Int[ J[ Solids Structures Vol[ 24\ Nos 23Ð24\ pp[ 3574Ð3696\ 0887Þ 0887 Published by Elsevier Science Ltd[ All rights reserved\ Pergamon Printed in Great Britain

9919Ð6572:87:,*see front matterPII ] S9919Ð6572"87#99989Ð9

MODELLING THE EFFECT OF FLUIDCOMMUNICATION ON VELOCITIES IN

ANISOTROPIC POROUS ROCKS

SHIYU XU$Research School of Geological and Geophysical Sciences\ Birkbeck and University Colleges\

University of London\ London WC0E 5BT\ U[K[

"Received 14 April 0886 ^ in revised form 18 January 0887#

Abstract*An inclusion!based anisotropic poroelasticity model "IBAPM# has been developed fromthe J[ D[ Eshelby "0846\ Proceedin`s of the Royal Society of London Series A\ 130\ 265Ð285# and T[Mura "0876\ Micromechanics of Defects in Solids\ Martinus Nijho}\ Dordrecht# theories[ Eshelby|sinteraction energy approach is employed to calculate the e}ective elastic constants of a solidcontaining dilute pores\ since it o}ers two approximations ] one corresponding to the condition ofconstant load and the other to the condition of constant displacement[

The e}ect of pore ~uid communication on the poroelasticity of a porous rock is modelled viapore connectivity[ A totally isolated pore system implies that pore pressure is equilibrated withineach individual pore only and may vary from pore to pore\ depending on the shape and orientationof each pore[ In terms of the local ~ow mechanism\ this gives us a high!frequency estimate ofthe e}ective elastic moduli[ The low!frequency estimate\ on the other hand\ is modelled by acommunicating pore system[ In this case all pores are assumed to be connected and\ consequently\pore pressure is equilibrated within the whole pore system[

Other related problems discussed in the paper include ] "0# extension of the model to the caseof pores having arbitrary orientations\ "1# extension of the non!interacting model to higher poreconcentrations and "2# consistency checks of IBAPM with previous models[ In particular the low!frequency version of the model is compared with the anisotropic version of the Gassmann model"R[ J[ S[ Brown and J[ Korringa\ 0864\ Geophysics\ 39\ 597Ð505#[ Finally the model is applied tothe laboratory measurements reported by J[ S[ Rathore\ E[ Ejaer\ R[ M[ Holt and L[ Renlie "0884\Geophys[ Prosp[\ 32\ 600Ð617#[ Þ 0887 Published by Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

It is well!known that elastic!wave velocities are frequency dependent[ Considerable under!standing on the mechanisms of velocity dispersion has been gained from comprehensivelaboratory measurements "e[g[ Wyllie et al[\ 0851 ^ Gordon and Davis\ 0857 ^ Tokso�z et al[\0868 ^ Winkler and Nur\ 0871# and theoretical studies "e[g[ Gassmann\ 0840 ^ Biot\ 0845a\b ^ Walsh\ 0855 ^ Mavko and Nur\ 0864 ^ O|Connell and Budiansky\ 0866#[ The proposedmechanisms for intrinsic velocity dispersion and\ hence\ wave attenuation include ] "0# localand macro ~uid ~ow "Gassmann\ 0840 ^ Biot\ 0845a\ b ^ Mavko and Nur\ 0864 ^ O|Connelland Budiansky\ 0866 ^ White\ 0864 ^ Dutta and Ode�\ 0868a\ b# and "1# frictional sliding"Walsh\ 0855 ^ Johnston et al[\ 0868#[ Since ~uid ~ow mechanisms are considered the maincause of velocity dispersion in sedimentary rocks\ they have received considerable attentionfrom research scientists[

Gassmann "0840# proposed a model which simulates elastic bulk and shear moduli ofa porous medium containing a totally relaxed pore ~uid[ Later Biot "0845a\ b# developeda comprehensive theory for simulating elastic!wave propagation in ~uid!saturated porousmedia[ The BiotÐGassmann "BG# theory has been used to simulate the observed velocitydispersion and wave attenuation of well!consolidated and cemented rocks[ In cases wherecompliant pores "e[g[ micro!cracks# are present\ however\ the BG theory was found tounderestimate the velocity dispersion "e[g[ Mavko and Jizba\ 0883#[ The extra velocitydispersion was frequently explained as a result of the local ~uid ~ow between compliantand sti} pores[

$ Tel[ ] 9060 076 6949 X1279[ Fax ] 9060 272 9997[ E!mail ] xuÝgeophysics[bbk[ac[uk

3574

3575 S[ Xu

The idea of local ~ow "also called squirt ~ow# was _rst proposed by Mavko and Nur"0864#[ O|Connell and Budiansky "0866# systematically analyzed the problem and proposeda crack model for simulating the e}ect of local ~uid ~ow on the poroelasticity of a solidcontaining randomly oriented cracks[ Endres and Knight "0886# recently extended themodel to a more general case where a solid is permeated with randomly oriented inclusionsof arbitrary aspect ratios[ Thomsen "0884# took a step further and presented an anisotropicporoelasticity theory applicable to a composite containing two groups of pores ] equantinter!granular pores and perfectly aligned thin microcracks[

Xu and White "0884a\ b\ 0885# treat the problem in a di}erent manner[ They employinclusion!based models "e[g[ Kuster and Tokso�z\ 0863# to calculate the dry rock framemoduli "assuming a zero bulk modulus for the pore ~uid at this stage#[ The di}erentiale}ective medium "DEM# theory is incorporated into the model to overcome the limitationof dilute pore concentration in these theories[ They argue that in dry rock it does not matterif the pores are connected or isolated\ since pore pressure is zero[ They then use Gassmann|sequations to calculate the saturated moduli\ implying that all pores are connected and porepressure is equalized[ To model wave velocities at high frequencies\ the authors set asideGassmann|s theory and calculate the saturated elastic moduli by including the bulk modulusof the pore ~uid when applying the KusterÐTokso�z and DEM theories[ A similar approachwas also proposed by Ravalec and Gueguen "0885#[

Most poroelasticity models "e[g[ Gassmann\ 0840 ^ Brown and Korringa\ 0864 ^ Biot0845a\ b ^ Mavko and Jizba\ 0880\ 0883 ^ Mukerji and Mavko\ 0883# require the dry bulkand shear moduli as input parameters[ This does not cause a serious problem for simulatinglaboratory measurements\ since the dry rockframe moduli can be measured in the labora!tory[ When constructing seismic models from well logs\ however\ the dry rockframe moduliare not practically {{measurable||[ The inclusion!based poroelasticity models "e[g[ Mavkoand Nur\ 0864 ^ O|Connell and Budiansky\ 0866 ^ Endres and Knight\ 0886#\ on the otherhand\ predict the e}ect of the local ~uid ~ow on the poroelasticity of a porous rock frompore parameters[ The main advantage of inclusion!based models is that they allow us tostudy the e}ect of each individual pore on the poroelasticity[ Unfortunately the inclusion!based poroelasticity models mentioned above are restricted to composites having randomlyoriented pores[ Although Thomsen "0884# developed an anisotropic poroelasticity model\it is limited to a special case where the composite is permeated with two groups of pores ]"0# equant pores and "1# perfectly aligned thin cracks[ There is considerable potentialapplication for a poroelasticity model applicable to rocks having arbitrary pore orientationsand arbitrary aspect ratios[

In this paper\ an inclusion!based anisotropic poroelasticity model "IBAPM# ispresented[ Following the work by O|Connell and Budiansky "0866# and Endres and Knight"0886#\ two special cases are considered\ one corresponding to the isolated pore system andthe other to the communicating "undrained# pore system[ Numerical examples are thengiven to demonstrate the impact of pore ~uid communication on the poroelasticity of aporous rock[ Other related problems\ such as the extension of the non!interacting model tohigher pore concentrations and consistencies between the IBAPM and previous models\are also discussed[ Finally the model is applied to simulate the laboratory measurementsreported by Rathore et al[ "0884#[

1[ THE CONCEPT OF EIGENSTRAIN AND ITS DETERMINATION

The eigenstrain o� is a generic name given to such non!elastic strains\ such as thermalexpansion\ initial strains and plastic strains "Mura\ 0876#[ It was called the stress!freetransformation strain by Eshelby "0846#[ The determination of the eigenstrain o� is the keypart of estimating e}ective elastic constants of a composite[ For an isolated pore system\Eshelby "0846# has given formulae for calculating o� in an isotropic matrix\ whereas Mura"0876# has demonstrated how to calculate o� in an anisotropic matrix with hexagonal ororthorhombic symmetry[ In this section Mura|s formulation for isolated pores is reviewedand the formulation for communicating pores is then derived based on principles similarto those of Mura "0876#[

3576Fluid communication on velocities in anisotropic porous rocks

1[0[ o� for an isolated poreFor a solid containing isolated pores "or pores without ~uid communication#\ the pore

~uid pressure is equalized within each pore[ To a _rst order approximation\ the eigenstraino�ij for a particular pore can be calculated by embedding the pore into the matrix[ Let Cdenote the elastic sti}ness tensor of the matrix and CI that of the pore occupying a sub!domain V\

x1

a1¦

y1

a1¦

z1

c1� 0 "0#

We also de_ne

, the applied stress at in_nity ] s9ij

, the applied strain at in_nity ] e9ij �Aijkls

9kl

, the stress deviation from s9ij ] sij

, the strain deviation from e9ij ] eij[

Inside the pore\ Hooke|s law is written as

s9ij¦sij �CI

ijkl"e9kl¦ekl# "1#

Imagine now that the following transformation is taking place ] the pore is replacedby the solid material[ To maintain the total strain within V exactly the same as that beforethe transformation\ an extra strain o� must be introduced[ In this sense\ the eigenstrain o�is the extra strain caused by the presence of the pore replacing matrix material[ Accordingto Eshelby "0846# and Mura "0876#\ the stressÐstrain relationship in V can also be expressedas

s9ij¦sij �Cijkl"e9

kl¦ekl−o�kl# "2#

Combining eqns "1# and "2#\ we have

CIijkl"e9

kl¦ekl# �Cijkl"e9kl¦ekl−o�kl# "3#

Eshelby "0846# has shown that\ if s9ij is uniform\ the strain disturbance ekl inside V is

also uniform as long as V is ellipsoidal[ ekl can be related to o� using the equation given byEshelby "0846#[

ekl �Sklmno�mn "4#

where Sklmn is referred to as the Eshelby tensor\ which depends on pore shape and the elasticproperties of the matrix[ Eshelby "0846# provides the procedure for calculating Sklmn in anisotropic matrix and Lin and Mura "0862# and Mura "0876# for solids having hexagonalor orthorhombic symmetry "see Appendix A for Mura|s formulae for a solid havinghexagonal symmetry#[

Substituting eqns "4# into eqn "3# we have

CIijkl"e9

kl¦Sklmno�mn# �Cijkl"e9kl¦Sklmno�mn−o�kl# "5#

Applying the rules given in Appendix B and converting the second rank tensors to8×0 column vectors and the fourth rank tensors to 8×8 matrices\ the eigenstrain for anisolated pore "hereafter we name it o�iso# is given by

3577 S[ Xu

o�iso � ð"C−CI#S−CŁ−0 ðCI−CŁe9 "6#

1[1[ o� for a communicatin` poreThe assumption of a completely isolated pore system is plausible only at high frequenc!

ies[ In reality\ pores in sedimentary rocks are rarely totally isolated[ They are often connectedby pore throats such as microcracks and grain contacts[ At high frequencies\ there is notime for ~uid to ~ow from one pore to another and they may behave as {{isolated|| pores[At low frequencies\ on the other hand\ pore ~uid will have plenty of time to interact andthe pore pressure will eventually be equalised[ The term {{communicating|| here means thatpore pressure will be equilibrated after the sample is subjected to an external stress[ This isthe basic assumption made also by Gassmann "0840#[

The eigenstrain for a communicating pore depends not only on the compressibility ofthe current pore but also on the compressibilities of other pores within the system[ Therelationship between the stress in the pore ~uid "PÞ�s9¦s# and the eigenstrain for thisparticular pore is given by eqn "2#

PÞ�C"e9¦e−o�# "7#

Substituting eqn "4# and s9 �Ce9 into eqn "7#\ the eigenstrain can be expressed as afunction of the di}erential stress PÞ−s9 ]

o�� ðC"S−I#Ł−0"PÞ−s9# "8#

There are\ in general\ two ways of estimating the pore stress change PÞij as a functionof the applied stress or strain _eld[ The _rst is to estimate the stress _eld in each pore andthen average it over all pores[ Since the total strain in each pore is e9¦e\ the average porestress PÞ is

P� sN

n�0

nnCI"e9¦e# "09#

where nn is the fractional volume of the nth pore normalized by the total pore volume[Thus\ we have the relationship

sN

n�0

nn � 0 "00#

Gassmann "0840# and Brown and Korringa "0864#\ on the other hand\ have shownthat pore pressure\ pf\ can also be calculated from the total change in pore volume[Mathematically this can be written as

pf �−kf

DVp

Vp

"01#

where kf is the bulk modulus of the pore ~uid and DVp:Vp is the normalized pore volumechange "or dilatation# which can be expressed as a function of the total strain\

DVp

Vp

� sN

n�0

nn"e900¦e9

11¦e922¦e00¦e11¦e22#n "02#

The pore stresses\ PÞij\ can then be expressed as a function of pore pressure "Brown andKorringa\ 0864#\

3578Fluid communication on velocities in anisotropic porous rocks

PÞij �−pfdij "03#

Here dij is the Kronecker delta function[ Comparing eqns "01#Ð"03# with eqn "09#\ it is easyto prove that the pore stresses calculated using the above two schemes are identical[ Sincethe _rst scheme is much simpler than the second\ it will be used in the formulation below[

Substituting eqns "4# and "8# into eqn "09#\ PÞ can be expressed as

PÞ�CIe9¦ sN

n�0

nnCISn ðC"Sn−I#Ł−0"PÞ−s9# "04#

which is a function of the geometries of other pores Sn and PÞ itself[ Noting that both thepore stress PÞ and the applied stress s9 are constant\ the following equation is obtained aftersome manipulations\

PÞ�"I−RÞ#−0"CI−RÞC#As9 "05#

where

RÞ � sN

n�0

nnRn "06#

and

Rn �CISn ðC"Sn−I#Ł−0 "07#

Equations "05#Ð"07# show that the pore pressure depends not only on the geometryand orientation of the current pore but also on the geometries and orientations of the otherpores within the system[ It should be kept in mind\ however\ the distinction between theinteractions of pore ~uid and the mechanical interactions between pores[ Although thepore ~uid pressure is assumed to be equalised\ the formulations are valid for dilute poreconcentrations only since the mechanical interactions between pores are ignored[ Finallythe eigenstrain for a communicating pore "hereafter we name it o�com# can be obtained bysubstituting eqns "05#Ð"07# into eqn "8#\

o�com � ðC"S−I#Ł−0""I−RÞ#−0"CI−RÞC#−C#e9 "08#

2[ EFFECTIVE ELASTIC CONSTANTS OF A SOLID WITH DILUTE PORECONCENTRATIONS

Let us now estimate the e}ective elastic constants of a solid permeated with diluteellipsoidal pores having arbitrary aspect ratios and a perfect orientation[ The pores arefurther assumed to be either well!interconnected "communicating# or totally isolated\ rep!resenting\ respectively\ the low! and high!frequency cases in terms of the local ~ow mech!anism[ In reality\ pores in sedimentary rocks are rarely totally isolated[ At high frequencies\however\ there may be no time for ~uid to ~ow from one pore to another and they maybehave as {{isolated|| pores[

The e}ective elastic constants of the composite can be approximated by using eitherthe volumetric averaging approach "e[g[ Gassmann\ 0840 ^ Brown and Korringa\ 0864 ^Hill\ 0854 ^ Hornby et al[\ 0883# or the interaction energy approach "e[g[ Eshelby\ 0846 ^Nishizawa\ 0871#[ Both approaches provide identical results as long as the formulation isderived under the same boundary conditions and to the same order of approximation[The interaction energy approach is used here since it o}ers two approximations yielding\respectively\ the upper and lower bounds for the e}ective elastic constants "Nishizawa\0871#\ although the bounds are much wider than the HashinÐShtrikman "0852# bounds[

3589 S[ Xu

Eshelby "0846# has shown that\ for a dilute pore concentration\ the total energy of thee}ective medium can be calculated using the following two equations corresponding to twoboundary conditions ]

0[ constant stress applied at in_nity

01A�ijkls

9ijs

9kl �

01Aijkls

9ijs

9kl¦

01

sN

n�0

vns9ijo�ij "19#

1[ constant strain applied at in_nity

01C�ijkle

9ije

9kl �

01Cijkle

9ije

9kl−

01

sN

n�0

vnCijkle9klo�ij "10#

Here N denotes the number of pores in the solid ^ vn is the volume concentration of thenth pore ^ C and C� represent the elastic sti}ness tensors of the solid and the e}ectivemedium\ respectively\ and A and A� denote their corresponding compliance tensors[ Fol!lowing Nishizawa "0871# and above two conditions will hereafter be referred to as theconditions of constant load and constant displacement\ respectively[

Nishizawa "0871# has shown how to calculate the e}ective sti}ness and compliancetensors once the eigenstrain o� is known[ Substituting eqns "6# and "08# into eqns "19# and"10#\ respectively\ we have

A�ijkls9ijs

9kl �Aijkls

9ijs

9kl¦ s

N

n�0

vns9ijQijklAklmns

9mn "11#

for the condition of the constant load\ and

C�ijkle9ije

9kl �Cijkle

9ije

9kl− s

N

n�0

vnCijkle9klQijmne

9mn "12#

for the condition of the constant displacement[ Here

Q� ð"C−CI#S−CŁ−0 ðCI−CŁ "13#

for the isolated pores "hereafter referred to as Qiso#\ and

Q� ðC"S−I#Ł−0""I−RÞ#−0"CI−RÞC#−C# "14#

for communicating pores "hereafter referred to as Qcom#[ Analysis of eqns "6# and "08#reveals that Q is an operator which relates the eigenstrain inside a pore to the applied strainat in_nity[ It is thus a dimensionless measure of pore compressibility[

Since C is diagonally symmetric\ i[e[ Cijkl �Cklij\ eqns "11# and "12# may be written as

A�ijkls9ijs

9kl �Aijkls

9ijs

9kl¦ s

N

n�0

vn ðQAŁijkls9ijs

9kl "15#

and

C�ijkle9ije

9kl �Cijkle

9ije

9kl− s

N

n�0

vn ðCQŁijkle9ije

9kl "16#

To solve A�0000 from eqn "15#\ s900 is set to one and all the other elements of s9 are set

to zero[ Similarly A�0011 can be obtained by setting s900 and s9

11 to one and the other elements

3580Fluid communication on velocities in anisotropic porous rocks

to zero[ The same logic can be applied to calculate C�\ but in this case e9 is varied insteadof s9[

It must be kept in mind that these equations are valid only for dilute pore concen!trations[ In the case of isolated pores\ Nishizawa "0871# has shown that elastic constantscalculated using eqn "15# could di}er signi_cantly from those calculated using eqn "16#[ Infact\ eqn "15# gives an upper bound for the e}ective elastic constants\ whereas eqn "16#gives a lower bound[ The same may apply to a solid containing communicating pores[Extension of the model to higher pore concentrations will be discussed in detail later[

3[ PORE ORIENTATION

The above equations are ready to be applied to a solid containing perfectly alignedpores[ In reality\ however\ pores are often not perfectly aligned but have rather a preferredorientation which may be described\ say\ by a von Mises distribution[ In this section\ themodel is extended to a more general case of arbitrary orientations of pores[

3[0[ Isolated poresIn the case of isolated pores\ all that is necessary is to calculate QisoA in eqn "15# and

CQiso in eqn "16# in a local coordinate system\ with the z!axis parallel to the normal of thepore and then transfer them from the local coordinate system to the global coordinatesystem[ Mathematically\

ðQ isoAŁ"u\b#ijkl �Tijklmnpq ðQ isoAŁlocalmnpq "17#

and

ðCQ isoŁ"u\b#ijkl �Tijklmnpq ðCQ isoŁlocalmnpq "18#

where u is the angle between the global Z!axis and local z!axis and b is the azimuth\ theangle between the global X!axis and the projection of the local z on the global XOY plane[Tijklmnpq is the 7th rank transformation tensor which can be calculated using the followingequation[

Tijklmnpq �KimKjnKkpKlq "29#

For oblate spheroidal pores with the same long semi!axis\ Kij is represented by the matrix

K� &cos"u# cos"b# −sin"b# sin"u# cos"b#

cos"u# sin"b# cos"b# sin"u# sin"b#

−sin"u# 9 cos"u# ' "20#

3[1[ Communicatin` poresIn the case of communicating pores the procedure is more complicated than that for

isolated pores[ First the tensor R in eqn "06# should be rotated from the local system to theglobal system\ when calculating pore pressure[

R"u\b#ijkl �TijklmnpqRlocalmnpq "21#

As shown by eqn "8#\ the eigenstrain for a communicating pore is the inner product of"a# the compliance of the dry pore ðC"S−I#Ł−0 and "b# the di}erential stress PÞ−s9[ Underthe constant load condition\ the di}erential stress PÞ−s9 should remain constant throughoutthe pore system[ It should not\ therefore\ be rotated[ This does not mean that the porepressure is independent of the pore orientation[ Rather\ the e}ect of pore orientation on

3581 S[ Xu

the pore ~uid pressure has been considered when calculating RÞ[ Therefore\ only the termðC"S−I#Ł−0 in eqn "14# needs to be rotated\

ðC"S−I#Ł−0"u\b#ijkl �Tijklmnpq ððC"S−I#Ł−0Łlocalmnpq "22#

In the case of applied constant displacement\ ðCQcomŁ can be rewritten as

CQ com � ðCðC"S−I#Ł−0CŁ"A"I−RÞ#−0"CI−RÞC#−I# "23#

where the term CðC"S−I#Ł−0C represents the sti}ness of the dry pore[ Term A"I−RÞ#−0

"CI−RÞC#−I is a dimensionless measure of the di}erential strain "pore total strain minusthe applied strain#[ In this case\ only the term CðC"S−I#Ł−0C needs rotating ^ the otherterm remains unchanged[

ðCðC"S−I#Ł−0CŁ"u\b#ijkl �Tijklmnpq ðCðC"S−I#Ł−0CŁlocalmnpq "24#

Finally we have\

Q comA� ðC"S−I#Ł−0"u\b#"ð"I−RÞ#−0"CI−RÞC#ŁA−I# "25#

for the condition of constant load\ and

CQ com � ðCðC"S−I#Ł−0CŁ"u\b#"A"I−RÞ#−0"CI−RÞC#−I# "26#

for the condition of constant displacement[

4[ HIGHER PORE CONCENTRATIONS

As mentioned earlier\ the formulae given above are valid for dilute pore concentrationsonly[ Unrealistic results\ such as a negative bulk modulus\ may be obtained when theporosity exceeds a certain value[ In the case where the solid is permeated randomly orientedpores with a single aspect ratio\ Kuster and Tokso�z "0863# give the following limit for thepore concentration

f

a³ 0 "27#

where f is the porosity\ and a�"c:a# is the aspect ratio[ The non!interacting theories arehardly practical with this limitation[

Considerable e}orts have been made to overcome the problem[ The self!consistent"SC# scheme developed by Hill "0854# and Budiansky "0854# makes it possible to calculatethe e}ective elastic constants of a composite at higher pore concentrations[ In the SCscheme\ pores are embedded into the e}ective medium which already contains the pores[Although the scheme provides results within the HashinÐShtrikman bounds "Willis\ 0866#\it has been criticized for over!estimating the e}ect of interactions between the pores[Bruner "0865# comments that the SC scheme overestimates the e}ect of the interactions byunrealistically considering the interaction between a pore and itself[

The di}erential e}ective medium "DEM# scheme "Bruner\ 0865 ^ Henyey andPomphrey\ 0871#\ on the other hand\ treats the problem in a di}erent manner[ To avoiddoubling the e}ect of the interactions\ DEM divides the total pore space f into N portions[The pore concentration Df "Df�f:N# for each portion is so small that the assumptionof dilute concentration is valid[ In combination with a non!interacting theory\ DEM thenintroduces the pores into the system step by step[ During each iteration\ a portion of poresare introduced into the system and the elastic constants of the e}ective medium are updated[

3582Fluid communication on velocities in anisotropic porous rocks

The resulting e}ective medium is then used as the host medium of the next iteration[ Theprocedure is repeated until all the pores are introduced into the system[

It should be noted that the actual pore concentration DV to be put into the systemduring each iteration is larger than Df "Nishizawa\ 0871 ^ Sheng\ 0880 ^ Hornby et al[\0883#[ This is because\ when a certain amount of pore space is put into the system\ thesame volume of the matrix has to be taken out from the system to maintain the unit totalvolume[ As a result\ a fraction of the pore space already embedded in the matrix will betaken out\ which needs to be compensated for by increasing the pore concentration for thecurrent portion to be embedded ]

DV�0

0−VDf "28#

where V denotes the amount of the pore space embedded into the system so far[Germonovich and Dyskin "0883a\ b# developed a Virial "power# Expansion scheme

and used it to calculate the e}ective shear modulus of an isotropic material with parallelcylindrical inclusions[ Their results also demonstrate that the di}erential e}ective mediumscheme gives more accurate results than the self!consistent scheme[ This agrees with theconclusion drawn by Bruner "0865#[

It should be noted that both the di}erential e}ective medium and self!consistentschemes violate the basic assumption on the equalised pore pressure when they are appliedto the communicating pore system[ This is because the elastic properties of the host mediumvary from iteration to iteration\ resulting in a varying pore pressure during the calculation[More discussion about this problem will be given later[ But the preliminary results showthat more research is needed to solve the problem and this is beyond the scope of thispaper[

5[ NUMERICAL SIMULATIONS AND DISCUSSION

Numerical simulations have been performed to investigate the combined e}ects ofporosity\ pore shape\ pore orientation and pore pressure communication on the por!oelasticity of a porous rock[ The solid "background matrix# is assumed to be isotropic withelastic constants l�09[5 GPa\ m�31[3 GPa and a density of 1[54 g:cm2[ The pore ~uidis assumed to have a bulk modulus of 1[14 GPa and a density of 0 g:cm2[ It is furtherassumed that the total pore space "f�4)# is equally partitioned into two groups[ The_rst group has an aspect ratio of 9[4 and the second group 9[94[ Under these assumptions\two special cases have been studied ]

, all the pores are aligned perpendicular to the Z!axis and, the pores in the _rst group are aligned perpendicular to the Z!axis whereas the normals

of the pores in the second group are at an angle of 34> to the Z!axis but are randomlyoriented in the XOY plane[

In both cases the resulting e}ective medium is transversely isotropic[ The _ve elasticconstants\ compressibility and the constant!displacement bulk modulus "refer to AppendixC for the de_nition of the compressibility and bulk modulus for an anisotropic medium#are listed in Table 0[

Several interesting points can be drawn from the numerical results[

, The low!frequency "communicating pores# elastic constants are lower than the cor!responding high!frequency "isolated pores# ones[ This agrees with the results obtained byO|Connell and Budiansky "0866#\ Endres and Knight "0886#[ For a communicating poresystem\ pore ~uid will ~ow from compliant pores to sti} pores as the rock sample issubjected to a compression[ Consequently the equilibrated pore pressure of the com!municating pore system should be lower than the pressure in the compliant pores buthigher than that in the sti} pores of the same system with isolated pores\ making thecompliant pores more compliant and sti} pores sti}er[ Since the overall elastic constants

3583 S[ Xu

Table 0[ Tabulation of the numerical results[ "a# Case 0 ] all pores are perfectly aligned\ "b# Case 1 ] pores inGroup 0 are aligned perpendicular to z!axis but the normals of the pores in Group 1 have an angle of 34> to the

Z!axis[ Total porosity is 4) equally partitioned between the two groups

C00 C01 C02 C22 C33 Compres[ CDBM

"a#Const[ load LF 78[43 09[95 7[36 50[30 29[28 9[9205 21[615

HF 78[44 09[96 7[53 53[18 29[28 9[9298 22[012DRY 78[11 8[64 6[94 44[93 29[28 9[9227 20[135

Const[ disp[ LF 78[00 09[99 6[34 31[36 14[53 9[9265 29[946HF 78[02 09[91 6[71 38[04 14[53 9[9249 29[861DRY 77[51 8[40 3[45 14[28 14[53 9[9422 15[545

"b#Const[ load LF 68[96 8[32 8[69 69[02 21[20 9[9205 20[662

HF 68[82 8[43 09[25 60[93 22[53 9[9298 21[170DRY 66[43 6[89 6[34 55[70 21[20 9[9227 18[611

Const[ disp[ LF 64[57 8[23 8[89 59[75 18[04 9[9225 29[946HF 65[78 8[33 09[80 51[32 20[25 9[9214 29[861DRY 62[28 6[94 5[00 43[46 18[04 9[9272 15[545

Abbreviations ] Const[*constant ^ disp[*displacement ^ LF*low frequency "communicating pores# ^ HF*high frequency "isolated pores# ^ Compres[*compressibility "in 0:GPa# ^ CDBM*constant displacement bulkmodulus "in GPa# ^ C00ÐC33 are elastic constants "in GPa#[

are a}ected by the compliant pores more than by the sti} pores\ the composite withcommunicating pores exhibits more compliant behaviour in comparison with the samecomposite with isolated pores[

, The elastic constants estimated under the constant load conditions are signi_cantly higherthan those under the constant displacement condition[ As stated earlier\ the formulationsgiven above are valid only for dilute pore concentrations because the interactions betweenpores are ignored[ In the case of isolated pores\ Nishizawa "0871# has shown that theformer provides a higher bound for the e}ective elastic moduli and the latter provides alower bound[ More accurate results may be obtained by incorporating the DEM schemeinto the model[

A numerical calculation has been performed to investigate how the DEM schemehandles this problem[ In this numerical simulation\ the _rst group of pores are assumedto be randomly oriented and the second have a preferred orientation with a Gaussiandistribution having a standard deviation of 29>[ The pore aspect ratio is further assumedto be 9[01 for the _rst group and 9[92 for the second[ Other parameters are kept unchanged["Hereafter\ these parameters will be applied to all following numerical calculations unlessotherwise indicated[# Velocity dispersions "de_ned as "Viso−Vcom#:Viso# for P and S wavestravelling parallel to the symmetry axis were calculated[ The results in Fig[ 0 clearly showthe velocity dispersion estimated under the constant load condition is much lower thanthat estimated under the constant displacement condition[ Incorporating the DEM sch!eme with an iteration number equal to 0999\ the model provides nearly identical resultsregardless of the applied boundary condition[

, For perfectly aligned pores\ it is observed that

C iso33 �C com

33 �Cdry33 "39#

indicating that C33 is independent of pore ~uid\ regardless of the degree of pore ~uidcommunication[ This phenomenon has not been reported by O|Connell and Budiansky"0866# or Endres and Knight "0886#\ since their formulations can handle the randomly!oriented pore system only[

If the pores are not perfectly aligned\ C iso33 is generally not equal to C com

33 but C com33 is

always identical to Cdry33 [ This indicates that the deformation of a solid containing isolated

pores involves both shear and dilatation when a pure shear is applied at in_nity\ resulting

3584Fluid communication on velocities in anisotropic porous rocks

Fig[ 0[ Comparison of velocity dispersions calculated using various schemes as a function of porosity[Upper ] vertical P wave\ lower ] vertical S wave[ Solid line ] non!interacting constant load\ dashedline ] non!interacting constant displacement\ dotÐdashed line ] DEM under the condition of constant

load\ dotted line ] DEM under the condition of constant displacement[

in a ~uid!dependent shear modulus[ In the communicating pore system\ the overall shearstresses are decoupled from the compressional stresses[ As a result\ no dilatation is causedby applying a pure shear at in_nity[ The shear modulus of a communicating system istherefore identical to that for dry rocks\ independent of the pore ~uid properties[ Similarresults for a solid containing randomly oriented pores have been reported by Endres andKnight "0886#[

, It has been reported "e[g[ Endres and Knight\ 0886 ^ Ravalec and Gueguen\ 0885# thatthe volumetric compressibility of a composite should be independent of pore orientation[The results presented here show that this is true for the constant load condition[ Under

3585 S[ Xu

the condition of constant displacement\ however\ the calculated compressibilities arefound to vary with pore orientation[ The constant displacement bulk modulus "de_nedin Appendix C# of the composite\ on the other hand\ is found to be independent of thepore orientation[ This is understandable\ since in the former case the process involvesaveraging the compliances " from which the compressibility is calculated# whereas in thelatter it involves averaging the sti}nesses " from which the constant displacement bulkmodulus is calculated#[ The results also demonstrate that the constant displacement bulkmodulus\ unlike the conventional bulk modulus which is simply the reciprocal of thecompressibility\ provides additional information on the elastic behaviour of an anisotropicmaterial[

In addition to the issues discussed above\ the following points were also observedduring the numerical calculations\ although their values are not listed in Table 0[

, In both cases the estimated dry Ciso is found to be identical to the estimated dry Ccom\indicating the self consistency of the formulations[

According to the model\ the state of pore ~uid communication is important because ita}ects pore pressure which in turn controls the elastic behaviour of a porous rock[ In thedry case\ however\ the e}ect of pore ~uid communication vanishes since the pore pressureitself becomes zero[ In other words\ the model should provide identical results whetherthe pores are isolated or they are communicating[ This can easily be seen by setting CI inQiso ðeqn "13#Ł and Qcom ðeqns "14#\ "06# and "07#Ł to zero[

, There is no dispersion observed when pores with a single aspect ratio are assumed to beperfectly aligned[

This is obvious since pores having the same geometry and orientation should have thesame pore pressure when the composite is subjected to a uniform external stress or strain_eld[ Consequently there is no ~uid ~ow among the pores and\ hence\ no dispersionshould be observed[ In a special case where the solid is permeated by round pores\ thesame phenomenon has been reported by Thomsen "0874#\ Endres and Knight "0886#[

In order to demonstrate the e}ect of the pore ~uid communication on seismicanisotropy\ P! and S!wave velocities were calculated as a function of the incident angle[Figure 1 shows the results obtained under the condition of the constant load "left# andthose of constant displacement "right#[ In general\ the impact of the applied condition onthe estimation of velocities is high in comparison with the velocity variations caused by thepore ~uid communication[ In particular\ the angular dependence of velocities is also a}ectedby the applied boundary condition[

Figure 2 shows the calculated vertical P! "upper# and S!wave "lower# velocity dis!persions as a function of the concentration of the compliant pores "normalized by the totalpore space#[ The results demonstrate a combined e}ect of pore shape and pore orientationon the poroelasticity[ In two special cases where the concentration of the compliant poresapproaches zero or one "i[e[ all the pores now have the same shape#\ the resultant dispersionscan only be attributed to the e}ect of pore orientation[

6[ COMPARISON WITH BIOTÐGASSMANN|S THEORY

The BG theory has been widely used in the oil industry for estimating velocity dis!persion and ~uid substitution[ The BG theory considers the e}ect of the macro ~uid ~ow"at the scale of the mean wavelength# on velocity dispersion\ whereas this model considersthe e}ect of the local ~ow "or squirt ~ow# on velocity dispersion[ For a rock containingperfectly aligned pores with a single aspect ratio\ this model predicts no velocity dispersionbut the BG model does[ Even if the pore compressibility is the same everywhere in thesystem\ macro ~uid ~ow can be generated by a passing wave[ In any case\ the high!frequencyvelocities predicted by the two models are di}erent[

At the low frequency limit\ however\ both the IBAPM and BG models make the sameassumption about the pore pressure relaxation[ The low!frequency version "communicating

3586Fluid communication on velocities in anisotropic porous rocks

Fig[ 1[ Illustration of P! and S!wave velocities calculated under the conditions of constant load"upper# and constant displacement "lower# displacement as a function of the incident angle[ Dottedlines ] dry\ solid lines ] low!frequency "communicating pores#\ dashed lines ] high!frequency "isolated

pores#[

pores# of the IBAPM should\ therefore\ be consistent with the BG theory and its extensions"e[g[ Brown and Korringa\ 0864#[

To be consistent with the BG model\ two criteria must be satis_ed "Thomsen\ 0874# ]"0# the shear modulus of an unsaturated rock is identical to that of the same rock saturatedwith liquid\ and "1# the unsaturated bulk modulus di}ers from the saturated bulk modulusby a de_ned amount predicted by the BG theory "Brown and Korringa in the anisotropiccase#[ In the previous section it has been shown that the non!interacting shear modulus ofthe communicating system is identical to the dry shear modulus regardless of the conditionapplied[ In this sense the model is consistent with the BG model[ In this section moretheoretical analysis and numerical examples will be given to show the relationship betweenthe IBAPM and the BG model[

As stated earlier\ the IBAPM o}ers two means of estimating the low!frequency e}ectiveelastic constants ] one corresponding to the condition of constant load and the other to thecondition of constant displacement[ The formulations given by Brown and Korringa "0864#and Gassmann "0840# are based on the condition of applied constant load only[ I shall\therefore\ start with the comparison of the constant load formulation of the model discussedhere with that given by Brown and Korringa "0864#[

Since eqn "19# is valid for any applied stress s9\ it may be rewritten as

A�s9 �As9¦ sN

n�0

vno�com "30#

Substituting eqn "8# into eqn "30# and adding APÞ−APÞ\ we have

A�s9 �APÞ−6A− sN

n�0

vn ðC"S−I#Ł−07"PÞ−s9# "31#

Note that pore stresses PÞij can be expressed in terms of pore pressure pf using eqn "03#

3587 S[ Xu

Fig[ 2[ Illustration of the vertical P! "upper# and S!wave "lower# velocity dispersions as a functionof the concentration of second group of pores "normalized by the total pore space#[

"Brown and Korringa\ 0864#[ Also the e}ective elastic compliances of the dry rockframecan be obtained by setting CI in eqns "13# to zero

Adry �A− sN

n�0

vn ðC"S−I#Ł−0 "32#

Substituting eqns "03# and "32# into eqn "31# and changing the matrices back totensors\ we have

A�ijkls9kl �Adry

ijkl"s9kl¦pfdkl#−pfAijkldkl "33#

Equation "33# is exactly the same as eqn "11# in the BrownÐKorringa paper "0864#\from which they derived their anisotropic Gassmann equation[ This indicates that our non!interacting formulation under the condition of the constant load is strictly consistent withthat of BrownÐKorringa[ The di}erence between the two models is that they express thepore volume change DVp as a function of the dry rockframe moduli and the bulk modulus

3588Fluid communication on velocities in anisotropic porous rocks

of the pore ~uid\ while the IBAPM evaluates DVp from the speci_ed pore shape parametersand pore ~uid properties ðeqns "09#Ð"03#Ł[

Numerical studies have been carried out to check the consistency of the IBAPM withthe BrownÐKorringa model[ The e}ective elastic constants of porous rock with com!municating pores are _rstly calculated using IBAPM under the two speci_ed conditions[The same pore parameters are used to calculate the elastic constant of the dry rockframeby simply setting CI to zero[ The BrownÐKorringa equation is then used to calculate thee}ective elastic constants of the saturated rock from the dry rockframe moduli calculatedabove and the same pore ~uid properties[

Figure 3 shows Cwet22 −Cdry

22 as a function of porosity[ Note that at the low frequencylimit the non!interacting formulation of IBAPM under the condition of constant load"dashed line# gives results identical to that of BG "dotted line#\ con_rming the mathematicalveri_cations above[

However\ under the condition of constant displacement\ the dependence of the elasticmoduli on pore ~uid predicted by the IBAPM "dotÐdashed line# is much higher thanthe BrownÐKorringa prediction[ Incorporating the DEM scheme into the constant loadformulation provides results inconsistent with the BrownÐKorringa prediction\ but nicelylying between the two extremes predicted by IBAPM under the two conditions[

Figure 4 further shows the elastic constant C33 as a function of the ~uid bulk modulus[As observed earlier\ the non!interacting C com

33 was found to be independent of the bulkmodulus of the pore ~uid\ regardless of the applied boundary condition[ It also shows thatthe dispersion predicted under the condition of constant load is much smaller than that ofconstant displacement[ Again\ it was found that both the absolute value of C33 approximated

Fig[ 3[ Comparison of the dependencies of the elastic constant C22 on pore ~uid calculated usingvarious schemes[ Dashed line ] non!interacting constant load\ dotÐdashed line ] non!interactingconstant displacement\ solid line ] DEM under the condition of constant load\ dotted line ] aniso!

tropic Gassmann "Brown and Korringa\ 0864#[

3699 S[ Xu

Fig[ 4[ Comparison of the low! and high!frequency elastic constant C33 as a function of the bulkmodulus of pore ~uid[

by DEM and its dispersion lie within the two extremes mentioned above[ However the low!frequency C33 from the DEM exhibits a slight dependence on the bulk modulus of the pore~uid[

The inconsistency of the SC or DEM with the BG theory has been observed by anumber of authors "e[g[ Thomsen\ 0874 ^ Berryman\ 0881 ^ Endres and Knight\ 0886#[ Theinconsistency is interpreted as a result of violating the BG basic assumption of equalisedpore pressure by SC and DEM[ Although the pore pressure during each iteration is keptthe same by DEM\ it varies from iteration to iteration\ due to the change of the elasticconstants of the host medium[ Thomsen "0874# believes that the BG theory adequatelydescribes the dependence of elastic moduli on pore ~uid compressibility at low frequencies[Thomsen "0874# and Berryman "0881# have proposed methods to overcome the violationof the BG relationship by the SC and DEM schemes[

7[ APPLICATION TO LABORATORY MEASUREMENTS

Rathore et al[ "0884# developed a technique for manufacturing arti_cial porous rockspermeated with parallel cracks with known geometries[ Porous rock samples were con!structed by cementing together the sand grains using an epoxy resin[ Metal disks withknown positions\ orientations\ shape and aspect ratio were embedded into the arti_cialsamples during the construction process[ These disks were later leached out chemically\leaving disk voids representing a network of parallel cracks sitting in a porous rock[ Tworock samples were manufactured during their experiment ] one containing parallel cracks

3690Fluid communication on velocities in anisotropic porous rocks

Table 1[ Crack parameters

Crack shape circularCrack diameter 4[49 mmCrack thickness 9[91 mmCrack density 9[09Number of cracks in 9[990 m2 3797Number of layers 37

and the other being a crack!free dummy[ Table 1 shows the designed parameters for thecracks[ Ultrasonic velocities VP\ VSV and VSH were measured at eight angles to the symmetryaxis of the cracks with a 11[4> increment[

Parameters used by Rathore et al[ "0884# were adopted in simulating the laboratorymeasurements[ The solid grains are assumed to have a bulk modulus of 07[1 GPa\ a shearmodulus of 09[1 GPa and a density of 1[97 g:cm2[ The bulk modulus of the pore ~uid isassumed to be 1[14 GPa and a density of 0 g:cm2[ The total pore space comprises two parts ]equant inter!granular pores with a porosity of 23[5) and cracks with a porosity of 9[12)[The equant pores are assumed to be randomly oriented\ giving an isotropic porous matrix[Their aspect ratio "9[14# was derived by _tting the predictions to P! and S!wave velocitiesof the dummy sample[ The aspect ratio for the cracks "9[9925# was calculated from thecrack parameters[

Figure 5 compares the predictions with the measured P! and S!wave velocities of thedry sample as a function of the incident angle[ Despite a slight over!estimate in the S!wavevelocities\ the predictions simulate the measurements rather well[

Figure 6 compares the measurements with the low!frequency "solid lines# and high!frequency "dashed lines# predictions[ The results clearly show that the low!frequency versionsimulates the laboratory measurements much better than its high!frequency counterpart[The low!frequency behaviour of the measurement may be attributed to the relatively large

Fig[ 5[ Comparison of P! and S!wave velocities measured by Rathore et al[ "0884# of the drysynthetic sample with aligned cracks and the predictions from the IBAPM[

3691 S[ Xu

Fig[ 6[ Comparison of P! and S!wave velocities of the saturated synthetic sample "Rathore et al[\0884# and IBAPM|s low! "solid lines# and high!frequency "dashed lines# predictions[

size of the cracks "4[4 mm in diameter#\ which greatly enhances the permeability of thesynthetic sample[

The main advantage of the inclusion!based poroelasticity models over the traditionalones such as Biot|s "0845a\ b# is that the former do not require dry frame moduli as thekey input parameters[ The inclusion!based models calculate the dry moduli from poreparameters "aspect ratios# and the elastic properties of solid and pore ~uid[ This featuremakes the inclusion!based models very practical since in many cases we do not really knowthe dry frame moduli[ The goodness of _t between the measured dry and saturated P! andS!wave velocities and the predictions demonstrates the predictive power of the model\considering that nearly all the input parameters are de_ned from the independent measure!ments[

3692Fluid communication on velocities in anisotropic porous rocks

8[ CONCLUSIONS

"0# An inclusion!based anisotropic poroelasticity model "IBAPM# has been developed[The model simulates the low!frequency response of a porous rock by assuming acommunicating pore system and the high!frequency response by an isolated poresystem[

"1# The local ~ow within a communicating pore system and\ hence\ the velocity dispersionare controlled by two key factors ] pore geometry and pore orientation[ Unlike manyother poroelasticity models which use the dry rockframe moduli as input parameters\IBAPM calculates them from pore parameters and properties of the matrix and ~uid[This feature makes the model very useful since in many practical applications the dryframe moduli are not measurable[

"2# A further major di}erence between IBAPM and the previous poroelasticity models isthat IBAPM o}ers two approximations\ one corresponding to the condition of constantload and the other to the condition of constant displacement[ Numerical results showthat velocity dispersion calculated by applying the constant load boundary conditionis much lower than that calculated by applying that of constant displacement[ Inco!rporating the DEM theory into IBAPM provides results lying between the two extremes[

"3# The non!interacting constant load formulation of IBAPM is found to be completelyconsistent with the BG theory[ At higher pore concentrations\ although DEM violatesthe BG assumption on the equalised pore pressure\ it probably predicts the pore ~uiddependence of the elastic moduli more accurately than BG[

"4# The IBAPM has been applied to simulate the laboratory measurement made by Rathoreet al[ "0884#[ The results clearly show that the low!frequency version simulates thelaboratory measurements much better than its high!frequency counterpart[ The low!frequency behaviour of the measurements may be attributed to the relative large sizeof the cracks "4[4 mm in diameter#\ which greatly enhances the permeability of thesynthetic sample[

Acknowled`ements*The author is indebted to the sponsors of the London University Research Programme inSeismic Lithology\ Amoco "UK# Exploration Company\ Elf UK plc\ Enterprise Oil plc\ Fina Exploration Ltd\Mobil North Sea Ltd and Texaco Britain Ltd\ for their support of this research[ I especially thank Professors RoyWhite and Michael King\ both from University of London for reviewing the manuscript and many usefuldiscussions[ I would also like to thank Dr Leonid Germanovich and an anonymous referee for their constructivereview of the manuscript[

REFERENCES

Berryman\ J[ G[ "0881# Single!scattering approximations for coe.cients in Biot|s equations of poroelasticity[Journal of Acoustic Society of America 80\ 440Ð460[

Biot\ M[ A[ "0845a# Theory of propagation of elastic waves in a ~uid saturated porous solid[ I[ Low!frequencyrange[ Journal of Acoustic Society of America 17\ 057Ð067[

Biot\ M[ A[ "0845b# Theory of propagation of elastic waves in a ~uid saturated porous solid[ II[ Higher!frequencyrange[ Journal of Acoustic Society of America 17\ 068Ð080[

Blangy\ J[ P[\ Strandenes\ S[\ Moos\ D[ and Nur\ A[ "0882# Ultrasonic velocities in sands*revisited[ Geophysics47\ 233Ð245[

Brown\ R[ J[ S[ and Korringa\ J[ "0864# On the dependence of elastic properties of a porous rock on thecompressibility of the pore ~uid[ Geophysics 39\ 597Ð505[

Bruner\ W[ M[ "0865# Comment on {{seismic velocities in dry and saturated cracked solids|| by Richard J[O|Connell and Bernard Budiansky[ J[ Geophys[ Res[ 70\ 1462Ð1465[

Budiansky\ B[ "0854# On the elastic moduli of some heterogeneous materials[ Journal of Mechanics and Physicsof Solids 02\ 112Ð116[

Dutta\ N[ C[ and Ode�\ H[ "0868a# Attenuation and dispersion of compressional wave in ~uid!_lled rocks withpartial gas saturation "White model#*Part I[ Biot theory[ Geophysics 33\ 0666Ð0677[

Dutta\ N[ C[ and Ode�\ H[ "0868b# Attenuation and dispersion of compressional wave in ~uid!_lled rocks withpartial gas saturation "White model#*Part II[ Results[ Geophysics 33\ 0678Ð0794[

Endres\ A[ L[ and Knight\ R[ J[ "0886# Incorporating pore geometry and ~uid pressure communication intomodelling the elastic behavior of porous rocks[ Geophysics 51\ 095Ð006[

Eshelby\ J[ D[ "0846# The determination of the elastic _eld of an ellipsoidal inclusion\ and related problems[Proceedin`s of the Royal Society of London\ Series A 130\ 265Ð285[

Gassmann\ F[ "0840# Elasticity of porous media[ Vierteljahrschrift der Naturforschenden Gesellschaft in Zu�rich 85\

0Ð10[

3693 S[ Xu

Germonovich\ L[ N[ and Dyskin\ A[ V[ "0883a# Virial expansions in problems of e}ective characteristics[ 0[General concepts[ Mech[ Composite Mater[ 29\ 046Ð056[

Germonovich\ L[ N[ and Dyskin\ A[ V[ "0883b# Virial expansions in problems of e}ective characteristics[ 1[Antiplannar deformation of a _ber composite[ Analysis of self!consistent methods[ Mech[ Composite Mater[29\ 123Ð132[

Gordon\ R[ B[ and Davis\ L[ A[ "0857# Velocity and attenuation of seismic waves in imperfectly elastic rocks[ J[Geophys[ Res[ 62\ 2806Ð2814[

Hashin\ Z[ and Shtrikman\ S[ "0852# A variation approach to the theory of the elastic behaviour of multiphasematerials[ Journal of Mechanics and Physics of Solids 00\ 016Ð039[

Henyey\ F[ S[ and Pomphrey\ N[ "0871# Self!consistent elastic moduli of a crack solid[ Geophys[ Res[ Lett[ 8\ 892Ð895[

Hill\ R[ "0854# A self!consistent mechanics of composite materials[ Journal of Mechanics and Physics of Solids 02\

102Ð111[Hoenig\ A[ "0867# The behaviour of a ~at elliptical crack in an anisotropic elastic body[ International Journal of

Solids and Structures 03\ 814Ð823[Hoenig\ A[ "0868# Elastic moduli of the non!randomly cracked body[ International Journal of Solids and Structures

04\ 026Ð043[Hornby\ B[ E[\ Schwartz\ L[ M[ and Hudson\ J[ A[ "0883# Anisotropic e}ective!medium modeling of the elastic

properties of shales[ Geophysics 48\ 0469Ð0472[Hudson\ J[ A[ "0870# Wave speed and attenuation of elastic waves in material containing cracks[ Geophys[ J[ Roy[

Astr[ Soc[ 53\ 022Ð049[Johnston\ D[ H[\ Tokso�z\ M[ N[ and Timur\ A[ "0868# Attenuation of seismic waves in dry and saturated rocks[

II ] Mechanisms[ Geophysics 33\ 580Ð600[Kuster\ G[ T[ and Tokso�z\ M[ N[ "0863# Velocity and attenuation of seismic waves in two!phase media[ Part 0 ]

Theoretical formulation[ Geophysics 28\ 476Ð595[Lin\ S[ and Mura\ T[ "0862# Elastic _elds of inclusions in anisotropic media[ Phys[ Status Solidi "a#04\ 170Ð174[Mavko\ G[ and Jizba\ D[ "0880# Estimating grain!scale ~uid e}ects on velocity dispersion in rocks[ Geophysics

45\ 0839Ð0838[Mavko\ G[ and Jizba\ D[ "0883# Relation between seismic P! and S!wave velocity dispersion in saturated rocks[

Geophysics 48\ 76Ð81[Mavko\ G[ and Nur\ A[ "0864# Melt squirt in the asthenosphere[ J[ Geophys[ Res[ 79\ 0333Ð0337[Mavko\ G[ and Nur\ A[ "0864# Wave attenuation in partially saturated rocks[ Geophysics 33\ 050Ð067[Mukerji\ T[ and Mavko\ G[ "0883# Pore ~uid e}ects on seismic velocity in anisotropic rocks[ Geophysics 48\ 122Ð

133[Mura\ T[ "0876# Micromechanics of Defects in Solids[ Martinus Nijho}\ Dordrecht[Nishizawa\ O[ "0871# Seismic velocity anisotropy in a medium containing oriented cracks*transversely isotropic

case[ J[ Phys[ Earth 29\ 220Ð236[O|Connell\ R[ J[ and Budiansky\ B[ "0866# Viscoelastic properties of ~uid!saturated cracked solids[ J[ Geophys[

Res[ 71\ 4608Ð4625[Rathore\ J[ S[\ Ejaer\ E[\ Holt\ R[ M[ and Renlie\ L[ "0884# P! and S!wave anisotropy of a synthetic sandstone

with controlled crack geometry[ Geophys[ Prosp[ 32\ 600Ð617[Ravalec\ M[ L[ and Gue�guen\ Y[ "0885# High! and low!frequency elastic moduli for saturated porous:cracked

rock!di}erential self!consistent and poroelastic theories[ Geophysics 50\ 0979Ð0983[Shams\ M[ K[\ King\ M[ S[ and Worthington\ M[ H[ "0882# Whitchester seismic cross!hole test site*petrophysical

studies of cores[ In Expanded Abstracts 44th EAGE Mt`[ Stanvanger\ Norway[Sheng\ P[ "0880# Consistent modeling of the electrical and elastic properties of sedimentary rocks[ Geophysics 45\

0125Ð0132[Thomsen\ L[ "0874# Biot!consistent elastic moduli of porous rocks ] low!frequency limit[ Geophysics 49\ 1686Ð

1796[Thomsen\ L[ "0884# Elastic anistropy due to aligned cracks in porous rocks[ Geophys[ Prosp[ 32\ 794Ð718[Tokso�z\ M[ N[\ Johnston\ D[ H[ and Timur\ A[ "0868# Attenuation of seismic waves in dry and saturated rocks ]

I[ Laboratory measurements[ Geophysics 33\ 570Ð589[Walsh\ J[ B[ "0855# Seismic attenuation in rock due to friction[ J[ Geophys[ Res[ 60\ 1480Ð1488[White\ J[ E[ "0864# Computed seismic speeds and attenuation in rocks with partial gas saturation[ Geophysics 39\

113Ð121[Winkler\ K[ W[ and Nur\ A[ "0871# Seismic attenuation ] e}ects of pore ~uids and frictional sliding[ Geophysics

36\ 0Ð04[Willis\ J[ R[ "0866# Bounds and self!consistent estimates for the overall properties of anisotropic composite[

Journal of Mechanics and Physics of Solids 14\ 074Ð191[Wyllie\ M[ R[ J[\ Gardner\ G[ H[ F[ and Gregory\ A[ R[ "0851# Studies of elastic wave attenuation in porous

media[ Geophysics 16\ 458Ð478[Xu\ S[ and White\ R[ E[ "0884a# A new velocity model for clay!sand mixtures[ Geophys[ Prosp[ 32\ 80Ð007[Xu\ S[ and White\ R[ E[ "0884b# Poro!elasticity of clastic rocks ] a uni_ed model[ In Trans[ 25th Ann[ SPWLA

Symp[\ Paris\ Paper V[Xu\ S[ and White\ R[ E[ "0885# A physical model for shear!wave velocity prediction[ Geophys[ Prosp[ 33\ 576Ð

606[

APPENDIX A ] ESHELBY TENSOR Sijkl FOR A PORE IN A HEXAGONAL SOLID

Lin and Mura "0862# and Mura "0876# have shown how to calculate the Eshelby tensor Sijkl for a poreembedded in a solid with hexagonal symmetry[

3694Fluid communication on velocities in anisotropic porous rocks

Sijmn �07p

"Gipjq¦Gjpiq#Cpqmn "34#

where C is the sti}ness tensor of the solid[ G is a four!rank geometry tensor which depends on the pore shape andthe elastic properties of the solid[ For an oblate spheroidal pore with an aspect ratio a � c:a\ Lin and Mura "0862#claim that there are 01 non!zero elements for Gijkl which can be calculated using the following integrals[

G0000 �p

1 g0

9

"0−x1#ð" f"0−x1#¦hr1x1#""2e¦d#"0−x1#¦3fr1x1#−`1r1"0−x1#x1ŁD dx "35#

G1111 � G0000 "36#

G2222 � 3p g0

9

x1r1"d"0−x1#¦fr1x1#"e"0−x1#¦fr1x1#D dx "37#

G0011 �p

1 g0

9

"0−x1#ð" f"0−x1#¦hr1x1#""e¦2d#"0−x1#¦3fr1x1#−2`1r1"0−x1#x1ŁD dx "38#

G1100 � G0011 "49#

G0022 � 1p g0

9

x1r1 ð""d¦e#"0−x1#¦1fr1x1#" f"0−x1#¦hr1x1#−`1r1"0−x1#x1ŁD dx "40#

G1122 � G0022 "41#

G2200 � 1p g0

9

"0−x1#"d"0−x1#¦fr1x1#"e"0−x1#¦fr1x1#D dx "42#

G2211 � G2200 "43#

G0101 �p

1 g0

9

"0−x1#1 ð`1x1r1−"d−e#" f"0−x1#¦hr1x1#ŁD dx "44#

G0202 � −1p g0

9

"0−x1#x1r1"e"0−x1#¦fr1x1#D dx "45#

G1212 � G0202 "46#

where

D � "ðe"0−x1#¦fr1x1Łð"d"0−x1#¦fr1x1#" f"0−x1#¦hr1x1#−`1r1"0−x1#x1Ł#−0

d � C0000

e � "C0000−C0011#:1

f � C1212

` � C0022¦C1212

h � C2222

r1 � "0:a1# "47#

APPENDIX B ] CONVERTING TENSORS TO MATRICES

Following the notations given by Hill "0854#\ a 8×0 column vector is used to represent a second!rank tensor\e[g[ e and s\ and a 8×8 matrix to represent a fourth!rank tensor\ e[g[ A and C[ The tensor components arearranged using the following sequence ] 00c 0\ 11c 1\ 22c 2\ 12c 3\ 02c 4\ 01c 5\ 21c 6\ 20c 7\ and 10c 8[The leading pair of indices corresponds to rows and the terminal pair to columns so that the inner product oftensor Aijklekl can be written as Ae[ Similarly AijklCklmn can be written as AC and AijklCklmnemn as ACe[

The inverse of a fourth!rank tensor A is de_ned as "Hill\ 0854 ^ Hornby et al[\ 0883#

AA−0 � A−0A � I "48#

where A−0 is the inverse of tensor A and I is a unit tensor

3695 S[ Xu

Iijkl �0

1"dikdjl¦dildjk# "59#

and dij is the Kronecker delta[All the fourth!rank tensors we deal with in this paper have the following symmetry

Aijkl � Ajikl � Aijlk "50#

Their corresponding matrices are consequently singular with a rank less than or equal to six[

APPENDIX C ] COMPRESSIBILITY AND BULK MODULUS OF AN ANISOTROPICMEDIUM

The compressibility of an isotropic medium is de_ned as the ratio of the normalized bulk volume change tothe applied hydrostatic stress

c �

DVVs9

"51#

where DV:V denotes the normalized bulk volume change in the elementary volume V\ which is a function of thetotal strain e\

DVV

� ex¦ey¦ez "52#

and s9 is the applied hydrostatic stress with

sx � sy � sz � s9 "53#

The de_nition of the bulk modulus is the inverse of the compressibility\ i[e[ the ratio of the hydrostatic stressto the normalized bulk volume change "dilatation# under simple hydrostatic pressure\

k �s9

DVV

"54#

The de_nition of the compressibility ðeqn "51#Ł and that of the bulk modulus ðeqn "54#Ł can be applied to theanisotropic case without modi_cation[ Assuming the rock has a compliance tensor A\ the normalized volumechange due to the applied hydrostatic stress s9 is

DVV

� A00sx¦A01sy¦A02sz¦A10sx¦A11sy¦A12sz¦A20sx¦A21sy¦A22sz "55#

Since sx � sy � sz � s9\ the compressibility for an anisotropic medium is given by

c � s2

i�0

s2

j�0

Aij "56#

Equation "56# is exactly the same as the de_nition of the compressibility given by Brown and Korringa "0864#[Since the bulk modulus is de_ned under the condition of constant load with sx � sy � sz � s9\ it may be

referred to as the constant load bulk modulus "CLBM#[ The CLBM for an anisotropic medium is always thereciprocal of the compressibility and consequently carries no more information on the elastic behaviour of thematerial than does the compressibility[ A new bulk modulus is de_ned as the ratio of the average stress s9 to thenormalized bulk volume change as the material undergoes a uniform volume deformation with

ex � ey � ez � e9 "57#

Mathematically\ the new bulk modulus has the same form as eqn "54#[ But s9 now denotes the average stressof sx\ sy and sz\ i[e[

s9 � 0

2"sx¦sy¦sz# "58#

Under the condition of uniform volume deformation as speci_ed by eqn "57#\ the normalized volume change isgiven by

DVV

� ex¦ey¦ez � 2e9 "69#

Since the new bulk modulus is de_ned under the condition of constant strain\ it may be referred to as theconstant displacement bulk modulus "CDBM#[ Applying Hooke|s law\ the constant displacement bulk modulusfor an anisotropic medium can be expressed as a function of the elastic constants

3696Fluid communication on velocities in anisotropic porous rocks

k �08

s2

i�0

s2

j�0

Cij "60#

where C is the sti}ness tensor of the material[It should be borne in mind that the constant displacement bulk modulus de_ned above is generally not the

reciprocal of the compressibility[ Unlike the CLBK it provides extra information about the elastic behaviour ofan anisotropic material[ However\ in the isotropic case\ the CDBM is exactly the same as the conventional bulkmodulus CLBK[ This is understandable because\ in an isotropic material\ eqn "57# still holds when a hydrostaticpressure applies ðeqn"53#Ł and vice versa[