acoustic wave propagation through porous media revisited

Acoustic wave propagation through porous media revisitedTim W. Geeritsa)

Faculty of Mining and Petroleum Engineering, Delft University of Technology, P.O. Box 5028,2600 GA Delft, The Netherlands

~Received 2 June 1994; revised 15 March 1996; accepted 8 April 1996!

In this article a new macroscopic theory for acoustic wave propagation through fluid-saturatedporous media will be derived in which losses of the interface type are incorporated. It is argued thatin order to explain the occurrence of a slow compressional wave~which is typical for fluid-saturatedporous media!, a discontinuity in the tangential components of traction and particle velocity at themicroscopic fluid/solid interface is essential. The method ofspatial volume-averagingwill beapplied in order to translate the microscopic field equations pertaining to the constituents of theporous medium to the macroscopic level~i.e., several times the wavelength!. This method clearlydemonstrates the interaction between the fluid and the solid phase at the microscopic fluid/solidinterface by means of surface integrals which arise as a consequence of the averaging procedure.These surface integrals are assumed to be linearly related to the volume-averaged state quantities ofthe phases involved. In this way the interface-type losses are also incorporated. These losses areassumed to be of the inertial type and are triggered by the possible strong fluctuations in the spatialgradients of the microscopic fluid/solid interface~tortuosity!. The resulting macroscopic theory,which is assumed to be valid at relatively high frequencies~the frequency range where we observea slowP wave!, mathematically has the same appearance as the most general Biot theory but witha different physical interpretation. ©1996 Acoustical Society of America.

PACS numbers: 43.20.Gp, 43.20.Jr@JEG#

INTRODUCTION

In this article amacroscopicacoustic wave propagationtheory for fluid-saturated porous media will be derived. Inthis derivation we start from the fundamentals with thosefield equations which pertain to the constituents of the fluid-saturated porous medium, and subsequently we will translatethe corresponding field equations to a macroscopic level bymeans of the method ofspatial volume-averaging. In acous-tic wave propagation problems this method was first appliedby De Vries ~1989! and later by Prideet al. ~1992!. Thisarticle will be closely linked to the work of De Vries~1989!but will, in contrast, be more precise about the choice of themicroscopic field equations that pertain to the constituents ofthe fluid-saturated porous medium. Furthermore, whereas DeVries’ theory does not incorporate losses of any kind, thepresent theory will. The type of losses presented in this ar-ticle are due to the interaction between the fluid phase andthe solid phase at the microscopic fluid/solid interface andare assumed to be due to the strongly changing geometry ofthe microscopic fluid/solid interface. Depending on fre-quency and viscosity this may give rise to inertial losses. Inthe inviscid limit ~at very high frequencies!, where the lossesdisappear, the presented theory changes into De Vries’theory~1989!. De Vries’ theory is compatible with the high-frequency range of the Biot theory~Biot, 1956b!, but has adifferent physical interpretation. The theory presented in thisarticle mathematically has the same appearance as the low-frequency range of the Biot theory~Biot, 1956a!, but with adifferent physical interpretation.

Several theoreticians tried to derive an acoustic wavepropagation theory for fluid-saturated porous media on amacroscopic level, starting from first principles, i.e., the mi-croscopic field equations that pertain to the constituents ofthe fluid-saturated porous medium. In this respect the articleof Burridge and Keller~1981! is very well known and isoften referred to as the justification of Biot’s theory. In thistheory the two-space method of homogenizationis used~Keller, 1977; Bensoussanet al., 1978! in order to translatethe microscopic behavior of the fluid-saturated porous me-dium from a microscopic level to a macroscopic level. Thebasic idea of the two-space method is that each physicalquantity associated with a porous material functionally varieson two independent length scales, one being the grain scaleand the other being the seismic-wave scale. The ratio ofthese two length scales~say, r ! is a fundamental small pa-rameter that can be used to expand the microscopic fieldequations into a truncated asymptotic power series inr . Bycollecting terms of like powers ofr and then averaging overthe grain scale, equations are left that vary only on the seis-mic scale. The macroscopic field equations obtained in thisway are identified as those derived by Biot~1956a, 1962!provided the dimensionless viscosity of the fluid pertainingto the seismic-wave scale~as introduced by Burridge andKeller, 1981! is small. However, when the dimensionlessviscosity of the pore fluid is large, different macroscopicfield equations are obtained. They characterize the porousmedium as a viscoelastic solid, in agreement with the sug-gestion of Cleary~1975, 1978!. The author of this article hasone important objection against the theory of Burridge andKeller ~1981!, stated as follows.

Although it is known from experimental observationsa!Present address: Western Atlas International, 10201 Westheimer, Houston,TX 77042.

2949 2949J. Acoust. Soc. Am. 100 (5), November 1996 0001-4966/96/100(5)/2949/11/$10.00 © 1996 Acoustical Society of America

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that acoustic waves propagating through a fluid-saturated po-rous medium are influenced by the viscosity of the consistingfluid, such a fluid should not be modeled as a linear viscous~Newtonian! fluid ~Burridge and Keller, 1981, p. 1141!.When theP-wave Green’s function in an unbounded linearviscous fluid is considered, a physical/mathematical incon-sistency is observed. On physical grounds it can be arguedthat in the inviscid limit~when frequency goes to infinity!theP-wave Green’s function of a linear viscous fluid shouldchange in that of an ideal frictionless fluid. A rigorous proofof this is impossible. This makes the linear viscous fluidequations inconsistent and can therefore not be used to de-scribe the acoustical behavior of a viscous fluid, i.e., at highfrequencies. However, there is a sound argument to circum-vent this problem.

If there is no observable experimental evidence of anytype of losses in the constituents of the fluid-saturated porousmedium~even in the fluid phase, although it has a nonzeroviscosity!, it makes no sense to incorporate viscosity typelosses in the fluid phase itself. In such a case it must beconcluded that losses observed in the fluid-saturated porousmedium are due to the interaction between the fluid and thesolid phase at the microscopic fluid/solid interface. The lossmechanism, which is triggered by the interface, will manifestitself in a thin boundary layer adjacent to the microscopicfluid/solid interface and will depend on frequency and vis-cosity. The method of spatial volume-averaging offers theopportunity to incorporate such an interface-type loss mecha-nism most straightforwardly by means of surface integralsover the microscopic fluid/solid interface. These surface in-tegrals are the direct consequence of the averaging proce-dure.

Concerning the physical reason behind the existence of aslow compressional wave in fluid-saturated porous media ev-eryone is unanimous: it exists due to the out phase motionbetween the fluid and the solid phase. Mathematically this isa rather vague statement because the question that immedi-ately arises is how to define this relative motion. If we wantto observe relative motion in a fluid/solid composite, ithighly depends on the magnifying glass~the wavelength!with which we look to the porous medium, i.e., whether adistinction can be made between the~macroscopic! fluid andsolid motion. Consequently, a consistent mathematical tech-nique should be employed to deal with this very aspect. Atthis moment two such techniques are available:the methodof spatial volume-averaging~Slaterly, 1962; Whitaker, 1969;De Vries, 1989; Prideet al., 1992! and the method of spacehomogenization~Burridge and Keller, 1981!. In the end bothtechniques share the basic idea that some kind ofvolume-averagedfield quantities are descriptive of correspondingmeasurable physical field quantities~e.g., stress and particlevelocity!. However, the way volume-averaged field quanti-ties are defined in either method~volume-averaging methodor space homogenization method! is different. At this mo-ment it is still illusive to prove which definition coincideswith physical reality. In the present article it will be arguedthat according to the method of spatial volume-averaging, itis essential to introduce a discontinuity in the tangential com-ponents of the microscopic fluid and solid particle velocity at

the microscopic fluid/solid interface in order to guaranteemacroscopic out phase motion, i.e., to guarantee the exist-ence of a slow compressional wave on the macroscopic level.According to the theory of Burridge and Keller~1981! sucha microscopic discontinuity isnot essentialto guaranteemacroscopic out phase motion, i.e., to guarantee the exist-ence of a slow compressional wave on the macroscopic level.This is due to the mathematical technique of space homo-genization.

However, criticism should be addressed to the theory ofPrideet al. ~1992!, who use the method of spatial volume-averaging to derive the macroscopic behavior of a fluid/solidcomposite consisting of a linear viscous fluid and a perfectlyelastic solid. In the present article it will be argued that thistheory is in contradiction with the physical reason behind theexistence of a slow compressional wave.

In Sec. I an overview will be presented concerning themicroscopic field equations, which are descriptive of theconstituents of the fluid-saturated porous medium. The fieldequations will be presented in a format that is generic forcontinuous media. We will distinguish between the equationof motion, the deformation rate equation, and the constitutiverelations. Whereas the equation of motion is generic for con-tinuous media, the deformation rate equations and constitu-tive relations may differ. In these aspects a distinction will bemade between fluids and solids, with or without losses. Atthe end of Sec. I the microscopic boundary conditions will bediscussed, when two different types of continuous media arein contact with each other.

In Sec. II a summary of the method of spatial volume-averaging will be presented. In Sec. III the method of spatialvolume-averaging will be applied to essentially two types offluid-saturated porous media. The first one is illustrative ofthe case where the traction and particle velocity are predomi-nantly continuous at the microscopic fluid/solid interface~e.g., a linear viscous fluid adjacent to a perfectly elasticsolid!, while the second is illustrative of the case where boththese quantities are discontinuous across the microscopicfluid/solid interface~e.g., an ideal frictionless fluid adjacentto a perfectly elastic solid!. In both cases the method ofspatial volume-averaging clearly demonstrates the impor-tance of the microscopic boundary conditions in relation tothe presence of a slow compressional wave. For the first caseit will be shown that on the macroscopic level there is fullcoupling between the fluid phase and the solid phase, i.e., theporous medium becomes an effective continuous mediumand no distinction can be made between the macroscopicfluid and solid motion~no slowP wave!. This bulk behaviorcan be described byoneequation of motion andonedefor-mation rate equation. In the second case the fluid and thesolid phase are only partially coupled. The degree of uncou-pling is determined through the predominant presence of adiscontinuity in the tangential components of the particle ve-locity and traction at the microscopic fluid/solid interface.Due to the fact that the fluid and the solid phase are onlypartially coupled, the porous medium can no longer be con-sidered an effective medium~microscopically as well asmacroscopically!, but a distinction needs to be made betweenthe fluid and the solid phase.

2950 2950J. Acoust. Soc. Am., Vol. 100, No. 5, November 1996 Tim W. Geerits: Propagation through porous media


As argued above, there are two main differences be-tween the present theory and the theory of Burridge andKeller ~1981!. First, the present theory is physically/mathematically mutual consistent and has a different physi-cal interpretation. Second, the present theory is much easierto comprehend than the theory of Burridge and Keller~1981!. The entire~possibly confusing! notion of having allquantities functionally depend on two independent lengthscales and of performing asymptotic expansions is unneces-sary when direct volume averaging proceeds so directly. Asfar as the differences in physical interpretation are con-cerned, the following remarks are in order: In classical Biottheory, but also in the theory of Burridge and Keller~1981!,the occurrence of a thin viscous boundary layer adjacent tothe microscopic fluid/solid interface is postulated and shearstresses are assumed to be present within this boundary layer.Outside the boundary layer, fluid shear stresses are assumedto be negligible and the fluid essentially is assumed to beideal frictionless~although the fluid is still modeled as alinear viscous one!. It is conjectured that the measurable vis-cous losses occur in this boundary layer. Further, it is con-jectured that losses might also occur due to the pore geom-etry. According to the method of spatial volume-averaging, itwould be inconsistent to model the fluid phase by a linearviscous fluid because this would be in contradiction with thephysical reason behind the existence of a slow compressionalwave, i.e., macroscopic out phase motion between the fluidand the solid would be impossible. Consequently, a bound-ary layer adjacent to the microscopic fluid/solid interface, inwhich shear deformation occurs, would be in contradictionwith this very fact. Therefore, in the present theory the fluidphase is modeled as an ideal frictionless fluid and losses areassumed to occur due to strong fluctuations in the spatialgradients of the microscopic fluid/solid interface. It is con-jectured that these losses are triggered by the interface andoccur in a thin boundary layer adjacent to the microscopicfluid/solid interface. Quantitatively, just as in the Biot theory,it is assumed that these losses will depend on fluid viscosityand frequency. These parameters~which quantify the losses!can be incorporated in the earlier mentioned surface~cou-pling! integrals.

I. THE GENERALIZED CONTINUUM EQUATIONS

In this section we will summarize the linearized fieldequations which are descriptive of the acoustical behavior ofcontinuous media in general. In this respect we distinguishbetween the equation of motion and the deformation rateequation. The local form of the equation of motion~Cauchy’s equation of motion! is given by

] jt i j2Fi1 f i50, ~1!

where t i j is Cauchy’s stress tensor~Pa!, f i is the volumesource density of body force~N/m3!, andF i is the mass flowdensity rate~kg/m2 s2!. When we incorporate inertial losses,the mass flow density rate in the left-hand side of Eq.~1!may be generalized as

F5r i j ] tn j1r i jn j , ~2!

wherer i j is the tensorial volume density of mass~kg/m3!, r i jis the dissipation tensor~kg/m3 s!, andn i is particle velocity~m/s!. In Eq. ~2! we have incorporated the most general kindof anisotropy and an inertial loss term which is proportionalwith the particle velocity.

The most general appearance of the local linearized de-formation rate equation is given by the relation

1/2~] jn i1] in j !2ei j2hi j50, ~3!

whereei j is the continuum deformation rate~s21! andhi j isthe volume source density of strain rate~s21!. In Eq. ~3! thecontinuum deformation rate is a function of the stress tensort i j . This functional dependence is given by the constitutiverelation. This relation highly depends on the type of con-tinuum that is considered~e.g., fluid or solid! and has thegeneral appearance

ei j5 f i j ~tpq!, ~4!

wheref i j is a tensor valued function of the stress tensortpq .A special case of Eqs.~1! and~3! occurs when we con-

sider the class of continuous media who are unable to sustainshear stresses, even when they are in motion, and are isotro-pic in their deformation behavior~e.g., an ideal frictionlessfluid!. In such a case, the off-diagonal elements of the stresstensor vanish and it is assumed that the stress in the deform-ing continuum is omnidirectional. The following mathemati-cal expression makes this phenomenon more precise

t i j5d i js5d i jtkk3, ~5!

wheres is the omnidirectional traction~Pa!, which is as-sumed to be equal to the mean of the trace of the stresstensor. Further, when we assume the medium to be isotropicin its deformation behavior~this is the case for most fluids!,the medium is unable to exhibit shear deformation. In thatcase we can take the trace of Eqs.~3! and ~4! to give

]knk2ekk2hkk50, ~6!

where

ekk5g~s!5gS tkk3 D . ~7!

The corresponding equation of motion can be found by sub-stitution of Eq.~5! into Eq. ~1!. In this way we obtain

] is2Fi1 f i50. ~8!

Next we will consider those constitutive relations which areof interest for the microscopic, acoustical description of theconstituents of a fluid-saturated porous medium, i.e., thefluid phase and the solid phase.

The perfectly elastic solid:The deformation rate equa-tion for a perfectly elastic solid is given by Eq.~3! and thecorresponding constitutive relation is given by

ei j5Si jpq ] ttpq , ~9!

whereSi jpq is solid compliance~Pa21!. In Eq. ~9! we haveassumed the most general kind of anisotropy. A medium thatexhibits such type of behavior@cf. Eq. ~9!# is called instan-taneously reacting and is lossless in its deformation behavior.



The linear viscous (Newtonian) fluid:The deformationrate equation for a linear viscous fluid is given by Eq.~3!.For all practical purposes it is sufficient to assume fluids tobe isotropic in their deformation behavior, and consequentlythe constitutive relation for an isotropic, linear viscous fluidmay be expressed as~Geerits, 1993, pp. 184!:

ei j51

2m*t i j2d i j F 1

2m*21

3k ] t@11k* k ] t#

21G tkk3,

~10!

wherem* is kinematic viscosity~Pa21 s21!, k* is bulk vis-cosity ~Pa21 s21!, andk is compressibility~Pa21!. When weconsider Eq.~3! together with Eq.~10!, it is observed that alinear viscous fluid is not instantaneously reacting and exhib-its deformation type losses.

The ideal (frictionless) fluid:A special case occurs whenwe consider an isotropic linear viscous fluid@cf. Eq. ~10!#which can sustain no shear stresses and where the fluid stressis omnidirectional. In such a case the deformation rate equa-tion is given by Eq.~6! and the constitutive equation followsfrom taking the trace of Eq.~10!. This yields

ekk5@11k* k ] t#21k ] ts, ~11!

where we have replacedtkk/3 by s and where the operator[11k* k ] t]

21 is the inverse of [11k* k ] t]. However,when we consider the fact that for almost all practical cir-cumstances it can be argued that at frequencies far below 1GHz thek* k ] t term ~which is proportional with frequency!is small compared to unity, Eq.~11! can be written as

ekk5k ] ts, ~12!

which is the constitutive relation for an ideal~frictionless!fluid. As can be observed from Eq.~6! and Eq.~12!, an ideal~frictionless! fluid is instantaneously reacting and conse-quently free of deformation losses.

So far we have discussed two classes of continuous me-dia:

~1! Class 1: Those who exhibit shear deformation@Eqs.~1!and ~3!# and can sustain shear stresses.

~2! Class 2: Those who are isotropic and can only sustainomnidirectional stresses~no shear stresses!. As a conse-quence such a medium is unable to generate shear defor-mation @Eqs.~8! and ~6!#.

When we consider the microscopic fluid/solid interfaceof a fluid-saturated porous medium, two of these classes ofcontinuous media are in contact with each other. For thesolid it is straightforward to say that it is of class 1 and forthe fluid it may be either of class 1 or class 2. Therefore, wewill give the boundary conditions at the microscopic fluid/solid interface there where class 1 is in contact with class 1and there where class 1 is in contact with class 2. The bound-ary conditions follow from the consideration that the spatialderivative perpendicular to the microscopic fluid/solid inter-face, with respect to any field quantity~traction or particlevelocity!, should exist. This means that the field quantitieswhich are operated upon by the normal component of thespatial operator should be continuous at the microscopicfluid/solid interface. For example, if there would be a discon-

tinuity in the normal components of the particle velocitywhen traversing the microscopic fluid/solid interface perpen-dicular to it, local cavities could occur. This, of course,would be in contradiction with physical reality.

Mathematically, the boundary conditions for the differ-ent configurations can now be established by decomposingthe spatial operators occurring in Eqs.~1!, ~3!, ~6!, and~8! ina part perpendicular and a part parallel to the microscopicfluid/solid interface. Applying this procedure for the pro-posed configurations and requiring the existence of the spa-tial derivative perpendicular to the microscopic fluid/solidinterface, the following boundary conditions follow~Geerits,1993, pp. 21–24!:

Class 1 adjacent to class 1:

yjt i jf 5yjt i j

s , ~13!

n if5n i

s , ~14!

where y represents the unit normal vector at an arbitraryposition at the microscopic fluid/solid interface. The super-scriptsf ands occurring in Eqs.~13! and~14! are descriptiveof the fluid and the solid phase, respectively.

Class 1 adjacent to class 2:

yisf5yjt i j

s , ~15!

yknkf5yknk

s . ~16!

Note that the boundary conditions Eqs.~13!–~16! are inde-pendent of the constitutive relations chosen for the constitu-ents of the fluid-saturated porous medium.

The boundary conditions Eqs.~13! and ~14! state thatthe traction and particle velocity are continuous at the micro-scopic fluid/solid interface. These boundary conditions alsoapply at an arbitrary interface inside a continuum that cansustain shear stresses~class 1!. Consequently, a fluid-saturated porous medium consisting of two class 1 constitu-ents effectively behaves like a class 1 constituent. Of course,this is only true if the wavelength of the transmitted signal ismuch larger than the dominant pore diameter, and if the con-stitutive and volume density of mass parameters of bothphases do not differ too much.

From the boundary conditions Eqs.~15! and ~16! it canbe deduced that there is a discontinuity in the tangentialcomponents of the traction and particle velocity at the micro-scopic fluid/solid interface. Consequently, a fluid-saturatedporous medium consisting of a class 1 and a class 2 constitu-ent can, effectively, never be replaced by either a class 1 or aclass 2 continuous medium, even if the wavelength is muchlarger than the dominant pore diameter.

From what was said above it can be concluded that thenumber of bulk waves existing in the first mentioned porousmedium configuration~1 adjacent to 1! is equal to the num-ber of bulk waves existing in a class 1 continuous medium,i.e., oneP wave and oneSwave. With respect to the secondporous medium configuration~1 adjacent to 2!, it can beargued that the number of bulk waves is restricted to a maxi-mum of three, i.e., twoP waves~one in the fluid phase andone in the solid phase! and oneS wave.

Next the method of spatial volume-averaging will besummarized in order to translate pore scale acoustical behav-



ior of the two different porous medium configurations to themacroscopic level, i.e., the scale of observation~e.g., thewavelength!.

II. THE METHOD OF SPATIAL VOLUME-AVERAGING

On a microscopic scale we distinguish two types of fieldquantities: one for the fluid phase@c f~x,t!# and one for thesolid phase@cs~x,t!#. The averages of these quantities, takenover a certain subdomain of porous material, may be definedin two ways~Fig. 1!:

^c f&~x,t !51

Vf Ex8PDf ~x!

c f~x8,t ! dV,

~17!

^cs&~x,t !51

Vs Ex8PDs~x!

cs~x8,t ! dV,

and

^c f&~x,t !51

V Ex8PDf ~x!

c f~x8,t ! dV,

~18!

^cs&~x,t !51

V Ex8PDs~x!

cs~x8,t ! dV.

Note that^•••& denotes the fact that the corresponding fieldquantity is averaged and thatV, Vf , andVs represent the totalvolume, fluid volume, and solid volume of the constitutingphases, respectively.

From Fig. 1 it follows thatDf~x!, Ds~x! represent thefluid and solid domain, respectively, which are the constitu-ents of the total porous domainD~x!. It also follows fromFig. 1 that the vectorx represents the position of the bary-center ofD~x! and thatx8 represents any arbitrary positionvector insideD~x!. From Eqs.~17! and ~18! it then followsthat the value of any averaged field quantity is assigned tothe barycenterx of D~x!. The quantitiesy f , ys represent theoutward directed unit normals, atS~x! ~the microscopicfluid/solid interface!, of the fluid and the solid phase, respec-tively.

The first type of averaging@Eq. ~17!# is calledintrinsicvolume-averaging, because the respective integrals are di-vided by the true~intrinsic! volume of the correspondingconstituting phases. The second type of averaging@Eqs.~18!#is called total volume-averaging, because the respective in-tegrals are divided by the total volume ofD~x!. Multiplying

Eqs. ~17! by the total porosity (f tf) and the complement of

total porosity (f ts), respectively, results in Eqs.~18!. Note

that the total porosity/complement of total porosity, is de-fined as

f tf ,s5

Vf ,s

V, ~19a!

and that

f tf1f t

s51. ~19b!

In order to calculate the averaged values of certain fieldquantities at different space-time coordinates, one may shiftthe representative domainD~x! through space. In this wayany intrinsic point of the fluid-saturated porous medium canbecome a barycenter, in a continuous way. The dimensionsof the representative domain should be such that the aver-aged field quantities obtained in this way vary in a continu-ous ~smooth! manner. Otherwise, the averaging procedurewould be meaningless, because it would be impossible toderive a set of partial differential equations~to describe theacoustic wave motion! in terms of volume-averaged fieldquantities. Therefore, ifDdom represents the dominant grainsize andl represents the characteristic dimension of the av-eraging domainD~x!, one must requireDdom!l for averag-ing to be useful. Additionally, as acoustic waves propagatethrough the porous material, the particle motion at any giveninstance will be directionally out of phase over length scalesof l/2, wherel is the wavelength. Therefore, for averagingto have any value in wave problems, one must requirel!l,because otherwise the average of a microscopic field quantitywould tend to zero asl→l. One therefore has the conditionDdom!l!l. From this it follows thatDdom!,l, thus con-straining the maximum allowable wave frequency and rulingout wave phenomena such as scattering from the grains.

In order to rewrite the microscopic fluid and solid equa-tions in terms of their macroscopic equivalents, one mayintegrate these equations over the fluid and solid phase ofD~x!, respectively. However, in this process, we encounterthe following question: How to express] I •••& in terms of] I^•••& and ^] t•••& in terms of] t^•••&? This question is an-swered by the averaging theorems:

^] icf&~x,t !5] i^c

f&~x,t !

11

V Ex8PS~x!

y ifc f~x8,t ! dA,

~20!

^] ics&~x,t !5] i^c

s&~x,t !11

V Ex8PS~x!

y iscs~x8,t ! dA,

and

^] tcf&~x,t !5] t^c

f&~x,t !,~21!

^] tcs&~x,t !5] t^c

s&~x,t !.

Note that Eqs.~20! have been derived by Slatterly~1962!and Whitaker~1969!. Equations~21! follow directly from thegeneralized transport theorem~Truesdell and Toupin, 1965!.In deriving the averaging theorems, we have chosen for theconcept oftotal volume-averaging@Eqs.~18!#, instead ofin-trinsic volume-averaging. Although this choice is arbitrary, it

FIG. 1. Important details in a representative elementary domain.



can be argued that the state quantities that describe theacoustic wave phenomena on a macroscopic level should inthe corresponding wave equations be treated in an equalmanner. In the next section we will apply the method ofspatial volume-averaging to the two different types of porousmedium configurations discussed in Sec. I.

III. FIELD EQUATIONS ON THE MACROSCOPICLEVEL

First we consider the porous medium configurationwhere the fluid phase as well as the solid phase is off class 1~both can sustain shear stresses!. In that case the fluid phaseand the solid phase are described by the same equation ofmotion@Eq. ~1!# and the same deformation rate equation@Eq.~3!#. Next we integrate the relevant microscopic field equa-tions@Eqs.~1! and~3!# over the fluid and solid domain of thefluid-saturated porous medium, and we divide the result bythe total volumeV of a representative porous medium do-main. Applying the spatial averaging theorems@Eqs. ~20!#,we arrive at

] j^t i jf &2^Fi

f&1^ f if&52

1

V Ex8P~x!

y jft i j

f dA, ~22!

12 ~] j^n i

f&1] i^n jf&!2êi j

f &2^hi jf &

521

V Ex8PS~x!

1

2~y j

fn if1y i

fn jf ! dA, ~23!

and

] j^t i js &2^Fi

s&1^ f is&52

1

V Ex8PS~x!

y jst i j

s dA, ~24!

12 ~] j^n i

s&1] i^n js&!2êi j

s &2^hi js &

521

V Ex8PS~x!

1

2~y j

sn is1y i

sn js! dA, ~25!

In Eqs. ~22!–~25! the superscriptsf , s denote the fluid andthe solid phase, respectively. Note that Eqs.~22!–~25! arestill analytical and that the choice of the mass flow densityrate and the constitutive relation for the fluid and the solidphase are still arbitrary. When an arbitrary interface is con-sidered through the macroscopic continuum described byEqs.~22!–~25! ~which are analytical!, it follows immediatelythat both the traction and particle velocity are continuouseverywhere in the porous medium, i.e.,y j^t i j

f &5y j^t i js & and

^n if&5^n i

s&. So, even on the macroscopic level it must beconcluded there is no out phase motion between the fluid andthe solid phase. Because macroscopic out phase motion is anessential requirement to guarantee the existence of a slowcompressional wave on the macroscopic level, it must beconcluded that the method of spatial volume-averaging cannever be used in combination with the linear viscous fluidequations in order to explain the occurrence of a slow com-pressional wave in fluid-saturated porous media. This wouldbe a contradiction in terms. Therefore, it must be concludedthat any theory which uses Eqs.~22!–~25! as a point of de-parture in order to derive the original Biot equations~Biot,1956a and/or Biot, 1956b!, is inconsistent. On these grounds

the theory of Prideet al. ~1992! is inconsistent.The surface terms occurring in Eqs.~22!–~25! are rep-

resentative of the coupling between the fluid phase and thesolid phase at the microscopic fluid/solid interface. In thisparticular case the fluid and the solid phase are fully coupled,due to the boundary conditions, Eqs.~13! and ~14!. Thephysical consequence of this follows when we add Eqs.~22!and~24! and Eqs.~23! and~25!, respectively, and employ theboundary conditions, Eqs.~13! and ~14!

] j^t i j &2^Fi&1^ f i&50, ~26!

1/2~] j^n i&1] i^n j&!2êi j &2^hi j &50, ~27!

where

^t i j &5^t i jf &1^t i j

s &, ~28!

^n i&5^n if&1^n i

s&, ~29!

^Fi&5^Fif&1^Fi

s&, ~30!

êi j &5êi jf &1êi j

s &, ~31!

^ f i&5^ f if&1^ f i

s&, ~32!

^hi j &5^hi jf &1^hi j

s &. ~33!

First it is mentioned that Eqs.~26! and ~27! are analyticaland mathematically have the same appearance as Eqs.~1!and ~3!, respectively. Consequently, if we consider an arbi-trary interface somewhere in themacroscopicporous me-dium described by Eqs.~26! and ~27!, the boundary condi-tions follow from inspection of Eqs.~13! and~14!, i.e., bothy j^t i j &, ^n i& are continuous everywhere in the macroscopiccontinuum.

The appearance of Eqs.~26! and~27! restrict the numberof propagating bulkwaves to two, i.e., oneP wave and oneSwave ~with or without losses!. Note that Eqs.~26! and ~27!occur, for example, when we assume the solid to be perfectlyelastic and the fluid linear viscous~Newtonian!.

Next we consider the porous medium configurationwhere the solid phase is of class 1 and the fluid phase is ofclass 2. In that case the microscopic solid equations are givenby Eqs.~1! and ~3!, and the microscopic fluid equations byEqs.~8! and~6!. The boundary conditions at the microscopicfluid/solid interface are given by Eqs.~15! and ~16!. Subse-quently, the microscopic fluid equations are integrated over aso-called representative elementary [email protected] D~x!#,and the result is divided by the total volumeV of the R.E.D.When the spatial averaging theorems@cf. Eq. ~20!# are ap-plied, we arrive at

] i^s&2^Fif&1^ f i

f&521

V Ex8PSx)

y ifs dA, ~34!

]k^vkf &2êkk

f &2^hf&521

V Ex8PS~x!

ykfnk

f dA, ~35!

where we have replacedhkkf by hf .

The volume-averaged solid equations are given by Eqs.~24! and ~25!:



] j^t i js &2^Fi

s&1^ f is&52

1

V Ex8PS~x!

y jst i j

s dA, ~36!

1

2~] j^n i

s&1] i^n js&!2^ei j

s &2^hi js &

521

V Ex8PS~x!

1

2~y j

sn is1y i

sn js! dA. ~37!

When an arbitrary interface is considered through the mac-roscopic continuum described by Eqs.~34!–~37! ~which areanalytical!, it follows immediately that both the traction andparticle velocity are discontinuous everywhere in the porousmedium, i.e.,y i^s&5y j^t i j

s & andyk^nkf &5yk^nk

s&. So there isa discontinuity in the tangential components of the macro-scopic fluid and solid traction and the macroscopic fluid andsolid particle velocity everywhere in the porous medium.This is essential to explain the occurrence of the slow com-pressional wave on the macroscopic level. When we wouldadd Eqs.~34! and ~36! and Eqs.~35! and ~37!, respectively,it is observed that not all coupling between the fluid and thesolid phase disappears. Due to the discontinuity in the tan-gential components of the particle velocity at the micro-scopic fluid/solid interface, the off-diagonal part of the right-hand side tensor in Eq.~37! will never vanish. Consequently,it is impossible to arrive at partial differential equations interms of$^t i j &,^n i&% as defined by Eqs.~28! and ~29! ~notethat in this caset i j &5d i j ^s&1^t i j

s &! and it is unavoidableto treat the fluid and the solid phase separately. Of course,the fluid and the solid phase are coupled by means of thecontinuity of the normal components of traction and particlevelocity.

Our ultimate goal is to derive a set of linear partial dif-ferential equations that is descriptive of the acoustic waves inthe composite at a macroscopic level. Therefore, we have tomake a fundamental assumption concerning the surface inte-grals occurring in Eqs.~34!–~37!.

When we consider the surface integrals occurring inEqs.~34! and~36!, it is observed that they are representativeof the rate of momentum transfer viaS~x! through which thefluid and the solid phase are coupled~by means of the mi-croscopic boundary conditions at the microscopic fluid/solidinterface!. Macroscopically the coupling is established by thefundamental assumptionthat these surface terms are linearlyrelated to the volume-averaged accelerations and/or particlevelocities of the fluid and the solid phase. The type of rela-tionship depends on physical information and is purely phe-nomenological. For example, ifF i

f andF is only consist of an

inertial term@first term in Eq.~2!# and no losses occur due tothe interaction between the fluid and the solid phase at themicroscopic fluid/solid interface, the surface [email protected].~34! and ~36!# are written as

1

V Ex8PS~x!

y ifs dA52

1

V Ex8PS~x!

y jst i j

s dA

5mi js f ] t^n j

f&2mi jf s ] t^n j

s&, ~38!

wheremi js f, mi j

f s are mutually induced tensorial volume den-sities of mass~kg/m3!. If, on the other hand, losses do occurdue to the interaction between the fluid and the solid phase at

the microscopic fluid/solid interface, it is assumed that therate of momentum transfer through the microscopic fluid/solid interface does not only consist of an inertial [email protected]. ~38!#, but also of a lossy part. This is exhibited in thefollowing assumption:

1

V Ex8PS~x!

y ifs dA52

1

V Ex8PS~x!

y jst i j

s dA

5mi js f ] t^n j

f&2mi jf s ] t^n j

s&2Ri js f^n j

f&

1Ri jf s^n j

s&, ~39!

where Ri js f, Ri j

f s are mutually induced dissipation tensors~kg/m3 s!. It follows directly from Eqs.~38! and ~39! thatwhen symmetry is present in theMs f, f s andRs f, f s tensors,i.e.,Ms f5M f s andRs f5Rf s ~Sec. IV!, the rate of momentumtransfer through the microscopic fluid/solid interface is pro-portional with the volume-averaged relative fluid accelera-tion and/or the volume-averaged relative fluid particle veloc-ity. The physical implication of this is that there is nocoupling between the fluid and the solid phase when there isno relative motion between the fluid and the solid~on themacroscopic level!, and consequently the losses also disap-pear. This physical information is also present in all Biot-liketheories~Burridge and Keller, 1981!.

It should be noted that if we know from ultrasonic ex-periments that there are no observable losses in the constitu-ents of a fluid-saturated porous medium while we observelosses in the porous medium itself~Geerits, 1993!, it isstraightforward to conclude that the observed losses must bedue to an interaction mechanism between the fluid and thesolid phase at the microscopic fluid/solid interface. This isexhibited in the linearization of the coupling integrals in Eq.~39!. Physically it is assumed that inertial losses at the mi-croscopic fluid/solid interface occur due to the strongly vari-able pore geometry of the microscopic fluid/solid interface~losses due to tortuosity!. This loss mechanism, of course,may also depend on fluid viscosity and frequency. Quantita-tively, these parameters can be incorporated in the dissipa-tion tensorRs f, f s. In this respect the results of the Johnsonet al. ~1987! model can be used.

In this context it should be stressed that on the micro-scopic level the fluid and the solid phase are assumed to befree of inertial losses. Therefore, after spatial volume-averaging$F i

f ,F is%, we have the additional relations

^Fif&5r i j

f ] t^n jf&, ~40!

^Fis&5r i j

s ] t^n js&, ~41!

wherer i jf , r i j

s are tensorial volume densities of mass of fluidand solid phase, respectively~kg/m3!.

Next we focus on the surface integrals Eqs.~35! and~37!, which are representative of the total time rate of defor-mation transfer viaS~x!, through which the fluid and thesolid phase are coupled. The surface term occurring in Eq.~35! is representative of the time rate of omnidirectional po-rosity change, whereas the surface term occurring in Eq.~37!is not only representative of the time rate of omnidirectionalporosity change (I5 j ) but also the time rate of deviatoricporosity change (IÞ j ). Due to the boundary condition Eq.



~16!, the surface term occurring in Eq.~35! should be equalbut opposite in sign to the trace of the surface integral oc-curring in Eq.~37!. Just as for the inertial interaction termswe have to make a fundamental assumption for the deforma-tion interaction terms. In general it is plausible to assumethese surface integrals to be linearly related to the time rateof the volume-averaged stresses, and the volume-averagedstresses of the phases involved. This kind of relationshipdepends on the constitutive relations that hold for the fluidand the solid phase on the microscopic level. Just as for theinertial terms, it is possible to take deformation losses intoaccount, for example, if we incorporate the bulk viscositylosses of the fluid phase@cf. Eq. ~11!#. However, in contrastto the inertial terms, it is not plausible to assume a deforma-tion loss mechanism purely due to the presence of the micro-scopic fluid/solid interface. Because we do not observe de-formation losses in the constituents of the fluid-saturatedporous medium, it is straightforward to assume the fluid aswell as the solid to be instantaneously reacting. This meansthat Eqs.~12! and~9!, respectively, are the representatives ofthe constitutive relations of the fluid and the solid phase.

In this context the surface integrals occurring in Eqs.~35! and ~37! may be linearized as

1

V Ex8PS~x!

ykfnk

f dA521

V Ex8PSx

yksnk

s dA

5ks f ] t^s&2k f s ] t^tkk

s &3

, ~42!

and

1

V Ex8PS~x!

1

2~y j

sn js1y i

sn js! dA

5Ki jpq ] t^tpqs &2ks f

d i j3

] t^s&, ~43!

where

Kiipq5k f sdpq

3, ~44!

and Ki jpq to the mutually induced tensorial compliance~Pa21!.

From Eqs.~43! and ~44! it follows that the trace of Eq.~43! equals minus the right-hand side of Eq.~42!. Note thatthe off-diagonal part of Eq.~43! is only coupled with thesolid phase and not with the fluid phase. This is due to thefact that the considered fluid phase cannot sustain any shearstresses, even when it is in motion.

In order to arrive at the partial differential equationswhich are descriptive of the acoustic behavior of a fluid-saturated porous medium, Eqs.~39!–~43! are substituted inEqs.~34!–~37!. As a consequence we obtain

] i^s&2mi jf f ] t^n j

f&2mi jf s ] t^n j

s&2Ri js f^n j

f&1Ri jf s^n j

s&

52^ f if&, ~45!

]k^nkf &2k f f ] t^s&2k f s ] t

^tkks &3

5^hf&, ~46!

and

] j^t i js &2mi j

ss ] t^n js&2mi j

s f ] t^n jf&1Ri j

s f^n jf&2Ri j

f s^n js&

52^ f is&, ~47!

12 ~] j^n i

s&1] i^n js&!2k i jpq

ss ] t^tpqs &2ks f~d i j /3! ] t^s&5^hi j

s &.~48!

In order to obtain Eqs.~46! and ~48!, respectively, thevolume-averaged equivalents of Eqs.~12! and~9! were sub-stituted in Eqs.~35! and ~37!, respectively. The volume-averaged equivalents of Eqs.~12! and~9! can be obtained byintegrating them over an elementary representative domainD~x!, dividing the result by the total volumeV of D~x!, andemploying the averaging theorems@cf. Eqs. ~21!#. In thisway we obtain

^ekkf &5k f ] t^s&, ~49!

and

^ei js &5Si jpq ] t^tpq

s &. ~50!

Concerning Eqs.~45!–~48! we have the following additionalrelations:

mi jf f5r i j

f 2mi js f ~51!

is the self-induced tensorial fluid volume density of mass~kg/m3!;

k f f5k f2ks f ~52!

is the self-induced scalar fluid compressibility~Pa21!;

mi jss5r i j

s 2mi jf s ~53!

is the self-induced tensorial solid volume density of mass~kg/m3!; and

k i jpqss 5Si jpq2Ki jpq ~54!

is the self-induced tensorial solid compliance~Pa21!.Equations~45!–~48! are generic for fluid-saturated po-

rous media which have an inhomogeneous, anisotropic con-tinuous pore structure on the macroscopic level. Losses ofthe interface type~intertial losses at the interface! are incor-porated and are assumed to be due to strong fluctuations inthe spatial geometry of the microscopic fluid/solid interface.A special case of Eqs.~45!–~48! occurs when we assumethere are no losses. The resulting equations were first derivedby De Vries~1989!. Mathematically, Eqs.~45!–~48! have thesame appearance as the low-frequency range Biot equations~Biot, 1956a!; however, with a different physical interpreta-tion. In absence of the dissipation terms~De Vries, 1989!,Eqs. ~45!–~48! have the same appearance as the high-frequency range Biot equations~Biot, 1956b! but with a dif-ferent physical interpretation.

To create order in the large number of phenomenologi-cal coefficients occurring in Eqs.~45!–~48!, and to get in-sight in the acoustical energy distribution, the local form ofthe acoustic energy balance will be derived in the next sec-tion.



IV. LOCAL FORM OF THE ENERGY BALANCE ANDSOME OF ITS CONSEQUENCES

In this section we investigate the exchange of acousticenergy between a certain portion of porous medium and itssurroundings. Let the portion of the medium under consider-ation be located in some bounded domainD interior to thepiecewise smooth, closed surface]D, and let the unit vectoralong the normal to]D and pointing away fromD, be de-noted byy i . A subdomainDSRC of D is occupied by theelastodynamic sources. Furthermore, the complement ofDø]D in the total domain occupied by the porous mediumis denoted byD8 ~see Fig. 2!.

Multiplying Eqs. ~45!–~48! by the macroscopic fieldquantities2^n i

f&, 2^s&, 2^n is&, and 2^t i j

s &, respectively,and adding the resulting equations, we obtain

ESRC5EKIN1EDEF1EDIS1]kSk , ~55!

in which

ESRC5^ f kf &^nk

f &2^hf&^s&1^ f ks&^nk

s&2^hi js &^t i j

s & ~56!

denotes the volume density of time rate at which the sourcesdeliver mechanical work to the acoustic disturbance~J/m3 s!;

EKIN5mkrf f ^nk

f & ] t^n rf&1mkr

fs^nkf & ] t^n r

s&

1mkrs f^nk

s& ] t^n rf&1mkr

ss^nks& ] t^n r

s& ~57!

represents the volume density of time rate of kinetic energy~J/m3 s!;

EDEF5k f f^s& ] t^s&1k f s^s& ] t^tkk

s &3

1ks f^tkk

s &3

] t^s&1k i jpqss ^t i j

s & ] t^tpqs & ~58!

is the volume density of time rate of deformation~potential!energy~J/m3 s!;

EDIS5Ri js f^n j

f&^n if&2Ri j

f s^n js&^n i

f&2Ri js f^n j

f&^n is&

1Ri jf s^n j

s&^n is& ~59!

is the time rate of volume density of dissipated energy~J/m3 s!; and

Sk52^nkf &^s&2^n j

s&^tk js & ~60!

denotes the area density of acoustic power flow@acousticPoynting vector~W/m2!#. Equation~55! is the local form ofthe acoustic power balance in a porous medium.

Next we investigate the conditions under which the vol-ume density of the time rate of kinetic energy@Eq. ~57!# canbe written as the time derivative of the kinetic energy den-sity, where the latter is a state quantity, i.e., a quantity thatonly depends on the instantaneous values of the averagedparticle velocities of the fluid and the solid. The relevantconditions are

mkrf f5mrk

f f , ~61!

mkrfs5mrk

s f , ~62!

mkrss5mrk

ss . ~63!

On the microscopic level the same argument holds, where ithas the consequence that the volume density of mass tensors%krf and%kr

s are symmetric. With this and the symmetry ofmkr

f f andmkrss, it follows from Eqs.~51! and~53! thatmkr

fs andmkrs f are symmetrical tensors, and therefore Eq.~62! can be

extended to

mkrfs5mrk

fs5mkrs f5mrk

s f . ~64!

Similarly, we investigate the conditions under which the vol-ume density of the time rate of deformation energy@Eq.~58!#can be written as the time derivative of the deformation en-ergy density, where the latter is also a state quantity, i.e., aquantity that only depends on the instantaneous values of theaveraged fluid traction and the averaged solid stress. Therelevant conditions are

k f s5ks f, ~65!

k i jpqss 5kpqi j

ss . ~66!

On the microscopic level the same argument holds, where ithas the consequence that the compliance tensorSi jpq is sym-metric. With this and the symmetry ofk i jpq

ss , it follows fromEq. ~54! that the mutually induced complianceKi jpq is asymmetrical tensor, i.e.,

Ki jpq5Kpqi j . ~67!

According to the Onsager–Casimir reciprocal relations~On-sager, 1931! the following symmetry relations hold for thedissipation tensorsRi j

s f, Ri jf s, occurring in the dissipation

function @cf. Eq. ~59!#:

Ri js f5Ri j

f s . ~68!

With the aid of Eq.~68!, Eq. ~59! can be written as a qua-dratic form, i.e.,

EDIS5^wT&R^w&, ~69!

where

^w&5^vf&2^vs&, ~70!

and

R5Rs f5Rf s. ~71!

Because on physical grounds it should be required thatEDIS>0 ~the dissipation function is said to be definite posi-

FIG. 2. Relevant domains with respect to energy distribution.



tive!, certain restrictions should be imposed onR. One ofthese restrictions is thatR is symmetric, i.e.,

Ri j5Rji5Ri js f5Rji

s f5Ri js f5Rji

f s . ~72!

Equations~61!–~68! and~72! are the symmetry relations per-taining to a so-called reciprocal, or self-adjoint, porous me-dium. With the aid of Eqs.~61!–~67!, Eqs.~57! and~58! canbe rewritten as

EKIN5] tEKIN , ~73!

EDEF5] tEDEF, ~74!

in which

EKIN5 12mkr

f f ^nkf &^n r

f&1 12mkr

sss^nks&^n r

s&1mkrfs^nk

f &^n rs&,

~75!

EDEF512K

f f^s&^s&1 12k i jpq

ss ^t i js &^tpq

s &1k f s^s&^tkk

s &3

,

~76!

whereEKIN and EDEF denote the volume densities of thekinetic and the deformation energy, respectively~J/m3!. Fur-ther, it is assumed that the kinetic and deformation volumedensities of energy are non-negative definite quadratic forms,i.e., the right-hand side of Eqs.~75! and~76! is non-negativefor any choice of n r

f&, ^n rs&, ^s&, and ^tpq

s &. This imposescertain constraints on the phenomenological coefficients oc-curring in Eqs.~75! and ~76! ~Geerits, 1993, p. 60!.

A relatively simple situation occurs when we consider ahomogeneous, isotropic, fluid-saturated porous medium,where the tensor valued coefficients have changed into sca-lars. In the particular case, where the porous medium is as-sumed to be lossless (R50) we are left with three phenom-enological coefficients, which can be determined, indirectly,by means of the porous media wave speeds~Geerits, 1993!.This method will be explained and applied in another article.

V. DISCUSSION

There are two main reasons for writing this article. First,the author’s disagree with using the linear viscous fluid equa-tions in order to model the fluid phase of a fluid-saturatedporous medium. When theP-wave Green’s function in anunbounded linear viscous fluid is considered a physical/mathematical inconsistency is observed. On physicalgrounds it can be argued that in the inviscid limit~whenfrequency goes to infinity! theP-wave Green’s function of alinear viscous fluid should change in that of an ideal friction-less fluid. A rigorous proof of this is impossible. This makesthe linear viscous fluid equations inconsistent and can there-fore not be used to describe a viscous fluid~at high frequen-cies!. However, there is a sound argument to circumvent theproblem of using the inconsistent linear viscous fluid equa-tions in order to model the viscous fluid that constitutes partof the porous medium. If there is no observable experimentalevidence of any type of losses in the constituents of the fluid-saturated porous medium~even in the fluid phase although ithas a nonzero viscosity!, it makes no sense to incorporateviscosity type losses in the fluid phase itself. In such a case itmust be concluded that losses observed in the fluid-saturated

porous medium are due to the interaction between the fluidand the solid phase at the microscopic fluid/solid interface.The loss mechanism, which is triggered by the interface, willmanifest itself in a thin boundary layer adjacent to the mi-croscopic fluid/solid interface and may depend on frequencyand viscosity. The method of spatial volume-averaging of-fers the opportunity to incorporate such an interface-typeloss mechanism most straightforwardly by means of surfaceintegrals over the microscopic fluid/solid interface@cf. Eq.~39!#. These surface integrals are the direct consequence ofthe averaging procedure. The second, and maybe most im-portant argument to write this article, is due to the new math-ematical insight concerning the reason behind the existenceof a slow compressional wave. If one accepts the method ofspatial volume-averaging as outlined in Sec. II to be an ad-equate mathematical technique in order to translate theacoustical behavior of a fluid-saturated porous medium fromthe microscopic level to the macroscopic level, it follows~Sec. III! that a discontinuity in the tangential components ofthe microscopic traction and particle velocity at the micro-scopic fluid/solid interface is essential to guarantee macro-scopic out phase motion. Since macroscopic out phase mo-tion is the physical reason for the existence of a slowcompressional wave, it must be concluded that the linearviscous fluid equations can never be used in combinationwith the method of spatial volume-averaging in order to ex-plain the existence of a slow compressional wave in fluid-saturated porous media~on the macroscopic level!. Theabove statement, which is proven in Sec. III, forms the sec-ond argument for using the ideal fluid equations in order tomodel the fluid phase of the fluid/solid composite, becausethey guarantee a discontinuity in the tangential componentsof the microscopic traction and particle velocity at the mi-croscopic fluid/solid interface~i.e., if the solid phase cansustain shear stresses!. The above argument has physicalconsequences when a momentum transfer loss mechanismadjacent to the microscopic fluid/solid interface should beincorporated, because it requires a discontinuity in the tan-gential components of the microscopic fluid and solidtraction/particle velocity at the microscopic fluid/solid inter-face. This very fact implies that, in contrast to Biot-like theo-ries, losses cannot occur as a consequence of fluid sheardeformation being a dominant factor in a thin boundary layeradjacent to the microscopic fluid/solid interface. In such acase, the microscopic fluid and solid traction and particlevelocity would be continuous across the microscopic fluid/solid interface, which is in contradiction with the physicalrequirement for the existence of a slow compressional wave,i.e., macroscopic out phase motion. Therefore, the presenttheory assumes inertial losses to occur due to the stronglychanging spatial gradients of the microscopic fluid/solid in-terface. Thus, it is conjectured that the loss mechanism istriggered by the microscopic fluid/solid interface and willtake place in a boundary layer adjacent to the microscopicfluid/solid interface. Although the exact physical process oc-curring in the boundary layer is still unknown, it is certainthat fluid shear deformation can only be of negligible impor-



tance. However, the possibility still remains that the lossesdo depend on fluid viscosity~although shear deformation isof negligible importance!.

The most fundamental assumption in the presentedtheory is that the surface integrals, which are representativeof the coupling between the fluid and the solid phase, lin-early depend on the volume-averaged state quantities of thephases involved. Unfortunately, the linearization of the sur-face integrals is purely phenomenological, but it is moststraightforward and simple. Assumptions of this kind arealso made in all previously mentioned theories, but on an-other level.

The occurrence of these coupling integrals offers anideal opportunity to incorporate any type of loss mechanism~inertial losses or deformation losses! that is triggered by themicroscopic fluid/solid interface. Of course, bulk losses oc-curring in the individual components of the porous mediumcould also have been incorporated but they are assumed to benegligible.

However, with respect to other theories, there is also adisadvantage, i.e., at this moment it is still illusive to expressthe phenomenological coefficients occurring in the theory@Eqs. ~45!–~48!# in terms of microscopic properties of theindividual constituents~e.g., mass densities and constitutiveparameters! and microstructural properties of the porous me-dium ~porosity, tortuosity, permeability, etc.!. However, it isconjectured that this can be achieved by a combination ofstatistical physical methods and laboratory experiments. Forthe time being, the relations derived by Biot~1956a, 1956b!,Burridge and Keller~1981!, and Johnsonet al. ~1987! can beused.

VI. CONCLUSIONS

A physical/mathematical framework was developedwhich is consistent with the reason of existence of a slowcompressional wave. On a phenomenological level theframework is flexible to take any kind of interfacial lossmechanism into account, although the present treatment onlyincorporates the physically most plausible loss mechanism,i.e., inertial losses that might occur due to strong changes inpore geometry~although no discontinuities may occur!. Thetheory is assumed to be applicable at relatively high frequen-cies, when there is predominant slip at the microscopic fluid/solid interface~i.e., the frequency range in which the slowP-wave is observed!. At lower frequencies the situation willoccur where there is a predominant ‘‘no slip’’ condition atthe microscopic fluid/solid interface. In that case the visco-elastic solid theory@cf. Eqs.~26! and~27!# is assumed to bevalid: on the macroscopic level the porous medium has be-come a single phase~effective! medium. This is in agree-ment with the theory of Burridge and Keller~1981!.

At this moment it is still illusive to express the phenom-

enological coefficients occurring in the theory@Eqs. ~45!–~48!# in terms of microscopic properties of the individualconstituents~e.g., mass densities and constitutive param-eters! and microstructural properties of the porous medium~porosity, tortuosity, permeability, etc.!. However, it is con-jectured that this can be achieved by a combination of statis-tical physical methods and laboratory experiments. For thetime being the relations derived by Biot~1956a, 1956b!,Burridge and Keller~1981!, and Johnsonet al. ~1987! can beused.

ACKNOWLEDGMENTS

The author is very grateful to Prof. Dr. A. T. de Hoop,Dr. S. M. de Vries, and Dr. D. M. J. Smeulders for the manyfruitful discussions concerning the work reported in this ar-ticle. Further, the author is indebted to the European Com-munity for sponsoring this research.

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acoustic wave propagation through porous media revisited

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