Mechanics of porous and fractured media

8

SERIES IN THEORETICAL AND APPLIED MECHANICS Edited by RKTHsieh

SERIES IN THEORETICAL AND APPLIED MECHANICS Editor: R. K. T. Hsieh

Published Volume 1: Nonlinear Electromechanical Effects and Applications by G. A. Maugin Volume 2: Lattice Dynamical Foundations of Continuum Theories by A. Askar Volume 3: Heat and Mass Transfer in MHD Flows by E. Blums, Yu. Mikhailov, and R. Ozols Volume 5: Inelastic Mesomechanics by V. Kafka Volume 9: Aspects of Non-Equilibrium Thermodynamics by W. Muschik

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Mechanics of Porous and Fractured Media V. N. INikolaevskij

World Scientific „ . uopyngiTwa Material ,,

Singapore * New Jersey • Hong Kong

Author V. N. Nikolaevskij Institute of Physics of the Earth U.S.S.R. Academy of Sciences Boul. Gruzinskaya 10, Moscow D-242 U.S.S.R. Series Editor-in-Chief R. K. T. Hsieh Department of Mechanics Royal Institute of Technology S-10044 Stockholm, Sweden Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Revised translation of the original Russian edition of "Mekhanika poristykh i treshinnykh sred" published by Nedra, Moscow © 1984.

Library of Congress Cataloging-in-Publication data is available. MECHANICS OF POROUS AND FRACTURED MEDIA Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo­ copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISSN 0218-0235 ISBN 9971-50-383-2

Printed in Singapore by JBW Printers & Binders Pte. Ltd.

PREFACE TO THE RUSSIAN EDITION

The book is devoted to the theory of deformation, rapture and flow of porous and fractured media. Recently the theory has become very important for many applications mainly because of the increase of human influence on the environ­ ment. The creation of grandiose constructions, intensification of oil recovery by enhancement of reservoir permeability and by changes of fluid compositions, construction of deep mines and superdeep wells, preservation of rock bursts, earthquake prediction problems and many aspects of geodynamics, earthquake engineering, seismic exploration, underground explosion works, problems of underground storage of gas and of environment protection against pollution — all need the basic concepts of rapture of rock masses and of underground hydro­ dynamics. Many problems of chemical technology, powder metallurgy, etc., are close to the themes mentioned above. During many years of work in the USSR Academy of Sciences, the author has been involved in the development of new chapters of continuum mechanics which were connected directly with applied geomechanics. Now the results of these studies are gathered in this book. Here the main attention is devoted to the for­ mulation of basic physical concepts of deformation and flow processes and of their presentation in the form of continuum equations. The resulting theory is supplied by solution of illustrative examples to show the key peculiarities of the theory which can be checked by direct experiments. Although the choice of such examples was made on account of their applications, the author always thought of the prolongation of the studies by the researchers, involved with creation of industrial projects. In many cases the author's studies were developed further and used in applied sciences indeed, and it gave him great satisfaction. The reader will meet some names of specialists which took part in these works at different

V

vi Preface to the Russian Edition

stages. Some of their personal results, but not all of them and not in the full scale are included in the book and is mainly because of the limited space. Be­ cause of the same reason, the list of scientists which made prominent con­ tributions to the science of porous media and of underground hydrodynamics is also not complete. One can find their names in the review paper written by G. K. Mikhailov and the author of "Mechanics in the USSR for 50 years", and also in the "Mechanics Review Journal" (VINITI, Moscow) in the editing of which the author has taken part for many years. The author's postdoctorate fellowship (1974) and further lecturing (1977) at the John Hopkins University and Brown University (USA) were very important in the selection of some themes of study. The reader will find the description of some of the achievements of American scientists and will mention the wide use of their experimental results and first of all of the data of experiments with geomaterials under high pressures and tem­ peratures. The reader will find here some chapters and results, which now are belonging truly to the classical theory of flow through the porous media. However, the book is written in such a manner, which avoids the repetition of the well-known mono­ graphs and textbooks (by A. P. Krylov, P. Ya. Polubarinova-Kochina, Yu.P. Borisov, V. M. Dobrynin, Yu. P. Zeltov, Yu. P. Korotaev, B. B. Lapuk, A. H. Mizzadjanzade, M.D.Rosenberg, F. A. Trebin, M. I. Shvidler, V. N. Zchelkachev, etc.), many of which were published by "Nedra". The author could avoid the repetitions of these books mainly because of the more general approach and because of the unification of themes which were traditionally belonging to different branches of science — to geophysics, oil engineering, mechanics of soils or rocks, explosion processes, etc. This approach may break the information barriers between these branches of knowledge and may lead to their mutual enrichment. The bibliography given in the book does not claim completeness but contains mainly the review papers where the full scope of literature was given or the scienti­ fic papers directly connected with the presented results.

V.N

FOREWORD

Mechanics of porous media has many applications which are mainly connected with geomaterials and their behavior in nature. The vast territory of the USSR and its boundless mineral resources were the reasons for Russian scientists to pay more attention to the development of underground hydrodynamics and geomechanics. It brought forth the strong Russian school in this branch of mechanics. Of course, the area of porous media is playing an important role in chemical technology, in storage and vibrotransport of granulated materials, in powder technology, etc. The blast technics and geological processes had led us to investi­ gations of the peculiarities of fractured media. The author of this book was happy to give a course of lectures on the matter at the Royal Technological Institute (Stockholm, Sweden) in the autumn of 1985. For lectures he used his book which had been published by the printing house, "Nedra" (Moscow, 1984) in Russian and reflected the experience and the main directions of the art. It happened that there was no equivalent to the course of lectures mentioned above in the English language. Therefore Professor Richard K. T. Hsieh made a kind suggestion to publish the English translation of the book in the Series in Theoretical and Applied Mechanics of which he is Editor-in-charge for World Scientific Publishing Com­ pany. This suggestion was accepted by the author with gratitude. Because this field of knowledge is developing intensively it was necessary to add some new principal results achieved for the interval of 1984-1987. Besides it was found that some chapters of the Russian edition were too brief and they had to be enlarged. Correspondingly, the bibliography now includes 330 titles instead of 50. The aim was to represent to a foreign reader the large scope of Russian scientific literature.

vii

viii

Foreword

The author is hoping that the exchange of ideas and experience between countries and scientists will help to further the progress of the technology and to the growth of the standards of living of all nations.

Viktor N. Nikolaevskij

CONTENTS

Preface to the Russian Edition

v

Foreword

CHAPTER 1

vii

BASIC CONCEPTS OF CONTINUUM MECHANICS

1

§1.1. Equations of Continuum Mechanics §1.1.1. Ideology of continuum approach §1.1.2. Balances of mass and momentum §1.1.3. Continuum thermodynamics § 1.1.4. Elastoplastic models §1.1.5. Viscoelastic models § 1.1.6. Model of viscoelasticity with internal oscillators §1.1.7. Conditions at discontinuities §1.1.8. Shock adiabats in hydrodynamical approximation

1 1 2 4 8 9 10 13 15

§ 1.2. Space Averaging and Scale of Description §1.2.1. Integral form of mass, impulse and energy balances § 1.2.2. Averaged surface and volume parameters § 1.2.3. Difference of macrostresses and volume averaged stresses § 1.2.4. Chain of macroscopic equilibrium equations §1.2.5. Problem of computation of averaging and differentiation

16 16 18 24 24 27

§1.3. Fracture § 1.3.1. § 1.3.2. § 1.3.3.

28 28 31 32

of Solid Materials Thermodynamics of viscothermoelastic body with crack Criterium of crack growth in rate form Crack growth in viscoelastic plane

ix

x Contents

§1.3.4. §1.3.5. §1.3.6. § 1.3.7.

CHAPTER 2

Thermodynamics of steady state vicinity of a crack tip Path independent contour J-integrals Path-independent rate contour integrals of fracturing Particular forms of contour rate integrals

DILATANCY OF GEOMATERIALS

34 38 40 45

50

§2.1. Inelasticity Mechanisms and their Modelling Presentation §2.1.1. Description of continuous fracture of the medium §2.1.2. Plastic flow into pores § 2.1.3. Irreversible repacking of particles in contact §2.1.4. Relative microslip with solid friction for dilatant media §2.1.5. Coulomb slip surface as strong tangential velocity jump

50 50 51 54 56 60

§2.2. Elastoplastic Theory of Dilatancy §2.2.1. Nonassociated plastic flow rule § 2.2.2. Comparison with triaxial tests of sands § 2.2.3. Variants of hardening laws for geomaterials §2.2.4. Elastoplastic dilatancy of rocks §2.2.5. Sophisticated mathematical models

63 63 66 71 75 79

§ 2.3. Limit Equilibrium and Flow of Dilatant Masses §2.3.1. Plane plastic flows §2.3.2. Characteristical equations for plane plastic flow §2.3.3. Centered plastic flows §2.3.4. Density and stress characteristics §2.3.5. Flow of particulated masses from plane hopper § 2.3.6. Noncoinciding of stress and velocity characteristics §2.3.7. Coulomb slip surface structure §2.3.8. Strain localization

82 82 84 86 88 89 92 92 94

§2.4. Cracks, Earth Crust Structure and Earthquakes §2.4.1. Fractured states of Earth crust rocks §2.4.2. Shallow seismic boundaries as levels of fracturization §2.4.3. Water permeability of Earth crust §2.4.4. Ocean crust and olivines §2.4.5. Earth continental crust types §2.4.6. Structure of Earth crust faults

97 97 98 99 101 101 102

Contents xi §2.4.7. §2.4.8. §2.4.9. §2.4.10. §2.4.11.

CHAPTER 3

Crust waveguides as porous reservoirs Tectonic waves Brittle states and earthquake origins Dilatant precursors of earthquakes Earthquake total energy and plate fragment size

DYNAMICS OF DILATING AND BRITTLE MATERIALS

§3.1. One - Dimensional Dilatant Stress Waves §3.1.1. Equations of one - dimensional dynamics § 3.1.2. Shock waves in nonholonomic media §3.1.3. Numerical calculation of underground contained explosion § 3.1.4. Role of initial porosity

103 104 104 105 106

110 110 110 114 118 123

§ 3.2. Elementary Theory and Experimental Modelling of Underground Explosions §3.2.1. Mass velocity field in cavity vicinity §3.2.2. Density integral for cavity vicinity §3.2.3. Zones of dilatant densification and loosening §3.2.4. Nonmonotonous post explosion density distribution in porous rock §3.2.5. Attenuation of blast waves §3.2.6. Limit velocity effect in explosion fracturing §3.2.7. Limit slippage velocity effect

131 136 138 142

§ 3.3. Limit Velocity, Dynamical Strength and Compressibility §3.3.1. Problem of two - front structure of fracturization wave §3.3.2. Theory of limit fracturing stress § 3.3.3. Theory of limit fracturing velocity §3.3.4. Dynamic overloading phenomenon §3.3.5. Hugoniot elastic limit as dynamic overloading § 3.3.6. Dynamic microcrack appearance §3.3.7. Similarity of blast motion § 3.3.8. Dynamic compressibility of particulated media §3.3.9. Dynamic pseudoviscosity

142 142 145 146 147 148 151 152 153 157

125 125 127 129

xii Contents §3.4. Sublimit §3.4.1. §3.4.2. §3.4.3. §3.4.4. §3.4.5.

Solid Friction and Attenuation of Seismic Waves Generalized model of dilatant elastoplasticity Harmonic linearization Attenuation of the P-waves Attenuation of shear waves The Rozhdestvensky-Yanenko method usage

158 158 162 163 165 167

§3.5. Nonlinear Transformation of Seismic Waves §3.5.1. Experiments with stress waves in sand § 3.5.2. Continuum dynamics of fragmentary media §3.5.3. Evolution equations for P-wave and S-wave §3.5.4. Nonlinear elastic and viscous effects §3.5.5. Dominant frequencies of seismic waves

172 172 175 179 180 182

CHAPTER 4

183

MECHANICS OF SATURATED ELASTIC MEDIA

§4.1. Equations of Saturated Porous Media Motion §4.1.1. Averaging of momentum for heterogeneous system §4.1.2. Thermodynamics of saturated porous media §4.1.3. Phenomenological kinetic coefficients §4.1.4. Permeability of porous medium §4.1.5. The Darcy law deviations §4.1.6. Possible boundary conditions

183 183 187 192 192 193 197

§4.2. Waves in Porethermoelastic Materials §4.2.1. Linear forms of equations of motion §4.2.2. Linear thermodynamics of saturated porous media §4.2.3. Constitutative elastic law for saturated media §4.2.4. Equations of isothermal linear dynamics of saturated porous media §4.2.5. Inertial relaxation in shear waves §4.2.6. Two-types of P-waves §4.2.7. Waves in soft saturated media §4.2.8. Thermoelastic waves in porous media §4.2.9. Velocity dispersion in water and oil reservoirs

198 198 199 201 203 205 207 208 210 216

§4.3. Dynamical Elasticity of Saturated Soils §4.3.1. Inspectional analysis of elastic dynamics

218 218

Contents xiii §4.3.2. §4.3.3. §4.3.4. §4.3.5. §4.3.6. §4.3.7. §4.3.8. §4.3.9.

Relaxation of pore pressure in the first P-wave P-waves of the II type in soils Seismic velocity discontinuity at the ground water level Load distribution over different phases Plane impact in poroelastic soft medium Dynamics of medium composed of incompressible phases Pore pressure relaxation immediately after impact Volume viscosity approximation

§4.4. Attenuation, Reflection and Transformation of Waves in Rigid Porous Media §4.4.1. Amplitude of two types of P-waves in isothermal gas-saturated medium §4.4.2. Thermoelastic damping of waves § 4.4.3. Sound damping in rigid gas - saturate d porous medium §4.4.4. Reflection of waves from the boundaries §4.5. Quasistatic Deformation of Poroelastic Saturated Materials §4.5.1. Distribution of load over phases in samples §4.5.2. Quasistatic loadings of saturated rocks §4.5.3. Undrained Poisson coefficient §4.5.4. Quasistatic deformation of soft media §4.5.5. One-dimensional consolidation §4.5.6. Two-dimensional consolidation §4.5.7. Mandel-Crayer effect §4.5.8. Airy's stress function

CHAPTER 5

NONELASTICITY AND MULTIPHASE SATURATION OF POROUS MEDIA

§5.1. Nonelasticity and Fracture of Saturated Porous Media §5.1.1. Yield criterium for saturated media §5.1.2. Dilatant elastoplasticity of saturated media § 5.1.3. Permeability dependence on stresses §5.1.4. Dilatant corrosion plasticity §5.1.5. Stability of corrosion dilatant homogeneous process §5.1.6. Water effects in Earth crust dynamics

220 222 226 226 227 235 237 237 238 238 242 243 246 252 252 254 255 257 258 260 262 264

266 266 266 268 270 271 273 274

xiv Contents

§5.1.7 §5.1.8 §5.1.9 §5.1.10.

Explosion deformation of dilating saturated media Dissipation of explosion wave in close vicinity Liquifaction of saturated soils Explosion action at contact of two geomaterials

276 277 278 280

§5.2. Shock Compression of Saturated Materials §5.2.1 Hydrodynamical approximation of shock front flows §5.2.2 Shock balances in two-phase media §5.2.3 Shock wave structure in soft soils §5.2.4 Nonequal phase shock pressures §5.2.5 Thermal effects of intensive shock compression §5.2.6 Resistance of pore vicinities to shock compression §5.2.7 Shock overheating of media with vacuumed pores §5.2.8 Shock states with subheating of phase material §5.2.9 Shock phase transitions in mixtures

280 280 282 284 287 290 293 296 298 302

§ 5.3. Mechanics of Media with Multiphase and Multicomponent Saturation §5.3.1 Dynamics equations for multiphase mixtures §5.3.2 Thermodynamics of multiphase motion §5.3.3 Surface capillary layer §5.3.4 Elastic deformation of multisaturated media §5.3.5 Generalized Darcy law §5.3.6 Phase energy balance §5.3.7 Waves in partly saturated soil §5.3.8 Cracks in soils at drought

302 302 306 309 310 313 316 317 322

CHAPTER 6

FLUID AND GAS MOTION IN DEFORMABLE RESERVOIRS

§6.1. Elastic Regime of Underground Fluid and Gas Flows ' §6.1.1 Basic equation of piezoconductivity §6.1.2 Nonlinear piezoconductivity §6.1.3 Stationary underground flows in well vicinity §6.1.4 Well productivity determination §6.1.5 Problem of well spacing over reservoir Nonstationary fluid flows to wells §6.1.6 §6.1.7 Pore pressure built-up curve

324 324 324 326 329 330 333 335 338

Contents xv §6.1.8. §6.1.9.

Reservoir boundaries effects Conditions at movable boundaries

341 343

§6.2. Nonstationary Underground Flows with Hydraulic Relaxation §6.2.1. Stresses in media with double porosity §6.2.2. Equations of flow through double porous media §6.2.3. Dimension analysis of piezoconductivity equations §6.2.4. Changes of boundary and initial conditions §6.2.5. Effective system of flow in fractured-porous media § 6.2.6. One - dimensional flow to gallery draining fractured-porous medium §6.2.7. Non-stationary flow to well in fractured-porous reservoir §6.2.8. Pressure build-up in fractured-porous reservoir §6.2.9. Nonlinear flows in fractured-porous medium

344 344 346 347 349 351

354 357 358

§6.3. Stress State of the Well-Bottom Vicinity and Nonlocal Elastic Effects §6.3.1. Formulation of problem in frame of electricity theory §6.3.2. General poroelastic solution for well vicinity §6.3.3. Stresses acting in well vicinity §6.3.4. Flow load at well column §6.3.5. Flow in elastic thin layer surrounded by elastic rocks §6.3.6. Nonlocal effects in underground flows §6.3.7. Plane nonlocal piezoconductivity §6.3.8. Axis-symmetrical nonlocal piezoconductivity §6.3.9. Nonlocal effects of pressure build-up process §6.3.10. Remark on irreversible deformation of reservoirs

361 361 364 366 369 370 373 374 375 376 381

CHAPTER 7

PHYSICO-CHEMICAL UNDERGROUND HYDRODYNAMICS

§7.1. Equations and Flows of Multicomponent Heterogeneous Mixtures in Well Vicinity §7.1.1. Thermodynamics of flow with phase transitions §7.1.2. Relative phase permeabilities and mixture composition §7.1.3. Conditions of local thermodynamical equilibrium

351

383 383 383 388 389

xvi Contents

§7.1.4. §7.1.5. § 7.1.6. §7.1.7. §7.1.8. §7.1.9.

Binary mixture model Ternary mixture model Retrograde condensation in PVT - cell Stationary flows of multicomponent mixtures Quasistationary flows of multicomponent mixture Nonstationary flow of multicomponent mixture in well vicinity

§ 7.2. Multicomponent Flows with Essential Changes of Phase Compositions §7.2.1. Global characteristics of composition and saturation changes due to cycling process § 7.2.2. Multicomponent balances at discontinuities in mixture flows §7.2.3. Conditions at displacement front of ternary mixture § 7.2.4. Self- preserving problem of plane cycling gas process §7.2.5. Displacement of gas-condensate mixture by enriched gas §7.3. Theory of Convective Diffusion in Porous Media §7.3.1. Averaging of concentration field over pore space §7.3.2. Statistical theory of random walking of particles in porous space § 7.3.3. Dependence of convective diffusion coefficient of filter velocity § 7.3.4. Characteristic intervals for convective diffusion in porous media §7.3.5. Experimental data on longitudinal diffusion coefficients §7.3.6. Note on lateral diffusion §7.3.7. Absorption in flows through porous media §7.3.8. Chromatography effect §7.4. Problems of Convective Diffusion in Porous Media § 7.4.1. Dispersion in one - dimensional flows §7.4.2. Two-dimensional dispersion in uniform flows §7.4.3. Dispersion in nonuniform plane flows §7.4.4. Application of method of asymptotic expansions

391 392 393 395 398 400 402 402 404 405 409 412 414 414 417 419 421 423 426 428 430 430 430 435 436 438

Contents xvii

§7.4.5. §7.4.6. §7.4.7.

Adsorption problems for flows through porous media Plane anisotropic diffusion Estimation of radioactive danger from underground nuclear explosions

BIBLIOGRAPHY

441 444 446

452

CHAPTER 1 BASIC CONCEPTS OF CONTINUUM MECHANICS

§1.1. Equations of Continuum Mechanics §1.1.1. Ideology of continuum approach THE MASTER equations of continuum mechanics are the balance laws for mass, momentum and moment of momentum for elementary volume AV, linear scale / of the latter has to be smaller than the external scale L of the problem under con­ sideration, but at the same time larger than the characteristic scale X of micromotion of particles of which the whole media is composed, that is L »

/ »

X.

The balance equations for the volume AV can be simplified by the limit transition / ► 0

L which can be practically achieved by the choice of sufficient large external scale L. If this limit condition is valid, the elementary volume is the differential one: AF = dF . If the condition L » I is invalid then the balance equations for AV cannot be interpreted as continuum macroequations. The second limitation (7 » X) is the representative condition for intervals of averaging over the volume AV itself or over its cross-sections.208

1

2 Mechanics of Porous and Fractured Media If the condition / » X is invalid, then the space averaging does not give the regular mean values. So, under the strict condition L » I » X, where AV ~ Z 3 , the balance equation for AV can be interpreted as the differential ones with the internal linear scale X. The latter is the size of a crystalline or a grain or a pore. For nonstationary micromotions one can use averaging over the representative interval of time. This procedure gives the equation of correlation in the space element with the scale X, that is, it gives the averaged microstructure. Generally speaking, these equations do not coincide with the continuum macroequations. Microstructure of a porous fractured medium is presented by pores and cracks and by grains and blocks. The continuum equations can be constructed directly as the equations for mean parameters of the volume AV. Another approach needs the formulation of the microequations for the scale / « X and the following averaging procedure over the volume AV. Of course, under the first approach the averaging is fulfilled by implication and possible contradictions can be avoided by special rheological experiments. The latter gives so-called constitutive laws for the whole medium. By a secondary approach the type of constitutive laws and sometimes quantitative values of rheological parameters can be evaluated theore­ tically. §1.1.2. Balances of mass and

momentum

In both cases the equations are the differential equations of the mass balance 9/0

dp v.-

+

^T -^ e

(11)

the momentum balance 9

PV/ , ot

d v v

Pij dx,-

d =

°ij , dXj

R

(1.2)

and the balance of moment of momentum. In the simplest case the latter is known as the law of stress tensor symmetry, that is

"9 = °ii

(1.3)

Basic Concepts of Continuum Mechanics 3 Here p is the density, v{ is the displacement velocity, Q^fi^ is the mass source (sink), F{ is the body force and i,j = 1,2,3. The equations (1.1) and (1.2) are true balance laws for the volume dV = dx1 dx2dx3, whose mass center has coordinatesx l t x 2 ,x 3 . The functions p, vt,..., are continuous in d V. The stress component o« is the projection of the traction vector on the axis Xj. The traction is acting at the unity cross-section with the normal which coincides with the axis x.-. The strain e,y of a medium element, which fills in the space volume dV at the moment t is not small in general case and can be found by the rule 1 / bu,e

ii = —\—L+ 2 \bXj

aw

bill,

— + bxt

—bxt

dill, \

—-) bXj }

(1-4)

where u, is the displacement. On the other hand, one can introduce the strain rate e,y by velocity field: 1 /bv,bv,\ it,11 = - l - L + -L) 2 \bXjbxJ

.

(1.5)

The definitions (1.4) and (1.5) are compatible if one introduces299'307 the increment Deif = iqDt by the following manner

" (- ( i f + * 4 - *°«-

De

3 e ii

■J + *

^eii

^

~ ejk

-

*

Dt + e*ikhj ikhf ++ ee/Jtejt/ ,■ jk*ki\ J Dt

*'

1 /bvk

bv, \

2 \bXj

bxk j

O * - ejk

~

n

ki

nki 0-6) ,

x

One can see that the equations (1.5) and (1.6) are equivalent to the Oldroyd derivative which is more general than the following Jumamn derivative:

4 Mechanics of Porous and Fractured Media d.e,-,— dt

be;i = —^ bt

+

\ k

beu —~ bxk

"

e

~ eik^ki

ik&ki tK

k

,k

'

■

O- 8 )

ki

The complexity of these equations is connected with the fact that the Theologi­ cal laws are formulated for the fixed medium elements (that is, for the Lagrangian coordinates), but in the Eulerian space coordinate system they are corresponding to the increments De^. So, the simplest law of elasticity is the following one Dokl=EifklDeif

(1.9)

and E.-kl is the tensor of instantaneous elastic moduli: 2 ijki - (K - ~r

E

G

2G

) hi Hi +

hk hj ■

Here K is the bulk modulus, G is the shear modulus, 6;,- is the unit tensor. The law (1.9) corresponds to the model of linear hypoelastic material according to C. Truesdell. 299 If K and G are material constants, the integration of (1.9) will give the usual form of the Hook law for stress and strains themselves. Sometimes, the usage of the substantional derivatives

—lJbt

e.-,- = - 3 L = " At

+

vk

—t ?>xk

(1.10)

is sufficient. This derivative takes into account the translation with the particle velocity, but it neglects its distortion and rotation. §1.1.3. Continuum

thermodynamics

Let us consider now the thermodynamics of the medium element in the space volume dV. The total energy balance has the following form:

39

V-V-

V,V;

0b

2

bXj OX;

I I — P(e p(e++ -Li-) ') + —

bt ot

9ff;,V; bXj 0Xj

- + % vi

V,V;

v,{e + -i-i-) p „.(* ") = =

bOj dXj bXj

'

2

(1.11)

Basic Concepts of Continuum Mechanics 5

where e is the specific internal energy, q.- is the heat flux. If one multiplies the momentum equation (1.2) by the velocity vt, the equation of balance of kinetic energy will follow: b

viV{ p

b +

bt

2

=

b

vtvt p

bXj

a

2

ff

v.'

bv(

^ *"'- *V + ^ "

(L12)

The difference of the equations (1.12) and (1.11) gives the balance of internal energy a9 — pe +

*t

9 bXj

pe

bvt{ v.- = °u 11 —— '

bxj

bq.f bq bXj

(1.13)

If one uses now the mass balance (1.1) (with Q^p\ = 0) and the definition (1.5) of strain rate, then due to the stress symmetry (1.3) the internal energy equation will have the following form: p P

de . 9qy - == o,vy ev, -

*dr

*

^

ty

(1.14)

The total strain e

e

v

//

e

v

is the sum of two components — recoverable (elastic) ef. and irreversible eP.. The work of stresses a« at the increments def converts into elastic energy but the work Oj.de? at the irreversible strains is dissipated. The equation (1.14) is the first law of thermodynamics. The second law can be written as q

%l- = pp Tds Tds -- dW dW „ + - V) = p~(v-n - V ) , P+",+ «

- V)-oJ„

pHvJ - V)( C + +

= p-vr(v-

- V) - or

^±)-a-nV*

= p-(v-„ - V)(e" + ~~)-oJn

vj .

,

(1.46)

Basic Concepts of Continuum Mechanics 15 §1.1.8. Shock adiabats in hydrodynamical approximation If values of the parameters, for example, before the jump, are given and if the rheology of the medium, that is, the functions P(p, T), e(p, T), are known, the system of three equation (1.46) contains four unknown variables fi+, T+, v* and V. So, the system (1.46) can be resolved in the form of a shock (Hugoniot's) adiabat. It means the dependence of three variables on the fourth. Very often experimental measurements of the Hugoniot adiabats give the linear relations between v^ and V V = a + bv ,

v = v+

(1.47)

which are valid for very strong shock waves. Here a and b are the coefficients given in Table 1. The dependence (1.47) or other possible forms of the Hugoniot adiabats gives the possibility to solve inverse problems — to find the equation of state for geomaterials under such high dynamic pressure and temperature, which cannot be achieved under static conditions. Internal energy c of a material for pressures much higher than its strength can be presented323 as the sum of cold e°(V) and heat energy V

e = e°(V) +

(P - P°) .

(1.48)

It is the state equation in the Mui-Gruneisen form, V= \\p is a specific volume, e°(V) is specific energy at the 0°K isoterm, P° = -de0/dVis the cold pressure, r(V) is the Gruneisen coefficient. In many cases F(V) can be recalculated by the simple formulae:

r = r0 —

,

r0 = I \ F 0 ) .

(1.49)

0

In particular cases the cold pressure P° can be found on the base of the Hugoniot adiabats: PH = PH{VH) ,

eH{VH)

= jPf,{V0

- VH) .

(1.50)

16 Mechanics of Porous and Fractured Media

From (1.48) and (1.50) one can find the relation P° = = p°

= = P pHH-r— - r —

w bV

(1.51)

v

"

V

and from (1.47) the expression for PH follows

«2(v0 - vH)

H

"

PH

=

(1.52)

2

) 2 diV + fli«~0, -1-2

V«>

p v.dAj

'

+ QH> 4-4-■ ■ e / e/ = =\ J[fiVity }dA} f Q{% {PIV^- -v,)v,o- - o9if„)dA A

m m

G $ = {( Q\*\

AA

m m

^ {/»(« + "- y^ ^t t yy - V/) - «&»* + 9q^dAj /}d^

(i.55) (1.55)

18 Mechan ics of Porous and Fractured Media 0{«)

-.=

j {PS(Vj - v,) - -jr)

dA

f

+

Q(s)m ■

A

m

Here Q^m is an additional source of the entropy at the surface^ itself (because of the solid friction, for instance), Q<J> = G ( r t = - Q < g , G $ « G ( e ) = - f i j g Convective transfers of mass, impulse, energy and entropy through Am are determined, as usual, by the relative velocity v.- — M of the material particles and of the pore surface V.-. This difference is non-zero if there is a mass loss at the pore surface, for example, because of solution or vaporization of the solid matrix material. If the pore deformation is such that the same particles are remaining at their surface Am , then the quality Vi = Vi is valid for these particles. Such deformation (recoverable or irreversible) is developing without fracture of the solid matrix material. If v.- - V.- # 0 in any isolated point 0 of the pore surface Am , then it is a fracture point. The fracture will grow here from the pore surface into the matrix body, and the above mentioned difference

i = v / - vi is the component of the crack growth velocity (of the crack tip) relative to the matrix material. § 1.2.2. Averaged surface and volume parameters Let us assume that the volume V is the elementary one AV = A ^ A A ^ A ^ , corresponding to macroscopic coordinates AX{ and such that AX( S> X » dXf where X is the characteristic scale of pores and cracks. The balance integral relations (1.54) can be interpreted as finite-difference balance equations for the volume AV. Let us introduce a volume concentration AF(a) AV of the phase a and mean volume density

^-mW) A(p< a >i^>m/«>) at the pairs of faces AAp separated from each other by the distance AX, that is

20 Mechanics of Porous and Fractured Media

av/ idA=

L "

(pWv,iK)

^

AV

^

The sink Q(*) in the balance (1.54) will be equal to zero if at the interface Am(t) of phases inside the volume AV the velocity Vj of the particle displacement coincides with the velocity of the surface Am(t) itself. The nonequality Vj^Vj means the existence of a mass flux through boundary surface, that is, there is a phase transition of the following intensity:

Q Q

=

^ = ~AV ^ ~AV I[

'("/-v/)"/^. '("/-V/)"/) >) = = 0 . 3* dX ' ' 3* dXjf ' '

(1.58)

If AV » X3, then the volume - averaged variables, for example (1.56), and the surface-averaged variables, for example (1.57), are the regular functions of macrocoordinates, that is of the volume AV mass-center or of the cross-section masscenter A A.- (or of the volume AV, displaced at the distances ±AX-/2). In the mathematical models of multiphase mechanics the porosity m has to be used as the essential variable. The porosity m is the volume of pores per the unit volume of the whole medium. Therefore m = = 1\ - mS1' = mS2' For the media with a chaotic internal microstructure it is possible to claim that

m = = TT7 AV J\ AV

fat*

f

v

™i =

ft*•

f;

{f)

i

=

TT" AAj \

J

AA.

;

w

>dF

")

AA

Basic Concepts of Continuum Mechanics 21 where f(xit r; to) = 1, if the micropoint x{, t belongs to a pore space and /(*,•, t; to) = 0 if the point x{, t belongs to the matrix. The parameter co accounts the random belonging of the point x{, t to the phase a of the medium. The averaging of / over the ensemble of medium realizations141 marked by the parameter a>, or in other words, the statistical average is determined by

ff&i, (^ , f) f) = = jj faf, t; «) id) dw dw ,, /(*,., t; JJ

did = = 1 1 .. d«

CO CO

Let us assume that the volume AV is sufficiently large that the variables m and m}- do not depend on the random parameter 10. One can see19 that the surface porosity, averaged over all parallel cross-sections AA.- of the volume AV, is equal to volume porosity:

m=

I1 f

/

AX


* x- ^x ^ LL ) L. ' i * L ' H~ ^ "'^ > - ^^ )-

m =i£

m d

(L59)

(L59)

;

If the surface porosity w;- is a constant for all cross-sections, orthogonal to the axis Xj, then m = rhj If the surface porosity is continuously changing, then ~

rrijixj) == m,(A/) +

dw,'-

*

1

d2mf

,

2

2

*

9A*

and consequently the integral relation (1.59) gives

m(Xj) = mf(Xf) + 0 f ^ - J

, (1.60)

/ = 1,2,3

22 Mechanics of Porous and Fractured Media

The internal structure of a porous media can be anisotropic. Then the surface porosity thj can depend, generally speaking, on the orientation of the surface of its definition. The condition (1.59) will mean that the averaged surface porosity is invariant to the axis X choice and according to the equality (1.60) the surface porosity m- is a scalar, accurate to the second order infinitesimal value. Because terms of such order magnitude can be neglected in the continuum equations, then it is admissible to get m = m. Another situation appears during the averaging of equations of motion. Let us consider the particular case of the equilibrium equations of elastic microheterogeneous material which has no pores. Let in microscale, that is for differential elements dv = dx dx dx, the equilibrium equations take place do,-,— ^ + Ft = 0

(1.61)

dXj

and Ojj = Off. If one multiplies these equations by coordinate xk, then 9 Ob xk

(1.62)

OXj

The following multiplication of the equation (1.62) by the alternating tensor €lik will lead to balance equations of total moment of momentum in the volume dx dx dx because e^j aik = 0,

a g— (elki°ijXk)

+ emFiXk

= 0

(1.63)

Because of the assumed continuity of the field a;y with neglecting of inertial forces and without fluxes through the surface Am of pores, the second equation (1.54) can be simplified: I

OjjdA-

+ 1

F(dV

= 0

(1.64)

where d F = dx dx dx dA.-= n- dA. Dividing the equation (1.64) by AV and introducing variables, averaged over volume AV and surface AAj =AX j t AZ m in the following manner 205 ' 2 0 8

Basic Concepts of Continuum Mechanics 23 (...)

= — AV

'

( *AV

AXkAXm

(...)dV

3AA.y

'

k ¥= m # / one can get the resulting macroequation: Haull lJ-L- + (F) J-L- + (F)' 9Xy dXj '

= 0 . = 0 .

(1.65)

Here the macrodivergence is a sum of following expressions

j-( AV

\

Af V

a. AA ) = " / AX, ]

(o,, > JT,+A^ / 2

(1.66) ^ -r A X A*^, /2 /2

after the transition to the differentials, which is valid in asymptotic sense for

AXL L

-»■ 0

where L is the external scale of the problem under consideration. The averaged body force {Ft) is a regular function of macrocoordinate X. = x.- - f • of the mass-centre of the volume AV. Here f,- is the radius-vector between the mass-center and an arbitrary point xi inside AV The finite difference presentation (1.66) shows that the mean tractions (o,y)at the oriented cross-sections (at the faces of the volume AV) can be treated as regular functions of coordinates of the mass-center of AV displaced by the distance ±AXkl2.

24 Mechanics of Porous and Fractured Media § 1.2.3. Difference of macrostresses and volume averaged stresses Averaging of the equation (1.62) over the volume AV will give us the following result

(0ik)

—hr^ + iFiik) = {°ik)k

°-67)

where (oik) is the stress-tensor averaged over the volume. According to the equation (1.67) the macrostress is expressed (a;fc \ as the difference of the mean volume stress and divergence of the tensor of higher order. The variables M,y# = (ff,y £# ),• are known as the double stresses and (Fj f^ ) is a double volume force. The difference equation (1.66) shows that the stresses are functions of the mass-center, that is, they can be calculated by averaging over oriented sections, crossing the mass-center of AV. § 1.2.4. Chain of macroscopic equilibrium equations The double stresses are the moment of traction distribution over oriented crosssection AAj of the higher rank than macrotensor ( c,y).. They can be expressed also by its volume averaged value (a,-.- £% ) and by moments of higher rank, that is, by triple stresses. The proper equation can be developed by multiplication of equation (1.61) by the dyads Xjcxm and by following averaging. The repetitions of such a procedure will give the chain of macroequations, including the macrotensors of the rank n, which are nonsymmetrical because of the rule of their intro­ duction as a mean value over oriented cross-sections. The multiplication of the equation (1.67) by the alternating tensor 6/,^ will give the equation of moment of momentum: ■ ^ T + €Hk
(1.73)

In this case the second term of (1.72) is omitted and mean volume strain is a symmetrical part of a gradient of mean displacement field. The condition (1.73) can be replaced by a less limiting one (Ui*). = = (u?) = = 0,

i=j,

26 Mechanics of Porous and Fractured Media (u?).*(uf), (u?).*{uf),

it*j. ¥=f .

(1.74)

Then the second term of the right-hand part of (1.72) is not equal to zero. Physically it means that the circulation of tangential components of a displace­ ment along the closed contour is nonequal to zero, if there is a dislocation inside. Then one has to introduce additional kinematic state parameter. The number of such additional kinematical variables has to coincide with the number of the equations of equilibrium. If the number of state parameters is less then the number of chain equations, mentioned above, then one will get so called gradiental models ofcontinua.161'162 For example, if the rotation cjk- of a microparticle is introduced besides the mean rotation Slk.- then it is necessary to keep the balance of moment of momen­ tum (1.68) besides the balance of momentum (1.65). Such continuum models are known as the Cosserat media or, in other words, as models of asymmetrical or micropolar mechanics. One can assume that such complicated models are necessary for account of flows of granulated materials in narrow channels 122 ' 177 ' 185 ' 235 or to account physically essential details of some boundary condition (for example, the asperity of real rock contacts). Now, let us compare mean volume and surface values. From the equality = TAVF $J±v atfdF = rAXj r fJAx;. 2 M} \ 12 / v v

0.77)

2

" >

2

Mf

\

12 /

If AX would approach zero then one could get the result (1.76). However, in this case the right-hand side of (1.76) would coincide with local value e« identi­ cally, because the area of averaging AA.- is approaching the zero also as Xk -*■(). So due to the estimation (1.77) a possible deviation from the equality (1.76) has the order AA"2, that is, the order X2 of the square of microstructure length scale is a minimal. If one takes into account that mean volume stress tensor (ay ) is symmetrical, because the local stress tensor o;y is symmetrical, then the righthand side of (1.77) will be a main part of antisymmetrical component of macrostress tensor, which play the junction role in the balance equation of moment of momentum (1.68). § 1.2.5. Problem of computation of averaging and differentiation The commutation condition of averaging and differentiation is a consequence of integration procedure:

\bx \bxtt//

AXj Jixj-KAXj y2AXj x

bXf dXj

= * / 2 ) - -< /