model (F’) at the corresponding time points is also constant (dynamic similarity) :

F = z = const.

F‘

(5.5-3)

and this ?G constant is called the forces transforming factor. From the three baaic transforming factors the constant ratio of the interrelated values of any other original and model parameters can be calculated, as indicated in the first column of Table. 5.5-1. The role of model laws is to create contact between the basic factors, making the calculation of any transformation possible if one of the ratios (generally the A transforming factor of the length) is chosen arbitrarily and determined a priori. Only those relationships, which provide the determinaton of such contact between the basic factors can be called model law. The ratio of the forces acting on the moving particles in the original system and in the model respectively can be determined separately for each

716


Table 6.6-1. Transforming fwtors of some physical quantities moat frequently used in hydr8ulics on the basis of vdous model laws

Length 111' Area AIA' Volume VlV' Time tlt' Velocity v/v' Acceleration a/a'

&chsrBe QIQ' Force F/F' Work W/W' Power PIP'

type of force. Thus the numum of e q u a h n s describing the rate of forces, is equal to the number of forces having a dominating role in the system. These equations can be used to derive the relationships between the baaic transforming factors. As already mentioned, the three most important forces acting within the seepage field are gravity, internal friction and inertia. Even in the cam of other hydrodynamic processes these forces are generally regarded aa dominating wtions. Three forces provide three equations expressing the ratio of the corresponding forces within the two interrelated systems. The ratio of gravitational forces: (5.5-1)

assuming that acceleration due to gravity is the same in both systems, and expressing the rate of densities of the two fluids by a =.:e The ratio of forces caused by internal friction:

e

where = q/q' is the ratio of the dynamic viscosities of the two fluids (i.e. those percolating in the original system and those applied in the model, respectively). The ratio of inertia (5.5-6)


717

The equations give the following conditions for the relationships between the three main transforming factors, A, z and n if the mentioned three forces are dominant only in the system:

f G ( A , z, n,a ) = aA3 - n = 0 ; A2

fs(1, z , n , p ) = p - z f T ( A , z,

n = 0;;

(5.5-7)

14

n,a ) = a - - n = 0. 22

The three equations contain five variables, among which two, ( a and /l) depend on the physical properties of the fluids moving in the two systems. Their values are, therefore, determined by choosing the fluids. The other three can be calculated from the three equations. None of them, therefore, can be chosen arbitrarily, if the parameters of the fluids are given as a priori determined quantities. The most common case is where the two fluids are identical. The movement of water is investigated in the natural system, and water has also to be used as the model fluid, where the use of another medium would be too costly because of the great quantity required. Neglecting the di#erences in density and viscosity which may be caused by the temperature difference, the conditions of Eq. (5.5-7) can be simplified: fp&,

z, n) = A 3 - n = 0; A2

ps(2, z, n) =- - n = 0; z

&A,

(5.5-8)

14

z , n ) =-- n = 0. z

The solution of this system of equations gives the simplified result

A=z=n=l;

(5.5-9)

which is the mathematical expression of the statement, that in the case where the same fluid is used in both the prototype and the model and having three or more dominant forces, the effect of which cannot be neglected, the total hydromechanical similarity can only be achieved if the sizes of the two systems are equal to each other and naturally in this case the corresponding time periods, and the interrelated forces are also equal. Three or more dominant forces can be considered, therefore, in a small scale model, only if the fluid applied in the model is different from that of the prototype (Hank6, 1965). The application of small scale models using the same fluid is limited to the cases when the influences of only two forces are dominant in the system, and thus the effects of the others can be neglected or their actions can be considered in a special way. The similarity ensured by taking only two main

718

6 Movement equations describing seepege

forces into account is called partial mecbnicul similarity, and the wellknown model laws always give the relationships between A, z and n for these cases. Only two main forces being dominant, two conditions of Eq. (5.5-8) have to be considered simultaneously. One of the basic transforming factors can be chosen arbitrarily (generally that of the lengths, A) and the other two can be calculated as its functions [e.g. z(A) and 7441. Knowing these relationships the transforming factors of the other characteristic quantities can also be expressed, depending on the A value (see further columns of Table 5.5-1). The model laws describing the relationships between the basic transforming factors are generally given in the form of dimensionless numbers, and the requirement is that the dimensionless number valid in the case of the process in question, should have the same numerical value in the prototype and the model, if i t is calculated from corresponding quantities. The dimensionless numbers can be determined as the ratio of the two forces regarded clcs dominant in the development of the process investigated, as already shown in Chapter 2.1. In the caae of water movement in an open channel, the accelerating force is gravity and the most important retarding force is inertia. From the ratio of these two forces the Froude-number can be derived. Thus, partial mechanical similarity is ensured in this caae, if this value is the same calculated from the data of the prototype and the model respectively: ( 5.5-10)

Relationships determined from this equation between the basic transforming factors, fulfill both the h t and second members of Eq. (5.5-8) simultaneously (i.e. those describing the ratio of gravitational forces and the ratio of the forces caused by inertia respectively): z=

v>; and

7c

= 13.

(5.5-11)

I n the caae of a closed system of pipes the effect of gravity can be neglected, compared to the pressure differences along the pipes, and this latter accelarating force can be modelled arbitrarily by choosing suitable pressure values applied at the open sections of the pipes. However, in this case the geometrical similarity of the piezometric lines is not satisfied, and from this point of view the model is a distorted one. Because the main accelerating force can be freely modelled, two retarding forces, in this special case, can be considered aa dominating actions when calculating the transforming factors: i.e. friction and inertia. Their ratio results in the Reynolds' number, the equality of which haa to be achieved in the two systems to ensure the partial mechanical similarity: vl vtlt - - -= Re = const. (5.5-12) v

v

To express the transforming factors z and 7c as the function of A , the same relationship can be derived from Eq. (5.5-12) a~ from the combination of


719

the second and third conditions of Eq. (5.5-8), (i.e. conditions determined aa the ratios of forces created by friction and of those caused by inertia in the two similar systems): t = A2; and n: = 1. (5.5-13) Finally a third variation can also be formed from two of the conditions listed in Eq. (5.5-8), i.e. the consideration of gravity and friction. On the basis of the dynamic analysis of seepage given in Chapter 2.1 it is well known that the dominant role of these two forces is characteristic in the case of the investigation of laminar seepage (the theoretical derivation of Darcy's law is also based on this hypothesis). The dimensionless number achieved by dividing friction by gravity is the Mosonyi-Kov&cs number (Mosonyi and Kov&cs, 1952; 1956): (5.5-14)

Investigating this type of movement, the relationships between the basic transforming factors can be expressed by the following equations :

t i = - ;1 A

'

and

3 ~ ~ 1 3 .

(5.5-15)

Considering the analysis of the model laws it can be stated that the partial mechanical similarity i s ensured in two geometrically similar seepage spaces if the Mosonyi-Kovcics numbers calcwlated from corresponding data of the two systems are equal to each other. Here all sizes of the systems not only the length of the contours but the sizes of grains and pores as well, are proportional to each other, 1 being constant for any size. The transforming factors of the various parameters calculated from Eq. (5.5-15) are listed in the fourth column of Table 5.5-1.

5.5.2 Geometrically distorted sand box models

As explained in the previous section only partial mechanical similarity can be achieved between the prototype and the hydraulic model. The small scale process may differ considerably from the original one, if the effects of the forces not taken into account are not negligible. Thus the characteristics calculated from the data observed in the model may also differ from the actual parameters. It is also evident that better result can be achieved, if the size of the model is nearer that of the prototype (1tends to l ) , because in this case the effects of the neglected forces become more and more similar in the two systems. This phenomenon is called scale eflect. Moving in the other direction by increasing the ratio of the corresponding geometrical parameters, the differences caused by the change of the neglected forces become greater and greater. There exists a certain limit, and if the model i s decreased below i t , the data measured in the small scale process cannot be used

720


for the determination of the original parameters, because of the uncertainties arising from the various effects of the neglected forces. There are cases, when the proportional decrease in the size of the process investigated causes a sudden change in the basic physical properties of the phenomenon. The scale belonging to this state indicates an absolute limit of the model size, because above this limit not only are the possible errors increased by uncertainties but the character of the movement is also completely changed. These limits generally referred in classical hydromechanics as Eisner’s limits, are as follows: (a)Development of cavitation; (b) Development of capillary waves instead of gravitational ones; ( c )Limit between turbulent and laminar flow; and (d)Limit between tranquil and shooting flow.

These limits are generally supplemented in special cases by considering the effects of both surface roughness and the development of bed load movement. As is shown, however, by the character of the limits, these were determined for hydraulic models of open channel flow. It is necessary, therefore, to add, a further limit to the previous one, if seepage is investigated by using small scale sand box models. To ensure geometrical similarity each size of the prototype has to be decreased in the same proportion determined by the length transforming factor A. As already mentioned, this proportionality should also be ensured between the grain sizes (or the pore sizes) of the two systems. It is quite evident that the decrease in the grain size of the solid matrix applied in the model is physically impossible below a certain limit, the high ratio of colloid particles can cause the modification of the physical character of the seepage (development of microseepage instead of laminar flow; large stagnant parts of the field because of high threshold gradient). If the available facilities (place, water supplying system, etc.) require a smaller model than that prescribed by one of the limits, the general procedure is t o distort one or more geometrical parameters i.e. by building the model on a small scale, proportionally reducing all but one (or sometimes more) sizes of the system. In sand box models the geometry of the boundaries of the seepage space is always decreased proportionally, but the grain size is distorted applying generally coarser particles than those which would be needed on the basis of geometrical similarity. It is the usual practice, to fill the seepage space of the sand box with the same material (having the same size and distribution of grains and even the same porosity) as the prototype. T o apply the original porous medium in the model is a special cme of the distorted model, when the transforming factor of the grain size is equal to unity ( A D = 1). As discussed in Chapter 2.1 the characteristic velocity in Mosonyi-KovAcs number, can be either Darcy’s seepage velocity (v) or the effective mean velocity in the pores (vet,). There are similarly more than one choices when selecting the characteristic length. It can be either Koieny’s effectivediameter (Dh),or the average diameter of pipes hydraulically equivalent to the pores ( d o ) ,or the square root of intrinsic permeability (k).In the case of distorted

5.5 Model laws for send box models

721

models all geometrical parameters concerning the boundaries (and also boundary conditions) of the seepage space, have t o be proportionally decreased and, therefore, (5.5-16)

stating that the total pressure head has t o be determined, as does any other size of the seepage space. It follows from this condition, that the average hydraulic gradient has to be the same in both systems: (5.5-17)

At the same time the internal flow conditions of the seepage field depend on. the decrease in the grain size of that of the equivalent pipe diameter

hydraulic conductivity and intrinsic permeability being linearly proportional to the square of the grain size, or pipe diameter. Combining Eqs (5.5-17) and (5.5-18), the velocity transforming factor can be determined for distorted sand box models: V=

1

K I ; vefr=-KI; n

and V’

1

= K ’ I ’ ; v,ff = 7K’I’;

n

therefore -- %ff = A :. 0’

(5.5-19)

v:tr

The ratio of Mosonyi-KovBcs numbers calculated for the two systems can also be determined. The results will be the same, independent of the characteristic velocity and length, supposing that the latter has always t o be a length describing the i;iternal flow conditions (i.e. effective grain diameter, equivalent pipe diameter, or the square root of intrinsic permeability): (5.5-20)

It cari be stated, therefore, that the equality of Mosonyi-Kovdcs numbers calculated from the parameters of the prototype and the model, respectively, provides the model law for each type of sand box models. It ensures the partial mechanical similarity in the case of a geometrically similar model, while i t is thc condition of the required hydraulic similarity if the model is a distorted one. 46

722


From Eq. (5.5-19) the basic transforming factors can also be determined

therefore

A and A2

n =- (Eq. 5.5-8); therefore n = A A&.

(5.5-21)

2

Eq. (5.5-21) can be further simplifiedin thespecial case when the original material is applied in the model and, therefore, AD = 1:

_v -- 1 ; % = A ;

n=A;

vt

(5.5-22)

while the other transforming factors are listed in the last column of Table 5.5-1.

References to Chapter 5.5 BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier New York, London, Amsterdam. BEAR,J., ZASLAVSKY,D. and IRMAY, S. (1968): Physicd Principles of Water Percolation and Seepage. UNESCO, Arid Zone Reaearch, Peris. EISNER,R. (1926): Hydraulic Laboratories in Europe (in German). Leipzig. -6, Z. (1966): Fulfilment of Similarity when Investigating Seepage Process in Small Scale Models (in Hungarian). M T A MzZszaki T u d o d n y o k Oeztrilycirrak K o d e mdnyei, Tom. 36. 1-4. HORVLTH,J. (1961): Model Law for Considering Friction and Capillarity Simultaneously as Dominant Forces (in Hungarian). Viziigyi Kozlemdnyek, 4. HORV~TH, I. (1962): Similarity of Seepage Process Considerin the Simultaneous Effects of Ca illarity, Friction and Gravlty (in Hungarian). Hic$roMgiai Kodony,3. MOSONYI, and KOV~OS, a. (1962): Model Laws for Considering Gravity and Friction Simultaneously (in Hungarian). HidroMgiai K o a n y , 7-8. MOSONYI, E. and Kovbos, G. (1966): Model Law of Seepage (in French). IASH Symposium, Dijon, 1956. SPRONCH, R. (1932): Hydrodynamic Similarity and Investigation of Models (in French). An& dea Tmvaux Publiquerr de Belgique, 1932. WEBER,M. (1919): Basis of Mechanics of Similarity and its Evaluation in the Investigations of Models (in German). Jahrbuch der Schiffbautechmbchen Qeaellschaft, 1919.

8.

Subject index

absorbed water 18 accelerating forces 202 acceleration 206 - due to gravity 30, 203 accretion 180-181,184-186, 609, 613, 661 active clay content 67-70 active root zone 99 active surface 63, 211 adhesion 101, 123, 211, 304 adhesion V.S. moisture content relationship 125 adhesive flux 3 11 adhesive force 26, 211, 269 adhesive (hydraulic) conductivity 303, 307 adhesive porosity 113 adhesive saturation 124, 294, 307 adhesive water content 124 adhesive zone 102 aggregates 39, 48, 66-67 aggregate-size distribution 120 aggregation of fine grains 48, 121, 351 airbubbling pressure 94, 129-130 air compression 290-291 amorphous colloids 70 analogue models 667; 713 analytic complex function 664 andesite 416 angle of friction 364 anisotropy 338-348, 621 aquicludes 17 aquifers 17 8guifUgeS 17 aquitards 17 areal porosity 22-23, 427 artesian water 16 artesian well 16 atmospheric zone 99 attractive force 63 average shape coefficient 46 axial symmetrical seepage (flow, movement) 328, 694

basalt 412 basic mapping functions 676 bedding planes 41 8 Bingham’s plastic 21 1 binomial form of seepage law 261 body force 203 boiling effect 350, 362 Bondarenko’s constant 273 boundary condition 496, 605 - external 6 0 6 6 1 8 internal 6 2 2 6 2 8 Boussinesq’s equation 688 break of stream lines 623 Brownian movement 33 Buckingham’s potencial 287 Buckingham-Reiner’s equation 2 13, 270 bulk density 166

-

calcite 60 Capillarity 103, 217-224 capillary (hydraulic) conductivity 303,309 capillary contact angle 137, 219, 226 capillary exposed faces 611 capillary flux 311 capillary fringe 609, 648 capillary height 129, 224, 294, 611, 648 capillary meniscus 222 capillary porosity 113 capillary pressure 221 capillary rise 130 capillary saturation 136-136, 294, 309 capillary slit 224 capillary suction 286, 323 Capillary SUl‘faCe611 capillary tube 224 capillary tube diameter 89 capillary tube model 87-94, 302, 387 capillary water content 135 capillary water transport 612, 648-660 capillary zone 99, 103, 612 carbonate rocks 417-437

724

Subject index

Casagrande’s - Aline 70, 72 - apparatus 105 Cauchy-Riemann’s condition 490, 565 cave 418 cavitational point 484 cemented sediments 17 centrifuge moisture equivalent 169-173 change in grain-size distribution 75 characteristic curve of ground-water balance 188-199 characteristic grain diameters 44 characteristic length 231 characteristic velocity 231 chernozem soil 122 Chezy’s equation 229 classification - of hydraulic models 503 - of subsurface waters 13-18 clay content 53-69, 74 clay minerals 4 2 4 3 , 50-52 clogging 317, 380-395 closed capillary zone 99, 103, 111, 301 closed ground water 15 coagulation 39, 48 coefficient - of anisotropy 341 - of compressibility 154, 321 - of consolidation 156 of resistivity 251 - of saturation 106 - of uniformity 44, 47, 352 coefficient of uniformity V.B. porosity relationship 82 colloid particles 4 8 4 9 , 53-65, 73 colmatation 380 column drainage 162-168 combined application of mapping functions 596 complex number 563 complex potential 536, 563-567 complex variable 564 compressibility - of solid matrix 155-159, 480 - of water 154-155 compression 86, 155 - of air 290 - V.S. loading relationship 156 uoncentration - of dissolved salts 193 - of suspension 381 concentric joints 398 conceptual model for characterizing fissured rocks 454 conductivity of one slit 442 confined flow 495 confined seepage field (domain, space) 325, 497 confined water bearing layer 16 conformal mapping 565

-

conjugate directions 581 conjugate velocity 566, 568 connate water 18 consolidation 160, 205, 376 constant draw down 698, 702, 708 contact angle 137, 219, 226 continuity (equation) 476-478, 661 continuum approach 32-36, 424, 474 corner point 484 covering layer 682 critical gradient 360, 365, 378 critical velocity 360, 365, 373, 378 crystalline rocks 398, 401 Darcy ’s - hydraulic conductivity 30, 240-250 - law 28-31, 179 - veloci9 29-30, 239 - zone of seepage 231, 256 deep ground water 15 depression cone 187, 436 depth of seepage field in layered systenis 524 development of water table a t draining trenches 625 differential-thermo-analysis (DTA) 54, 73 diffuse duble layer 210 diffusion theory 283, 289 dimensionsless numbers 226-237 dipole (flow net) 579 dipole molecule 207 direct solution of Boussinesq’s equation 691 distance from the solid wall 210, 217, 223 distorted model 714, 721 disturbed samples 322 dolomite 36, 42, 417 dominant forces 203, 227 double infiltrometer 324 double layer 210 draining trench 603, 607, 625 drains along the bank of surface water 59 7 dry sieving 120 Dupuit ’8 - condition (hypothesis) 179, 628, 661 - equations 628-648 - parabola 632 dynamic analysis of seepage 202 dynamic balance (equilibrium) 98, 103, 110 dynamic similarity 715 dynamic viscosity 30 dynamic water resources 11 dynamics of soil moisture 98 effective dianieLer (Kozeny’s) 4 4 4 5 , 231 effective porosity 22

Subject index effective stress 158, 204, 376 effective velocity (mean velocity) 30 effect of free exit face 332, 517, 631 Eisner’s limits 720 elasticity of layers 206 electrostatic charge 52, 123, 210 electrostatic field 52-53 entry face (section) 508 equilibrium level 193 equipotential line 487 equipotential surface 481 evapotranspiration 98, 101, 150, 183188 exit - face, free 510, 630 - face, covered by water 508 - height 634, 649, 655, 656, 660 - point 629 - velocity 551, 644-647 expansion 159 expansion fissures 418 exponential function 581 exponential integral function 695 external boundary condition 506-518 external corner 484 extrusive volcanic rocks 17, 412417 fault zone 419 fictive starting time 335 field - capacity 107, 109-114 tests 322 filter law 370 finite difference method 558-559 first transition zone 231, 256, 258 fissured and fractured rocks 17, 397417 fissures 396 flow - conditions 31, 473 - function 488, 544 - line 482, 488, 544 - net 489, 568 - plane 485 - through saturated porous media 26 - through saturated solid rocks 26 - through unsaturated porous media 27, 283-284 - through unsaturated solid rocks 28 - tube 485 flux veator 478 foliation 401 Forchheimer’s equation 229 fossil water 18 fractional function 576, 578 fractures 396 free electrostatic charges 52 free exit face 510, 517 free porosity 109-114 free seepage (flow, movement) 616

-

725

friction 26, 206, 716 Froude’s law 718 - number 227 - zone 231, 256 fully penetrating drains 598, 604 generalized form of seepage law 266, 553 geometrical application of mapping 570 geometrical classification of seepage 493 geometrical condition, of suffusion 351, 370 geometrical similarity 229, 715 gradient 29-30, 338 gradient balanced by static shearing stress 273 gradual change in water level 700 grain - size (diameter) 39, 81 - size distribution 4 3 4 9 , 353, 371 - shape 3 9 4 3 grain size V.B. porosity relationship 81 granite 397 graphical approximation of flow nets 49 1 gravel 40, 50 gravimetric water (moisture) content 104, 106 gravitational flow 649 - ground water 111 - porosity 109-114 - seepage field 648 - storage 151 gravity 26, 203, 716 Green-Ampt’s equation 287, 562 ground-water 15-18 - balance 148-153, 193-194 - flow 26, 100, 150, 178-181 - resources 11 - surface 14 - table 14 - zone 99 halloysite 42, 66, 70 Hamel’s mapping function 538 Hazen’s design diameter 44, 89 H-bonds 214 head water 629 height of exit point 652 Hele-Show’s model 88, 440-442 heterodisperse sample 44, 83-84, 263 hodograph image 529-535 - contour 530 - field 531 - mapping 531, 368 - plane 529 homodisperse sample 45, 77, 83, 262 homogeneity 327 horizontal foundation 678

726

Subject index

horizontal moisture flux 148 Horton’s equation 283 hydraulic condition, of suffusion 369, 370 hydraulic conductivity (Darcy) 30 hydraulic conductivity (generalized) 3 1 hydraulic consuctivity, of solid rocks 437,

-

470 V.S.

tension relationship 288, 298,

intrusive rocks 17, 398-401 irrotationd movement 481 isolated porosity 22 jet, infiltrating 323 joints 398 juvenile water 18

312

- V.S.

water content relationship 288,

296, 312

hydraulic gradient 29-30, 229, 294 hydraulic head 644 hydraulic model 667, 713 hydraulic parameters of seepage 666 hydraulic similarity 721 hydrodynamic force 364, 377 hydrological classification of subsurface waters 14 hydrometric method 39, 64, 66, 68 hydromica 62 hydrostatic pressure 204, 287 hygroscopic moisture content 107-108,

Kamensky‘s equation 179-181 kaolinite 42, 6162, 64, 64, 277 karstic rocks 417 karstic water 14 Keuper marl 64 kinematic classification of seepage 493, 496

kinematic similarity 7 16 kinematic Viscosity 30 Kohny-Carman’s equation 243 Kohny’s effective diameter 44-46, 89 Kuron’s hygroscopicity 107-108, 127

127-128

hygroscopicity 107-108, 127-128 hypergeometrical function 636 hysteresis ,137, 141-146, 174-176, 294 ideal fluid 207 igneous rocks 17, 42 illite 42, 62 impervious boundary 607 index of consistency 107 of Dlaticitv 70. 106 indura&d clasdc &iments 17, 407-417 inertia 26, 206, 716 infiltration 98, 101,- 149-160, 183-188,

--

3R.? - -- basin 324 - curve189-190 - from canals 616, 672 - test 322 infiltrometer 322 infinite series of sources:677, 684 influenced seepage 616 infrared photos 106 inhomogeneous anisotro y 480 inhomogeneous layer 32,! 480 initial condition 639-641 ink-bottle effect 137 integral-exponential function 336 interaction of rechargingland draining can& 667 interfacial tension 218 intermediate (porphyric) rocks 17 internal boundary condition 6 2 2 6 2 8 internal corner 484 intrinsic permeability 30-31, 232 429,

laboratory tests 319 laminar seepa e (flow, movement) 26 laminar zone 827, 231, 266, 643 landsubsidence 163, 360, 362 Laplace’s equation 481, 660 lateral air flow 291 layered see age field (domaine, plane) 6 19-62f leaching 196 leaking aquifer 600, 661 Legendre’s elliptical integral 694 limestone 42, 417 limit of plasticity 64, 106 Lindquist’s zone 231, 266 linear extension of seepage field 346 linearization of Boussinesq’s equation 696

liquid limit 64, 106 liquidization of layers 360, 362-369 load v.8. compreseion relationship 166 locd resistance 670 logarithmic function 677, 680 loose clastic sediments 17, 20 Lugeon’s number 324, 429 lysimeter 191 macroscopic characterization of flow domains 34 mapping 667 functions 664 Mariotte’s bottle 123 mathematical model 668 maximum capillary height 103, 129, 177,

-

301

maximum flow rate 643

Subject index maximum molecular water capacity 74, 106 mechanical similarity 716 meniscus 222-223 metamorphic rocks 42, 401-407 mica 42, 46, 60 micro seepage 26, 227, 269-281, 643 microscopic flow pattern 33 mineral composition of samples 60-76 mineralogical analysis 48, 64 mineralogical character of grains 60, 211 mineralogical composition of sediments 60 minimum capillary height 103, 129, 301 Mitcherlich’s hygroscopicity 107-108, 127 modulus of compressibility 86, 167 moisture content 104-109 - of rocks 14 v.8. adhesion relationship 126 V.B. hydraulic conductivity relationship 288, 296, 312 V.S. tension relationship 116, 312 moisture flux 99, 148 molecular forces 217, 227 montmorillonite 42. 61-62, 64. 64. 277 morphological character of colldd particles 73-74 Mosonvi-KovBics - rn‘bdel law 719 number 227, 236-236 motion of fine grains 349 multi-layered filter 373 multi-storied cave 418 Muntz-LainB’s infiltrometer 322

--

-

Na-montmorillonite 42, 277 Navier-Stokes’ equation 478, 643 Nesterov’s infiltrometer 324 neutral stress 158, 204, 376 Newtonian fluid 206 nodal point 484 non-laminar seepage 642-664 non-Newtonian fluid 213 non-wetting fluid 220 numerical method 668 observation well 327, 341 one-dimensional seepage (flow, movement) 27, 477, 493 open capillary zone 99, 103, 112, 301 orientation of dipoles 209 parallel plates model 88, 440442 partial mechanical similarity 718 partially penetrating drains 600, 606 partially penetrating wells 329, 671 path 482 Pavlovsky’s analogy 667 mapping 692

-

727

pelJicular water 102 pending capillary zone 103 pendular water 102 periodic fluctuation 640, 703 permeabimeter with changing head 320 with constant head 319 pF curve 114-117 pF value 114 physical soil parameters 293 piezometric level 497 piezometric Line 498, 629 piping effect 360 piston flow 286, 323, 660 point value of field capmity 110 Poiseuille’s equation 240-241 polarization of molecules 210 pore diameter 94 pore-size distribution 120, 131-136, 308 porosity 22-26, 169, 366 of carbonate rocks 422-428 of loose cleetic sediments 76-87 - of solid rocks 399, 402, 408, 414 V.S. coefficient of uniformity relationship 82 V.S. g r b - s i z e relationship 81 V.S. shape coefficient relationship 80 potential difference 631 potential force 204 potential form of seepage law 261 potential function 482, 487, 644 potential line 487 potential seepage 481 potential surface 482 potential velocity 481 power function 676, 680 pressure 204 distribution 204, 674 head 168, 671 of overlying layers 203, 206 wave 496 primary porosity 119, 407, 417 propagation velocity 286, 498 of wetting front 289 protective filter 369-376 pumped well 326 pumping test 324, 326-337, 436

-

-

---

-

quartz 42, 60 quesi-steady free seepage 616 quasi-steady seepage (flow, movement) 36 quick sand 362 radial joints 398 raindrop effect 137, 226 rapid change in water level 689, 708 rate of consolidation 162

728

Subject index

rate of narrow and large openings 422 rate of saturation 106, 294 - v.8. hydraulic conductivity relationship 288, 296, 312 - V.S. tension relationship 115, 312 recession curve 421 recharge of bore holes 324, 401, 405, 429 redistribution of fine grains 349 Reimann’s surface 578 relative compression 155 release fractures 398 representative elementary - length 426, 466 - unit 34-35, 424 - volume 35, 424 repulsive force 53 resistivity coefficient, in pipes 251 retarding forces 202 retention curve 113, 123 Reynolds’ - model law 718 - nuniher 228, 231-235, 333 Richards’ equation 288 riparian well 611 root - function 576, 580 - zone 99 saddle point 484 safety coefficient 374 salt - accumulation 196 - content 196 sand 42, 50 sand-box model 557, 713 sandstone 42, 407 scale effect 719 schistosity 401, 407 Schlichter’s number 77 Schwartz-Christoffel’s mapping 586-595 sealing of fissures 41 8 seasonal fluctuation 183, 185, 198 secondary porosity 119, 407, 417 second transition zone 231, 256, 268 sedimentation 39, 54, 66, 68 seepage 28-37 - law ( D t ~ ~ y ’28-31 s) - plane485 - parallel to layers 520 - perpendicular to layers 520 - resistance 521 - space (domain, field) 473 - under impervious foundation 588, 678 velocity (Darcy’s) 29-30, 32, 231, 239 - zones 231 self filtering character of layers 356 semi-confined flow 496, 661

-

semi-confined seepage field (domain, space) 497, 693 series of riparian wells 613 shallow groundwater 15 shape coefficient 39-43, 80 - V.S. porosity relationship 80 shearing stress 206 sheet - joints 398 pile 545, 589 sieving 39 silt 43, 50 simplified solution of Boussinesq’s equution 696 sine function 577, 581 single well near the bank of a river 611 singular points 477, 484 sink - flow net 484, 581 - holes 418 - in uniform flow 585 slope of the terrain 195. soil - horizons 121 - moisture 14, 99-104 - - redistribution 98, 104, 287 - - retention curve 109-114, 123145, 166, 167 - - zone 99-104 - structure 418-422 - water diffusivity 288, 293, 300 solid matrix 20, 382 solid rocks 17, 21 source 484, 581 specific discharge 29, 574 specific flow rate 489 specific flux 32 specific soil moisture values 107 specific surface 40 specific water capacity 288 specific water (moisture) content 104-109 specific weight 30 specific yield 154, 286, 335 specific yielding capacity of wells 41 1, 429, 431 stability - of filters 370 - of layers 317, 350, 359 - of slopes 363 Stagnation points 484 starting section 501, 661, 675, 697 static shsaring stress 211, 217, 271 static water resources 11 statistical distribution - of fissures 455 - of pore-size 131 steady flow 29, 327, 479, 499 steady free infiltration 616 steady influenccd infiltration 616

-

Subject index steady seepage (flow, movement) 29, 327, 479, 499 storage - capacity 153-178 coefficient 154-159 stream - function 488, 544 - line 482, 488, 544 - tube 485 suction (tension) head 287, 294 suffusion 350, 351-362 superposition - of a source and a sink 577, 583 - of 6 uniform flow and 6 sink 585 - of infinite series of sources 584 - of mapping functions 681 surface - deposit 383 - runoff 98 - tension 218, 220 - volume ratio 40, 50, 89 suspended load 380 swelling 116-1 17

729

turbulent zone 227, 256, 543 two-dimensional seepage (flow, movement) 27, 339, 485, 493

-

tail water 630 tectonic: joints 398 tension (suction) 110, 123 - V.S. water content relationship 116, 312 - V.S. hydraulic conductivity relationshii, 288, 294, 312 Terzaghi’s - filter law 370 - equation of compression 86, 156 Theis’s equation 336, 694 three-dim~nsioniL1(flow, movement) 493 threshold gradic:rt 212, 216, 236, 273, 546 tortuosity 91-92 total rlogging 390 total draw down 333 total elliptic integral 594 total energy 287 total mechanicnl similarity 71F total porosity 22 total salt content 193 total stress (vertical) 158 transient flow 335 transition zone of seepage 26, 227, 251259, 543 transniiseibility 181, 432, 434 transporting capacity of water films 301, 304 transverse anisotropy 338, 621 trap 413 trigonometric functions 581 turbulent flow in fractures 468 turbulent hydraulic conductivity 260 turbulent seepage (flow, movement) 26, 260, 468

unconfined flow 478, 495, 661-669 unconfined seepage field (domain space) 325, 497 unconfined water bearing layer 16 undisturbed Sam les 322, 344 uniformity coefKcient 44, 47, 82, 352, 371 unsaturated (hydraulic) conductivity 31, 288, 293, 295, 301-313 unsaturated seepage (flow, movement) 27-28, 227 unsaturated zone 99, 102 unsteady seepage (flow, movement) 499, 516, 539, 560, 687-712 unsteady effect - of constant discharge 699, 710 - of constant recharge 699, 710 unsteady seepage in horizontally limited field 704 validity of Dupuit’s hypothesis 630 validity zones of seepage 226, 230 vapour flux 99, 284 van der Weals’ force 62, 123, 210 velocity - Darcy’s or virtual 29-30 - effective 30 gradient 206 - hodograph 629 potential 476-481 vertical distribution of soil moisture 98 vertical exit face 369 vertically drained seepage field 601, 666 vertically recharged seepage field 600, 662 virtual width of canals 620, 673 viscosity - dynamic 30, 208 - kinematic 30, 208 viscous fluid 206 volcanic rocks 412-417 volumetric porosity 22-23, 427 volumetric water (moisture) content 105, 106, 288, 293 vortex point 484

-

water - balance 148 - balance equation 149, 179 - content 104-109 - - v.8. hydraulic conductivity relationship 288, 296, 312 - - V.S. tension reIationship 115, 312 - exchange 101, 152, 178-188

730 water film 101-102 intake 401, 406, 429

Subject index

- mining 196 - molecules 208 - pressure 204-206 - resources 11 - retention capacity 162, 169, 177 - table 14, 99, 149, 189, 194, 609 - - between two canals 668

wave prop ation 496 weetherhg?bures 398, 406 well screen 3 7 6 3 7 8 wet sieving 120 wetting fluid 220 front 286

-

vetting - process 138-140, 143 wilting point 107

X-ray investigation 64, 73 yield of riparian w e b 610 Zhukovsky’s mapping 636, 570, 626 zone - of adhesion 99, 102, 123, 301 - of cultivation 99 of plants 99 - of saturation 99, 102

-

Seepage Hydraulics

Raqs Media Collective: Seepage

Concise Hydraulics

Practical Hydraulics

Fluvial Hydraulics

Concise Hydraulics

Concise Hydraulics

Gas Pipeline Hydraulics

Hydraulics of Pipeline Systems

Fractured Rock Hydraulics

Liquid Pipeline Hydraulics

Open Channel Hydraulics

Hydraulics of Pipeline Systems

Civil engineering hydraulics

Open-Channel Hydraulics

Open-Channel Hydraulics

Liquid Pipeline Hydraulics

Practical Hydraulics Handbook, Second Edition

Wastewater Hydraulics: Theory and Practice

Overland Flow: Hydraulics And Erosion Mechanics

Multiphase Flow Dynamics 4: Nuclear Thermal Hydraulics

Liquid Pipeline Hydraulics (Dekker Mechanical Engineering)

Hydraulics of Dams and River Structures

Computational Hydraulics and Hydrology: An Illustrated Dictionary

Computational Hydraulics and Hydrology: An Illustrated Dictionary

Hydraulics of Pipelines: Pumps, Valves, Cavitation, Transients

Computational Fluid Dynamics: Applications in Environmental Hydraulics

Groundwater Hydraulics (Developments in Water Science)

Environmental Hydraulics for Open Channel Flows

Seepage Hydraulics

Raqs Media Collective: Seepage

Concise Hydraulics

Practical Hydraulics

Fluvial Hydraulics

Concise Hydraulics

Concise Hydraulics

Gas Pipeline Hydraulics

Hydraulics of Pipeline Systems

Fractured Rock Hydraulics

Liquid Pipeline Hydraulics

Open Channel Hydraulics

Hydraulics of Pipeline Systems

Civil engineering hydraulics

Open-Channel Hydraulics

Open-Channel Hydraulics

Liquid Pipeline Hydraulics

Practical Hydraulics Handbook, Second Edition

Wastewater Hydraulics: Theory and Practice

Overland Flow: Hydraulics And Erosion Mechanics

Multiphase Flow Dynamics 4: Nuclear Thermal Hydraulics

Liquid Pipeline Hydraulics (Dekker Mechanical Engineering)

Hydraulics of Dams and River Structures

Computational Hydraulics and Hydrology: An Illustrated Dictionary

Computational Hydraulics and Hydrology: An Illustrated Dictionary

Hydraulics of Pipelines: Pumps, Valves, Cavitation, Transients

Computational Fluid Dynamics: Applications in Environmental Hydraulics

Groundwater Hydraulics (Developments in Water Science)

Environmental Hydraulics for Open Channel Flows

Recommend Documents