Fractured Rock Hydraulics

Fractured Rock Hydraulics © 2010 Taylor & Francis Group, London, UK Fractured Rock Hydraulics F.O. Franciss Progeo –...

Author: Fernando Olavo Franciss

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Fractured Rock Hydraulics

© 2010 Taylor & Francis Group, London, UK

Fractured Rock Hydraulics

F.O. Franciss Progeo – Consultoria de Engenharia Ltda, Rio de Janeiro, Brazil


Taylor & Francis is an imprint of the Taylor & Francis Group, an informa business © 2010 Taylor & Francis Group, London, UK Typeset by Macmillan Publishing Solutions, Chennai, India Printed and bound in Great Britain by Antony Rowe (A CPI Group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Library of Congress Cataloging-in-Publication Data Franciss, F. O. Fractured rock hydraulics / Fernando Olavo Franciss. p. cm. Includes bibliographical references. ISBN 978-0-415-87418-2 (hardcover : alk. paper) – ISBN 978-0-203-85941-4 (eBook) 1. Seepage–Mathematical models. 2. Groundwater flow–Mathematical models. 3. Rocks. I. Title. TC176.F727 2010 627—dc22 2009036651 Published by:

CRC Press/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.co.uk – www.balkema.nl

ISBN: 978-0-415-87418-2 (Hbk) ISBN: 978-0-203-85941-4 (eBook)


Table of Contents

About the author Preface

IX XI

Introduction

1

Fractured rock hydraulics Scope

1 2

Chapter 1

3

Fundamentals

1.1

Basic concepts 1.1.1 Pseudo-continuity 1.1.2 Observation scale 1.1.3 Description at different scales 1.1.4 Representative elementary volume 1.1.5 Hydraulic variables 1.1.5.1 Introduction 1.1.5.2 Specific discharge 1.1.5.3 Hydraulic gradient 1.1.6 Hydraulic conductivity 1.1.6.1 Introduction 1.1.6.2 Fractures and conduits 1.2 Governing equations 1.2.1 Preliminaries 1.2.2 Energy conservation principle: Darcy’s law 1.2.3 Mass conservation principle: continuity equation 1.2.3.1 General equation 1.2.3.2 Dupuit’s approximation 1.2.4 Boundary and initial conditions 1.2.4.1 Main boundary types 1.2.4.2 Submerged boundaries 1.2.4.3 Impervious boundaries 1.2.4.4 Seepage boundaries 1.2.4.5 Unconfined groundwater-air interface


3 3 4 8 8 8 8 9 10 17 17 19 28 28 28 30 30 31 35 35 36 36 37 37

VI

1.3

Table of Contents

Addenda to Chapter 1 1.3.1 Addendum 1.1: Effective velocity and specific discharge 1.3.2 Addendum 1.2: Hydrodynamic gradient 1.3.3 Addendum 1.3: Hydraulic conductivity for randomly fractured subsystems 1.3.4 Addendum 1.4: Energy conservation principle 1.3.5 Addendum 1.5: Mass conservation principle

Chapter 2

Approximate solutions

2.1 2.2 2.3 2.4 2.5

Overview Differential operators Uniqueness of solutions Approximate solution errors Approximation methods 2.5.1 Preliminaries 2.5.2 Collocation method 2.5.3 Least squares method 2.5.4 Galerkin’s method 2.5.4.1 Orthogonality 2.5.4.2 Galerkin’s approach 2.5.4.3 “Weak solutions’’ 2.5.4.4 Variational notation 2.5.5 Time-dependent solutions 2.6 Addenda to Chapter 2 2.6.1 Addendum 2.1: Classification of second order linear partial differential equations 2.6.2 Addendum 2.2: Minimisation of the sum of the squared residuals 2.6.3 Addendum 2.3: Minimisation of the sum of the squared residuals transformed by the differential operators DV and DN 2.6.4 Addendum 2.4: The concept of “orthogonality’’ Chapter 3

Data analysis

3.1 3.2 3.3

Preliminaries Analysing geological features Handling of hydraulic head data 3.3.1 Variation in time 3.3.2 Variation in space 3.4 Handling of flow rate data 3.5 Handling of hydraulic conductivity data 3.5.1 Preliminaries 3.6 Hydraulic transmissivity and connectivity 3.6.1 Preliminaries 3.6.2 Hydraulic conductivity appraisal 3.6.2.1 Hydraulic tests at “core sample’’ scale 3.6.2.2 Hydraulic tests at “borehole integral core’’ scale 3.6.2.3 Hydraulic tests at “cluster of boreholes’’ scale 3.6.2.4 Hydraulic tests at “aquifer’’ scale


39 39 40 41 42 43 47 47 48 53 53 59 59 60 65 71 71 71 76 77 80 87 87 88 89 89 91 91 91 91 93 96 105 107 107 108 108 108 109 109 112 115

T a b l e o f C o n t e n t s VII

3.6.3

3.7

Hydraulic connectivity appraisal 3.6.3.1 Dynamic correlations of WT time series 3.6.3.2 Filtering WT contour maps Modelling hydrogeological systems 3.7.1 Concepts 3.7.2 Guidelines to conceptual models

Chapter 4

Finite differences

4.1 4.2

Preliminaries Finite difference basics 4.2.1 Difference equations 4.2.2 Finite differences 4.2.3 Difference equations for steady-state systems 4.2.4 Difference equations for unsteady-state systems 4.2.5 Difference equations for boundary conditions 4.2.6 Simultaneous difference equations 4.2.6.1 Preliminaries 4.2.6.2 Gauss-Seidel iterative routine 4.2.6.3 Crank-Nicholson iterative routine 4.3 Finite differences algorithms for fractured rock masses 4.3.1 Preliminaries 4.3.2 Steady-state solutions 4.3.2.1 Dupuit’s approximation 4.3.2.2 3D algorithms 4.3.3 Transient solutions


119 119 119 120 120 121 125 125 125 125 126 128 131 133 134 134 134 140 143 143 144 144 158 174

About the author

Born in 1935, Fernando Olavo Franciss grew up in Rio de Janeiro and was educated as a civil engineer in the Pontifical Catholic University of Rio de Janeiro, Brazil. He started his professional career in 1959 by his education in applied geology by the late and distinguished Prof. Reynold Barbier at the Institut Dolomieu of the University of Grenoble, France. Ten years later, in 1970, he obtained his doctoral degree from the same university. A leading rock engineer, he has gathered a lifetime of international experience in civil engineering practice, often while crossing with other fields such as engineering geology, underground mining and oil reservoir engineering. From 1964 to 1980, he worked as a part-time professor at the Pontifical Catholic University of Rio de Janeiro. Until 1991 he worked at Sondotecnica, a reputed Brazilian consulting bureau, and since then as an independent consultant. Many now well-known Brazilian experts in civil, earth and water engineering started their professional careers closely working with Prof. Franciss, a fact that pleases him very much. During his career, Dr. Franciss has had the chance to devote part of his time to an investigation of the hydraulics of fractured rocks related to civil works, mining, oil and gas storage caverns and the interactions of hydrothermal resources with dam reservoirs. He has accordingly developed a tensor approach to describe the hydraulic properties of fractured rocks and unique finite difference matrix-algorithms to model the hydraulic and hydrothermal behaviour of randomly fractured rock masses. He is member of the Brazilian Society for Soil Mechanics and Geotechnical Engineering, the Brazilian Society for Engineering and Environmental Geology, the National Academy of Engineering and the International Society for Rock Mechanics. He has won a number of prestigious awards in Brazil, and has written several papers and a number of books: Soil and Rock Hydraulics (Balkema, Rotterdam, 1985), Weak Rock Tunnelling (Balkema, Rotterdam, 1994) and a co-authored contribution with Manoel Rocha on rock mass permeability in Structural and Geotechnical Mechanics by W.J. Hall, ed. (Prentice Hall, New Jersey, 1976).


Preface

As for any other book, the history of this one helps to understand why it was written, despite the excellent works that were published on this subject already. In fact, my interest on fracture hydraulics arose when, challenged by practical problems, I developed in 1962 an electrical analogue by simply drawing fracture traces with special electro-resistive ink and plate condensers with electro-conductive ink on very thin polyester paper. Later, in 1969, while working on my thesis on the same subject at the National Civil Engineering Laboratory of Lisbon (LNEC), I was captivated by the possibilities of the tensorial language to correctly describe anisotropic physical properties. Inspired by Ferrandon’s model1 , I tried to fit second order tensors to the hydraulic transmissivity of isolated fractures aiming at the description of the equivalent permeability of a group of fractures. In 1972, the Mines and Applied Geology Division (DMGA) of the Technological Research Institute of São Paulo (IPT) invited me, encouraged by the favourable opinion of the late but ever prized Pierre Londe, to give a course on hard rock hydraulics at the Mines Department of the Polytechnic School of the University of São Paulo (EPUSP). At the same time, I had the honour to closely work with the late and distinguished Manoel Rocha. In that fertile period, I introduced in Brazil the LNEC integral sampling method and large-scale testing techniques for rock masses. After gaining a bit more of experience as a part-time professor at the Catholic University of Rio de Janeiro, I revised the original EPUSP course and offered it in 1978 at the Institut Dolomieu, University of Grenoble, on the kind invitation of Prof. Jean Sarrot. Except for the simplified electro-analogue, the Grenoble course, published by Balkema in 1984, was incomplete in several aspects, notably because an appropriate link between fundamental concepts and tailor-made solving techniques for randomly fractured hard rock was missing. However, due to private commitments, I could not spend more time thinking about the problem. To support my colleagues at Sondotecnica, a very reputed and traditional consulting bureau in Brazil, I left the Catholic University of Rio de Janeiro to devote myself to professional activities only. Since 1994 until presently now as an independent consultant, I have had the opportunity to be deeply involved in the complex dewatering of an underground mine in fractured, faulted and karstified dolomites. Moreover, thanks to Votorantim Metais Zinco from Brazil and to Intraconsult Associates from South Africa, I had the chance to

1 Ferrandon, J. (1948), Les lois de l’écoulement de filtration, Le Génie Civil, no 2, 125, 24–28.


XII P r e f a c e

compare the characteristics of this particular problem to some South African groundwater cases in dolomite land. During the same period, I faced many seepage problems related to dam foundation, interaction of dam reservoirs and hydrothermal resources and underground LPG storage. Thus, I was again compelled to think about fractured rock hydraulics. By fate, my clients offered me a new chance to complete my interrupted work, now finally presented in this current publication. Hydraulic behaviour of inhomogeneous and anisotropic randomly fractured rock masses is usually governed by major contacts, faults and discontinuities. On the other hand, fine discretisation of any hydrogeological system into small cuboids, a sort of 3-D bit-mapping, reasonably describes their complex discontinuous architecture and, as a natural consequence, encourages the use of finite differences to simulate their hydraulic behaviour, if their heterogeneity and the tensor character of the transmissivity of all their singularities are duly taken into account. Based on that approach, I developed and thoroughly tested specific finite difference algorithms to deal with common problems concerning the hydraulics of hard rock without restrictions to the 3D geometry and to the properties of their discontinuities. Their inclusion in the present book completes my previous but unfinished work with no other pretension than to share my personal experience with other professionals and students. To condense this information in a 12 lectures course, a concise explanation was adopted. Besides all the friends and organisations referred above, I wish to express my gratitude to many students, colleagues and clients who helped me with due criticism and sensible proposals. Finally, my thanks also go to Germaine Seijger, Senior Editor of Engineering, Water & Earth Sciences, CRC Press/Balkema and also to the Balkema team, for their patient support and incentive during the gestation period of this book.

F.O. Franciss Rio de Janeiro, August 2009


Introduction

Fractured rock hydraulics Acid igneous rocks, like granites, basalts and gabbros, as well as metamorphic rocks, like gneisses, schists and slates, form more than 95% of the earth’s crust and make up the rigid basements of the continents, attaining more than 70 km depth at the root of high mountain chains. Basic igneous rocks, like petrified seafloor basalt extrusions, pile up to 7 km thickness under the oceans. However, igneous and metamorphic rocks emerge as outcrops only over 15% of the earth’s surface and are frequently covered by layers of sedimentary rocks, like sandstones, siltstones and limestone, up to hundreds metres thick, and also by less resistant thinner veneers, such as weathered rocks and residual soils or unconsolidated sediments like gravels, sand and clays. Yet, rock masses near the earth’s surface, up to tens or hundreds metres depth, are not massive. Indeed, distinct groups of almost planar discontinuities split all igneous and metamorphic rock masses into contiguous and practically impervious blocks of quite resistant intact rock. From a strict mechanical point of view, even if genetically dissimilar, these rock masses, as well as some types of crystalline limestone and dolomites, can be collectively classified as “hard rocks’’ or, simply, as “fractured rocks’’. By custom, the generic term fracture stands for all kinds of restricted partitions or discontinuities within rock masses. Irrelevant and small amounts of groundwater, almost clogged, accumulate inside the minute voids and micro cracks of intact rock blocks and hardly drop under the action of gravity. This kind of water remains practically unavailable for exploitation. On the other hand, significant quantities of groundwater may saturate the fractures of rock masses and other sorts of open discontinuities. This kind of water, called free water, may drain and percolate under its self-weight. Only this type of “gravity-driven’’ groundwater is the focus of the present book. In the last 50 years, many technical publications have dealt with the theoretical principles and practical field and laboratory tests related to the appraisal of the groundwater flow through sedimentary rocks and unconsolidated sediments. In the same period, as “hard rocks’’ are not good aquifers by themselves, few publications have considered fractured rock hydraulics. In fact, groundwater storage volume is proportional to the porosity of the rock reservoir, including interconnected interstitial voids, fractures and dissolution discontinuities. For hard rocks, porosities seldom attain 2% of their total volume while may range from 5 to 15% for consolidated clastic rocks and up to 25%


2

Fractured rock hydraulics

for unconsolidated sands and gravels. Yet, during the last decades, the growing need of groundwater supply in “hard rock’’ land, mainly in poor countries, coupled with a better valuation of the seasonal groundwater recharge and corresponding transient storage at the top of deep weathered “hard rock’’ land in wet inter-tropical climate, has altered attitudes. Moreover, in the same period, new investigation methods and new technologies based on sounder theoretical concepts have been devised to improve “hard-rock’’ hydraulics models, commonly motivated by attempts to enhance the structural and environmental safety factors of increasingly larger engineering works founded and excavated in these rocks. Civil or mining facilities under the sea level or under the groundwater table, such as access drifts, stopes, panels, galleries, tunnels, shafts, underground hydropower plants, oil and gas storage caverns, and classed nuclear waste deep disposal, are examples of excavation works where their structural behaviour highly interacts with their hydraulics.

Scope This book is written for readers with some knowledge of hydraulics, geology, hydrogeology and soil and rock mechanics. It has an introductory level. Chapter 1 covers the fundamentals of fractured rock hydraulics under a tensor approach. Chapter 2 presents some key concepts about approximate solutions. Chapter 3 discuss some data analysis techniques applied to groundwater modelling. Chapter 4 presents 3D finite difference matrix algorithms to simulate practical problems concerning heterogeneous and anisotropic fractured rock masses.


Chapter 1

Fundamentals

1.1 Basic concepts 1.1.1

Ps eu d o - c o n t i n u i t y

To construct uncomplicated models, the governing equations of groundwater movement must be based on the simplified assumptions of classical continuum mechanics. According to its fundamental postulate – the continuum concept of matter – the observation and the prediction of the groundwater behaviour in space and time must be described from a “macroscopic’’ point of view but taking implicitly into account all interactions, heterogeneities, anisotropies and discontinuities influencing the groundwater movement from a “microscopic’’ point of view. Boundaries separating the “macro-scale’’ from the “micro-scale’’ depend simultaneously on the purposes of the analysis and on the characteristics of the pervious media. In addition, the application of the continuum concept to any material system assumes an unbroken distribution of its matter in space and its uninterrupted existence in time. Consequently, to model a groundwater system according to these principles, two key conditions must be fulfilled: • •

First, the size of its subsystems must be set above an almost empirical minimum size demarcating the interface between the macro and the micro observation scales. Second, a fictitious continuous system, i.e. a pseudo-continuum, having an almost equivalent hydraulic behaviour, must substitute for the real discontinuous matter.

If these two conditions are satisfied, it is possible to construct an acceptable numerical model by observing the following preliminary steps: • • •

First, select the most adequate observation scale for the pseudo-continuous system, taking into account the compatibility between the model size and the desired level of detail. Second, split the pseudo-continuous system into several smaller pseudo-continuous subsystems. Third, describe the hydraulic properties of these subsystems and formulate their corresponding governing flow equations so as to approximately replicate the factual hydraulic behaviour of the system at the elected observation scale.


4


Crushed zones and major fracture zones Fracture zones or shear zones 500 m

N

Figure 1.1 Example of a typical mosaic pattern delineated by shear zones and fractures affecting the underlying hard rock basement of Norway, Sweden, Finland, and Northern Denmark, over 3.1 billion years old, dubbed Fennoscandia (figure adapted from S, Johansson, Large Rock Caverns at Neste Oy’s Porvoo Works, Large Rock Caverns Proceedings of the International Symposium, Helsinki, 1986, structural features adapted from fig. 2 of Volume 1).

For most numerical models, the modal (most observed or most frequent class of values) linear dimension of the pseudo-continuous subsystems generally do not exceed 1/30 to 1/300 of the greatest linear dimension of the model. 1.1.2

Ob s er va t i o n s c a l e

Natural discontinuities occurring in hard rock basements delineate several interlaced and hierarchical nets. Based on their relevance, it is possible to distinguish a number of well-defined families of discontinuities, from major lineaments, as tension fractures and shear zones, to minor joint systems. The major discontinuities outline contiguous hard rock mosaics, from high order size, between 10 to 100 km, to low order size, between 100 to 1000 m. Several minor discontinuities, classified by frequency and magnitude, still crack these mosaics into smaller and less relevant units, imparting structural complexity to hard rock domains (see see fig. 1.1). As a result, it is difficult to construct adequate pseudo-continuous models for these intermingled lithologic and structural features. However, there are rough guidelines for setting the minimum size for their pseudo-continuous subsystems. This minimum is related to the extent of the model, the length and frequency of the discontinuities and the desired simulation details. In plain words: it depends on the observation scale. Common observation scales may be classified as follows: •

Class I – “infilling scale’’: Consider, for example, a major discontinuity plugged by a pervious granular soil (see fig. 1.2). To simulate the hydraulic behavior of that confined layer, the size of their pseudo-continuous subsystems must be greater than 10 to 30 times the soil modal pore diameter. Their hydraulic properties must


Fundamentals

Intact rock

5

Fault gauge ≈2m

Pseudo-continuous subsystem

Figure 1.2 Some simulations require a detailed appraisal of the hydraulic influence of the infillings of intersecting major discontinuities under high differential pressures. An assemblage of pseudo-continuous subsystems at a very small observation scale may model these pervious granular layers adequately (adapted illustration from NTNU-Project Report 1D-98).

•

•

•

reflect the integrated effect of all individual grains and pores they contain. Civil engineers or hydrogeologists hardly ever perform this type of analysis. However, this very small-scale approach may be useful to evaluate and predict effects of burial pressure changes on the permeability and porosity of major discontinuities in oil reservoirs. Class II – “intact rock scale’’: Consider, for example, the tension gashes associated with stylolites in carbonate rock blocks. These small structural features play an important role in some oil reservoirs (see fig. 1.3). Their equivalent hydraulic properties are sensibly anisotropic. The smallest size of their pseudo-continuous subsystems must be about 10 to 30 times the modal discontinuity spacing. Class III – “discontinuity scale’’: Consider, for example, the observation of the hydraulic behavior of partially filled and partially sealed discontinuities taken one by one. In this case, the major dimension of their flat rectangular pseudocontinuous subsystems must be greater than 10 to 30 times the modal discontinuity thickness (see fig. 1.4). Their hydraulic properties reflect the integrated effect of their incomplete infilling, imperfect sealing and erratic voids. This type of analysis may be employed to predict the hydraulic behaviour of discontinuities in close proximity to underground civil engineering works or groups of wells. Class IV – “rock mass scale’’: Consider, for example, a pseudo-continuous cuboid subsystem enclosing a volume of rock mass partitioned by several discontinuities at random, separating intact rock blocks (see fig. 1.5). For one or more erratic discontinuities, the equivalent hydraulic property of that pseudo-continuous subsystem results from specific construction rules depending on the hydraulic properties of every discontinuity, as discussed in section 2.5.4. In this case, the minimum subsystem size must be greater than 3 to 10 times the thickness of the most relevant discontinuity. For quasi-systematic fracture sets, the equivalent hydraulic property


6


Figure 1.3 Stylolites associated with secondary fissures. These features are jagged discontinuities but are not structural fractures. They occur in limestone and other sedimentary rocks and typically delineate irregular and interlocked small columns, pits and teeth-like projections, probably formed diagenetically by differential movement under pressure, accompanied by solution.They may be widened by subsequent groundwater flow. A valuable discussion about the 3D effect of intersecting stylolites on the permeability of whole cores can be found in Nelson, 2001.

Pseudocontinuous subsystem size

Vast discontinuity with variable thickness and infillings with variable hydraulic properties

N

Figure 1.4 Some simulations require the detailed appraisal of the hydraulic influence of major discontinuities that crosses the flow domain. In these very special cases, an assemblage of pseudo-continuous subsystems with variable hydraulic properties may model the pervious discontinuity adequately (Canadian Shield, figure adapted from Les Eaux Souterraines des Roched Dures du Socle, UNESCO, 1987, approximate scale 1:8000).


Fundamentals

7

Erratically discontinuous rock mass subsystem

Random fractures Pseudo-continuous subsystem hydraulically equivalent

Figure 1.5 An erratically discontinuous rock mass subsystem may be substituted by a pseudocontinuous subsystem having an almost equivalent hydraulic behaviour.

Bearing II

Bearing I

Bearing IV

Bearing III

Kmin E-8 m/s

Kint 1.2E-8 m/s

Kmax 1.7E-8 m/s

Bearing V 7.367.000 N 300 m 400 m 200 m

20 m

7.366.00 N 100 m 454.000 E

455.000 E 0m

Atlantic Ocean

Figure 1.6 A mixed model may adequately simulate the hydraulic behaviour of the small rectangular area enclosing well-defined discontinuities but correctly coupled to the hard rock mosaics (South Atlantic Brazilian coast).

•

may also be statistically inferred. In this case, the subsystem must be greater than 10 to 30 times their modal spacing. Class V – “hard rock basement scale’’: Consider, for example, a hard rock basement depicting mosaics separated by several lineaments. In this case, more than one observation scale may be combined into a single model: smaller and restricted models for a group of lineaments and a greater integrated model for all mosaics (see fig. 1.6).


8


In practice, the minimum size of the pseudo-continuous subsystems for numerical simulations normally exceeds the minimum imposed by the rules set out on pages 4–7. There are no simple criteria to define the best size. Usually, a satisfactory limit is elected after judging the most convenient size for the model, the costs of getting the geological data, the desired output details, the trade-off between truncation errors and round-off errors, costs of processing time and storage capacity, etc. To reduce costs, it is always possible to refine a previous simulation only within certain subdomains where this is essential. It is important to note that the size of a pseudo-continuous subsystem may range from few centimetres (for a granular seam) to tens or hundreds of metres (for a huge aquifer). In transient processes, the system variables change from an unbalanced condition to a final state of equilibrium. In this case, the observation time interval may vary from a few seconds or minutes (for example, during the equalisation of neutral pressure transients in saturated cores tested in a triaxial chamber) to several hours or days (for example, during the progressive dewatering of an underground mine). 1.1.3

Des c r i p t i o n a t d i f f e r e n t s c a l e s

All variables describing the hydraulic properties and the physical state of a pseudocontinuous subsystem, such as porosities, pressure heads, flow rates and hydraulic conductivities, not only must be referred to its geometrical centre and to the midpoint of the time interval but also must be, in some way, properly averaged in space and time. Therefore, if the output of a measuring device can only be considered as a fair average for a volume sensibly smaller than the volume of the modelled subsystem, this single output cannot be extended to the whole subsystem. Similar considerations apply to an observation time interval. To be consistent, the size of the rock mass affected by the measurement and the size of the subsystem must be quite comparable. If not, a coherent handling of the measurement data is imperative. These remarks must be kept in mind when calibrating numerical models against a few field measurements or test results located in strategic points of the flow domain. 1.1.4

Rep r e s e n t a t i ve e l e m e n t a r y vo l um e

Bear introduced the elegant concept of the representative elementary volume, abbreviated as REV (Bear, 1972). For quasi-homogeneous rock masses, this concept can be recognized in a physical or a statistical sense, but always associated to an elected class of lithologic or structural features. In a physical sense, REV corresponds to the minimum size of a test volume that will probably give significant average outcomes that can be extended to the whole volume of a quasi-homogeneous fractured rock mass. In a statistical sense, REV corresponds to the smallest sampling volume allowing a confident statistical description of the character and physical properties of the whole volume. Depending on the chosen observation scale, the REV size and the pseudo-continuous subsystem size may be of the same order. However, for an erratically inhomogeneous fractured rock mass the REV concept can hardly be applied. 1.1.5

H yd r au l i c va r i a b l e s

1.1.5.1 Int ro d u c ti on In brief, a pseudo-continuous groundwater model simulates the space and time variation of two dependent measurable physical quantities within the flow domain. The first


Fundamentals

9

x3

x2 x1

Pseudo-continuous subsystem defined by the extremity of a position vector r

Figure 1.7 Coordinates (x1 , x2 , x3 ) of the extremity of a position vector r defines the geometric centre of a pseudo-continuous subsystem.

one, dubbed hydraulic gradient, measures the spatial decay rate of the energy carried by a percolating water particle. The second one, called the specific discharge, measures the intensity of the resulting flow. Both variables are interconnected by a third parameter, called the hydraulic conductivity. Jointly, these three interrelated variables adequately describe the hydraulics of a groundwater system. They must be referred to the geometrical centre of each pseudo-continuous subsystem (see fig. 1.7). The hydraulic head is described by a scalar quantity, i.e. by a single number, but the hydraulic gradient and the associated specific discharge are described by vector quantities. A vector quantity may be split into three components referred to a rectangular Cartesian coordinate system. These three components can be converted to magnitude and direction measurements. The hydraulic conductivity that typifies an isotropic pervious media is condensed in a simple scalar quantity. However, for an anisotropic media this quantity can only be described by a second order tensor. Generally, the description of a second order tensor requires nine components properly referred to an arbitrary rectangular Cartesian coordinate system. These nine components can be reduced to three if suitably referred to the particular directions of three privileged axes. 1.1.5.2 S pe c i f ic d i s c h a r g e As for soil mechanics, the void volume Vv contained in the interior of a volume V of a pseudo-continuous subsystem corresponds to the total volume that is occupied by the non-solid phase, i.e. gas and water. It may be associated with conduits, fractures and other types of discontinuities, including minute openings. The effective void volume Ve excludes from the void volume Vv not only all hydraulically isolated pores or closed discontinuities but also all immobilised and very thin water layers bonded to the solid phase surface. Consequently, Ve is somewhat smaller than Vv . The effective porosity ne equals the ratio between the effective volume Ve and the total volume V, i.e. ne = Ve /V. It has no units and for hard rocks effective porosities seldom attain 1 to 2% of total volume, while they may range from 5 to 15% for consolidated clastic rocks and up to 25% for unconsolidated sands and gravels (see fig. 1.8). As formulated by Bear, the groundwater average effective velocity ve ascribed to the centre of the pseudo-continuous subsystem corresponds to the vector average of the


10


Free liquid phase: flowing water Solid phase: intact rock

Adherent liquid phase: pellicle water

Figure 1.8 Schematic association of the intact rock, the percolating water and the immobilised pellicle water. Free water particles seep with varying speeds. The average effective velocity ve of the moving water ascribed to the centre of the pseudo-continuous subsystem corresponds to the vector-average of the velocities of all water particles vP percolating throughout the effective voids during an observation time interval δt.

velocities of all water particles vP percolating throughout the effective voids during an observation time interval δt (Bear et al., 1968). The specific discharge q, also called Darcy’s velocity (Darcy, 1856), may be determined by the product of the effective velocity ve and the effective porosity ne , i.e. q = ne · ve (see addendum 1.1 on page 39). It is important to note that the specific discharge q is always smaller than the effective velocity ve . The effective velocity ve and the specific discharge q are collinear vector quantities, carrying the units of a velocity LT−1 . Both are tangent vectors to the pseudostreamlines. They describe vector fields q(r, t) and ve (r, t) referred to an arbitrary Cartesian frame that are functionally related to the space-time coordinates (x1 , x2 , x3 , t) or, in compact symbolic lettering, to (r, t). Percolating water, entering or leaving the boundaries of a pseudo-continuous system, can only seep across the voids of the boundary elements δS (see fig. 1.9). Then, the elementary discharge δQ traversing an area element δS result from the scalar product of the effective voids area ne δS by the effective velocity ve : δQ = ve ne δ S = |q| |δ S| cos (ve , us )

(1.1)

In the above expression (ve , uS ) denotes the angle between the vector ve and the exterior normal uS . By the convention adopted in this book, δQ is positive for inflows and negative for outflows.

1.1.5.3 H y d ra u l i c g r a d i en t A percolating water particle consumes part of its own energy as it moves and as time elapses. The total stored energy associated to a percolating water particle, dubbed hydraulic head, customarily symbolised by H, is a scalar quantity that results from


Fundamentals

δS ⫽ us·δS

11

Ve

δS

δS

Figure 1.9 Any area element δS of a pseudo-continuous subsystem boundary S is a vector quantity defined by the product of its area δS and its exterior normal uS . Table 1.1 Common normalised expressions of the total hydraulic head H. Normalised hydraulic head H

Type of energy

Referred to the density of water ρ (L2 T−2 )

Referred to the unit weight of water ρg (L)

Referred to the unit volume L3 (ML−1 T−2 )

Gravitational Piezometric Kinetic

gZ p/ρ v2e /2

Z p/ρg v2e /2g

ρgZ p ρ v2e /2

the sum of three kinds of energy: gravitational, piezometric and kinetic. The gravitational head corresponds to the potential energy acquired by a water particle when it is raised from an arbitrary reference level (such as the mean sea level) to an elevation Z. The piezometric head measures the elastic energy stored in a water particle when compressed from the atmospheric pressure Patm to a confining pressure P. To make things easier, a relative scale p is adopted for the piezometric head. In that scale, the barometric pressure rises and falls are bypassed by reckoning the absolute pressure P from the atmospheric pressure Patm , i.e. the absolute atmospheric pressure Patm is taken as the zero origin of the relative pressure scale of p (gauge pressure). Therefore, relative patm = 0 always and the absolute pressure P = p + Patm . The kinetic head measures the additional energy due to its motion. As typical effective velocities ve are very small, usually from 10−10 to 1 m/s, kinetic energies carried by percolating water particles in soils and rocks are commonly neglected. In this case, the total head H is only a potential head. To be homogeneously written, as in table 1.1, these energies should be normalised by the density or unit weight or unit volume of water. A transient or unsteady-state head field H(r, t) characterises a flow domain where the potential head H varies both in space and time. For any point in this field, its


12


potential head H may be described by a functional relationship involving its spacecoordinate r and time-coordinate t. A permanent or a steady-state head field H(r) characterises a flow domain where H solely varies in space and may be described by the space-coordinates r only. Points on both types of fields storing the same potential head H configure an equipotential surface: a moving one for a transient condition or a stable one for a permanent condition. In a zenithal orthogonal coordinate system, the vertical Z-axis points to the zenith. When written in L units, the head H at a generic point (E, N, Z) results from the sum of the gravitational head Z plus the piezometric head p/ρg: H=Z+

p ρg

(1.2)

In another reference frame xi (i = 1, 2, 3) the head H at a generic point (x1 , x2 , x3 ) is expressed as: H = Z0 + x1 cos (Z, x1 ) + x2 cos (Z, x2 ) + x3 cos (Z, x3 )

(1.3)

In the above expression, Z0 denotes the origin elevation and cos(Z, xi ) corresponds to direction cosines of the axes xi referred to the zenithal axis Z. They are measured in the trigonometric sense. The head decay per unit length of a moving water particle, its spatial head decay rate, may be measured by the space gradient of the hydraulic head. Consider two points A and B of a head field H(r, t) joined by a straight line AB to which are respectively associated two position vectors rA and rB and two heads HA and HB . Their separation δr and their differential head δH can be respectively calculated by the vector distance (rB − rA ) and the scalar difference (HB − HA ). Now, assume that A and B are selected in such a manner that the straight line AB is perpendicular to the equipotential surface HA through the point A. In this particular configuration, the scalar-to-vector ratio δH/δr measures the average spatial head decay rate between these two points. Although it is mathematically inappropriate to derive a scalar quantity H by a vector quantity r, or, more generally, to derive a tensor P by another tensor Q, it is convenient to define the symbolic partial derivative ∂P/∂Q as a tensor R whose rank equals the sum of the ranks of P and Q (Koerber, 1962). Then, the scalar-to-vector ratio δH/δr is a vector quantity and has the same direction as δr but pointing towards B if HB > HA and vice versa. As the point B gradually recedes toward the point A all along the normal AB, this vector rate-of-change continuously tends to a limit and when B finally converges to A it defines the local spatial head decay rate. To be in agreement with the general sense of the groundwater flow, practical formulas for flow calculation substitute −∂H/∂r (note the negative sign) for the hydraulic gradient J (see fig. 1.10). For any other point B of any other straight line AB but not perpendicular to the equipotential surface at the point A it is also possible to apply a similar reasoning and define another directional gradient J A . However, the absolute value of J A will always be smaller than the absolute value of JA . Indeed, J A = JA cos (α) where α is the angle BAB . So JA > J A . Being a vector, J results from the vector sum [(∂H/∂x1 )u1 + (∂H/∂x2 )u2 + (∂H/∂x3 )u3 ] where the unit vectors (u1 , u2 , u3 ) are referred to an arbitrary reference frame


Fundamentals

13

Equipotential surface by point A Head: HA Coordinate: rA

Equipotential surface by point B Head: HB Coordinate: rB Pseudo-streamline

Figure 1.10 For homogeneous and isotropic pseudo-systems, all pseudo-streamlines traverse all equipotential surfaces perpendicularly. In this case, the limit of the symbolic ratio −δH/δr when B recedes toA defines the hydraulic gradient JA atA,tangent to the pseudo-streamline.

(x1 , x2 , x3 ). Writing H in L units, these components if referred to a zenithal frame ENZ are:  ∂H  

 1 ∂p − − cos (x, Z) +    ∂x   ρ g ∂x    Jx  ∂H  

   = Jy  = − 1 ∂p   ∂y   − cos (y, Z) +     Jz  ρ g ∂y   ∂H − cos (z, Z) − ∂z

(1.4)

In the above expressions, the terms −[(∂p/∂x)/ρg] and −[(∂p/∂x)/ρg] correspond to the components of the piezometric gradient and the term −cos(z, Z) = −1 in the last row of the column vector of the right member corresponds to the vertical Z-component of the gravitational pull. However, in an arbitrary non-zenithal frame (x1 , x2 , x3 ), these three components of the gravitational gradient are [−cos(z, x1 ), −cos(z, x2 ), −cos(z, x3 )]T . It must be kept in mind that the hydraulic gradient has no units only for heads H written in L units. The head decay per unit time of a moving water particle, its temporal head decay rate, may be measured by the hydrodynamic gradient DH/dt resulting from the summation of the time gradient ∂H/∂t and the product of spatial gradient ∂H/∂r by the specific discharge q, i.e. DH/dt = ∂H/∂t + (∂H/∂r)·q (see fig. 1.11). The scalar product (∂H/∂r)·q measures the head spatial decay as the water particle moves during a unit time. In addition, for q = 1·LT−1 the negative value of this product


14


H t

t

t

Water particle

t t

t

(∂H/∂ t )δ t

Pseudo-streamline

(∂H/∂r)qδ t

r

δt

t

Figure 1.11 This schematic figure shows a transient 4D head field H(r, t) and helps to understand the meaning of DH/dt = ∂H/∂t + (∂H/∂r) · q. Since head decays in both time and space, the observer must keep track of two kinds of motion: he must stand still in a fixed point to watch the head time decay or he must go after the moving water particle to watch the head spatial decay.

Cell centre: i, j, k z

z y x x

Figure 1.12 3D cubic cell (left) and 2D slab lattices (right) referred to an arbitrary coordinate system xyz. Hydraulic gradients J at the centre of these cell types can be numerically approximated as linear functions of the total head values at its neighbourhood.

is the hydraulic gradient J. Then, J may formally be defined as a measure of the consumption of the potential energy stored in a percolating water particle when it travels a unit length during a unit time with a unit apparent velocity (see addendum 1.2 on page 40). Numerical approximations of hydraulic gradients are relatively simple but require attention. For 3D cubic lattices referred to an arbitrary frame xyz (see fig. 1.12 left), devised to simulate the hydraulic behaviour of pseudo-continuous pervious media, the hydraulic gradient J components at the lattice centre, defined by the subscripts (i, j, k),


Fundamentals

15

can be estimated by the head values H at its neighbourhood: (Jx )i,j,k = −

Hi+1,j,k − Hi−1,j,k xi+1,j,k − xi−1,j,k

(Jy )i,j,k = −

Hi,j+1,k − Hi,j−1,k yi,j+1,k − yi,j−1,k

(Jz )i,j,k = −

Hi,j,k+1 − Hi,j,k−1 zi,j,k+1 − zi,j,k−1

(1.5)

As the head H corresponds to the sum of the piezometric head p and the gravitational head Z (elevation measured in the zenithal frame):

Zi+1,j,k − Zi−1,j,k 1 pi+1,j,k − pi−1,j,k + (Jx )i,j,k = − ρ g xi+1,j,k − xi−1,j,k xi+1,j,k − xi−1,j,k

Zi,j+1,k − Zi,j−1,k 1 pi,j+1,k − pi,j−1,k (1.6) (Jy )i,j,k = − + ρ g yi,j+1,k − yi,j−1,k yi,j+1,k − yi,j−1,k

Zi,j,k+1 − Zi,j,k−1 1 pi,j,k+1 − pi,j,k−1 (Jz )i,j,k = − + ρ g zi,j,k+1 − zi,j,k,1 zi,j,k+1 − zi,j,k,1 Expressing the components of the gravitational gradient by the direction cosines of the axes xi referred to the zenithal axis Z:

1 pi+1,j,k − pi−1,j,k + cos (x, Z) (Jx )i,j,k = − ρ g xi+1,j,k − xi−1,j,k

1 pi,j+1,k − pi,j−1,k (Jy )i,j,k = − + cos (y, Z) (1.7) ρ g yi,j+1,k − yi,j−1,k

1 pi,j,k+1 − pi,j,k−1 (Jz )i,j,k = − + cos (z, Z) ρ g zi,j,k+1 − zi,j,k,1 The components of the gravitational gradient, i.e. cos(x, Z), cos(y, Z) and cos(z, Z), only depend on the direction of the z-axis with respect to the Z-axis of the zenithal frame. If, instead of an arbitrary xyz coordinate system, the spatial lattice is consistent with the ENZ zenithal frame, these components are: (JE )i,j,k = −

1 pi+1,j,k − pi−1,j,k ρ g Ei+1,j,k − Ei−1,j,k

1 pi,j+1,k − pi,j−1,k ρ g Ni,j+1,k − Ni,j−1,k

1 pi,j,k+1 − pi,j,k−1 =− +1 ρ g Zi,j,k+1 − Zi,j,k,1

(JN )i,j,k = − (JZ )i,j,k

(1.8)

For 2D slab lattices referred to an arbitrary xyz frame, as very thin flattened cells attached to the plane xy, as shown in fig. 1.12 right, it is assumed that the confined


16


water pressure p in between the fracture walls may be taken as unvarying in the z-direction. Indeed, the fracture aperture δz is relatively very small if compared to its x and y dimensions. In this case, conceived to simulate the hydraulic behaviour of a planar fracture attached to the plane xy, the hydraulic gradient J components at the cell centre (i, j, k) are:

1 pi+1,j,k − pi−1,j,k + cos (x, Z) ρ g xi+1,j,k − xi−1,j,k

1 pi,j+1,k − pi,j−1,k =− + cos (y, Z) ρ g yi,j+1,k − yi,j−1,k

δz − cos (z, Z) =− δz

(Jx )i,j,k = − (Jy )i,j,k (Jz )i,j,k

(1.9)

For a horizontal fracture, referred to the zenithal axis Z: (JE )i,j,k = −

1 pi+1,j,k − pi−1,j,k ρ g Ei+1,j,k − Ei−1,j,k

(JN )i,j,k = −

1 pi,j+1,k − pi,j−1,k ρ g Ni,j+1,k − Ni,j−1,k

(1.10)

(JZ )i,j,k = −1 For linear tubes conceived to simulate the hydraulic behaviour of straight elements of solution conduits, shafts and galleries and even rivers, all previous procedures can be simply adapted. A hydraulic gradient J referred to an arbitrary frame xyz may be described in the zenithal frame ENZ by the following matrix operation: 

   JE Jx JN  = R Jy  JZ Jz

(1.11)

The rotation matrix [R] is defined by: 

 cos (x, E) cos (y, E) cos(z, E) R = cos (x, N) cos (y, N) cos(z, N) cos (x, Z) cos (y, Z) cos(z, Z)

(1.12)

The angular measures (x, E), (x, N) . . . (z, Z) comply with the trigonometric rules. The magnitude and direction of J remains invariant under this rotation.


Fundamentals

17

Pseudo-streamline

Water particle Hydraulic gradient J

Specific discharge q

Figure 1.13 Specific discharge q and hydraulic gradient J at an arbitrary point. Except for homogeneous and isotropic pervious soils and rock masses, both vectors q and J are normally noncollinear.

1.1.6

H yd r a ul i c c o n d u c t i vi t y

1.1.6.1 In t ro d u c ti on To any pervious pseudo-continuous subsystem can be attributed a physical property, called its hydraulic conductivity, that measures its ability to transfer groundwater from one point to another through its interconnected pores, fractures, conduits and other open discontinuities. Assuming a laminar flow, the specific discharge q, the hydraulic gradient J and the hydraulic conductivity [k] are linearly interrelated: [hydraulic conductivity] α [specific discharge]/[hydraulic gradient]

(1.13)

This linear relationship was empirically discovered in 1856 by the French engineer Henry Philibert Gaspard Darcy and bears his name as Darcy’s law. It gives a scalar quantity, i.e. a zero order tensor, for homogeneous and isotropic pervious media. However, as the specific discharge q and the hydraulic gradient J are vectors, i.e. first order tensors, their vector-to-vector ratio implies a second order tensor, generally written as a matrix [k] (see fig. 1.13). Referred to an arbitrary orthogonal frame xyz, the relationship between the specific discharge q, the hydraulic gradient J and the general hydraulic conductivity [k] is written in full matrix lettering as: 

  qx kxx qy  = kyx qz kzx

kxy kyy kzy


  Jx kxz kyz  Jy  kzz Jz

(1.14)

18


Expanding the right-hand product:    kxx Jx + kxy Jy + kxz Jz qx qy  = kyx Jx + kyy Jy + kyx Jz  qz kzx Jx + kzy Jy + kzz Jz 

(1.15)

Note that each component qi of the specific discharge q comes from the sum of three summands qij = kij Jj each one parallel to an i-direction. However, each group of the three summands parallel to the same direction is driven by all components of the hydraulic gradient J:    qxx + qxy + qxz qx qy  = qyx + qyy + qyz  qz qzx + qzy + qzz 

(1.16)

The matrix [k] is symmetric, i.e. kij = kji (i or j = x, y, z). The diagonal elements kii are always positive but the off-diagonal kij elements can be positive or negative. The magnitude of each component kij is proportional to the size of the interconnected discontinuities (pores and/or fractures and/or conduits) of the pervious soil or rock mass. The quantity, the value, the algebraic signal and the location of each off-diagonal element kij depend on the geometric symmetry of the 3D pattern and the size of these discontinuities. As the main diagonal components kii are always positive quantities, all diagonal ii-summands qii point to the same direction of Ji . However, to match the correct flow direction, all off-diagonal ij-summands qij make ±90◦ with respect to Jj depending on the algebraic signal of the off-diagonal components kij (+90◦ for positive kij and −90◦ for negative kij ). A positive signal indicates that the strike and dip of the fracture constrain the direction of the groundwater flow to point to the same direction of the hydraulic gradient. A negative signal indicates the contrary. In a particular frame, called principal axes, only the elements kii of the main diagonals are not equal to zero. These non-zero elements are called the eigenvalues of [k] (normally lettered as ki , i.e. as k1 , k2 and k3 ) and they measure the maximum, in between and minimum hydraulic conductivities along the principal axes of [k]. Each eigenvalue ki is defined by a set of three direction cosines, i.e. ai1 , ai2 and ai3 , called its eigenvectors (see matrix diagonalisation techniques for finding eingenvalues and corresponding eigenvectors). Using matrix notation: 

k1 eigenvalues of [k] =  0 0

0 k2 0

 0 0 k3

 ai1 eigenvectors of [ki ] [i = 1, 2, 3] = ai2  ai3

(1.17)




(1.18)

Fundamentals

19

1.1.6.2 F ra c t u r es a n d c on d u i ts The concept of transmissivity applied to a horizontal and isotropic fracture is quite similar to the one applied, for example, to a flat bed of pervious sedimentary rock. It is applied to a unit wide strip and its dimension is L2 T−1 . By simple premises, it is possible to presume that the magnitude of the transmissivity of a fracture depends on its “equivalent’’ hydraulic aperture e, on the interstitial permeability k of its infilling and on the extent κ of the distribution of its infilling over the fracture area. The parameter κ is the ratio between the partially filled area Aκ and the total area A, i.e. κ = Aκ /A. As the partially filled area Aκ varies between 0 and A, the parameter κ varies between 0 and 1, i.e. 0 ≤ κ ≤ 1. For completely filled fractures, i.e. defined by κ = 1, their average transmissivity T1 may be estimated by: T1 = k e

(1.19)

For completely clean fractures, i.e. for κ = zero, their average transmissivity T0 may be estimated by: T0 = c e3

(1.20)

In formula 1.13, c is an empirical coefficient that must be dimensionally consistent with the employed units (see Louis, 1969 and Quadros, 1982). As frequently observed at leaking rock outcrops or at underground excavations, groundwater usually percolates along a network of irregular and braided flow paths within the fracture itself. Moreover, the thickness of the open fractures and fracture zones varies from place to place, normally from 0.01 mm to 1 mm for single fractures and from 1 mm to 100 mm for narrow fracture zones. Consequently, the so-called “equivalent’’ hydraulic aperture e of a fracture or fracture zone is an ideal concept and corresponds to a value that may be cautiously employed to estimate the average fracture transmissivity T, as defined by the above functional relationships. For partially filled fractures, i.e. for 0 < κ < 1, the average transmissivity Tκ may be approximated by the Maxwell mixture rule: Tk = (T0 )1−κ (T1 )κ

(1.21)

In fact, more or less reliable estimates of fracture transmissivities for laminar flows require ingenious field tests accurately performed, as discussed later. Actually, the hydraulic regime of the groundwater flow throughout a pervious rough and uniform fracture depends on the magnitude of the hydraulic gradient J, on the fracture average aperture e and on its relative roughness δe/e, where δe is the modal height of the fracture asperities (see fig. 1.14). For clean fractured rocks, normally defined by “equivalent’’ hydraulic apertures of e < 1 mm and δe/e around 1/3, the groundwater flow far from wells or springs may normally be taken as laminar because the natural hydraulic gradients J seldom exceed 1 (Franciss, 1970; for a detailed account about these influences, see Louis, 1969).


20


Average fracture aperture (mm)

6 5 TURBULENT

4 3 2 LAMINAR

1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hydraulic gradient (-)

Figure 1.14 Approximate empirical boundary, as a function of the average fracture aperture e and the hydraulic gradients J, separating the laminar from the turbulent flow domain for relative roughness δe/e around 1/3 (numerical data from Louis, 1969).

The hydraulic transmissivity of a non-horizontal and non-isotropic fracture must be treated as a second order tensor [T]. For instance, the anisotropic hydraulic transmissivity [T] of an almost planar fracture (referred to an orthogonal coordinate system xyz, whose axis z is normal to the fracture plane and the axes x and y coincide with the maximum and minimum transmissivity directions) is (see fig. 1.15): 

Tx [T] =  0 0

0 Ty 0

 0 0 0

(1.22)

In this case, referred to the fracture proper axes xyz, the relationship between the fracture discharge θ per unit width, the hydraulic gradient J and the hydraulic transmissivity [T] is:    θx Tx  θy  =  0 θz 0

0 Ty 0

  0 Jx 0 Jy  Jz 0

(1.23)

As intuitively expected, the z-component Tz of the tensor [T] referred to the fracture proper frame xyz must be zero. As a result, the z-component θz of the fracture discharge vector referred to this frame is also zero, as results from the right-hand product, despite the fact that the z-component of the gravity driving pull Jz is not zero:     θx Jx Tx θy  =  Jy Ty  θz 0


(1.24)

Fundamentals

21

z Z-zenithal axis

x Fracture discharge per unit width (in the fracture plane) Hydraulic gradient (not in the fracture plane)

Component of the hydraulic gradient (in the fracture plane)

y

Almost planar fracture Gravitational Z-component of the hydraulic gradient

Figure 1.15 Almost planar and hydraulically anisotropic fracture.The orthogonal coordinate system xyz has its z-axis normal to the fracture plane and axes x and y defined by the maximum and minimum transmissivity directions.The fracture discharge θ per unit width stays confined to the fracture plane and is only driven by the component of the hydraulic gradient J projected on the fracture plane. For a hydraulically isotropic fracture, the specific discharge and the projected gradient are aligned. Note the out-of-plane gravitational components of the hydraulic gradient referred to the proper axes xyz and to the zenithal axis Z.

The components of the transmissivity tensor [T] with respect to a zenithal frame ENZ come from the following matrix rotation:     TEE TEN TEZ Tx 0 0 TNE TNN TNZ  = R  0 Ty 0 RT (1.25) 0 0 0 TZE TZN TZZ The rotation matrix [R] is the same previously defined (see equation 1.17). The resulting matrix is symmetric, i.e., TEN = TNE , TEZ = TZE and TNZ = TZN . This matrix is an orthogonal square matrix and the elements of its three columns correspond respectively to the direction cosines of the unit vectors of the x-axis, y-axis and z-axis with respect to the zenithal reference frame ENZ. Then, its transpose equals its inverse, i.e. RT = R−1 . Consequently, the matrix transformation R{q} = R[T]RT· R{J} holds. As a result, referred to the zenithal frame ENZ, the relationship between the discharge θ per unit width, the hydraulic gradient J and the hydraulic transmissivity [T] can be written as (Rocha and Franciss, 1977):      θE TEE TEN TEZ JE θN  = TNE TNN TNZ  JN  (1.26) θZ TZE TZN TZZ JZ It is important to keep in mind that all matrix equations of the type θ = [T] · J, either referred to the fracture “proper’’ coordinate system xyz or to the zenithal frame ENZ, relate the same vectors θ and J via the same tensor [T].


22


Modelling groundwater flow by taking into account the interplay of many arbitrary fractures and applying for each one a particular relationship θ = [T]·J is difficult. However, if less accuracy and less detail are accepted and if Darcy’s law is adapted to discontinuous systems, it is relatively simple to construct a pseudo-continuous model for a rock mass implicitly incorporating several arbitrary discontinuities (Franciss, 1970). To do this, the transmissivity [T] of an arbitrary discontinuity traversing a subsystem of volume δV = δE·δN·δZ may be substituted by a hydraulically equivalent conductivity tensor [k] with the following expression:   TEE TEN TEZ √ √ √    δZδE δEδN    δNδZ k11 k12 k13  T TNN TNZ  NE   √ √ |k| = k21 k22 k23  =  √ (1.27)    δEδZ δZδE δEδN k31 k31 k33    TZE TZN TZZ  √ √ √ δNδE δNδE δEδN Then, a more tractable relationship q = [k]·J holds. Example 1.1 helps to understand the above concepts. Example 1.1 If referred to the fracture “proper’’ coordinate system, the anisotropic transmissivity tensor [T] of an almost planar and anisotropic fracture attached to the plane xy is described by (see fig. 1.16):     Txx 0 0 0 0 1 × 10−4 2    m T =  0 Tyy 0 =  (1.28) 0 5 × 10−5 0 s 0 0 0 0 0 0 z

y Tyy

Txx

x

Figure 1.16 Planar fracture referred to its “proper’’ reference frame xyz. The Txx and Tyy transmissivity directions match the x-axis and y-axis. The z-axis is normal to the fracture plane. The hydraulic conductivity anisotropy may be measured by the ratio |Txx |/|Tyy |.


F u n d a m e n t a l s 23

As shown in fig. 1.17 the attitude of this fracture, referred to the zenithal ENZ frame, may be defined by its dip φ and dip direction χ. Then, the transmissivity tensor T referred to this frame can be determined by a simple matrix rotation operation. For dip φ = 30◦ and dip direction χ = 130◦ , the rotation operator is: Z

Fracture dip

N

Rotation angle: + from X to N Dip direction: + from N to X E

Figure 1.17 Planar fracture referred to the zenithal ENZ frame. The dip and dip direction are indicated.



cos (χ)  R = −sin (χ) 0

cos (φ)sin (χ) cos (χ)cos (φ) −sin (φ)

  sin (χ)sin (φ) −0.643 0.663   cos (χ)sin (φ) = −0.766 −0.557 cos (φ) 0 −0.5

 0.383  −0.321 0.866

(1.29) Then, the transmissivity tensor T referred to the ENZ frame is:   TEE TEN TEZ   T TNE TNN TNZ  = R T R TZE TZN TZZ   6.332 × 10−5 3.078 × 10−5 −1.659 × 10−5  m =  3.078 × 10−5 7.418 × 10−5 1.392 × 10−5  s −1.659 × 10−5 1.392 × 10−5 1.25 × 10−5 (1.30) Considering a pseudo-continuous cuboid subsystem of volume δV = δE · δN · δZ defined by edge lengths δE = 100 m, δN = 125 m and δZ = 150 m, the above hydraulic transmissivity tensor can be substituted by an equivalent hydraulic conductivity [k]:   TEE TEN TEZ √ √ √  δNδZ   δZδE δEδN    4.72 × 10−7 2.513 × 10−7 −1.514 × 10−7   T TNN TNZ   NE   √ √  =  2.513 × 10−7 6.057 × 10−7 1.271 × 10−7  √   δEδZ δZδE δEδN −7 −7 −7   −1.514 × 10 1.271 × 10 1.141 × 10  TZE TZN TZZ  √ √ √ δNδE δNδE δEδN (1.31)


24


The physical meaning of the positive or negative signs of the off-diagonal parameters of the transmissivity tensor k is as follows: • • •

kEN = 2.513 · 10−7 : positive JN induce positive flow qEN = kEN · JN along the E-axis and vice versa. kEZ = −1.514 · 10−7 : positive JZ induce negative flow qEZ = kEZ · JZ along the E-axis and vice versa. kNZ = 1.271 · 10−7 : positive JZ induce positive flow qNZ = kNZ · JZ along the N-axis and vice versa.

Observation of the geometric relationships depicted in fig. 1.17 helps how to be sure about these conclusions.

The same arguments previously applied to planar fractures hold for tubular conduits filled or partially filled with a pervious matrix depicting a linear hydraulic behaviour. To keep the same rotation formula, its axis must be attached to the y-axis of its “proper’’ reference frame and, in this case, its hydraulic transmissivity is (see fig. 1.18): 

0 0  |T| = 0 Tyy 0 0

 0 0 0

(1.32)

Using appropriate and specific coefficients, these tubular tensors may also model rivers segments or inclined shafts and galleries, after avoiding non-linearity by using approximate solutions and adequate inner boundary conditions. As shown example 1.2, the same theoretical principles applied to fractures may be used to describe the transmissivity tensors of tubular conduits.

y

Figure 1.18 A tubular conduit. To keep the same rotation formula, its axis must be attached to the y-axis.


F u n d a m e n t a l s 25

Example 1.2 The transmissivity tensor [T] of a tubular conduit attached to the y-axis of the plane xy is described by (see fig. 1.19):     0 0 0 0 0 0 2    m T = 0 Tyy 0 = 0 1 × 10−3 0 (1.33) s 0 0 0 0 0 0

z

Tyy y x

Figure 1.19 Tubular conduit referred to its “proper’’ xyz reference frame. The Tyy transmissivity direction matches the y-axis.

The conduit geometry referred to the zenithal ENZ frame may also be defined by its dip φ and dip direction χ, as shown in fig.1.20. The transmissivity tensor T referred to this frame can be determined by a matrix rotation operation. In this case, dip φ = 20◦ and dip direction χ = 160◦ . Then, the rotation operator is: 

   cos (χ) cos (φ) · sin (χ) sin (χ) · sin (φ) −0.94 0.321 0.117     R = −sin (χ) cos (χ) · cos (φ) cos (χ) · sin (φ) = −0.342 −0.883 −0.321 0 −sin (φ) cos (φ) 0 −0.342 0.94 (1.34) Z

Conduit dip

N Rotation angle: ⫹from X to N

E

Dip direction: ⫹from N to X

Figure 1.20 Tubular conduit referred to the zenithal ENZ frame. The dip and dip direction are indicated.


26


The transmissivity tensor T referred to the ENZ frame is:   TEE TEN TEZ   T TNE TNN TNZ  = R · T · R TZE TZN TZZ  1.033 × 10−4 −2.838 × 10−4  = −2.838 × 10−4 7.797 × 10−4 −1.099 × 10−4 3.02 × 10−4

 −1.099 × 10−4 2 m 3.02 × 10−4  s 1.17 × 10−4 (1.35)

Considering a pseudo-continuous subsystem of volume δV = δE · δN · δZ delimited by δE = 100 m, δN = 125 m and δZ = 150 m, the hydraulic transmissivity can be substituted by an equivalent hydraulic conductivity [k]: 

√

TEE

TEN √ δZ · δE TNN √ δZ · δE TZN √ δN · δE

 δN · δZ   T NE  √  δE · δZ   TZE √ δN · δE  7.7 × 10−7  = −2.317 × 10−6 −1.003 × 10−6

√

TEZ



δE · δN   TNZ   √  δE · δN   TZZ  √ δE · δN −2.317 × 10−6 6.366 × 10−6 2.757 × 10−6

 −1.003 × 10−6 m 2.757 × 10−6  s 1.068 × 10−6

(1.36)

For erratic discontinuities labelled a, b, c . . . defined by their equivalent hydraulic conductivities |ka |, |kb |, |kc | . . ., the resulting equivalent hydraulic conductivity for a pseudo-subsystem can be approached by a “parallel coupling’’ if the hydraulic gradients within the subsystem are substituted by an average value (see addendum 1.3 on page 41): |k| = |ka | + |kb | + |kc | + . . .

(1.37)

Written in full matrix lettering: 

[(k11 )a + (k11 )b + (k11 )c + . . . ] |k| = [(k21 )a + (k21 )b + (k21 )c + . . . ] [(k31 )a + (k31 )b + (k31 )c + . . . ]

[(k12 )a + . . . ] [(k22 )a + . . . ] [(k32 )a + . . . ]

 [(k13 )a + . . . ] [(k23 )a + . . . ] [(k33 )a + . . . ] (1.38)

It is implied in the above summation rule that minor hydraulic head losses at the discontinuity intersections can be neglected and the “modeller’’ accepts that the accuracy and detail of the simulation results suit his needs at the selected scale of observation.


Fundamentals

27

Discontinuity to exclude

Discontinuity to include

Figure 1.21 All discontinuities that do not traverse the subsystem from side to side must be excluded from the summation matrix. The influence of these discontinuities may be approximately incorporated in the intact rock permeability.

The hydraulic conductivity tensor |kr | of the rock mass must be added to the resulting tensor |k|. Then, the final hydraulic conductivity tensor |K| for the fractured rock mass results from: |K| = |kr | + |k|

(1.39)

To work with a stable algorithm, do not neglect the influence of |kr |. To give consistent results, the summation rule given above is restricted to uninterrupted discontinuities that cut the subsystem from side to side (see fig. 1.21). Occasionally an almost impervious discontinuity, such as a thin diabase dike, a fracture completely sealed with calcite or clay material, can restrain the percolation. In this case, the eigenvalues of the permeability tensor of the rock block may be approximately constructed as follows: •

•

First hypothesis: all pervious discontinuities that traverse the subsystem maintain their connectivity as they cross the impervious discontinuity. In this case, only the eigenvalues of the permeability tensor [kr ] of the intact rock matrix are reduced before applying the summation rule. This may be calculated approximately by multiplying [kr ] by (kf /kr )p /kr , where kr is the average rock matrix original permeability, kf V,t − wS · < [ai hi (r, t) − H(r, t)], hi (r, t) >S,t = 0

(2.58)

For the weighted inner products, neither the minimum constraint within the flow domain nor the minimum constraints at the flow domain boundaries are exactly satisfied. How well they approximate the solution in each location depends on the relative magnitude of the weighting coefficients wV and wS . The least squares method also holds when the differential operators DV and DN are applied to the residuals [h(r, t) − H(r, t)]. In this case, the unknown coefficients aj , after minimising the sum of the squared residuals ε2V and ε2N , correspond to the solution of the following simultaneous equations (see addendum 2.3 on page 89): tf V DV 0

n

[ai hi (r, t) − Q(r, t) [hj (r, t)]dv dt = 0

i=o

0

for all j elementary solution hj (r, t)

tf S DN 0

0

n

[ai hi (r, t) − qn (r, t)

hj (r, t)

i=o

3

(2.59)

αk dS dt = 0

k=1

for all j elementary solution hj (r, t)

(2.60)

Both requirements can also be combined (occasionally using weighting coefficients). Then, using the convenient inner product notation, these conditions can be simply written as: < DV · ai hi (r, t) − Q(r, t), hj (r, t) >V,t − < DN · ai hi (r, t) − qn (r, t), hj (r, t) >S,t = 0

(2.61)

Example 2.5 clarifies this method.

Example 2.5 Parallel linear drains buried under an embankment must be designed to keep the rising water table, caused by the infiltration of 75% of a continuous and extremely exceptional 100 mm rain fall over a 4 period, below a predetermined maximum level of 0.5 m. This gives an average infiltration rate of 18.75 mm/day (see fig. 2.7).


A p p r o x i m a t e s o l u t i o n s 69

The rising water table may be approximated by a quadratic polynomial in x (x-distance measured from the drains’ centreline) multiplied by a sine function in t (elapsed time since the start of the continuous rain):

H(x,t,T) = (a0 + a1 x + a2 x2 ) sin

πt 2T

+ HL

(2.62)

In this formula, HL is the vertical distance from the embankment’s impervious base to the drain level. Obviously, the units of the numerical coefficients must be compatible, i.e. a0 : [L], a1 : [−], a2 : [L−1 ], HL : [L] and x: [L]. Flow symmetry implies zero horizontal discharge at the drains’ centreline, i.e. qn = 0, at x = 0, corresponding to the following Neumann condition at x = 0:

−k

∂ (H + HL ) ∂x

=0

(2.63)

x=0

Substituting H for the selected approximation:

πt ∂ 2 =0 −k (a0 + a1 x + a2 x ) sin + HL ∂x 2T x=0

(2.64)

Deriving:

−k(a1 + 2a2 x) sin

1 t π 2 T

=0

(2.65)

x=0

From which a1 = 0. The draining capacity is designed to assure zero hydraulic pressure the drains’ centres. Then, the following Dirichlet condition at x = L is:

(a0 + a2 L2 ) sin

πt 2T

= HL

+ HL

(2.66)

x=L

From which a2 = −a0 /L2 . Taking into account the preceding values for a1 and a2 , the expression for the approximant is reduced to:

1 1 t H(x,t,T) = a0 1 − 2 x2 sin π + HL L 2 T

(2.67)

Now, the least squares method can be employed to find a0 . The unconfined Dupuit’s continuity equation in (x, t) and the Neumann condition at (L, t) for a 1 m wide strip are,


70


respectively: ∂ ∂ ∂ ρH −k (H + HL ) + (nρH) − ρω = 0 ∂x ∂x ∂t ∂ =0 −k [(H + HL )m] − qn ∂x x=L

(2.68)

To effectively drain, qn must balance all water accretion ωL, i.e. qn = ωL. Substituting the last expression of H(x, t,T) in these two conditions and evaluating:

(a0 )2 2 1 t 2 1 2 1 t 2 k 2 x sin k + 2ρ(a ) π 1 − sin x ... π 0 L4 2 T L2 2 T L2

1 2 1 t π 1 π − ρω = 0. + nρa0 1 − 2 x cos 2 L 2 T T

−4ρ

a0 m 1 t 2ρk 2 × sin π − ρωL = 0. L 2 T

(2.69)

(2.70)

Then, the squared residuals can be minimised within the flow domain and the drain boundaries. Making hj = 1 and combining these two requisites considering equal weighting coefficients: 



  (a0 )2 2 1 t 2 1 2 2 −4ρ x sin k + 2ρ(a ) π 1 − x 0     4 2 2 T   T L  L 2

L     1 2 1 k   sin 1 π t  dx dt . . . . . . + nρa0 1 − 2 x     2 2 T L 2 L    

0 0    1 t π cos π − ρω 2 T T T

a0 1 t 2ρk 2 L sin π m − ρωL dt = 0 L 2 T

+

(2.71)

0

Evaluating symbolically these two integrals gives: 4 a0 m 2 ρLna0 − 2ρωLT + Tρk =0 3 π L

(2.72)

Solving for a0 : a0 = 3ωL2 T

π L2 nπ + 6T k m

(2.73)

Finally, substituting a0 in the last expression for H(x, t,T):

πt π 1 2 + HL H(x,t,T) = 3ωL T 2 1 − 2 x sin L nπ + 6T k m L 2T 2


(2.74)


71

Results for different periods are shown in fig. 2.7. WT rise: 100 mm rain in 4 days Static WT before 4 days of 18.75 mm/day/m² infiltration WT rise after 1 day of rain infiltration WT rise after 2 days of rain infiltration WT rise after 3 days of rain infiltration WT rise after 4 days of rain infiltration Embankment top Linear parallel drain Linear parallel drain

WT rising heigth (m)

0.5 0.4 0.3 0.2 0.1 6

4

2

0

2

4

6

Distance to centre line (m)

Figure 2.7 Water table rising controlled by parallel linear buried drains.

2.5.4

Ga ler ki n’s m e t h o d

2.5.4.1 O rt ho g on a l i ty A “vector’’ of an n-dimensional vector space has n components. Similarly, a linear function h(x) can also be regarded as a “vector’’ of a ∞-dimensional function space, having an infinite number of components h(x) defined for x varying from a to b. This concept can be easily extended from the one-dimensional argument x to the fourdimensional argument (r, t). Hence, the solution H(r, t) of a linear partial differential equation and its approximant h(r, t) can be considered as “vectors’’ of “function spaces’’ in which a “plane’’ P can be defined by the “vector’’ h(r, t), for random changes of its numerical coefficients aj . If the “vector’’ H does not belong to this “plane’’ P, then there exists a unique “vector’’ h(r, t) on P, called the “projection’’ of H(r, t) on P, such that the “approximant error vector’’ ε is perpendicular to P and takes its lowest minimum, i.e. its infimum (see fig. 2.8). Then, h(r, t) is the best approximant to H(r, t) in the sense of the minimum least squares method (see addendum 2.4 on page 89). A similar reasoning can be applied for the transformed “error vectors’’ εV , εN and εD . However, the least squares method does not guarantee that the “error vectors’’ ε, εV , εN and εD are orthogonal to the same “plane’’ P nor that the “orthogonalisation’’ of one kind of error implies the “orthogonalisation’’ of the others. Additionally, the least squares method requires a continuous approximant up to the second derivative. These insufficiencies are resolved by Galerkin’s method.

2.5.4.2 Gal e rk i n’s a p p r oa c h

In Galerkin’s method, the coefficients aj of the best approximant aj hj (r, t) are given by a set of simultaneous equations expressing the orthogonality condition for ε, εV , εN and εD to a “plane’’ defined by an arbitrary set of linearly independent base


72


H h–H h

h+bjhj

bjhj

Figure 2.8 “Visualisation’’ of the orthogonality requirement to find the best approximant h(r, t) to a function H(r, t) in the sense of the minimum least squares method. Let the “plane’’ P be defined in the “function space’’ by the “vectors’’ h(r, t) or, cast as a polynomial approximant, by the “vectors’’ ah hj (r, t) as its numerical coefficients aj change arbitrarily. For a particular set of values aj the “vector’’ h(r, t) may match the “orthogonal projection’’ of H(r, t) on P, and in this moment, the “approximant error vector’’ ε = [h(r, t) − H(r, t)] takes its lowest minimum, i.e. its infimum value.

“vectors’’ φj (r, t): < ai hi (r, t) − H(r, t) , φj (r, t) >V,t − < ai hi (r, t) − H(r, t) , φj (r, t) >S,t = 0 (2.75) < DV · ai hi (r, t) − Q(r, t) , φj (r, t) >V,t − < DN · ai hi (r, t) − qn (r, t) , φj (r, t) >S,t = 0

(2.76)

The “plane’’ does not necessarily match the “plane’’ P defined by the approximant “vectors’’ aj hj (r, t) and its transforms by the operators DV and DN (see fig. 2.9). However, that “geometric congruence’’ may be forced if the set of “base vectors’’ {φj (r, t)} that define the “plane’’ is constructed by sequentially integrating the elementary solutions hj (r, t). In addition, the forcing sinks or sources Q(r, t) and the boundary inflows or outflows q(r, t)n can also be approximated by linear combinations of these solutions: DV · ai φj (r, t) − Q(r, t) ≈ DV · ai φj (r, t) − bk φk (r, t)

(2.77)

DN · ai φj (r, t) − qn (r, t) ≈ DN · ai φj (r, t) − ck φk (r, t)

(2.78)

In this case, it can be proved that if the orthogonality condition holds for the approximant error ε, then it also holds for the transformed errors εV , εN and εD . Example 2.6 clarifies Galerkin’s method.



73

hΦ–H hP–H H

hΦ hP

Figure 2.9 “Visualisation’’ of the orthogonality requirement to find the best approximant h(r, t) to a function H(r, t) by Galerkin’s method. The “plane’’ does not necessarily coincide with the “plane’’ P defined by the approximant “vectors’’ h(r, t) and its transforms by the operators DV and DN .

Example 2.6 The variation of the hydraulic head inside a homogenous core of a rock-fill dam may be fairly described by a bilinear approximant referred to a non-zenithal frame (see fig. 2.10): h = a0 x + a 1 z + a z x

(2.79)

The narrow free surface practically matches the x-axis where the falling head can be approximated by −x · cos(i). Then, a simple orthogonality requirement for the Dirichlet condition may be applied along the free surface: < [(a0 x + a1 z + a2 xz) − (−x · cos(i))],1 >free surface = 0 Alternatively, written in full: x1 [a0 x + a1 z + a2 xz − x(−cos(i))]dx = 0

(2.80)

(2.81)

0

Integrating and taking into account that z = 0 at the free surface gives a0 = −cos(i). Applying a similar reasoning to the core upstream face where the hydraulic head is zero (equal to the head reference level): < [(a0 x + a1 z + a2 xz − 0],1 >core upstream face = 0 Alternatively, written in full: z3 (−cos(i)x + a1 z + a2 xz)dz = 0 0


(2.82)

(2.83)

74


0 1 0

z(m )

2 0 3 0

0

4 0

1

5 0

0

6 0 0

2

0

Z zenith H0

z(

m)

x ( 10 m) 20 D Eq ata c u Eq ipo ore u t Eq ipo en Eq uipo tent tial 0 Eq uipo tenti ial 0 .2 H uip ten al .4 H m ote tia 0.6 nti l 0. H mm 8 al H Hm m

4

0

z

5 0

(0, 0)

6

0 0

(x1, 0)

nin (0, z2)

dS

(0, z3)

nout βx

i

(x2, z2)

3 0

x βz

H -H

10 x( Da m 20 Eq m co ) Eq uipo re Eq uipot tentia Eq uipo enti l 0. Eq uipo tenti al 0.42 H m uip ten al 0 H ote tia .6 m ntia l 0. H m lH 8H m m

Figure 2.10 Top left sketch: approximated head h contours inside the core. Bottom left sketch: adopted non-zenithal frame xz. The head zero reference level coincides with the water table. The total head difference at the downstream bottom of the core equals −H. The points (0, 0), (x 1 , 0), (0, z3 ) and (x 2 , z2 ) define the core geometry. The upstream core outward normal equals nin . The outward normal of the seepage face equals nout . The upstream core inclination equals i. The seepage inclination equals β. Right sketch: detail of the head h contours for this example.

Integrating and making x = 0 at the upstream core face gives a1 = 0. Then, the polynomial approximation reduces to: h = −cos(i) x + a2 x z

(2.84)

In this non-zenithal frame, the total head at any point (x, z) is (see Chapter 1, section 1.1.5.3 Hydraulic Gradient): h = ax x + αz z +

P ρg

αx = cos(Z,x) = cos(π − i) = −cos(i) π αz = cos(Z,z) = cos − i = sin(i) 2


(2.85)


75

At the seepage face, the water seeps out into the free atmosphere. Then, the directional gradient ∂P/∂r must be parallel to the outward normal nout . Thus, the components of its direction cosines are βx = cos (nout , x) and βz = cos (nout , z): ∂ ∂ [(−cos(i)x + a·2 xz − α·x x − α·z z)ρg] [(−cos(i)x + a·2 xz − α·x x − α·z z)ρg] ∂x ∂z − =0 β ·x β ·z (2.86) Taking the derivatives and simplifying: (−cos(i) + a2 z − αx )

1 1 − (a2 x − αz ) = 0 βx βz

(2.87)

The head at the upstream seepage face is z · sin(i). Applying a mixed orthogonality requirement at that boundary: < [(a0 x + a1 z + a2 xz) − (z · sin(i))],1 >seepage face + . . . + < [(−cos(i) + a2 z + αx )/βx − (a2 x + αz )/βz ],1 >seepage face = 0

(2.88)

Alternatively, written in full:

S2 S1

[(−cos(i)x + a2 xz) − (z sin(i)))dS]

+

S2

S1

1 1 (−cos(i)x + a2 z − αx ) − (a2 x − αz ) dS = 0 βx βz

(2.89)

Integrating and solving for a2 (taking into account the geometrical relationships between units, levels, point coordinates, direction cosines of the upstream outward normal, face inclinations, etc.):

a2 = 3

sin(i)(z3 )2 + 2z3 cos(i)x1 βx + 2z2 cos(i) + 2z2 αx (βz )3 +(2x1 αz − 2x2 αz )(βx )2 2(z3 )3 (βz )4 + 3(z3 )2 x1 βx + 3(z22 ) (βz )3 + −3(x2 )2 + 3(x1 )2 ) (βx )2

(z3 )2 (βz )4 cos(i) +

(2.90) Substituting a2 in the approximate head gives a fairly analytic solution. The derivative ∂h/∂x of the analytic expression gives the exit gradient in the x direction (see fig. 2.11): sin(i)(z3 )2 + 2z3 cos(i)x1 βx + 2z2 cos(i) + 2z2 αx (βz )3 + (2x1 αz − 2x2 αz )(βx )2 h = −cos(i)x + 3 xz 2(z3 )3 (βz )4 + 3(z3 )2 x1 βx + 3(z22 ) (βz )3 + −3(x2 )2 + 3(x1 )2 ) (βx )2 (z3 )2 (βz )4 cos(i) +

(2.91)


76


2.5

0

2 1.5 1 0.5

10

20

z(m)

z(m)

20

30

40

40

50 Hydraulic gradient (-) 60

Exit gradient 0

5

10 x(m)

15

20

Dam core Equipotential 0.2 H m Equipotential 0.4 H m Equipotential 0.6 H m Equipotential 0.8 H m Equipotential H m

Figure 2.11 Exit gradient in the x-direction and estimated head contours for (x 1 = 15 m, z1 = 0 m), (x 1 = 20 m, z1 = −45 m) and (x 1 = 0 m, z1 = −56.55 m).

2.5.4.3 “We a k s ol u ti on s’’ In the “finite element method’’, the flow domain is partioned in contiguous small subdomains. To each subdomain, appropriately called a “finite element’’, corresponds a local approximant of the solution of the governing equation, subjected to the finite element inherent boundary conditions. To guarantee convergence, as the size of the finite elements decrease, the inter-element continuity for their first and second order derivatives of the local approximants should be preserved. However, the full continuity constraint cannot be applied to approximants that are only continuous up to the first derivative. To avoid that difficulty, it is convenient to transform the differential form



of this constraint so as to eliminate the second order derivative from the first inner product. To do this, consider the full orthogonality constraint: < [DV · ai hi − Q], hj >V,t − < [DN · ai hi − qn ], hj >S,t = 0

(2.92)

Then, integrating by parts the terms DV · ai hi and DN · ai hi of the above requirement, the resulting transformed orthogonality condition can be applied to approximants that need to be continuous only up to the first derivative: < kk · ∂h/∂xk , grad · (hj ) >V,t − < SV · ∂h/∂t, hj >S,t + < Q, hj >V,t − < qn , hj >S,t = 0

(2.93)

These types of approximants are called “weak solutions’’ because they do not fulfil the partial differential equation itself but only its equivalent integral formulation. Moreover, the Dirichlet condition is “strongly’’ imposed by the above requirement but the Neumann condition is only approximately satisfied in the limit, in terms of an average. 2.5.4.4 Va ri a t ion a l n ota ti on If δh(r, t) denotes the total variation of an approximant h(r, t) due to arbitrary variations of its unknown coefficients ai , it follows that: δh = δa0 h0 + δa1 h1 + · · · + δan hn =

n

(δai hi )

(2.94)

i=0

On the other hand, the first orthogonality constraint, as shown below, remains unaffected if it is multiplied by an arbitrary variation δ-aj of any coefficient aj : δaj · < [ai hi − H], hj >=< [ai hi − H], δaj · hj >= 0 Adding up all these j constraints and recalling that δh =

(2.95) δaj · hj :

δaj · < [ai hi −H], hj >=< [ai hi −H], δaj ·hj >=< [ai hi −H], δh >= 0

(2.96)

This sort of integrated notation can be applied to all types of orthogonality requirements. For “weak solutions’’ it is written usually as follows: < kk · ∂h/∂xk , grad · (δh) >V,t − < SV · ∂h/∂t, δh >S,t + < Q, δh >V,t − < qn , δh >S,t = 0

(2.97)

“Variational notation’’ is employed for orthogonality requirements formulated in terms of maxima and minima of functionals (functions of functions), i.e. in terms of the principles of the calculus of variations, one of the oldest mathematical disciplines and developed almost simultaneous with differential and integral calculus. Example 2.7 clarifies the use of the variational notation.


78


Example 2.7 The hydraulic behaviour of an artesian horizontal leaking aquifer (see fig. 2.12) can be described by a one-dimensional steady flow continuity equation and two Dirichlet boundary conditions: −T

∂2 h + ch = 0 ∂x2

0<xL − < C,δH >V − < qk ,δH >k=i,j = 0

(2.105)

Developing in full: δHi (1 − X) (−Hi + Hj ) dX dx − Ci,j [Hi (1 − X) + Hj X] δHj X 0 0 [∂Hi (1 − X)]X=1 [∂Hi (1 − X)]X=0 − q.i =0 (2.106) − q.j (∂Hj x)X=1 (∂Hj x)X=1

1

−δHi δHj


1

80


Integrating the above condition gives a system of two simultaneous equations for each subdomain i-j:

 

Ci , j Ci , j 1− − 1+  3 6  −qi   Hi − =0 (2.107) 

  qj Ci , j  Hj Ci , j 1− − 1+ 6 3 However, as a first-degree approximant gives an average discharge q for each finite element, the terms −qi and qj cancel out. Then, applying that result to the four subdomains A-1, 1-2, 2-3 and 3-B, gives a system of three simultaneous equations for the unknowns H1 , H2 and H3 , after some matrix algebra manipulation. Instead of two Dirichlet boundary conditions, the same problem can be stated for one Dirichlet HA at A and one Neumann condition qB at B. Then, Darcy’s law could determine the missing Dirichlet condition HB : qB = −T

HB − H3 xB − x 3

(2.108)

This example illustrates the essence of the “finite element method’’.

2.5.5 Time-d e p e n d e n t s o l u t i o n s Approximate solutions also simulate unsteady groundwater flow. However, as time boundaries are usually open, with the exception of cyclic behaviour, an approximate functional relationship between time and head variation is not self evident, except in some circumstances. A common approach is to apply a first order linear approximant for very short time intervals and to establish a recursive solution. Example 2.8 illustrates this approach.

Example 2.8 The excess pore pressure dissipation within a nearly saturated and plastic clay core of a rock-fill dam can be reasonably modelled by the product of a quadratic space-dependent polynomial and a time-dependent polynomial (see fig. 2.13): (a + a x + a x2 ) (b + b t) 0 1 2 0 1 u = −L ≤ x ≤ L,t ≥ 0

(2.109)

The starting excess pore pressure u0 at any point of the centreline of the impervious core can be estimated by: u0 = ux=0, t=0 = Bγh


(2.110)


C.L.

Excess pore pressure dissipation

h

Figure 2.13 Excess pore pressure dissipation within a nearly saturated plastic clay core of a rock-fill dam following vertical consolidation inducing horizontal drainage.

Where, B (b-bar) denotes an empirical coefficient (ranging from 0.25 to 0.75 for typical core materials), γ the fill saturated unit weight and h the soil height above the point being considered. The chosen approximant can be written as: u = A0 + A1 + x + A2 x2 + A3 t + A4 xt + A5 x2 t

(2.111)

Where the new coefficients Ai are related to the old coefficients aj and bk as: A0 = a0 b0 A 1 = a1 b 0 A 2 = a2 b 0 A 3 = a0 b 1

(2.112)

A 4 = a1 b 1 A 5 = a2 b 1 Known boundaries and initial conditions help determine four coefficients Ai : –

Initial conditions at x = 0 and t = 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)x=0, t=0 = u0

–

Dirichlet boundary condition at x = −L and t ≥ 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)(x=−L), t≥0 = 0

–

(2.113)

(2.114)

Dirichlet boundary condition at x = L and t ≥ 0: (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t)(x=−L), t≥0 = 0


(2.115)

81

82


–

Neumann boundary condition at x = 0 (horizontal slope):

∂ (A0 + A1 x + A2 x2 + A3 t + A4 xt + A5 x2 t) ∂x

=0

(2.116)

x=0, t≥0

Solving the preceding equations, substituting the expressions for Ai (i = 0, 1, 2, 3, 4) in the new approximant and manipulating algebraically yields:

x2 t u= 1− 2 u0 − A 5 2 L L

(2.117)

Now, the Galerkin method may be employed to determine the remaining unknown coefficient A5 . If cr symbolises the experimental radial consolidation coefficient, then the continuity equation for vertical consolidation and horizontal drainage can be written as: 1 ∂2 u+ 2 ∂x cr

∂ u =0 ∂t

(2.118)

Substituting u for its approximation in the continuity equation gives: ∂2 ∂x2

1−

x2 L2

u0 − A5

t L2

+

1 cr

∂ ∂t

1−

x2 L2

u0 − A5

t L2

= 0 (2.119)

Deriving yields: 2 L2

u0 − A 5

t L2

+

1 cr

x2 L2

1−

A5 =0 L2

(2.120)

Applying a simple orthogonality constraint:

t.f 0

0

L

2 L2

u0 − A 5

t L2

+

1 cr

1−

x2 L2

A5 dx dt = 0 L2

(2.121)

Integrating, solving for A5 and substituting in the new approximant, gives the approximate closed solution:

x2 6 u= 1− 2 1− c t u0 (2.122) r L 3cr tf + 2L2 In this formula, t denotes time from 0 to tf . For a very short interval δt, the average excess pore pressure u can be estimated by making t = δt/2:

x2 1 u= 1− 2 u0 1− (2.123) 2 L 1 + 23 cLδt r

Assuming for each succeeding short time interval δt that the pore pressure at the centreline (i.e. at x = 0) can be taken as the previous average and applying the above solution



recursively yields an expression for u after n intervals δt/2:

x2 1 u1 = 1 − 2 u0 1− 2 L 1 + 23 cLr δt 2

x2 1 u2 = 1 − 2 1− u0 2 L 1 + 23 cLδt r

(2.124)

..................................................... ..................................................... ..................................................... n

x2 1 1− u0 un = 1 − 2 2 L 1 + 23 cLδt r

Then, the excess pore pressure dissipation Un at time step n can be computed as: n un 1 u=1− = 1− (2.125) 2 u0 1 + 23 cLδt r

The graph below shows the excess pressure dissipation U at a horizontal level and at the corresponding centreline. Core width at that level is L = 10 m; the x coordinate is normalised as X = x/L; the radial coefficient of consolidation exemplified is cr = 10−3 cm/s (see fig. 2.14):

U (%)

Pore pressure dissipation 100 80 60 40 20 1

0.5

0 X ()

0.5

initial excess pore pressure

% dissipation after 180 days



1


U (%)

Pore pressure dissipation at centreline 100 80 60 40 20 0

100

200 days

300

400

Figure 2.14 The pore pressure decay. The second-degree approximant in x gives a parabolic pore pressure graph.


83

84


Another alternative to cope with unsteady problems is to combine a time-marching “finite difference’’ algorithm with a “finite element’’ spatial solution. Example 2.9 illustrates this common technique.

Example 2.9 The half-width of the dam core shown in example 2.8 can be partitioned into four finite elements 0-1, 1-2, 2-3 and 3-4. Using matrix notation, the excess pore pressure dissipation at each finite element can be approximated by a “weak polynomial interpolation’’ (see fig. 2.15): Un =

T

fj fj+1

Uj ,n Uj+1,n

(2.126)

C.L.

Excess pore pressure dissipation

h

j

j1

Figure 2.15 Excess pore pressure dissipation within the core of a rock-fill dam. Due to the centreline symmetry, the core half-width can be partitioned into four finite elements 0–1, 1–2, 2–3 and 3–4.

In this approximant, the symbols fj and fj+1 denote the interpolating functions at the end points j and j + 1 of each finite element:

fj = 1 − fj+1 =

x − xj xj+1 − xj

x − xj xj+1 − xj

(2.127)

The capital letters Uj,n and Uj,n denote the excess pore pressure dissipation at the points j and j + 1 at a certain moment of time defined by n · δt where δt is a short time interval and n the number of time steps elapsed since the start of the dissipation process (initial conditions).



The orthogonality requirement (equation 2.93) applied to the finite element (j, j + 1) gives:   T  Uj,n  ∂ fj,n fj xj+1  ∂  dx xj ∂x ∂x fj+1 Uj+1,n fj+1  =

xj+1 1 ∂ xj cr ∂t





T

fj fj+1

+ qj+1,n

  Uj,n  fj  dx . . . fj+1 Uj+1,n

fj fj+1



fj

+ qj,n

fj+1

x=xj+1



(2.128)

x=xj

As a first-degree approximant only gives an average discharge q for each finite element, the terms −qj,n and qj+1,n cancel out. Since the two nodal parameters Uj,n and Uj+1,n are independent of x, they can be taken out of the integrand. Then, considering the values of fj and fj+1 at x = xj+1 and x = xj and integrating, the above constraint yields:

M

Uj,n Uj+1,n

∂ +N ∂t

Uj,n Uj+1,n

=0

(2.129)

Where the time derivative multiplying N can be approximated by a “finite difference’’ quotient: ∂ 1 Uj,n Uj,n+1 − Uj,n = =0 (2.130) ∂t Uj+1,n δt Uj+1,n+1 − Uj+1,n The coefficients M and N are computed as:

xj+1

M = xj

T    x−xj x−xj ∂  1 − xj+1 −xj  ∂  1 − xj+1 −xj  2 dx = x−x x−x j j ∂x ∂x xj+1 − x xj+1 −xj



xj+1

N= xj

1 1 cr

x−x − xj+1 −xj j x−xj xj+1 −xj

xj+1 −xj

T   

1

x−x − xj+1 −xj j x−xj xj+1 −xj



(2.131)

 dx = 2 1 (xj+1 − xj ) 3 cr

Substituting the time derivative by its finite difference analogue and manipulating algebraically yields a time-marching algorithm:

M Uj,n+1 Uj,n = 1 − δt . (2.132) Uj+1,n+1 Uj+1,n N Then, applying this result to the four subdomains 0–1, 1–2, 2–3 and 3–4 gives a system of three simultaneous time-marching equations for the unknowns H0 , H1 , H2 , and H3 after some matrix algebra handling.


86


If, instead of four finite elements, the time-marching algorithm is applied to only one large finite element, defined from the core centreline to the core edge, it is possible to compare the results to those obtained in example 2.8. Now, by a similar reasoning, the recurring formula is:

3 δt n U(n) : = 1 − 2 cr L 2

(2.133)

The graph below (see fig. 2.16) shows the excess pressure dissipation U at a horizontal level and at the corresponding centreline for L = 10 m; x = 0 m to 10 m; cr = 10−3 cm/s, the same parameters used in example 2.8 (see fig. 2.14):

Pore pressure dissipation

U (%)

100 80 60 40 20 10

5

0

5

10

x (m) initial excess pore pressure





Pore pressure dissipation at centreline

U (%)

100 80 60 40 20 0

100

200 Days

300

400

Figure 2.16 The pore pressure decay is almost the same in both examples. However, in this example, the first-degree finite element approximant gives a linear pore pressure graph instead of the parabolic one, as in example 2.8 (see fig. 2.14).



2.6 Addenda to Chapter 2 2.6.1 A d d en d u m 2. 1: C l a s s i f i c a t i o n o f s e c o nd o r de r lin ea r p a r t i a l d i f f e r e n t i a l e q u a t i o ns Consider that a dependent variable U is related to n independent coordinates (x1 , x2 … xn ) by a linear second order partial differential equation. By an appropriate coordinate transform (defined by the solution of an eigenvalue problem), this partial equation loses its mixed derivatives ∂2 U/∂xi ∂xj (i = j) and then can be written as: n

Ai

i=1

n

∂2 ∂ B U + U +CU+D=0 i ∂xi ∂xi2 i=1

(2.134)

Through any point (x1 , x2 … xn ) of the independent variable space pass one or two peculiar hyper-surfaces, called characteristics. For points resting on these hypersurfaces the second order derivatives of U can be mathematically defined but are indeterminate and finite (0/0 indetermination). For 3D (or 2D) systems, these hypersurfaces are associated with the form of a quadric defined by the coefficients Ai . These quadrics can have the shape of an ellipsoid (or an ellipse), a paraboloid (or a parabola) and a hyperboloid (or a hyperbola). This categorisation is generalised for an n-dimensional space. These surfaces propagate within the solution domain all information controlled by the boundary conditions. Then, at any moment of time, these characteristic surfaces separate into two distinct but continuously changing subspaces: the controlled and the controlling domains. Values of U on the controlled domain depend on the values taken by U on the controlling domain at that moment. Any change of U at one or more points on the controlling domain will affect the U values on the controlled domain. The class of a differential equation is revealed by the values and signals of the coefficients Ai of the quadric surfaces: – – –

Elliptic if all Ai are non-zero and of same signal. Hyperbolic if all Ai are non-zero and of the same signal but with one exception. Parabolic if one (or more) Ak is (are) zero but the corresponding Bk is (are) non-zero and the remaining Ai are non-zero and of the same signal.

As the coefficients Ai , Bi , C and D may be functions of the space coordinates xi and of the derivatives ∂U/∂xi but not of ∂2 U/∂xi , the classification of a linear second order partial differential equation may also change more than once within the definition domain, mainly in complex systems. For 3D groundwater flow systems, the dependent variable is the head H and the independent variables are the time-coordinate t and the space-coordinates (x, y, z). A time-independent solution can be described by an elliptic equation that has no real characteristics and no directional restrictions. In this case, its unique “controlling domain’’ merges with its unique “controlled domain’’. As a simple example, consider the steady flow induced by a penetrating well in a confined pervious layer. If this well pumps continuously but preserves a constant drawdown s at the well axis, its pumping rate Q is also constant. However, any change δs in the drawdown not only implies a change δQ in the pumping rate but also alters the heads H of all points of the confined


88


pervious layer. On the other hand, if s never changes, any head disturbance δH at any point of the confined layer will affect all the other heads and also the pumping rate Q. For a transient flow regime, described by a parabolic equation, two identical families of real characteristics separate the “controlling domain’’ from the “controlled domain’’ by parallel planes. Consider the previous example in unsteady state. Then, at any moment tn , the “controlled domain’’ corresponds to points located above the characteristic plane (x, y, z, tn ), i.e. all future heads for t > tn . On the other hand, the “controlling domain’’ corresponds to points located below this plane, i.e. past heads for t < tn defining the history of the system, including its initial values and all precedent boundary values. Therefore, at any moment tn , past values “under’’ the characteristic plane (x, y, z, tn ) control all future values “above’’ this plane. Hyperbolic 3D equations have two real and distinct characteristics that define two crossing surfaces, separating both influencing and conditioned but confined domains. This type of equation does not occur in groundwater flow.

2.6.2 A d d en d u m 2. 2: M i n i m i s a t i o n o f t he s um o f t he squared residuals After electing the most promising approximant, the set of coefficients aj that minimises the sum of the squared residuals ε2v within the flow domain satisfies the following condition:

∂ ∂aj

tf

V

(εV )2 dV dt = 0

for all j

(2.135)

0

0

Or:

tf

0

V

εV

0

∂ εV dV dt = 0 ∂aj

for all j

(2.136)

Replacing, in the last expression, εV with [h(r, t) − H(r, t)], ∂εV / ∂aj with hj (r,t) and simplifying the resulting equation, one obtains:

tf

0

V

[h(r, t) − H(r, t)][hj (r, t)] dV dt = 0

0

for all j elementary solutions hj (r, t).

(2.137)

Writing h(r, t) as a complete polynomial approximant: 0

tf

0

V

n

[ai hi (r, t) − H(r, t) [hj (r, t)] dV dt = 0

i=0

for all j elementary solutions hj (r, t).

(2.138)

The solution of this system of (n + 1) equations defines the unknown coefficients aj .



89

If applied to the flow domain boundaries, this method implies another set of (n + 1) simultaneous equations: 0

tf S

0

n

[ai hi (r, t) − H(r, t)

hj (r, t)

i=0

3

ak dS dt = 0

k=1

for all j elementary solutions hj (r, t)

(2.139)

In the last constraint, αk denote the direction cosines of the outward normal n to the boundary element dS. 2.6.3 A d d en d u m 2. 3: Mi n i m i s a t i o n o f t he s um o f t he s q u a r e d r e s i d u a l s t r a n s f o r m e d by t he di f f e r e nt i al oper a t o r s DV a n d DN Transforming the residuals εV within the flow domain by the differential operator DV and the residuals εS at its boundaries by differential operator DS gives: DV · [h(r, t) − H(r, t)] = DV · h(r, t) − Q DN · [h(r, t) − H(r, t)] = DN · h(r, t) − qn

(2.140)

Then the set of coefficients aj minimising the sum of the squared transformed residuals satisfies two sets of simultaneous equations: tf V n DV · [ai · hi (r, t) − Q(r, t) · [hj (r, t)]dV dt = 0 0

0

i=0

for all j elementary solutions hj (r,t) 0

tf S 0

DN ·

n

[ai · hi (r, t) − qn (r, t) · hj (r, t) ·

i=0

(2.141) 3

αk

dS dt = 0

k=1

for all j elementary solutions hj (r, t)

(2.142)

2.6.4 A d d en d u m 2. 4: T h e c o n c e p t o f “o r t ho g o nal i t y’’ Let the “plane’’ P be defined in the “function space’’ by the “vectors’’ h(r, t) or, cast as a polynomial approximant, by the “vectors’’ aj hj (r, t) as its numerical coefficients aj change arbitrarily (see fig. 2.17). Let (h + bj hj ) be any other “vector’’ on the “plane’’ P. If (h − H) is the infimum, then the “length’’ of the “error vector’’ (h − H) is smaller than the “length’’ of the vector [(h + bj hj ) − H] and the following inequality prevails: < [(h + bj hj ) − H], [(h + bj hj ) − H] > − < (h − H), (h − H) > ≥ 0

(2.143)

Taken into account that ||x||2 = <x, x > and simplifying the preceding inequality: b2j < hj , hj > −2bj < (h − −H), hj > ≥ 0


(2.144)

90


h-H

H h

h + bjhj

bjhj

Figure 2.17 For a particular set of values aj the “vector’’ h(r, t) may match the “orthogonal projection’’ of H(r, t) on P, and in this moment, the “approximant error vector’’ ε = [h(r, t) − H(r, t)] takes its lowest minimum, i.e. its infimum value.

As that result must hold for arbitrary values of bj it necessarily implies a formal orthogonality constraint defined by a null inner product: < (h − H), hj > = 0

(2.145)

References Band, W., 1960, Introduction to Mathematical Phyisics, D. Van Nostrand, Princeton. Chung-Yau, L., 1994, Applied Numerical Methods for Partial Differential Equations, Prentice Hall. Linz, P., 1979, Theoretical Numerical Analysis, New York, Dover.


Chapter 3

Data analysis

3.1 Preliminaries Modelling the spatial distribution of the hydraulic properties of a flow domain is not an easy task as it reflects the distribution of 3D geological features, commonly inferred from insufficient combinations of 2D field mapping and one dimensional logs. The non-conservative nature of geological features rule out simple conservation laws, as for mass and energy, which could guide the evaluation of their spatial distribution. Whenever possible, point inferences for the eigenvalues and eigenvectors of hydraulic conductivities may allow interpolations or correlation to descriptors of lithologic and structural features outside the sampling points. Given a dense spatial distribution of the hydraulic head, it may sometimes be possible to infer some characteristics of these tensor components. However, hydraulic heads are seldom monitored at many sampling points. Despite these difficulties, there are few techniques to approximate the spatial distribution of hydraulic attributes and properties within a flow domain. If the “modeller’’ bypasses this important step and substitutes the actual flow domain with a simple homogeneous and isotropic one, the resulting numerical model has little value despite its computational merits. This is particularly true regarding groundwater simulation for fractured rock hydraulics.

3.2 Analysing geological features For groundwater modelling, the traditional tools employed by geologists to picture and evaluate the chief parameters of structural features are generally adequate to correlate these attributes to the results of hydraulic tests. For example, traditional histograms that depict orientation and spacing distributions of several kinds of discontinuities may be correlated to observed head contours inferred from monitored data (see fig. 3.1 and fig. 3.2; Franciss, 1994).

3.3 Handling of hydraulic head data Hydraulic heads measurements are associated to specific times and observation points. Observation times must be chronologically recorded: year, month, day and hour of the events. Observation places must be referenced to their map coordinates, duly related to a known reference frame system. Therefore, despite the natural continuity of all implicit physical processes, head measurements H(r, t) may be considered as discrete dependent variables because they are only defined at specific locations and at some times.


92


Statistical distribution of the orientation of the major structures 12

10

8

6

4

2

0

90

75

60

45

30

15

Megafractures

0 Faults

15

30

Dykes

45

60

75

90

Slikensides

Figure 3.1 Statistical distribution of the orientation of the major structures affecting the Brazilian Precambrian Shield resulting from field and desk studies for a planned underground LPG refrigerated cavern system (Almirante Barroso Maritime Terminal-TEBAR, São Sebastião, SP, Brazil). A vast gneissic complex is traversed by recent sub-vertical diabase dykes. Most rock masses conform to typical migmatites. The general foliation plunge is N 324◦ /27◦ (dip/dip direction). The depth of the weathered cover attains 30 to 40 m. Occasionally, deep altered zones, locally related to crossings of megafractures, faults and/or dykes, are found bellow the 100 m depth.

Relative frequence (%)

Inferred joint spacing statistical distribution 30

20

10

0

Joint spacing

0

1

2

3 Spacing (m)

4

5

6

Figure 3.2 Statistical distribution of the general joint spacing based on scan line counts, for the site in fig. 3.1. Modal value around 2 m. Two dominant joint trends, one parallel and other normal to the strike of the foliation, are N 27◦ /86◦ and N 136◦ /82◦ (dip/dip direction). Secondary joints traverse foliation at low angles.

Different interpolation and extrapolation techniques extend these data to other times and positions. Obviously, interpolated data within the observation time span and/or contained in the monitored area are more consistent than extrapolated data.


Data analysis

0.01 1

0.10

Porosity (%) 1.00

10.00

93

100.00

Depth (m)

Porosity (%) = 8.9865[Depth (m)]-0.5559

10

100

Figure 3.3 Plot of porosity against depth. Samples extracted from a very thick soil-rock transition zone in a deep weathered quartzite 120 km2 plateau in a tropical climate. A significant amount of infiltrated rainfall is temporarily stored each year at the soil-rock interface and slowly transferred to the fractured aquifer below. Annual recharge averages, statistically inferred from four-year period, varies from 500 to 1000 litre/s.

3.3.1 V a r ia t i o n i n t i m e The rise and fall of the water table elevation in the saturated zone of a fractured rock aquifer mostly depends on the intensity and duration of the groundwater recharge and on the value of the effective specific porosity of the aquifer. Bare outcrops of fractured rocks in cold and temperate climates restrict water infiltration from direct precipitation runoff. However, groundwater recharge from long lasting snow and ice melts are important. In wet tropical climates, typified by deep chemical weathering, the “soil-rock’’ transition zone may be very thick and may acquire an open and significant secondary porosity. These horizons are able to retain enough infiltrated rainwater to give rise to perched aquifers that feed the fractured aquifer underneath. These perched aquifers may be fully depleted before the start of the next rainy season or may be recharged several times a year (see fig. 3.3). A rough estimate of the seasonal amount of the “rise and fall’’ of the water table in the saturated zone may be obtained by dividing the infiltrated rainfall height by the effective porosity (the infiltrated rainfall height, is only a small fraction of the total rainfall height, about 5 to 25%). As the effective porosity of fractured rocks is 10 to 100 times smaller than that of sedimentary rocks, water table fluctuations on fractured rocks are correspondingly greater than on sedimentary rocks (see fig. 3.4). Whenever possible, the variation of the water table elevation over time should be monitored by self-powered and fully automatic data loggers. If this is not possible,



WT elevation (m)

94

800

750

3.45104

3.5104

3.55104 3.6104 3.65104 Day (excel serial date)

3.7104

3.75104

Observed at P6 Observed at P7 Observed at P8

WT elevation (m)

Figure 3.4 Water table rise and fall during seven years at three observation points in the saturated zone of a moderately weathered quartz-schist monocline ridge, defined by porosities lower than 1%. Yearly oscillations, only due to variations of seasonal rainfall, attained more than 10 m in two observation points. 745

740

735

3.48104 3.5104 3.52104 3.54104 3.56104 3.58104 3.6104 3.62104 3.64104 3.66104 3.68104 3.7104

Day (excel serial date) WT observed Fast Fourier transform cycles

Figure 3.5 Fast Fourier transform techniques applied to a time series of water table elevation in saturated schist not only strongly removed all irregularities and secondary cycles but also emphasised the main aspects of the seasonal cycles.

recorded levels measured at different dates may be interpolated to allow synchronous comparisons. Almost every commercial spreadsheet program incorporates friendly piecewise polynomial interpolation routines for time series analysis. Linear interpolation gives sharp corners but cubic spline interpolation at regular time intervals gives smooth curves continuous up to the second derivative. Moving average or exponential smoothing, also available in commercial spreadsheets programs, may concomitantly be used to reduce data irregularities. Fast Fourier transform algorithms, included in some mathematical software, can not only filter but also describe time series data in terms of “frequency domain’’ thus revealing all significant temporal cycles if they exist (see fig. 3.5). Future or past values, even missing gaps, of time series may be approximated by traditional interpolation or extrapolation algorithms. For almost periodic sequences, it may be possible to get plausible results using some type of forecasting technique; these


WT elevation (m)

Data analysis

95

745

740

735 3.45104

3.5104

3.55104 3.6104 3.65104 Day (excel serial date)

3.7104

3.75104

Observed WT elevations Interpolated data

WT level (m)

Figure 3.6 Time series of continuously monitored water table levels. The missing gap was filled using Burger’s linear prediction method. In fact, predicted values only “mimic’’ past values, correctly shifted ahead in time. Nevertheless, this type of adjusted time series may allow some statistical analysis.

585 580 575 570 0

5

10

15

20

25

Observed values Mean value Mean value standard deviation Mean value standard deviation

30

35 40 Month

45

50

55

60

65

70

75

Figure 3.7 Time series of monthly water table elevations. It is possible to discern five complete repetitive cycles of 12 months each (a regular hydrologic year). Observed irregularities are due to several causes, including uneven rainfall intensity. For each month, it is possible to compute five-year statistics, as averages and standard deviations. At the end of every month, the predicted levels may be systematically re-evaluated.

techniques are included in some mathematical software (Press et al. 1994, Sophocles, 1992; see fig. 3.6). However, despite their mathematical cleverness, no prediction technique actually matches factual outcomes. If a sequence of predicted values is not required, it is sufficient to use traditional statistics to picture the probable range of future values (see fig. 3.7). Assessment of dynamic correlation between different time series may confirm their mutual interdependence or prove their cause and effect connection. Lag correlation or non-parametric correlation techniques, also available in some mathematical software, may give good results (see fig. 3.8).



80

0.88 0.86

0.85954

0.84 0.82 0.8

0 20 40 60 80100

normalized WT elevation

Spearman correlation

96

2 Piezometer SR3 Piezometer NEB

1 0 1 2 3.4104

Time lag (days)

3.45104

3.5104

3.55104

3.6104

Excel serial date

Figure 3.8 Left graph:“Spearman’s correlation coefficient’’ x“time lag’’. Coefficients calculated for growing time lags to compare the time evolution pattern of two WT time series: piezometer NEB closed to deep pumping wells on fractured quartzite and piezometer SR3 far from the pumping wells, located at the top of a 350 m high plateau, more than 25 km apart. Right graph: Comparison of the time series of NEB and SR3, both normalised,“lagged’’ by 80 days at peak correlation. Normalisation is only useful for graphical display but unnecessary for correlation calculation and comparisons.

3.3.2 V a r ia t i o n i n s p a c e The spatial interpolation (and extrapolation) of observed quantities monitored at regular or erratic sampling locations may be achieved by different types of built-in interpolation algorithms in commercial software specifically designed to produce contour maps from rectangular arrays (“grids’’) of estimated values. Check, for example, the capabilities of Surfer – from Golden Software, Inc. – a traditional and well-proven product, or GW Contour – from Waterloo Hydrogeologic, Inc. – a less traditional product but one specifically designed for hydrogeologists. In a certain sense, almost all interpolation algorithms are weighted averages, i.e. finite interpolation polynomials where the interpolating functions fi (r) play the role of averaging weights. However, instead of being constructed as functions of the coordinates r of the estimation point, as for traditional piecewise approximants, these averaging weights are constructed as functions of the radial separation δrij (and sometimes of the azimuth αij ) between the estimation point i and a few points j around it, forming a cluster of influencing values (see fig. 3.9). The main difference between weighted averages algorithms resides in the way their weights are determined. Different weighting criteria yield comparable but unequal estimations for the same data and grid geometry. Usually, for each type of interpolation algorithm, the influence of the size, spatial distribution and inherent variability of the input data on the character of the resulting grid is properly considered in the “help section’’ of commercial software. Inverse distance is one of the simplest and intuitive weighted average methods. Each observed value j is weighted by the inverse of its separation from the estimation point i (frequently, distance squared). The closer the observation point to the estimation point, the greater its contribution to the estimated value.


Data analysis

97

8000 Observation well j

Radial separation ij 6000 Y (m)

Estimation point i 4000

2000

2000

4000

6000

8000

10000

12000

14000

X (m)

Figure 3.9 Example of a cluster of five observations wells j around an arbitrary estimation point i.

Natural neighbour is a fine and robust weighted average suggested for sparse and irregular observation points. Optimal averaging weights are proportional to the “influence area’’ of each observation point j clustered around the estimation point i (see, for example, the fundamentals of the Thiessen method – source of the natural neighbour method – that are described on sections devoted to the subject of descriptive hydrology in common textbooks). Kriging is a more elaborated weighted average based on two basic assumptions. The first one assumes that all observations can be decomposed in two parts. One part is a deterministic value, a global average or a trend value calculated at the observation point. The other part is a “residual’’ value, defined as the difference between the observed value and the deterministic value. The second assumption presumes that it is possible to statistically infer the character of a “measure’’ of the dissimilarity of two “residuals’’ as a function of their radial separation δrij up to a certain limit (and sometimes, as a function of the radial separation δrij and the azimuth αij ). Then, based on an empirically constructed “dissimilarity measure’’ between closer observations, local averaging weights are optimised for every cluster of P points around an estimation point i (fig. 3.9). The corresponding “objective function’’ to be minimised for each cluster is a statistical measure of the local dissimilarity between predicted and observed values (error variance).


98


Table 3.1 Input data set. Order

E(m)

N(m)

Z(m)

H(m)

1 2 3 4 5 6 … … …

741749.92 712850.46 708794.42 711900.74 745198.91 810899.43 … … …

7786118.54 7785151.50 7783985.81 7780999.76 7771800.78 7769598.54 … … …

670.00 568.00 568.00 640.00 660.00 502.00 … … …

615.00 530.00 530.00 604.00 648.00 502.00 … … …

159 160 161 162 163 164

706249.19 704899.72 655100.57 710000.14 703599.72 689750.07

7430951.04 7425898.83 7425700.20 7423701.23 7423150.53 7422082.14

580.00 575.00 750.00 600.00 610.00 650.00

577.00 564.00 742.00 565.00 587.00 604.00

Besides the set of observed values, additional information about the sample domain may be just ignored (simple kriging), treated as complementary random variables (cokriging) or incorporated as a paired spatial bias. Finally, it is important to note that kriging algorithms may be used to produce point estimations or to generate the averages of point estimations around cell centres, smoothing the corresponding map contours in the latter case. Excellent introductory texts on geostatistics and kriging are available, for example, Kosakowski and Bohling, 2005. Example 3.1 helps to grasp the fundamentals of almost all griding methods and illustrates the construction of a variant of an inverse distance squared algorithm.

Example 3.1 The stabilised values of the piezometric elevations Hweel at the roof of a half-confined groundwater body trapped in a sandstone formation were recorded during the perforation of 164 deep wells for oil prospecting (see table 3.1). Based on this data record it was possible to picture the most probable configuration of the piezometric surface over a vast area of nearly 70,000 km2 . An optimised inverse distance squared algorithm was chosen to interpolate (and extrapolate) the observed piezometric elevations H over a rectangular grid of 40 × 64 equidistant points, centering squares of 5000 m side. Based on that grid, a contour map of H


Data analysis

equal values was graphed (see fig. 3.10). To get this result the following procedure was adopted:

7750000

800 m 780 m 760 m 740 m 720 m 700 m 680 m 660 m 640 m 620 m 600 m 580 m 560 m 540 m 520 m 500 m 480 m 460 m 440 m 420 m 400 m

7700000

N (m)

7650000

7600000

7550000

7500000

7450000

650000

700000

750000

800000

E (m)

Figure 3.10 Location of the observation points (X symbols) and head H contours based on the grid estimates.

Step A: Detrending The coefficients of a deterministic 3D planar trend Htrend through the cloud of the m observed heads (Ewell , Nwell , Hwell ) were found by the traditional least squares method. The resulting best-fit plane was: Htrend = −2.124223 · 10−4 · E + 2.257132 · 10−5 · N + 540.19


(3.1)

99

100


Having defined this plane, the head residuals δHwell at each well with respect to the trend plane Htrend were computed as the difference between the observed head Hwell and the calculated trend Htrend at the well location (see fig. 3.11): δHwell = (−2.124223 · 10−4 · Ewell + 2.257132 · 10−5 · Nwell + 540.19) − Hwell (3.2)

600

H (m)

800

400

E (k

)

N (km

m)

700

7700

7600 800

7500

Figure 3.11 Plot of the observed heads and of the planar trend.

Step B: Immediacy of observed wells The immediacy or closeness of each pair of wells s and t was measured by two quantities, their separation δrs,t and corresponding azimuth αs,t : 1

δrs,t = [(E·wells − E·wellt )2 + (N·wells − N·wellt )2 ] 2

N·wells − N·wellt αs,t = atan E·wells − E·wellt

(3.3)

Care must be exercise when computing αs,t to avoid indeterminate answers. Step C: Parameters optimisation The residual head δHi at each grid node i – defined by its coordinates Ei and Ni – was estimated by the inverse distance squares algorithm taking only into account the observed residual heads δHj at P nearby wells – defined by its coordinates Ej and Nj – around the estimation node i. This algorithm required four parameters to be optimised: a) b) c)

The quantity P of neighboring wells around estimation nodes. One constant scale factor A to adapt the value of the distance squared. Two constant parameters B and C respectively related to the eccentricity and the orientation of the domain anisotropy.


Data analysis

101

The resulting algorithm expression is:  j ∈ cluster of P nearby wells around estimation point i

δHj =

 

 1+A(δri, j )2

1 sin(αi, j +c)2 B2 + 1 cos(αi, j+C )2 B2

This expression can be concisely written as: δHj =

δHj 

1

j ∈ cluster of P nearby wells around estimation point i 1+A(δri,j )2



1

(3.4)

1 sin(αi, j +C)2 B2 + 1 cos(αi, j+C )2 B2

(wi, j δHk )

(3.5)

j ∈ cluster of P nearby wells around estimation point i

The averaging weights wi, j of the above approximant are calculated by: 1

  1 + A(δri, j )2  wi, j =

   1 2 + 2 cos(αi, j + C) B 1

sin(αi, j +

C)2 B2

(3.6)

j ∈ cluster of P nearby wells around estimation point i

×

1

  1 + A(δri, j ) 

   1 2 + 2 cos(αi, j + C) B 1

sin(αi, j +

C)2 B2

It easy to understand that the averaging weights wi,j are between zero and one. In addition, their total sum is always unity. The values rapidly fall with separation (see fig. 3.12): 1

Weight

0.1 0.01

1103 1104

5104

0

1105

1.5105

2105

Separation between observation and estimation points (m)

Figure 3.12 Weight value declines as radial separation increases. Graph plotted after optimisation of the four parameters P, A, B and C.

To optimise the four parameters P, A,B and C,a tractable approach was used to minimise each parameter separately, going from simple to successively more complex expressions. This approach leads to some loss of mathematical rigour, but it is not too compromising.


102


The objective function was the sum of all squared interpolation errors between observed and estimated residuals at all observation points precisely, i.e. at all the wells. As the first parameter to be optimised was A, the sum of squared errors A to be minimised was: A =

m

(estimated dhs − observed dHs )2

(3.7)

s=0

To estimate the residual heads at points t based on the observed heads at all points s, except at the target well, the following expression was used: m 1 (s = t) δH s 1 + A(δrs, t )2 s=0 δHt = (3.8) m 1 2 s=o 1 + A(δrs, t ) However, to enhance the influence of the neighbourhood of the estimation points, the squared errors sum A was modified as follows:

m average δr 2 2 A = (estimated dHs − observed dHs ) average drs s=0

(3.9)

Squared error sum

In this expression,“average δr’’ corresponds to the average of the separations of all pairs of wells s and t, as defined above, and “average δrs ’’ corresponds to the average of the separations of the well s and all the wells t. The graphical expression of the A value that minimises the squared error sum A is shown bellow (see fig. 3.13):

2.05106 2106 1.95106

A 158.094109

1.9106 1.85106 0

1107

2107 Parameter A

3107

4107

Figure 3.13 Variation of the squared errors sum A with the parameter A. The A value that minimises this sum was found by a non-linear optimisation method (conjugate gradient in this case).

The next parameters to be optimised were B and C. These parameters control the inherent anisotropy of the observed values concurrently due to the geometric pattern of the sampling points and to the geologic heterogeneity of the monitored domain. However, to estimate residual heads at points t based on the observed heads at all points s, except at


Data analysis

103

the target well, a more complex expression was adopted, including not only the separation δrs, t but also the corresponding azimuth αs, t :       m    (s = t)    s=0   1 + A(δrs,t )2 

δHt =

 m    s=0   1 + A(δrs, t )2

   δHs ·  

1 1 sin(αi,j + C)2 B2 +

1 cos(αi, j + C)2 B2 

1 1 sin(αi, j + C)2 B2 +

1 cos(αi, j + C)2 B2

(3.10)

     

To enhance the influence of the observations in the far field and eventually detect an existing anisotropic pattern, the squared errors sum BC was again modified as follows:

m average δrs 2 2 (estimated dHs − observed dHs ) (3.11) BC = average dr s=0

Squared error sum

Compared to the last expression for A it is important to note that in the above expression for BC the ratio [average δrs ]/[average δr] is inverted. The graphical expressions of the B and C values that simultaneously minimise the squared error sum BC are shown bellow (see fig. 3.14): 1.5106 1.45106 1.4106 1.3510

B 0.163

6

1.3106

0

2

4 Parameter B

6

8

Squared error sum

1.38106 1.36106

C 26.029 deg

1.34106 1.32106

0

100

200

300

Parameter C

Figure 3.14 Variation of the squared errors sum BC with the parameters B and C. The B and C values that concomitantly minimise this sum were found by a non-linear optimisation method (conjugate gradient in this case). Optimum values can also be found graphically after painstaking trials.


104


Anisotropy ellipse 120

90

60

2

7.7106

30

150 1 180

0

0 7.6106

330

210 240

300

270

7.5106

Best fit ellipse Anisotropy factor

1 sin(α C) B 2

2

1 cos (α C)2 B2

6.5105 7105 7.5105

Figure 3.15 Best fit ellipse that reflects the inherent anisotropy of the observed values. It is not mathematically possible to separate what is due to the geometric pattern of the sampling points from what is due to the geologic heterogeneity of the monitored domain.

The polar graph of the optimised anisotropy ellipse is shown above (see fig. 3.15). However, this plot may give a false impression of the effects of the detected anisotropy. In fact, as the anisotropy factors multiply the distance squared in the denominator of the algorithm’s expression, their effects on the grid values are inverted. In other words, the grid-based contours “shrink in’’ and “stretch out’’ in the opposite directions of the major and minor axis of the best fit ellipsis, respectively. Besides the influence of the sample pattern, a qualified hydrogeologist may infer additional geological influences on the detected anisotropy. The last parameter to be optimised was the quantity P of neighbouring wells around estimation points. Then, applying the same objective function (by now defined by the optimised parameters A, B and C) for each cluster of points around the estimation points and successively increasing P from the nearest observation well to those less closer, it was possible to clearly see that eight (8) was the best number (see fig. 3.16).

Squared error sum

1.54106 1.52106

P values Best P value

1.5106 P8

1.48106 1.46106 1.44106 1.42106

0

5

10

15

20

25

P (-)

Figure 3.16 Variation of the squared errors sum P with the quantity P of neighbouring wells around estimation points. Eight (8) determines the first minimum.


D a t a a n a l y s i s 105

After griding all residuals using the completely optimised algorithm, corresponding piezometric levels were found by adding to each inferred residual the related deterministic trend value. To reduce the “bull’s-eye’’ aspect inevitably associated with inverse distance algorithms, a smoothing filter may be applied to the grid nodes (i, j).

3.4 Handling of flow rate data Flow rate data from springs, watercourses, pumping wells etc. deserve a similar treatment to that applied to hydraulic head data. However, it is sometimes possible to draw practical conclusions using hydrology techniques when, for instance, using extreme value distributions. In fact, one of the most annoying accidents during gallery developments for underground mining is sudden water inrushes before cementation is done. Example 3.2 shows how to adapt hydrologic techniques to analyse flow rate data.

Example 3.2 Fig. 3.17 shows a typical but relatively small inrush observed a few weeks after the gallery head crossed an apparently sealed fracture in dolomite land.

Figure 3.17 Relatively small inrush observed a few days after completing the gallery excavation in dolomite land (Vazante underground mine, State of minas Gerais, Brazil).

The coordinates of the location of the significant inrushes, their flow rates and dynamic hydraulic heads, measured immediately after water irruption, were systematically registered during the excavations of the development galleries forVazante mine, State of Minas


106


Number of observed events (-)

Gerais, Brazil, which is located in dolomite land. As these inflows were not laminar, the measured discharges were normalised by the square root of the hydraulic head: L3 /T/L1/2 or L2.5 /T (see fig. 3.18).

1103 100 10 1 1

10

1103

100 2.5

Normalised inrush (m

/hr)

Figure 3.18 Histogram for 266 normalised inrushes (m2.5 /hr) systematically registered from December 1982 to December 2002.

It was possible to fit these data to a Gumbel extreme values distribution defined by: G(A,B) = (1 − e−e

A·ln(Q)+B

)

(3.12)

Cumulative frequency ( % )

In the above equation A and B are the fitted parameters and Q corresponds to the normalized inrush. Minimization of squared error gave A = 0.316 and B = −0.114 (see fig. 3.19). 100

50 Gumbel distribution Cumulative frequency of observed events

0 1103

0.01

0.1

1

10

100

1103

1104

Normalized inrush (m2.5/s)

Figure 3.19 Fitted Gumbel distribution of extreme values to the observed events.

Consider a gallery segment of length B as a baseline. If P is the probability of an occurrence of a normalised inrush along B less than Q, the probability of an occurrence of an inrush exceeding this value is 1−P. Now consider the length L of the whole gallery defined by the product kB. Then the probability of an occurrence along L of a specific flow less than Q is (1−P)kB . The probability of an occurrence greater than Q is [1−(1−P)kB ]. In the case of extreme flows, this complementary probability measures the expected risk (see fig. 3.20).


Cumulative risc of one inrush (%)

Data analysis

107

100

< 0.1 m^2.5/hr < 0.3 m^2.5/hr < 1 m^2.5/hr < 3 m^2.5/hr < 10 m^2.5/hr < 30 m^2.5/hr

50

0 10

1103

100

1104

Gallery length (m)

Figure 3.20 Expected risk of at least one inrush. Six curves defined from 0.1 m2.5 /hr to 30 m2.5 /hr during the excavation of more 10000 m of galleries.

This type of analysis can also be made considering different lithologies and depths.

3.5 Handling of hydraulic conductivity data 3.5.1

Pr elim i n a r i e s

As previously discussed, the size of a pseudo-continuous subsystem depends on the observation scale. At the selected scale, a preliminary estimate of the eigenvalues and eigenvectors of the hydraulic conductivity of a subsystem may be inferred from its lithologic and structural geologic features. In some cases, previous experience from similar places may also allow first assessments. However, to get useful results from numerical models, these parameters must be adjusted by solving inverse problems, applied either to large-scale field tests or to comprehensive monitoring data. It is important to always keep in mind that the hydraulic conductivity of a rock mass subsystem is, in fact, a scale dependent property to be applied to a fictitious pseudocontinuous subsystem. Its eigenvalues and eigenvectors range over many orders of magnitude from place to place. Furthermore, they reflect the intrinsic assumptions concerning anisotropy and boundary conditions of the analytical model employed in their determination. In other words, different models applied to the same tested volume may yield different, and sometimes quite dissimilar, values. Moreover, as these closed solutions are normally applied to ideal homogeneous and anisotropic domains, never encountered in nature, their results lose meaning as the heterogeneity of the test space increases. In view of these observations, the modeller must exercise judgment, considering costs and time limits, to appropriately decide what type of approach best fits his needs. For an example of how the choice of approach affects the result, see table 3.2 concerning the results of a seven days test involving two very deep thermal wells, 540 m apart, performed for the Brazilian Department of Mineral Production in 2000: well P (E 422.547 m, N 6.966.167 m, Z 436.8 m, 700 m of total depth, 160 m of bottomfilter length, artesian discharge of 320 m3 /h, external diameter 121/4 to 8 3/8 ) and well T (E 421.957 m, N 6. 966.273 m, Z 423.26 m, 680 m of total depth, 140 m of


108


Table 3.2 Example of how the choice of the analytical solution affects computed test results. Analytical solution

Confined sandstones Confined sandstones Basalt cap average permeability (m/s) transmissivity (m2 /s) storativity (-)

Copper-Jacob: steady-state solution 1.018·10−3 Hantush: transient solution for 3.616·10−4 leaking aquifers

1.282·10−4

1.485·10−7

bottom-filter length, artesian discharge of 268.95 m3 /h, external diameter 121/4 to 8 3/8 ). Both wells traverse a 540 m basalt cap and penetrate the confined sandstone (Mercosur aquifer, structured by Triassic-Jurassic sandstones confined by Cretaceous basalt flows, spreading over 1,194,000 km2 and including the Paraná Bassin and part of the Chaco-Paraná Bassin).

3.6 Hydraulic transmissivity and connectivity 3.6.1

P r elim i n a r i e s

The effective hydraulic transmissivity of discontinuities and their hydraulic connectivity at a chosen observation scale deserve attention when modelling a pseudo-continuous network for a fractured rock mass. 3.6.2

H yd r au l i c c o n d u c t i vi t y a p p r a i s al

The hydraulic transmissivity of a fracture may be partially blocked by secondary mineralisation, chemical weathering or other kinds of infillings. A crude appraisal of the areal extent of these obstructions may result from detailed fieldwork on bare outcrops or on excavated slopes. Usually, it is assumed that statistics derived from a planar support, as a planar discontinuity, may approximately match those resulting from several random linear supports on that same planar support. Then, the relative areal extent κ of the fracture blockage may be estimated from exposed fracture traces on outcrops by the division of the sum of the lengths of clogged segments δlc by the sum of the lengths of all traces δl, i.e.: κ = δlc / δl. This implies that for each group of open fractures observed on undisturbed borehole core samples the apparent average spacing ea roughly results from the product of the true average spacing e by the relative areal extent of κ obstructions, i.e. ea = κ·e. Therefore, for partially blocked fractures, i.e. for 0 < κ < 1, their average transmissivity T may be approximated by the Maxwell (1 − κ) κ) mixture rule: T = T0 · Tκ , where T0 and Tκ are, respectively, the non obstructed and obstructed fracture transmissivity. From these considerations, it follows that it is very difficult to directly appraise the true hydraulic transmissivity of fractures at a chosen observation scale. To complicate the problem, one must remember that in most cases seepage along partially opened fractures may be partially concentrated on anastomosed preferential small conduits within the fracture plane.



Figure 3.21 Oriented plug for permeability tests extracted from a large diameter whole-core sample.

Modelling strategies to evaluate the hydraulic conductivity of intact rock or rock masses depend on the desired observation scale as exemplified in the following sections. 3.6.2.1 H y d ra u l i c tes ts a t “c or e s a m p l e’’ s c a le The production and the storage of the oil reserves on fractured and porous sandstone reservoirs depend on the amount of connections between several groups of small discontinuities a few centimetres apart one another. Indeed, the commercial extraction of the hydrocarbons trapped in the interstitial pores of the porous matrix may be held back if these small fractures are almost clogged by secondary mineralisation. However, if these small fractures remain pervious and interconnected, extraction may be very efficient and effective. To have an idea of how much such fractures can improve exploitation, the average transmissivity of the relevant fracture groups may be measured by permeability tests made under different confined pressures on small samples (plugs) extracted from large diameter cores from prospecting wells (see fig. 3.21). If these fractures are partially blocked, the nature and areal extent of their sealing may be judged by visual inspection. The matrix permeability and corresponding effective porosity may be concomitantly measured in the laboratory. If the rock matrix is anisotropic, the evaluation of its hydraulic conductivity tensor requires many measurements of the oriented permeability Kd (n) on many “plugs’’ from the same whole core along different orientations. In fact, Kd (n) along a given direction defined by its unit vector n is related to the tensor |K| by the relationship Kd (n) = 1/[nT ·|K|−1 ·n]. Measuring more than six directional conductivities in many directions allows the determination of the components of |K| by inverse methods (Hsieh and Neuman, 1985). 3.6.2.2 H y drau l i c tes ts a t “b or eh ol e i n teg r a l c o r e’’ s c a le Integral sampling methods have been described in several papers (Rocha, M. and Barroso, M., 1971). The method consists of drilling a rock core already stiffened by a


110


coaxially oriented rigid rod previously cemented in a smaller inner hole to ensure the integrity of the sampled material. Example 3.3 shows how the anisotropic character of the permeability of rock masses can be approximately determined from water pressure tests and information about the structural attitude and the nature of their open and connected discontinuities as revealed by integral samples (Rocha, M. and Franciss, F. O., 1977).

Example 3.3 The foundation of the AguaVermelha Dam, in the State of São Paulo, Brazil, was investigated with help of 13 HW integral samples and 128 water pressure tests. The average eigenvalues and anisotropy of the permeabilities estimated are summarised below and the corresponding eigenvectors are depicted in fig. 3.22.

Compact basalt

Eigenvalue m/s

K max /K min

group CB1 2.95E-05 16 eigenvalues 2.81E-05 1.81E-06 group CB2 8.51E-07 7 eigenvalues 7.76E-07 1.28E-07 group CB3 4.78E-07 6 eigenvalues 4.26E-07 7.41E-08 group CB4 3.16E-07 27 eigenvalues 3.01E-07 1.19E-08 group CB5 6.30E-06 675 eigenvalues 6.19E-06 9.33E-09 group CB6 Impervious

Vesicular amygdaloidal Eigenvalue basalt m/s group VA1 eigenvalues group VA2 eigenvalues group VA3 eigenvalues group VA4 eigenvalues group VA5

3.89E-06 6 3.31E-06 6.03E-07 6.60E-06 55 5.62E-06 1.20E-07 1.69E-06 1070 1.69E-06 1.58E-09 4.78E-07 24 4.57E-07 1.97E-08 Impervious

N 30

30

60

E

KIII

120

120 KI

N 30

60

KIII

W

KIII E

120

120

W

KII

120

150

150

E 120

KI 150

30

60

60 KII

K max /K min

group BB1 3.31E-06 12 eigenvalues 3.16E-06 2.69E-07 group BB2 1.00E-07 2 eigenvalues 5.25E-07 5.25E-08 group BB3 2.13E-07 18 eigenvalues 2.04E-07 1.20E-08 group BB4 Impervious

30

60

60 KII

150

Eigenvalue m/s

N

30

W

Basaltic K max /K min breccia

150

KI

150

S

S

S

(a)

(b)

(c)

Figure 3.22 Spread of the eigenvectors of the compact basalt group (a), vesicular-amygdaloidal basalt group (b) and basaltic breccias group (c).

For each test length L, the hydraulic conductivity tensor is evaluated by taking into consideration all open and supposedly connected fractures inside and around the volume L3 .


Data analysis

111

To tentatively minimise the sampling bias (Terzaghi, 1965), it is strongly recommended to include all fractures occurring above the top and below the bottom of the integral core that satisfy the condition (L + δ L)·cos(θ) < L/2, where θ is the fracture dip (see fig. 3.23). Fracture included in the sample

N 30

θ

30

60

L

60

δL

6

KIII 0.18 10

m/s

W

KI 0.16 10

L

4

m/s

E

L 120

120

150

KII 0.13 104 m/s

150

S

L

(2) Fracture not included in the sample

– (1)

(3)

0.1 – 0.5 mm 0.5 – 1.0 mm 1.0 – 5.0 mm

Figure 3.23 Left side: criteria for including fractures above and bellow the sampling volume. Right side: Schmidt diagram of fracture poles from an integral core as well as the associated hydraulic conductivity tensor.

To evaluate the hydraulic conductivity tensor for each test interval L, associated with i fractures (i = 1, 2, 3. . . N) within the sampling volume L3 , one must proceed as follows: Measure the average dip θi , dip orientation λi and the hydraulic aperture ei of each fracture i. b) For each fracture i, roughly evaluate its transmissivity Ti (derived from its hydraulic aperture ei and from the character of existing infillings) and the corresponding equivalent hydraulic conductivity ki (≈Ti /L). As eigenvalues are tentatively calibrated later, based on the water pressure tests results, reliable relative estimates are much more important than absolute values. c) Compute the components of the hydraulic conductivity:   sin(λi )2 + cos(θi )2 cos(λi )2 −sin(θi )2 cos(λi ) sin(λi ) −sin(θi ) cos(θi ) cos(λi )   2 2 2 2  ·|Ki | = ki   −sin(θi ) cos(λi ) sin(λi ) sin(θi ) cos(λi ) + cos(θi ) −sin(θi ) cos(θi ) sin(λi ) a)

−sin(θi ) cos(θi ) cos(λi )

−sin(θi ) cos(θi )sin(λi )2

sin(θi )2 (3.13)


112


d)

Compute the resulting tensor |K| by the summation rule:   (k11,1 + k21,1 + · · · + k11,1 + · · · + kN1,1 ) (k11,2 + k21,2 + · · · ) (k11,3 + · · · )   |K| =  (k12,1 + k22,1 + · · · ) (k12,2 + k22,2 + · · · ) (k12,3 + · · · ) (k13,1 + · · · ) (k13,2 + · · · ) (k13,3 + · · · ) (3.14)

e) f)

Compute the eigenvalues Kj and the corresponding eigenvectors αzj of the resulting second order tensor |K|. Determine the homogeneous transformation factor f to convert pressure tests results from an anisotropic to an isotropic condition (a homogeneous transformation guarantees volume invariance for elementary subsystems thus preserving source strengths in water pressure tests):   13  13 3 3 2 ! (αzj )   f =  Kj  (3.15)  Kj j=1 j=1

g)

As the borehole radius r almost equals the geometric mean of the transformed borehole cross-section, compute the average permeability Kmean for each water pressure test, H being the test pressure:

Q fL k·mean = ln 1.6 (3.16) 2πfLH r

h)

For each test, multiply the previously estimated eigenvalues Kj by a calibrating factor: ρ=

kmean 13 3 " Kj

(3.17)

j=1

These correcting factors ρ may depart more than one order from unity.Thus, to get reasonable results, the first estimate of the transmissivity of each fracture observed in an integral sample in step b) must be made carefully but in relative terms, i.e. in comparison to the transmissivity of the most prominent observed fracture in the sample.

3.6.2.3 H y dra u l i c tes ts at “c l u s ter of b or e h o le s’’ s c a le Traditional pumping tests may be adapted to measure the anisotropic character of pervious media (see, for example, Hantush, 1956). For a confined volume of a fractured rock mass, taken as quasi-homogeneous but anisotropic, there is an elegant inverse solution to evaluate its equivalent hydraulic conductivity and specific storage (Hsieh and Neuman, 1985). Example 3.4, abridged from Appendix 5B, “Using a multiple-borehole test to determine the hydraulic conductivity tensor of a rock mass’’, in Rock Fractures and Fluid


Data analysis

113

Flow: Contemporary Understanding and Applications, National Academic Press, Washington, 1996, exemplifies this methodology (Hsieh and Neuman, 1985).

Example 3.4 The hydraulic conductivity of a sample volume of Oracle granite, north of Tucson, Arizona, USA, was estimated with help of multiple pumping tests in three boreholes H-2, H-3 and H-6, drilled in a triangular pattern and separated by distances between 7 to 11 m (see fig. 3.24). 0

N

5

10

Scale (m)

H-6

H-3

H-2

Figure 3.24 Location of the boreholes drilled to perform the 3D tests.

To minimise the influence of the boundary conditions, as a rule located far from the test domain, only short transient pumping (or injection) tests are performed. In each test, water is pumped from a packer-bounded interval of one borehole and corresponding head drops are measured on other packer-bounded intervals of the remaining boreholes (see fig. 3.25). Six or more six short transient tests give sufficient observation data to write a set of mathematical equations relating to the sought hydraulic parameters. Between any pumping point i and any observation point j prevails a relationship involving the distance Rij between these two points, the time t, the pumping rate Q, the head drop hij , the directional conductivity Kij , the determinant of the hydraulic conductivity tensor |K| and specific storage Ss :

·hij =

Q(Kij )



1 2 1

erfc 

4πRij (|K|) 2

2

(Rij ) Ss 4Kij t

12  

(3.18)

Solving this equation system (usually by a weighted least squares method) gives the six components of the second-order symmetric and positive-definite tensor that best describes the anisotropic conductivity K of the tested volume and the specific storage Ss . If the tested rock mass volume can be taken as a statistically homogeneous but anisotropic pervious medium, a three-dimensional plot of the square root of the directional hydraulic diffusivity versus direction should depict an ellipsoid, as shown in fig. 3.26. The eigenvalues and corresponding eigenvectors of the best-fit K are shown fig. 3.27.


114


N

0

5

Sca

H-6 H-3

le (m

10

)

H-2

Observation interval

Point O

Point C

Pumping interval

Observation interval Point B

Figure 3.25 Spatial relationship of one of the several short transient pumping tests performed. Water is pumped from the packer-bounded interval O of borehole H-6 and corresponding head drops are measured in the packer-bounded intervals B and C of boreholes H-2 and H-3. N

E

Figure 3.26 Best-fitted ellipsoid of the square root of the directional hydraulic diffusivities estimated from transient pumping tests results.


Data analysis

115

Eigenvectors (lower hemisphere projection) 90 120

60

150

30

180

0

330

210

300

240 270 Latitude 60 Latitude 30 Equator

K max = 1.6E-07 m/s; bearing 75 N; plunge 39 K int = 6.9E-08 m/s; bearing 247 N; plunge 51 K min = 2.2E-08 m/s; bearing 342 N; plunge 4

Figure 3.27 Estimated eigenvalues and eigenvectors of the anisotropic average hydraulic conductivity of Oracle granite rock masses. Most probable specific storage is 5.1E-06 m−1 .

3.6.2.4 H y d ra u l i c tes ts a t “a q u i fer’’ s c a l e Having in mind the obvious limitations of extending its applicability to other scales and flow regimes, the methodology suggested by Hsieh and Neuman may be adapted to measure the average hydraulic conductivity of delimited volumes within larger and almost homogeneous units of a flow domain. These results may be extrapolated to the whole domain by appropriate geostatistical models. One of the main advantages of this methodology is that it takes into account, implicitly and collectively, all complex and 3D-varying fractures characteristics that are hard to identify and parameterise by field work and core inspection, such as true shape, persistence, hydraulic transmissivity, interconnectivity, etc. Example 3.5 exemplifies one variant of this methodology.


116


Example 3.5 A 60◦ dip tabular ore body is traversed by two major structures (see fig. 3.28). The direction and dip of neighbouring discontinuities of the host rock mass are closely related to the attitudes of the ore body and these two major structures. Lightly inclined original WT

Tabular 60 dip ore body 600 400

3000 2000 Inclined impervious boundary

1000

Two subparallel and subvertical major faults

0 1000

2000

0

200 0

2000

Figure 3.28 Perspective of the tabular ore body and the two major structures. The impervious bottom together with the slightly inclined water table defines the main boundaries of the natural flow domain.

The footwall of the tabular ore body and the two major structures are highly permeable. Their intercrossings behave as very efficient line drains able to induce the local gravitational lowering of the originally lightly inclined water table (see fig. 3.29). Lowered WT by gravitational drainage

600 400

3000 Two subparallel and subvertical drainage axis

2000 Dipping impervious boundary

1000

0

0 1000

200

2000

0

2000

Figure 3.29 Contour map of the lowered water table based on the readings of 16 piezometers.


Data analysis

117

Assuming an infinite and anisotropic pervious medium, Hsieh and Neuman gave a closed steady-state solution to determine the head drop at an arbitrary point k induced by a line source (or line drain) of length 2Li centered at the origin i of a Cartesian coordinate system (see fig. 3.30). Z ∆Ri,k vector components

Li vector components

∆Xi, k

Li

X

Li

LiY

∆Ri,k

∆Yi, k ∆Zi, k

LiZ Y

∆Ri,k : distance between points i and k

X

Line source (or line drain) of length 2Li centred at origin i of coordinate system

Figure 3.30 Coordinate system referred to a line drain of length 2Li .

For more than one line drain Li placed at i different locations (i = 1, 2, 3. . .), the resultant head drop Hk at an arbitrary point k may be calculated by:   12 RTi,k A Ri,k RTi,k A Li RTi,k A Li  +2 T +1 + T + 1   LiT A Li Li A Li Li A Li   Qi ln   1 2   T T T Ri,k A Li Ri,k A Li   Ri,k A Ri,k −2 T +1 + T −1 LiT A Li Li A Li Li A Li ·Hk = 1 8π(LiT A Li ) 2 i (3.19) The other variables in this formula have the following meanings: – – – –

Ri,k : distance between the i line drain center and an arbitrary point k described by its Cartesian components (Xi,k , Yi,k , Zi,k ). Qi : the total drainage rate percolating to the line drain i. A: the adjoint matrix A corresponding to |K| K−1 , where |K| is the determinant of the average hydraulic conductivity tensor K. Li : the Cartesian components of the half length Li of the i line drain.


118


For the situation depicted, the existing field data, 16 piezometers readings and two almost steady-state discharge measurements at the bottom of the two line drains (0.99 and 0.48 m3 /s), provided enough data to fit an approximation to the average hydraulic conductivity tensor K of the surrounding rock mass. As the closed solution was deduced for an infinite domain, the image method was applied. However, only the two most influencing image drains were added to the problem: one for the WT boundary and one for the impervious bottom boundary. The quadratic error sum concerning the difference between the calculated and the measured head drops was minimised by the Levenberg-Marquadt method. However, to start the search algorithm at the neighbourhood of the lowest minimum of the quadratic error sum 7-D space, the first approximation of K was based on the geometry and the known hydraulic characteristics of the tabular ore body and the major structures. The resulting best-fit average K was:   2.855 · 10−5 −1.33 · 10−5 −2.97 · 10−6  m  −5 (3.20) K= 1.28 · 10−5 2.44 · 10−6  · s  −1.33 · 10 −2.97 · 10−6 2.44 · 10−6 1.442 · 10−6 The corresponding eigenvalues and eigenvectors are depicted in fig. 3.31. Eigenvectors (lower hemispher projection) 90 120

60

150

30

180

0

330

210

300

240 270

Latitude 60 Latitude 30 Equador K max = 3.655E-5 m/s; bearing 120 N; plunge 6.2 K int = 5.343E-6 m/s; bearing 211 N; plunge 9.5 K min = 8.973E-7 m/s; bearing 357 N; plunge 78.5

Figure 3.31 Best-fit average K. Besides a pronounced change in the first approximation eigenvectors, the optimised minimum quadratic error sum was 71 times lower than its value for the starting tensor eigenvalues.

Recycling the image method based on the best-fit solution did not improve appreciably its value.


Data analysis

119

10 Piezometers Dynamic correlation: 0.85 a 0.90 Dynamic correlation: 0.90 a 0.95 Dynamic correlation: 0.95 a 1.00

Y (km)

8

6

4

2 2

4

6

8

10

12

14

16

X (km)

Figure 3.32 Hydraulic interconnection between piezometers as suggested by their mutually dynamic correlation coefficient.

3.6.3

H yd r a ul i c c o n n e c t i vi t y a p p r a i s al

Poor hydraulic connectivity between apparently persistent discontinuities may also hamper groundwater flow. Indeed, a model exclusively based on the geometry of the discontinuities without taking into consideration the extent of their hydraulic connectivity may overestimate the hydraulic behaviour of the rock mass. Normally, intercrossed major discontinuities (topographic trace lengths of more than a few kilometres) are mutually connected and are not blocked by much smaller ones. Modelling strategies to appraise the hydraulic connectivity of discontinuities are exemplified below. 3.6.3.1 D y na m i c c or r el a ti on s of WT ti m e s e r ie s Preferential corridors of groundwater movement normally induce similar and quasisimultaneous response of water table measurements in piezometers located on their flow paths. Therefore, for any pair of piezometers that behaviour may be approximately judged by the dynamic correlation of the fluctuation of their water table time series. As comparisons of time ordered data are essential to get realistic results, one must use non-parametric correlation measures such as, Spearmann’s or Kendall’s’ τ tests. High dynamic correlation values may imply high hydraulic connectivity. These tests must be performed, one by one, for all pairs of piezometers under analysis. Results from testing the dynamic correlation of 172 × 172 pairs of piezometers more or less erratically scattered over an area of 200 km2 are mapped in fig. 3.32. 3.6.3.2

Fi l t e ri n g WT c on tou r m a p s

Water table contour grids may be filtered by different types of algorithms (low-pass, high-pass, contrast amplification, etc.) to enhance some of their peculiarities. A simple


120


6000 N (m)

abs

hi1,j hi1,j hj1 hi,j1 4hi,j hi,j

4000

4000

6000

8000

10000

12000

14000

16000

18000

E (m)

Figure 3.33 A normalised Laplacian operator (as tagged on this figure) applied to the hydraulic head contour map (100 m × 100 m grid) of fig. 3.32 may enhance the anisotropic trends of fracture hydraulic connectivity (some of them marked out by straight lines).

filter applied to a node calculates a weighted average of its hydraulic head and the hydraulic heads at its neighbouring nodes. The size and shape of the neighbourhood and the specific weights used will have an impact on the effect of each type of filter. If there is enough data, and it is reasonably well distributed, a simple normalised Laplacian operator may reveal some fracture hydraulic connectivity trends as shown in fig. 3.33.

3.7 Modelling hydrogeological systems 3.7.1

C on ce p t s

An arbitrary bounded rock mass may be viewed as an isolated geological system formed by an assemblage of interrelated lithologic and structural features, from minute to huge scales, making up an integrated whole, containing around 90 to 99% of solid matter and a small proportion of fluid matter in the remaining space. If its fluid phase is in focus, it is labelled a hydrogeological system, and this is generally too complex to be fully appreciated. However, simplified models may describe some of its main aspects and behaviour from different points of view. For example, a “groundwater management model’’ may be formulated to inform decisions about cost-effective operation granting environmental safety. A numerical simulation devised to predict groundwater flow under different boundary conditions requires two paired models. The first model, called “conceptual’’ and necessarily conceived before the second, explains the various modes of occurrence and properties of the fluid phase by means of verbal, iconic and quantified descriptors based on the interpretation of many kinds of field and laboratory investigations, such as geological mapping, borehole core samples, geochemical indicators, geophysical parameters, hydrological data, etc. The second one, the “mathematical’’ model, results from the applications of the principles of hydraulics to the conceptual model. It describes, in purely mathematical terms, the physical and chemical processes involving the percolation of the interconnected and unrestrained fluid phase, including its



carrying capacity. It is utilised to clarify the past and to predict the future behaviour of the fluid phase for different scenarios. Its retrospective and predictive ability must be always measured against some reference space-time values and if detected errors surpass predefined levels, both models must be improved in a cyclic way or replaced by better representations. However, depending on the required levels of accuracy, the improved models may become increasingly sophisticated and complex. Construction of a conceptual model is an inductive process involving inferences and correlations from a finite set of field observations and investigations. It is not a deterministic outcome of an encoded procedure, 100% correct, as with a mathematical model. Its degree of realism depends on the inherent complexity of the hydrogeological system as well as on the economic constrains on time and energy required to build it. It may be more or less acceptable according to the needed output and level of detail. In contrast, the mathematical model is a formal deductive process, presumably well formulated, whose reliability is entirely reliant on the premises of the conceptual model. Consequently, it is simple to recognise that the conceptual construct is by far the most important part of a numerical simulation process. Conceptual models vary in how the main structural features that convey groundwater are treated, how their properties are estimated and how their hydraulic performances are described. Conductive fractures and conduits within a subsystem may be modelled as a network of separated but interconnected discontinuities or grouped together in a single entity. Their hydraulic properties may derive from the integration of small-scale statistical data or from the interpretation of large-scale tests results. Corresponding hydraulic behaviour may be portrayed for each discontinuity or for groups of fractures, or by a combination of both sorts of descriptions. A well organized and commented anthology on the subject can be found in Rock Fractures and Fluid Flow, National Academic Press, Washington, 1966. 3.7.2

Gu id el i n e s t o c o n c e p t u a l m o d e l s

Generally, hard rock masses are inhomogeneous at many scales and their conceptual models, depending on the type of the numerical approach to be followed, may be difficult to construct. However, conceptual models for pseudo-continuous solutions compliant to permeabilities described by second order tensors are relatively simple to implement for most practical applications. There are no fixed rules to build them but only a few guidelines. Indeed, taking into account all comments and observations presented in Chapter 1, one may proceed as follows: a)

b)

c)

If a dependable topographic map is not available at a scale judged appropriate for the intended simulation, including watercourses, ponds, springs, seeps and other hydrogeological features of interest, compile cartographic and surveys data to generate the model base map, preferably as a suitable digital elevation model. If not available or not adequate, produce a hydrogeological map compliant to the compiled base map relying on existing geological and hydraulic information properly supplemented by fieldwork, geophysics, geochemistry, borehole sampling, hydraulic field tests, etc. To cope with the frequent scarcity of data far from the area of interest of the simulation, the model boundaries are preferably placed on “natural’’ Dirichlet


122

d)

e)

f)

g)

h)

i)

j)


and Neumann conditions, such as lakes, rivers, practically impervious geological contacts or deep impervious rock masses. Agree on the network parameters and on the cell dimensions to accommodate the details of the numerical simulation and subdivide the flow domain into distinct units having almost similar hydraulic properties based on their hydrogeological features and properties. For the intact rock matrix within each cell or group of cells, evaluate the absolute or relative average 3D permeability tensor based on laboratory tests, published data or experience. For the fractured rock mass within each cell or group of cells, evaluate the absolute or relative average 3D permeability tensor taking into account the properties of the intact rock matrix and of the minor discontinuities of the rock mass. As before, this evaluation can rely on field tests, published data or experience. Field tests, as those proposed by Hshieh et al., takes into account the hidden contributions of all unseen discontinuities, without exception, and implicitly integrate the effect of the intact rock matrix. All major discontinuities whose trace length exceeds around a quarter to a half of the smaller horizontal dimension of the base map will be considered in the next step. Based on geological and structural features, field tests, published data, experience etc., evaluate the absolute or relative average 3D permeability tensor of the major discontinuities not included in previous steps. It is important to keep in mind that their eigenvalues and eigenvectors must implicitly translate their hydraulic transmissivity and interconnectivity at the simulation scale. To take into account the influence of major discontinuities, apply the summation rule (see Chapter 1, section 1.1.6.2, Fractures and conduits) to explicitly integrate the 3D permeability tensor of each cell or group of cells. Calibrate the 3D permeability tensors of the grid cells by optimising the numerical simulation based on known time series of hydraulic head values at observation points in the flow domain and its boundaries as well as on known time series of discharge values at sources, sinks and installed wells. This calibration becomes easier if based on relative rather than absolute previously inferred values. Recalibrate the simulation based on a time series of the gross mass balance of the flow domain, but only if properly known.

To assure a relative consistency between all these evaluations and calibrations, it is important to make use of the same premises and choice criteria for all sensible parameters of the whole model. After a few progressive adjustments to factual data, the calibrated simulation may replicate the probable groundwater flow with some dependability. In case of disagreements between the calibration results and the factual observations of an experienced hydrogeologist, decisions about the conceptual model should be inclined to the interpretation of the hydrogeologist.

References Bohling, G., 2005, “Kriging’’, Kansas Geological Survey, C&PE 940, October. Franciss, F. O., 1994, First Underground Refrigerated LPG Caverns in Brazil, Eurock 94, Rock Mechanics in Petroleum Engineering, Delft, Netherlands.


D a t a a n a l y s i s 123 Hantush, M. S., 1956, “Analysis of Data from Pumping Tests in Anisotropic Aquifers’’, Journal of Geophysics Research, no. 72. Hsieh, P.A. and Neuman, S.P., 1985, “Field determination of the three-dimensional hydraulic conductivity tensor of anisotropic media’’, Water Resources Research, Vol. 21, no. 11. Kosakowski, G., “Introduction to Applied Geostatistics’’, Waste Management Laboratory, Paul Scherrer Institut, 5200 Villigen PSI, Switzerland. Press, W. H. et alia, 1994, Linear Prediction and Linear Predictive Coding, Chapter 13, in Numerical Recipes, Cambridge University Press, 2nd Ed. Rocha, M. and Barroso, M., 1971, “A method of integral sampling of rock masses’’, Rock Mechanics, Vol. 3, 1. Rocha, M. and Franciss, F. O., 1977, “Determination of Permeability in Anisotropic Rock Masses from Integral Samples’’, Rock Mechanics, Vol. 9/2–3. Sophocles, J. O., 1992, “Yule-Walker algorithm and Burg’s method in Optimum Signal Processing’’, Macmillan. Terzaghi, R., 1965, “Sources of errors in joint surveys’’, Geotechnique. 15: 287–303.


Chapter 4

Finite differences

4.1 Preliminaries Actual groundwater flow systems concerning fractured rock masses are generally inhomogeneous, irregularly anisotropic and particularised by mixed boundary conditions. Therefore, the numerical simulation of the spatial and temporal variation of their hydraulic heads by a single polynomial approximant may bear unacceptable errors. However, fragmenting these complex systems into simple adjoining subsystems, each one described by its own polynomial approximant, may reduce the solution errors to acceptable levels. Evidently, these restricted “piecewise’’ approximants, properly allocated for each subsystem, must best fit the connecting boundary conditions to the adjacent subsystems. Finite difference and finite element methods make use of this approach. In the last decades, the decrease in size and cost concurrent with the increase of speed and memory of commercial computers as well as the development of user friendly programming languages has induced most universities, research centres and independent consultant engineers, geologists, geophysicists and hydrogeologists to explore the possibilities of numerical techniques more frequently. Indeed, working out practical problems requiring the solution of partial differential equations applied to relatively complex domains may only be acceptably realistic using numerical techniques. Among them, finite difference algorithms are very attractive since they are simple and intuitive as well as suited to spreadsheet programming. There is a vast literature on this subject including technical papers and textbooks. The reader is encouraged to gain more knowledge on this subject (Bear, 1979; Wang and Anderson, 1982; Lam, 1994; Press et al., 2007). However, given the introductory character of this book, only essential concepts and consistent finite difference algorithms are discussed in this chapter, including 3D applications for fractured rock masses.

4.2 Finite difference b asics 4.2.1

Differ e n c e e q u a t i o n s

The derivatives occurring in partial differential equations may be replaced by finite differences, thus converting these partial differential equations into simpler difference equations. As a result, continuity and boundary condition differential equations may be substituted by equivalent difference equations.


126


Figure 4.1 Example of rough discretisation of a horizontal aquifer.

Note that any 3D problem domain can be discretised, i.e. partitioned into rectangular cuboids defined by the size of their edges and the coordinates of their geometrical centres (see fig. 4.1). After this process, continuity difference equations may be applied at the centres of all cuboids within the problem domain. Additional supplementary difference equations may be formulated consisting of specified heads and/or flows at all cuboids fitted to the boundaries of the problem domain (Dirichlet and Neumann conditions). These equations, taken simultaneously, generate a set of algebraic equations whose numerical treatment gives the approximate solution of the partial differential equation at all cuboids’ centres of the problem domain. A similar argument applies to 2D or 1D problems. 4.2.2

Fin it e d i f f e r e n c e s

Numerical values of a continuous function may only be known at regularly spaced points in space. Even with this restriction, its first-order and second-order partial derivatives can be estimated by appropriate finite differences at these points. As an example, consider part of the x-trace of the sloping water table of a horizontal aquifer divided into four cells, 100 m wide (see fig. 4.2). The water table elevation, taken as equivalent to the hydraulic head h by Dupuit’s assumption, is only described at abscissas 150 m, 250 m, 350 m and 400 m. However, based on simple geometry, it is possible to estimate the first and the second-order partial x-derivatives of h at the cell boundaries and cell midpoints respectively. Denoting by δx the 100 m equal intervals, the first-order partial x-derivative at the cell boundary at xi−1/2 = 200 m, just in the middle of xi−1 = 150 m and xi = 250 m, may be estimated by:

hi+1 − hi ∂ h + O(δx2 ) = (4.1) ∂x i+ 1 δx 2


Finite differences

127

Head h (m)

540

530

(xi1 150 m, h 534.68 m)

(xi 250 m, h 532.72 m) (xi1 350 m, h 530.3 m)

100

200

300 x (m)

Continuous head h

400

Discrete head h values at cell mid-point

500

Cell boundary

dh/dx (m/m)

Figure 4.2 Trace of the sloping water table of a horizontal aquifer on an x-cross-section divided into four cells, 100 m wide. Three generic contiguous points are defined by (xi−1 = 150 m, h = 534.68 m), (xi = 250 m, h = 532.72 m) and (xi+1 = 350 m, h = 530.30 m). 0.015 x i

1 2

200 m, h 0.020 x

x i

0.02

1 2

300 m, h 0.024 x x i

0.025 100

200

300 x (m)

3 2

400

400 m, h 0.019 x

500

Numerical first derivative at cell boundary

Figure 4.3 x-∂h/∂x plot of the numerical of the first-order partial derivatives at cell boundaries xi−1/2 = 200 m, xi+1/2 = 300 m and xi+3/2 = 400 m.

The additional term O(δx2 ) in equation 4.1 stands for the approximation error. By a similar rule, the first-order partial x-derivative at xi+1/2 = 300 m, just in the middle of xi = 250 m and xi+1 = 350 m, may be estimated by:

∂ h ∂x

i− 12

=

hi − hi−1 + O(δx2 ) δx

(4.2)

For the above expressions (equation 4.1 and equation 4.2), called central approximation formulas, the error is proportional to δx2 . Consequently, dividing by two the interval δx, i.e. making new δx = δx/2, would decrease the truncation error to a quarter of the previous one. The x-∂h/∂x plot of the numerical first-order partial derivatives at the cell boundaries 200 m, 300 m and 400 m is depicted in fig. 4.3. The second-order partial x-derivative at the cell boundary 300 m, in the middle of xi = 250 m and xi+1 = 350 m, may also be estimated by a central approximation:

∂2 h ∂x2

= i

∂ ∂ h ∂x ∂x


i

hi+1 − hi hi − hi−1 − δx δx = + O(δx2 ) δx

(4.3)

d[dh/dx]/dx (m/m/m)

128


1·104 5·105

xi 250· m, 2 h 0.000037 x 2

0 5·105 100

200

xi1 350 m, 2 h 0.000058 x 2

300 x (m)

400

500

Numerical second derivative at cell mid point

Figure 4.4 x-∂2 h/∂x2 plot of the numerical second-order partial derivatives at cell mid-point xi = 250 m and xi+1 = 350 m.

As before, the term O(δx2 ) corresponds to the order of magnitude of the approximation error. The x-∂2 h/∂x2 plot of the numerical second-order partial derivatives at cell midpoints 250 m and 350 m is shown in fig. 4.4. In general, an n-order finite difference equation applied at a point xi may be formally deduced from a backward or a central or a forward Taylor series expansion about this point by ignoring the terms containing higher-order derivatives. The truncation error is proportional to the order of magnitude of the first term of the discarded part of the series. For a first-order finite difference equation, the truncation error is O(δx) for backward or forward expansions and O(δx2 ) for a central expansion. The first-order and second-order partial derivatives approximations estimated by appropriate finite differences, as discussed above, can be applied to any continuous parameter over the problem domain. However, the numerical algorithm must be constructed by keeping in mind that the hydraulic conductivity is implicitly considered a continuous parameter in the general continuity equation. As an important consequence, the hydraulic conductivity must be assumed as a continuous property, even if highly variable, to assure dependable algorithms. For an example of this possibility see fig. 4.5 which shows one of the results obtained from a numerical sensitivity analysis devised to simulate the head decay during the downward percolation of water through a layer of laminated sediments.

4.2.3

Differe n c e e q u a t i o n s f o r s t e ady-s t at e s y s t e m s

Finite difference equations that are algebraically equivalent to the continuity differential equations for steady-state systems can also be directly derived. As an example, consider a rectangular cuboid centered at point i defined by edges δx, δy and δz (see fig. 4.6). Assuming constant density ρ, the Poisson equation applied to a 1D steady-state system all along the x-axis can be written as:

∂ ∂ k h =Q ∂x ∂x


(4.4)

Finite differences

1

129

0.01 1·104 1·105

0.5

1·106

K (m/s)

H (m)

1·103

1·107 0

0

0.2

0.4

0.6

0.8

1·108

L (m) Head (m)

Permeability (m/s)

Figure 4.5 One of the numerical results of a sensitivity analysis of the head decay associated with the vertical percolation of water in an encapsulated integral core sample, measuring 1 m length, extracted from a thick layer of laminated sediments. Heads are referred to the left y-axis. Jagged vertical permeabilities are referred to the right y-axis (log scaled). The permeability of each 1 cm thick lamina varied randomly between 10−8 and 10−2 m/s.

z

i1

i x

i1 y

Figure 4.6 Cuboid with edges δx, δy and δz centered at point xi . Water enters left area δyδz at point xi−1/2 and leaves right area δyδz at point xi+1/2 under hydraulic gradients (hi − hi−1 )/δx and (hi+1 − hi )/δx, respectively.

Considering the hydraulic conductivity k as a continuous variable, the left-hand side of equation 4.4 can be expanded by the product rule and the expression takes the form:

∂ ∂ ∂ ∂ k h +k h =Q (4.5) ∂x ∂x ∂x ∂x Consider now three generic contiguous points: xi−1 , xi and xi+1 . To reduce the truncation error and assure a stable algorithm, the finite difference approximation for the first term of the left-hand side of the equation must be applied to the central point xi . This may be done by taking the average of the approximations at midpoints xi−1/2 and xi+1/2 .

∂ ∂ ∂ ∂ + k h k h

∂x ∂x ∂x ∂x ∂ ∂ i− 12 i+ 12 k h = (4.6) + O(δx2 ) ∂x ∂x 2 i


130


Then, the equivalent finite difference equation for above expression is:

∂ ∂ k h ∂x ∂x

i

ki+1 − ki hi+1 − hi ki − ki−1 hi − hi−1 + δx δx δx δx = + O(∂x2 ) 2 (4.7)

By the same arguments, the finite difference equation for the second product of the left-hand side of the equation is:

k

∂ ∂ h = ki ∂x ∂x

hi+1 − hi δx

−

hi − hi−1 δx

δx

+ O(δx2 )

(4.8)

Simplifying and collecting terms, the finite difference equivalent for the continuity equation (equation 4.4) is:

ki+1 + ki 2

hi+1 − hi δx

−

ki + ki−1 2

hi − hi−1 δx

δx

=Q

(4.9)

Multiplying both members of this expression by the subsystem volume, and taken into account that δδyδz = δS and δxδyδz = δV, yields:

ki+1 + ki 2

hi+1 − hi δx

−

ki + ki−1 2

hi − hi−1 δx

δS = Q δV

(4.10)

This expression has a clear physical meaning. The left-hand side measures the difference between the entering and leaving discharges along the x-axis through the two opposite faces of area δS. The right-hand side measures the amount of water volume Q · δV added or drained by an occasional source (or sink) of strength Q per unit volume. It is important to keep in mind that this algorithm considers the hydraulic conductivity as a continuous variable, but only valued at the cell centres. In this case, the hydraulic conductivity at midpoints xi−1/2 and xi+1/2 must be estimated by an arithmetic mean. However, if one assumes that the hydraulic conductivity takes constant values inside each cell, forming a series circuit along the x-axis, it is recommended to approximate the hydraulic conductivity at midpoints xi−1/2 and xi+1/2 by a harmonic mean (Bear, 1972). A simple exercise to compare the numerical effect of these two types of mean is shown in fig. 4.7. Considering the premises that support the pseudo-continuous concept for random fractures and, implicitly, for assuming averages between cell midpoints, the author has a preference for the arithmetic mean based on his modelling experience.


Finite differences

K assumed continuous but only defined at interval midpoints

K assumed constant in each interval (series circuit) 0.1

0.1

0.4

1105

0.2

1103

0.6 0.4

1105

0.2 1107

0 0

0.2

0.4

0.6

0.8

1107

0

1

0

0.2

0.4

L (m) Head (m)

0.6

0.8

L (m)

Permeability (m/s): max K/min K 10

Head (m)




0.1

0.6 0.4

1105

0.2

1103

0.6 0.4

1105

K (m/s)

110

3

H (m)

0.8 K (m/s)

0.8

0.2 1107

0 0

0.2

0.4

0.6

0.8

1107

0

1

0

0.2

0.4

L (m) Head (m)

0.6

0.8

L (m)


Head (m)




0.1 3

110

0.6 0.4

1105

0.2

H (m)

0.8 K (m/s)

0.8

1103

0.6 0.4

1105

K (m/s)

H (m)

K (m/s)

0.6

H (m)

0.8 1103

K (m/s)

H (m)

0.8

H (m)

131

0.2 1107

0 0

0.2

0.4

0.6

0.8

1

1107

0 0

0.2

L (m) Head (m)


0.4

0.6

0.8

L (m) Head (m)


Figure 4.7 These three pairs of L × H plots give an idea about the influence of the type of mean, arithmetic or harmonic, used in the finite difference algorithm. Compare the results of the numerical simulation of the head decay for a steady upward percolation of water from beginning to end of a homogenous soil column with height L = 1 m under a constant head of H = 1 m. The soil permeability is 10−6 m/s in all cases except for a 10 cm layer in the column centre that is N times more pervious than the rest of the soil. The left and right plots correspond respectively to the arithmetic and harmonic means for N equal to 10, 100 and 1000, as indicated in the plots. For N = 10, there is no appreciable difference between the left and right plots. For N = 100 or 1000 the left plots show negligible decays for the central cell, as expected, while the right plots remain practically unchanged.

4.2.4

Differ e n c e e q u a t i o n s f o r u n s t e ady-s t at e s y s t e m s

For unsteady-state systems, a 1D general continuity equation system all along the x-axis can be written as:

∂ ∂ ∂ k h = SV h +Q ∂x ∂x ∂t The term SV is the specific volumetric storage.


(4.11)

132


H(x,0) constant

1.5 H(x,t) surface 1 H (m)

n·t 1 n – ·t 2

(n1)·t 0.5

Drain 0

0

xi1 xi

10

8 xi1

6 4

X (m

)

2

20

in)

t (m

30

Figure 4.8 Transient surface H(x, t) estimated by a Crank-Nicolson scheme. The first-order partial t-derivative ∂h/∂t is approximated at point xi and at time step n + 1/2, i.e. just in the middle of the t-steps n and n + 1. The finite expression for the space continuity is taken as the arithmetic average of the finite expressions at point xi at t-steps n and n + 1.

As with the space-variable x, the time variable t can also be discretised into many steps of equal time-intervals δt counted from a starting time t0 . Then, the product n · δt measures the time interval (t − t0 ). The first-order partial t-derivative ∂h/∂t occurring in the right-hand side of the governing equation 4.11 may be approximated at point xi and at time step n + 1/2, i.e. just in the middle of the time steps n and n + 1, then:

SV

∂ h ∂t

n+ 12

+ Q = SV

(hi )n+1 − (hi )n + Q + O(δt2 ) δt

(4.12)

In this equation, the pseudo-exponents n and n + 1 stand for the values of the head hi at the moments (n)·δt and (n + 1)·δt (see fig. 4.8). The finite expression for space-continuity at xi , on the left-hand side of the equation 4.12 is undefined at the t-step n + 1/2. However, this value may be taken as the arithmetic average of the corresponding finite expressions at time steps n and n + 1. Then,


Finite differences

133

for each t-step n + 1/2, the equivalent finite difference equation for continuity is: 



(hi+1 )n − (hi )n ki+1 + ki ki + ki−1 (hi )n − (hi−1 )n − ···  2 δx 2 δx 1   

δS

2 ki+1 + ki (hi+1 )n+1 − (hi )n+1 ki + ki−1 (hi )n+1 − (hi−1 )n+1  + − 2 δx 2 δx   n+1 n (hi ) − (hi ) · · · SV δt = (4.13) δV +Q As before, the pseudo-exponents stand for the t-steps n and n + 1. This algorithm is unconditionally stable and second-order accurate in time and space. It generates for each t-step a set of simultaneous equations that must be solved to find the head values h corresponding to that t-step. This clever solution was jointly proposed by John Crank (1916–2006), a British mathematical physicist partly dedicated to numerical solutions of partial differential equations, and Phyllis Nicolson (1917–1968), also a British mathematician associated with the development of the Crank-Nicolson scheme in mid twentieth century.

4.2.5

Differ e n c e e q u a t i o n s f o r b o u n dar y c o ndi t i o ns

If the finite difference equations are applied to the interior nodes of a steady-state groundwater system for which all the heads at the exterior nodes, i.e. at the boundary nodes, are already specified (Dirichlet boundary conditions), then the number of unknown heads and algebraic equations are the same. However, the number of unknowns exceeds the number of algebraic equations if the discharge rates at one or more exterior nodes are specified as an alternative for heads (Neumann boundary conditions). In this case, supplementary equations are needed to solve the problem. For example, consider the Neumann prescription applied to a boundary point xi−1 . In this case, the hydraulic exit gradient ∂h/∂xi−1 at that point can be approximated by a second-order accurate backward difference:

∂ 3 · hi−1 − 4 · hi + hi+1 = h + O(δx2 ) (4.14) ∂x i−1 2 · δx Recall that a quadratic interpolation polynomial collocated at three successive and collinear points xi−1 , xi and xi+1 separated by equal intervals δx yields this same algorithm. In equation 4.14, the heads hi−1 , hi and h1+1 refer to the successive three collinear points xi−1 , xi and x1+1 . These points are located along an interior normal to the boundary at xi−1 . The corresponding Neumann condition is: k

3hi−1 − 4h1 + hi+1 δxδz = −qi−1 2δx


(4.15)

134


Then, it is possible to express hi−1 in terms of qi−1 , hi and hi+1 : hi−1

1 δx qi−1 = 4hi − hi+1 − 2 3 δxδz k

(4.16)

Replacing hi−1 occurring in all algebraic equations for the left-hand expression of equation 4.16 equates the number of equations and unknowns. After solving this set for the interior heads, the remaining border heads hi−1 are directly calculated by the Neumann condition. For unsteady simulations, both conditions may be time-dependent values, if this is the case. There are other ways to apply boundary conditions, including approximated solutions for refined and non-regular boundaries (see, for example, Lam, 1994). 4.2.6

S imu l t a n e o u s d i f f e r e n c e e q u a t i o ns

4.2.6.1 P re l i m i n a r i es For vast groundwater systems, described by inhomogeneous and anisotropic subsystems, full of different kinds of wells, drains, natural springs and sinks, particularised by many types of boundary and initial conditions, the resulting set of simultaneous difference equations may involve several thousands of unknowns. Solving it, according to their hydraulic properties and boundary and initial conditions, yields the head values at the centres of the discrete cuboids within the problem domain or fitted to its borders. Given the introductory character of this book, only two iterative techniques, known as the Jacobi and Gauss-Seidel methods will be considered next. Carl Gustav Jacob Jacobi (1804–1851), Johann Carl Friedrich Gauss (1777–1855) and Philipp Ludwig von Seidel (1821–1896) were outstanding and inspired nineteenth century German mathematicians. These methods are easily programmable with the help of commercial mathematical and graphical software. See some specialised sites: www.office.microsoft.com/excel, www.ibm.com/software/lotus, www.wolfram.com, www.ptc.com, www.mathworks.com. 4.2.6.2 Gau s s-Sei d el i ter a ti v e r ou ti n e Example 4.1 provides an easy way to grasp the essence of the Gauss-Seidel iteration routine.

Example 4.1 In this 2D elementary exercise, groundwater flows within a homogeneous, isotropic, horizontal and confined sandstone aquifer trapped by an upper and a bottom layer of impervious shale. In addition, two almost parallel and vertical diabase dykes, practically impervious, isolate a 1000 m wide sandstone band between them. A piece of that band, measuring 1000 m × 1000 m in plan view, may be taken as a simple groundwater system to be numerically simulated (see fig. 4.9).


Finite differences

135

Vertical almost impervious diabase dyke Impervious upper layer of shale

Vertical almost impervious diabase dyke

Confined sandstone aquifer

Impervious bottom layer of shale

Figure 4.9 Confined sandstone aquifer, “sandwiched’’ between an upper and a lower shale layers. Two parallel and vertical impervious diabase dykes, one in the front and the other in the rear of the perspective, isolate the aquifer laterally.

Now, consider a 2D square lattice defined by 20 × 20 squares of 50 m size (see fig. 4.10). Denoting by hi,j the head value at the centre (i, j) of each square and by δx its side dimension, the finite difference equation is simply: hi, j =

qi ,j 2 1 hi−1,j + hi+1 ,j + hi ,j−1 + hi ,j+1 + δx 4 T

(4.17)

This equation applied at the 400 interior nodes of the 2D square lattice generates a set of simultaneous equations characterised by a band diagonal matrix for the head coefficients. In this case, the Gauss-Seidel method may be used with success. The heads along the left and right sides of the grid, respectively defined by x1 = 0 m and x21 = 1000 m, are prescribed by Dirichlet (see fig. 4.11). The upper and lower sides of the grid, defined by y1 = 0 m and y20 = 1000 m, are no-flow borders except at point (x18 = 850 m, y1 = 0 m) where 50 m3 /hr of water leaks through a breach in the front dyke. The sandstone constant transmissivity is 3.5 · 10−4 m2 /s. Moreover, 150 m3 /hr and 75 m3 /hr of water are respectively pumped from and injected into the aquifer at points (x8 = 350 m, y14 = 650 m) and (x16 = 750 m, y10 = 450 m). The effect of the pumping or the injection well is simulated by a uniform withdrawal or accretion on top of the cell equal to the pumping or injection rate divided by the top area δx2 . For the breach in the front dyke, the uniform leakage corresponds to the total leak divided by the lateral area δxδz. To start a Gauss-Seidel iteration, an initial inexact solution array for all unknown heads hi,j must be advanced. There are many ways to do this. For example, it is possible to fit a surface h0 (x, y) to the specified boundary heads. Other approach is to find a simplified


136


1000

800

Qpump 150

m3 hr

600 Y (m)

Qinjection 75

m3 hr

400 Tsandstone 3.5 104

m2 s

200

Qleak 75

0

0

200

400

600

m3 hr 800

1000

X (m) Grid centers

Impervious upper border (Neumann)

Pumping well

Impervious lower border (Neumann)

Injection well

Pervious left border (Dirichlet)

Border leak

Pervious right border (Dirichlet)

Figure 4.10 2D square lattice formed by 400 squares of 50 m size. To each grid centre (i, j), defined by coordinates (xi , yj ), corresponds a generic head hi ,j . Four adjacent heads are described by four triplets: (xi−1 , yj , hi−1 ,j ), (xi+1 , yj , hi+1 ,j ), (xi , yj−1 , hi ,j−1 ) and (xi , yj+1 , hi ,j+1 ). Subscripts i and j run from 1 to 21.

solution employing an analytical method. Then, for all interior points (i, j) where qi = 0, except for the pump point (8, 14), injection point (16, 10) and near border points along rows (i, 2) and (i, 20), the general finite difference equation gives the head value hi,j : hi,j =

1 (hi−1,j + hi+1,j + hi,j−1 + hi,j+1 ) 4

(4.18)

The heads h8,14 at the pump point (8, 14) and h16,10 at injection point (16, 10) are given by: (Qpump )8,14 1 h8,14 = h7,14 + h9,14 + h8,13 + h8,15 + (4.19) 4 T


Finite differences

137

Dirichlet heads (m)

700

Left border Right border

600

500

400

0

200

400

600

800

1000

Y (m)

Figure 4.11 Specified head along the left and right borders (Dirichlet boundary conditions).

h16,10 =

(Qinjection )16,10 1 (h15,10 + h17,10 + h16,9 + h16,11 ) + 4 T

(4.20)

In these expressions, Qpump and Qinjection are negative and positive quantities, respectively. For the near border points along rows (i, 2) and (i, 20), the general expressions for hi,j include unknown heads along the boundaries (i, 1) and (i, 21). At these border points, the corresponding exit gradients may be approximated by three-point formulas:

∂ −hi,3 + 4hi,2 − 3hi,1 = h + O(δy2 ) ∂y i,1 2δy (4.21)

3hi,21 − 4hi,20 + hi,19 ∂ 2 = h + O(δy ) ∂y i,21 2δy Now, for impervious boundaries, all exit gradients normal to the no-flow borders are zero. Then, equating these approximations for ∂h/∂y to zero at points (i, 1) and (i, 21) it is possible to express hi,1 and hi,21 in terms of the interior heads, except for the leak at point (18, 1): 1 (4hi,2 − hi,3 ) 3 1 = (4hi,20 − hi,19 ) 3

hi,1 = hi,21

(4.22)

Substituting the right-hand side of these expressions for hi,1 and hi,21 the general finite difference equation at points along the rows (i, 2) and (i, 20) removes these unknowns, except near the leak point (18, 1): 1 1 hi,2 = hi−1,2 + hi+1,2 + (4hi,2 − hi,3 ) + hi,3 4 3 (4.23) 1 1 hi−1,20 + hi+1,20 + hi,19 + (4hi,20 − hi,19 ) hi,20 = 4 3


138


Simplifying and collecting: ·hi,2 = ·hi,20

3 3 1 hi−1,2 + hi+1,2 + hi,3 8 8 4

(4.24)

3 3 1 = hi−1,20 + hi+1,20 + hi,19 8 8 4

At the leak point (18, 1), the Neumann condition states that: (Qleak )18,1 −h18,3 + 4 h18,2 − 3 h18,1 δxδz = − 2δy k

(4.25)

Taking into account that δx = δy and that k · δz =T, this equation simplifies to: −h18,3 + 4h18,2 − 3h18,1 = −2

(Qleak )18,1 T

In this expression, Qleak is also a negative quantity, from which: 2(Qleak )18,1 1 4h18,2 − h18,3 + h18,1 = 3 T

(4.26)

(4.27)

Substituting for h18,1 the general finite difference equation at point (18, 2) removes this unknown: 2(Qleak )18,1 1 1 h18,2 = (4.28) h17,2 + h19,2 + 4h18,2 − h18,3 + + h18,3 4 3 T Simplifying and collecting: h18,2 =

3 3 1 1 Q.leak 18,1 h17,2 + h19,2 + h18,3 + 8 8 4 4 T

(4.29)

Having expressed all interior heads in terms of their interior neighbours, the iteration procedure starts from the initial solution estimate [hi,j ]0 , running from i = 1 to 21 and from j = 2 to 20 (because columns (1, j) and (21, j) are Dirichlet prescribed) as follows: •

At all interior points (i, j), except at the pump point (8, 14), injection point (16, 10) and near border points along rows (i, 2) and (i, 20): (hi,j )k+1 =

•

1 (hi−1,j )k+1 + (hi+1,j )k + (hi,j−1 )k+1 + (hi,j+1 )k 4

(4.30)

At pump point (8, 14) and injection point (16, 10): (Qpump )8,14 1 k+1 k+1 k k+1 k = + (h9,14 ) + (h8,13 ) + (h8,15 ) + (h8,14 ) (h7,14 ) 4 T (h16,10 )k+1 =

(Qinjection )16,10 1 (h15,10 )k+1 + (h17,10 )k + (h16,9 )k+1 + (h16,11 )k + 4 T (4.31)


Finite differences

•

139

At all near border points (i, 2) and (i, 20), except near the leak point (18, 1): (hi,2 )k+1 = k+1

(hi,20 ) •

3 3 1 (hi−1,2 )k+1 + (hi+1,2 )k + (hi,3 )k 8 8 4

(4.32)

3 3 1 = (hi−1,20 )k+1 + (hi+1,20 )k + (hi,19 )k+1 8 8 4

Near the leak point (18, 1): (h18,2 )k+1 =

3 3 1 1 (Qleak )18,1 (h17,2 )k+1 + (h19,2 )k + (h18,3 )k + 8 8 4 4 T

(4.33)

In the above iteration formulas, the pseudo-exponents k or k + 1 stand respectively for the approximations of the head values hi,j after iteration or k + 1. Moreover, it is take for granted that these systematic computations are performed in such a way that iterations k + 1 for hi−1,j and hi,j−1 are always calculated before the iteration k + 1 for hi,j , thus avoiding the need for extra storage.

1000

49

0

515

900

h 545 m h 540 m h 535 m h 530 m h 525 m h 520 m h 515 m h 510 m h 505 m h 500 m h 495 m h 490 m h 485 m h 480 m h 475 m h 470 m h 465 m h 460 m h 455 m h 450 m h 445 m h 440 m h 435 m h 430 m h 425 m h 420 m h 415 m

800 700

515

Y (m)

490

600 500

49

0

46

300

5

490

400

515

200 100 0 0

100

200

300

400

500 X (m)

600

700

800

900

Figure 4.12 Equipotentials distribution for the area modelled in example 4.1.


1000

140


Iteration stops when the greatest relative change after iteration k + 1, calculated by [((hi,j )k+1 − (hi,j )k )/(hi,j )k ], taking into account all iterated values hi,j or only some specific and critical points, is smaller than a prescribed limit ε: (hi,j )k+1 − (hi,j )k

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