Propagation of transient acoustic waves in porous media

(1)

ACOUSTIC WAVES IN

POROUS MEDIA

Sytze M. de Vries

TR diss

JL/«JIJ

(2)

PROPAGATION OF TRANSIENT

ACOUSTIC WAVES IN

POROUS MEDIA

P R O E F S C H R I F T

ter verkrijging van de graad van doctor aan

de Technische Universiteit Delft, op gezag van

de Rector Magnificus, prof. drs. P.A. Schenck,

in het openbaar te verdedigen ten overstaan van

een commissie aangewezen door het College van Dekanen

op dinsdag 30 mei 1989 te 16.00 uur

door

Sytze Marten de Vries

geboren te IJlst

elektrotechnisch ingenieur

TR diss

1 7 3 0

(3)

(4)

(5)

hospitality to carry out research, from July till November 1987. in the Geoacoustics Department.

(6)

Chapter 1 I N T R O D U C T I O N

Summary

Acoustic waves are standard diagnostic tools in determining the mechanical pa rameters (volume density of mass, compressibility, elastic stiffness) of fluids and solids. Ideal fluids and isotropic solids carry compressional waves, while, in ad dition, isotropic solids sustain shear waves. So the detection of a shear wave is indicative that the medium through which the waves propagate is a solid. As soon as the solid becomes porous and the pores are completely filled with an ideal fluid, an extra (slow) compressional wave occurs in case the pore sizes are small compared to the wavelength. So the existence of a slow compressional wave is indicative t h a t the solid is a porous one. A computational model study can possibly reveal, whether or not, and under which circumstances, such a slow compressional wave wave can be detected in a seismogram. The next question is how t h e properties of the slow compressional wave correspond quantitatively to the volume fraction and other geometrical properties of the pores. For this, insight is needed in the relationship between microscopic geometrical and macro scopic acoustic properties of porous solids, rock, for example. With this in mind, we investigate the scope of a macroscopic linear acoustic theory for impulsive wave propagation in a porous medium.

(11)

1.1 Introduction

In the search for natural gas and oil several branches of exploration geophysics (surface seismics, borehole seismics) make use of acoustic waves. In this type of application, rock with fluid-filled pores is a wave propagating medium of particular interest. For pore sizes much less than the acoustic wavelengths. the acoustic wave phenomena in this medium can be described in terms of certain macroscopic quantities that somehow result from an appropriate spatial averaging procedure applied to t h e acoustic equations at t h e microscopic (i.e.. poresize) level.

In t h e present monograph we investigate, starting from the fundamentals. the scope of a linear acoustic theory for impulsive wave propagation in a porous medium with the aid of a spatial volume averaging technique. This technique is simple, and offers great clarity in explaining what kind of averages occur in the macroscopic field quantities and what the different coupling terms in the acoustic wave equations of a two-phase (fluid and solid) model represent. The method also explicitates the kind of average that in a consistent manner occurs in these macroscopic acoustic wave equations. T h e method circumvents t h e construction of specific macroscopic energy functions (Biot 1956a. 1956b. 1962. Thigpen and Berryman 1985, K a t s u b e and Carroll 1987). t h e use of variational techniques (Berryman, Thigpen and Chin 1988, Berryman and Thigpen 1985). Lagrangian formulations (Johnson 1986), and the two-space method of homoge-nization (Levy 1979. Auriault 1980. Burridge and Keller 1981). which eventually are complicated intermediate steps in arriving in t h e end at linearizi J partial differential equations for describing wave motions of the same general kind as ours.

In our analysis we follow t h e kind of reasoning (spatial volume averaging technique) t h a t is used in the derivation of the equations that govern the steady flow of an incompressible, viscous fluid that permeates the connected pores in a perfectly rigid, immovable solid material (where it leads t o Darcy's law: see. e.g.. Hassanizadeh and Gray 1979a, 1979b, 1980a. 1980b. and van der Weiden 1988). In this field of application viscosity is important. Also de la Cruz and Spanos (1983, 1985) use the technique of volume averaging in describing low-frequency

(12)

1.1 INTRODUCTION 3

seismic wave phenomena in fluid-filled porous media. By making the assumption t h a t on the pore scale the inertia effects are dominated by the viscous effects, they end up with coupling in the viscosity terms of the equations of motion only. Walton and Digby (1987) combine the volume averaging technique with a low-frequency expansion technique for time-harmonic motion in a porous medium without viscosity effects. Our aim is, however, to do high-frequency seismics, i.e., the analysis of the wave shape of acoustic pulses generated by impulsive sources. Then, the fluid motion is considered at a high Reynolds number, i.e., the effects of viscosity are neglected with respect to the inertia effects.

The basic idea behind the method is the following. We assume t h a t on the scale of the geometry of the pores the continuum equations for elastic solids and compressible fluids hold. These "microscopic" continuum equations are next spatially averaged over a representative elementary domain of the fluid-solid composite. The diameter of this representative elementary domain should be both sufficiently small and sufficiently large, such t h a t both the macroscopic and the microscopic inhomogeneities do not affect the results of the averaging procedure.

At the microscopic level, the continuum equations under consideration have the shape of partial differential equations. Now, for the macroscopic partial dif ferential equations we need partial derivatives of spatially averaged field quanti ties (acoustic traction and particle velocity in the fluid phase, stress and particle velocity in the solid phase). This poses the question of how the partial deriva tives of a spatially averaged field quantity are related to the spatial average of the partial derivatives of this field quantity. The difference between the two is shown to be a surface interaction integral over the microscopic interface. In our theory the only fundamental assumption is that this interaction term is linearly related to the volume averaged acoustic state quantities of the phases involved. Such a general linear interaction is the most general one that is compatible with our ultimate goal: a system of linear partial differential equations t h a t is de scriptive of the acoustic waves in the composite at a macroscopic level. To arrive at such a system, the interaction terms of the indicated kind (that contain the fundamental coupling coefficients) can be most straightforwardly be postulated right away. Either explicitly or implicitly, an assumption of this kind is used in

(13)

all previous t r e a t m e n t s , e.g.. through the existence of a quadratic energy func tion in the averaged quantities or in the method of convex linear interpolation (two-space method of homogenization). but at a more complicated intermediate level.

The advantage of the present procedure is that it directly leads to explicit ex pressions for the coupling coefficients in terms of the microscopic field quantities along the interface between the two phases, which expressions could be used to draw further conclusions about their magnitude if further techniques of statisti cal physics were employed. Further, the concept of introducing coupling terms in the pertaining constitutive equations is illustrative for the physics behind all sorts of coupled phenomena, such as in electricity and magnetism the concept of self and mutual inductance of electric circuits, in mechanics the coupled mass-spring systems, and in molecular spectroscopy the spectral behavior of coupled atoms; the aspect seems, however, to be new in the acoustics of porous media. In a spatial averaging theory two kinds of averages show up. viz. averages over the single phases (denoted as ''intrinsic volume averages") and averages over a time-and shift invariant representative elementary spatial domain (denoted as "volume averages" as such). The two kinds of averages are interconnected by a factor t h a t is the volume fraction of the pertaining phase. Which of the two averages (or combination of them) is to be interpreted as a measurable quantity at a macroscopic level, is still a m a t t e r of discussion. It can. however. be argued that the state quantities that describe the acoustic wave phenomena at a macroscopic level should in the corresponding wave equations be treated in an equal manner. In this respect, the Biot theory shows an inconsistency. Biot defined the macroscopic particle displacements (and velocities) to be "intrinsic volume averaged" ones and the macroscopic acoustic pressure and stresses to be "volume averaged" ones. A criterion in selecting one of the two types of averages is the following: the space and time derivatives of acoustic wave field quantities should in the simplest possible way be transformed from the microscopic to the macroscopic level. The "volume average" satisfies this requirement, whereas the "intrinsic volume average" does not. Therefore, the "volume averaging" procedure is to be preferred in developing a consistent macroscopic theory of poro-elastic waves, and is taken as the point of departure in the present analysis.

(14)

1.1 INTRODUCTION 5

Once our final first-order partial differential equations (equations of mo tion, deformation equations, and constitutive equations) have been obtained, energy theorems are derived. Because the fundamental macroscopic acoustic field quantities are all consistently defined in terms of "volume averages", the kinetic energy density, the deformation (potential) energy density, and the acous tic Poynting vector, too, are expressed in terms of them. In this connection it is remarked that the energy theorems in the Biot theory (from which his first-order equations are deduced) are mutually inconsistent due to the inconsistent use of "intrinsic volume averages" and "volume averages".

The present analysis also deals with macroscopic sources and macroscopic boundary conditions at the interfaces. By introducing external sources a t the microscopic level we obtain via the "volume averaging" procedure macroscopic sources that can be taken as representatives for the action of monopole and dipole transducers. Such transducers are modeled through equivalent volume source densities of body force and strain rate for each of the two constituents. New boundary conditions are derived for configurations where two porous me dia with different acoustic properties are in contact with each other, as well as for the special cases where one of the two porous media degenerates into a pure fluid, a pure solid, or a vacuum. They follow from the consideration t h a t due to the non-zero volume of the representative elementary domain all volume averaged quantities change, in the macroscopic space, continuously with position. Across interfaces where they change rapidly due to rapid changes (or even jumps) in the microscopic configurational quantities, they can m a t h e m a t ically be modeled as j u m p discontinuities. Such a discontinuity may, however, nowhere lead to unbounded field values; in this manner, consistency with t h e differential equations with rapidly varying coefficients leads to our boundary conditions that have not been used in earlier treatments. In addition, our con ditions automatically guarantee the continuity of the normal component of the acoustic Poynting vector upon crossing an interface which is a prerequisite for a consistent macroscopic theory of energy transfer. The final system of differ ential equations with source terms and boundary conditions serves as the basis for the calculation of the acoustic wave motion generated by impulsive sources in certain model configurations (de Vries 1988, de Vries and de Hoop 1988).

(15)

There is a vast amount of literature on the subject: some authors take viscos ity into account, others do not. Biot (1956a. 1956b) developed a theory where the interaction between the solid matrix material and the fluid, assuming t h a t the latter completely fills the pores, is accounted for by certain coupling terms in the partial differential equations that govern the acoustic wave motion in the solid and fluid constituents. These coupling terms follow from the argument t h a t , in the lossless case, non-negative definite potential and kinetic energy den sities for the fluid/solid composite should exist. Later. Biot (1962] extended his theory, t h a t originally was set up for uniform porosity and isotropic composites. to the non-uniform porosity case, and further included anisotropy in the stiffness and the mutually-induced mass of the porous material. (The present analysis includes also anisotropy in the mass terms of the fluid and solid phases of the porous material as well.) Independently of Biot. de Josselin de Jong (1956) de rived coupled wave equations that describe acoustic wave propagation through sand saturated with water and predicts two compressional waves and a shear wave. Thigpen and Berryman (1985) present a continuum theory for (nonlinear) acoustic wave motion in a three-component porous medium with non-uniform volume fractions, after having postulated specific energy and interaction func tions. Katsube and Carroll (1987) extended the Biot theory by using a so called "mixture theory". Within the realm of this theory they allowed a shear interac tion between the viscous fluid and the solid at the microscopic level. Berryman. Thigpen and Chin (1988), and Berryman and Thigpen (1985) derived acoustic wave equations for partially saturated porous media by using variational tech niques. Berryman and Thigpen (1985) also present boundary co: "lions for partially saturated porous media. Johnson (1984) rederived Biot's equations in a phenomenological way by starting from a general continuum Lagrangian for mulation that could apply to a homogeneous, isotropic. two-component system with phenomena of a wavelike nature. He used the same type of spatial aver ages as Biot did. Levy (1979). Auriault (1980). and Burridge and Keller (1981) arrive, for time-harmonic motion, at coupled acoustic equations through the in termediate of the two-space method of homogenization t h a t is used in statistical physics. In this connection. Burridge and Keller made the following statement: *.... the actual physical values of the field quantities are obtained by restricting

(16)

1.2 OUTLINE OF THE DIFFERENT CHAPTERS 7

these functions to the physical diagonal..." which implies that their method

amounts to a convex linear interpolation between microscopic and macroscopic level, which is in fact the same principle as on which our equations are based.

Quite a different approach t h a t is used in statistical physics is ensemble averaging. There, first, the acoustic wave motion is solved in a representative configuration; next, appropriate weighted averages of the corresponding acoustic state quantities yield the macroscopic behavior of the configuration of the type investigated. In this category Schoenberg (1984) investigated the propagation of time-harmonic plane acoustic waves through a periodically stratified medium composed of alternating elastic solid and ideal fluid layers; in Schoenberg and Sen (1986) the relevant fluid is taken to be viscous.

1.2 Outline of t h e different chapters

In Chapter 2, basic relations for transient acoustic waves in a porous medium are derived. T h e derivation uses a volume averaging procedure t h a t is applied to the two constituents a perfectly elastic solid and an ideal fluid. Arbitrary anisotropy, including the pore-geometry induced one, is taken into account. Losses, includ ing the one due to viscosity, are neglected. The presence of sources is accounted for by the introduction of volume source densities of body force and of strain rate, for each of the constituents. The resulting equations have the appearance of the well-known Biot equations, but their interpretation is different, especially as far as the interaction between the two phases is concerned. At macroscopic interfaces where the constitutive coefficients have a j u m p discontinuity, bound ary conditions are derived. The corresponding energy relations are discussed and the time Laplace-transform domain convolution-type reciprocity theorem for acoustic wave fields in porous media is derived.

In Chapter 3, the acoustic radiation from sources in an unbounded, homoge neous, isotropic porous medium is investigated. Space-time domain expressions are presented for the quantities of the acoustic wave field that is causally re lated to the action of known sources of bounded extent in an unbounded porous medium t h a t is linear, homogeneous, isotropic, and lossless in its acoustic be havior. With the aid of a Laplace transformation with respect to time and a

(17)

spatial Fourier transformation, source-type integral representations for the vol ume averaged fluid traction and the volume averaged solid particle velocity of the acoustic wave field in a porous medium are obtained. Appropriate scalar. vector, and tensor potentials are introduced, and far-field integral representa tions are presented that show the different radiation characteristics for the fast and slow compressional-wave parts and the shear-wave part.

In Chapter 4, we calculate space-time domain expressions for the acoustic wave field radiated from an impulsive point source in a fluid porous-medium configuration with a plane boundary with the aid of the Cagniard-de Hoop technique. The source is located in the fluid part of the configuration. Numerical results are presented for the reflected-wave acoustic pressure due to a monopole source, especially in those regions of space where head waves occur. There is a marked difference in time response in the different regimes that exist for the wave speed in the fluid in relation to the different wave speeds in the porous medium. These differences are of importance to the situation where the reflected wave in the fluid is used to determine experimentally the acoustic properties of the porous medium.

(18)

C h a p t e r 2

B A S I C R E L A T I O N S F O R

T R A N S I E N T A C O U S T I C

WAVES I N A P O R O U S

M E D I U M

Summary

Fundamental differential equations are derived t h a t govern the macroscopic propagation of linear acoustic waves through a fluid-saturated porous solid. The derivation uses a volume averaging procedure t h a t is applied to the two con stituents, viz. a perfectly elastic solid and an ideal fluid. Arbitrary anisotropy, including the pore-geometry induced one, is taken into account. Losses, includ ing the one due to viscosity, are neglected. T h e presence of sources is accounted for by the introduction of volume source densities of body force and of strain rate, for each of the constituents. T h e resulting equations have the appearance of the well-known Biot equations, but their interpretation is different, especially as far as the interaction between the two phases is concerned. At macroscopic interfaces where the constitutive coefficients have a j u m p discontinuity, bound ary conditions are derived. The corresponding energy relations are discussed and the time Laplace-transform domain convolution-type reciprocity theorem for acoustic wave fields in porous media is derived.

(19)

2.1 I n t r o d u c t i o n

In the present chapter we develop, starting from the fundamentals, a linear acoustic theory for impulsive wave propagation in a fluid-filled porous solid on a "macroscopic" scale with the aid of a spatial volume averaging technique ap plied to the "microscopic" structure. This technique not only alters the scale of observation from the "microscopic" to the "macroscopic'" one. but also offers great clarity in explaining what kind of spatial volume averages shows up in the macroscopic field quantities and what the different coupling terms in the acoustic wave equations of a two-phase (fluid and solid) model stand for. The first-order partial differential equations (equations of motion, deformation equa tions. and constitutive equations) that describe the acoustic wave motion on a "microscopic" scale, are spatially averaged over a representative elementary domain of the fluid-solid composite. The surface integrals over the microscopic interface t h a t turn up in the procedure link, through the "microscopic" bound ary conditions, the acoustic quantities in the fluid and solid phases together. Relating the values of the surface integrals through convex linear interpolation to the asssociated "macroscopic" action terms of the two phases, a system of first-order partial differential equations (equations of motion, deformation equa tions, and constitutive equations) arises that describes the acoustic wave motion on a macroscopic scale. These equations are next used to derive energy relations and boundary conditions. Because the fundamental macroscopic acoustic field quantities are all consistently defined in terms of "volume averages", the energy relations and the boundary conditions, too, are expressd in terms of these. Fi nally, the time Laplace-transform domain convolution-type reciprocity theorem for acoustic wave fields in porous media is derived.

2.2 Spatial averaging considerations

To locate a point in the configuration, we employ the coordinates { z i , X j , 13} with respect to a given orthogonal Cartesian reference frame with origin 0 and the three mutually perpendicular base vectors {i1.i2.i3} of unit length each; in the given order, -{ii ,io-13} form a right-handed system. T h e subscript notation for

(20)

2 . 2 SPATIAL AVERAGING CONSIDERATIONS II

(Cartesian) vectors and tensors is used; for repeated subscripts the summation convention holds. All latin lower-case subscripts (except t) are to be assigned the values 1, 2 and 3, respectively. Occasionally, a direct notation will be used to denote a vectorial quantity; in particular x = Xkh will denote t h e position vector. The time of observation is denoted by t. Partial differentiation with respect to xk is denoted by d^, d, is a reserved symbol for partial differentiation

with respect to /. To distinguish between fluid and solid quantities we use, wherever necessary, the lower-case superscripts ƒ and s, respectively. Further, double lower-case superscripts denote macroscopic material quantities. Si-units are employed.

The basic assumption underlying our analysis is that the macroscopic prop erties of the porous medium under consideration follow from a procedure of spatial averaging applied to the local solid and fluid quantities over a so-called representative elementary domain De. The maximum diameter of the latter is

chosen such that the obtained average is insensitive to small changes in this diameter. If a small variation in the diameter of De would result into a varia

tion of the averaged quantity, the averaging procedure is not meaningful. The latter happens when the diameter of D€ is either so small that it approaches the

value of the diameter of the pores, or so large t h a t macroscopic inhomogeneities affect t h e averaging procedure. T h e domain £>6 is taken to be shift- and

time-invariant; its position is specified by the position vector x of its "center" (e.g., its barycenter) (see Figure 2.1). The macroscopic quantities are then assigned to the position of this "center".

In the two-component composite, De = De(x) is the union of the subdomain

£ ) | ( x ) where fluid is present and t h e subdomain Dse{x) where solid is present.

The volumes of De{x), -D^(x), £>*(x) are denoted by V6, V / ( x ) , and V*[x),

respectively. It is clear t h a t the volume fractions

* ' ( x ) = V / ( x ) / V€, (2.1)

*'(x) = Vl(x)/V& (2.2)

occupied by the fluid and solid, respectively, in general vary with position. Since

(21)

- 1 2

Figure 2.1: The position vector x' of fluid and solid particles in the sub-domains D{, and D^ inside the time- and shift- invariant representative elementary domain De centered around x .

For any quantity ip{ associated with the fluid phase in De we define the

corresponding "fluid average" as

(0')(x,O = i - / 1>'{x',t)dV.

(2.3)

Similarly, for any quantity p' associated with the solid phase in Dc we define the corresponding "solid average" as

(r>(x,z) = i - / y,{x',t)dV.*

(2.4)

To relate these volume averages to the intrinsic volume averages, we also intro duce for any quantity vJ associated with the fluid phase in Dc the "intrinsic

(22)

2 . 2 SPATIAL AVERAGING CONSIDERATIONS 13

fluid average" as

<^)'(x,t) = - i - ƒ" ^(x'.tJdV, (2.5)

Vg (x) Jx'eD'e{x)

and for any quantity ip' associated with the solid phase in £>g t h e "intrinsic solid

average" as

From Eqs. (2.3)-(2.6) it follows that

V},(x.t.) = óf(x)(v})f(x,t) (2.7)

for a fluid-phase quantity and

(P)[x>t) = ^(xKVTM) (2.8)

for a solid-phase quantity.

In deriving the expressions for the spatial derivatives of the volume averages, we also encounter surface integrals over the interface between fluid and solid as far as this is located in the interior of the representative elementary domain. This interface will be denoted by £e = Ee( x ) . The relevant relations are derived in

Appendix A; they are found to be

W)(x,0 = (W)(x,«) - i - / vfy

f

(x!,t)dA, (2.9)

diW)(x,t) = ( W M ) - i - / »;p{x',t)dA, (2.io)

dt(ip')(x,t) = (d,^)(x,t)., (2.11)

dt(P){x,t) = (dtr)(x,t), (2.12)

where f, and t £ are t h e unit vectors along the normal to Ee( x ) pointing away

from the respective subdomains D^ a n d D'€ (see Figure A-l) and where in

Eqs. (2.11) and (2.12) a linearization for small values of the particle velocities is carried out.

(23)

2.3 Volume averaging of t h e basic equations of

acoustic wave theory

To derive the basic equations that govern the macroscopic acoustical behavior of a porous medium, we apply the method of local volume averaging to the basic equations of acoustic wave theory for an ideal fluid and the basic equations of acoustic wave theory for a perfectly elastic solid. We start with the linearized equations of motion and the linearized deformation rate equations for a fluid and a solid. The equations are written in a specific form that is needed for our later considerations, i.e., the induced mass-flow density rate is expressed in terms of the gradient of t h e stress and an impressed source term, while the induced deformation rate is expressed in terms of the gradient of the particle velocity and an impressed source term. For the fluid phase we write

¥kt (2.13) 6. (2.14) H, (2.15) e„, (2.16) where <5*m a

fi

H

vi

h'

è

= = = = = = r r

symmetrical unit tensor of rank two (Kronecker tensor). fluid traction (Pa),

volume source density of fluid body force (N m3),

fluid mass flow density rate (kg m2s2),

fluid particle velocity ( m / s ) ,

volume source density of fluid injection rate (s~'). fluid cubic dilatation rate ( s_ 1) ,

and for the solid phase

6kmdmo + ƒ / = 6mrdnvfr - h'

(24)

2 . 3 VOLUME AVERAGING OF BASIC ACOUSTIC EQUATIONS 15

\{bkpt>mq + faqb-mp) = symmetrical unit tensor of rank four,

solid stress (Pa),

volume source density of solid body force ( N / m3) ,

solid mass flow density rate (kg/m2s2 ),

solid particle velocity ( m / s ) ,

volume source density of solid strain rate ( s_ 1) ,

solid deformation rate ( s_ 1) .

In parentheses, the corresponding Si-units are indicated. T h e solid stress rpq,

the deformation rate é,j, and the solid source density of strain rate /i*; are

symmetrical tensors, i.e., rpq = jqp = | ( rp, - Tqp), é,; = é,, = | ( « y + è; i),

and h'j = k"» = \{h'j + ^ « ) - To have the same distribution of algebraic signs in

Eqs. (2.13) and (2.15), we have introduced the fluid traction rather than the fluid pressure in Eq. (2.13). Also, to indicate the similarity in general structure of the fluid and the solid equations we have included the symmetrical unit tensors,

6km and Akmpq. Note that for the fluid phase we have

6kmdma = dka, (2.17)

6mrdmvi = dTvi, (2.18)

and for the solid phase

AkmPqdmTpq = \dm{rmk - Tkm), (2-19)

A,3 m römt/; = § ( * « ; + */»ƒ). (2.20)

The material properties of the fluid and the solid are specified by their con stitutive relations. We take these equations in their most general, linearized, form, i.e., we include the most general kind of anisotropv, but restrict ourselves to instantaneously reacting materials, i.e., we neglect losses (also the ones of the viscous type). Then we have

H =

PIAVI,

(2.21)

*kmpq 'pq = fk =

H =

Ki =

(25)

and

where

0 = Kfd,o, (2.22)

k = M*r, (2.23)

kj = s„pqd,Tpq, (2.24)

pkr — tensorial fluid volume density of mass (kg m3). K.' ~ fluid compressibility ( P a- 1) ,

P'kr = tensorial solid volume density of mass ( k g / m3) , s,m = solid compliance ( P a- 1) .

We consider media that are time invariant i.e., the constitutive coefficients are time independent. In general, the constitutive coefficients depend on posi tion; the medium is then inhomogeneous. In a homogeneous domain, the con stitutive coefficients are shift invariant. Because of the symmetry of the tensors

èij and rp,, the solid compliance s,ipq satisfies the symmetry relations

Sijpq — Sjipq = Sjiqp — sijqp- (2.2o)

The constitutive equation (2.24) can inversely be written as

At;p?<Vp, = c,J MéM, (2.26)

where

cijpi = s o' ' d stiffness (Pa).

From Eqs. (2.24) and (2.26) we observe that the stiffness and the compliance are each other's inverse according to

C,jpqSpqr/ = A0 r j. (2.27)

For an isotropic fluid we have

(26)

with

pf = scalar fluid volume density of mass (kg/m3),

while for an isotropic solid we have

Pi = P'hr- (2.29) c,Jpq = \6ij6„ + 2ttAijn, (2.30)

Hjpq = A6lj6pq + 2MAljpq.. (2.31)

with

p' = scalar solid volume density of mass (kg/m3),

A,/x = Lamé coefficients of the elastic solid (Pa),

A , M = isotropic compliance coefficients of the elastic solid ( P a- 1) .

The boundary conditions at the microscopic interface E€ between the fluid

and the solid phases, require the equality of the normal components of the traction, the vanishing of t h e tangential components of the traction of the solid, and the continuity of the normal components of the particle velocity. These conditions lead to the equations

v[rtjv) = a at E6, (2.32) "tra - ufTikuyé = 0 at E6, (2.33)

l/.'o/ = ~v[v[ at E€, (2.34)

Note that the tangential components of the particle velocities in the solid and the fluid may differ at either side of the interface E€ (slip can occur parallel to

the interface since viscosity has been neglected).

We start the averaging procedure by applying the volume averaging opera tor as defined by Eq. (2.3) to Eqs. (2.13) and (2.14) and applying the volume averaging operator as defined by Eq. (2.4) to Eqs. (2.15) and (2.16). In view of

(27)

the properties (2.9) and (2.10). we obtain 6kmdm{a)+(f[) = (${)-!-[ bkmu!modA. (2.35) 1 6mrdm{v{) - (h!) = 0- — tmru'mv!r dA. (2.36) and Af c m wam<rw> + <ƒ;> = * * } ■ - — / A *m Mi £ rM< / . 4 . (2.37) V6 ^ x ' e E e l x ) A,y m ram(w;> - (A?.-) = < v - ■ — / A ,; m^ ; > ; c M . (2.38)

respectively. The first terms on the left-hand sides in Eqs. (2.35) - (2.38) are the gradients of the volume averaged stresses or the gradients of the volume averaged particle velocities, while the second terms are the volume averaged im pressed source terms. T h e surface integrals on the right-hand sides of Eqs. (2.35) - (2.38) describe the acoustic interaction between the fluid and solid phases. In Eqs. (2.35) and (2.37) this interaction is related to traction and stresses and can, therefore, be expected to represent the coupled inertia properties of the system. In Eqs. (2.36) and (2.38) the interaction is related to the deformation rates and can, therefore, be expected to represent the coupled compliance prop erties of the system. The interaction terms in general consist of a self-induced part and a mutually-induced p a r t . In this respect we observe that in many branches of physics a linear coupling between subsystems occurs: we mention the magnetic coupling between circuits in electromagnetics, coupled springs in mechanics, and coupled atomic oscillators in molecular spectroscopy. In all these cases experience learns t h a t linear coupling terms can adequately explain the physical behaviour of such coupled systems. Therefore, such an approach seems promising in the analysis of acoustic wave motion in the coupled fluid solid system as well. The values of the coupling coefficients could, in principle, be deduced from a detailed statistical analysis of the geometry of the fluid solid interface at the pore-size level, and the mechanical properties of the fluid and the solid under consideration. This is beyond the scope of our analysis. Within the realm of a linear interaction theory we shall introduce the most general

(28)

phenomenological coupling coefficients that are compatible with such a theory, and leave aside how their values are related to the microscopic structure of the composite. The surface term in Eq. (2.35) represents the time rate of the net momentum transfer via the interface Ee through which the fluid and the solid

phases are coupled. As regards the surface integral in Eq. (2.35) it is assumed that it is linearly related to the volume averaged particle velocities of the fluid and solid phases. Accordingly, we write

1 / 6kmvfcdA = m£dt(v!)-m£dt(vtr), (2.39)

where

{mlr,m{'r} — mutually-induced tensorial volume densities of mass (kg/m3).

T h e choice of the sign of the right-hand side of Eq. (2.39) has been taken such that in the final first-order acoustic wave equations, and the associated energy relations as well, the distribution of signs is the most convenient one. (Note t h a t this sign convention differs from the one employed in De Vries (1988).) With the aid of the boundary conditions (2.32) and (2.33), Eq. (2.39) can be written as

i - f AkmM^rpqdA = mfcdM) ~ m£dt{v'r), (2.40)

where the left-hand side is the surface integral occurring at the right-hand side of Eq. (2.37).

The surface integral in Eq. (2.36) represents the time rate of the net defor mation rate (cubic dilatation rate) transfer via the interface Ee through which

the fluid and the solid phases are coupled. It is assumed t h a t this term is lin early related to the volume averaged traction of the fluid phase and the volume averaged stress of the solid phase. Accordingly, we write

i - / 6mTvfnv'rdA = K''dt(o) - Kf'dt(T-±), (2.41)

Ve Jx'eZcix) 3

where

{K>f,Kf>} - mutually-induced scalar compressibility ( P a- 1) .

The choice of the sign of the right-hand side of Eq. (2.41), too, has been taken such t h a t in the final first-order acoustic wave equations and the associated

(29)

energy relations, the distribution of signs is the most convenient one. (Note t h a t , here too. the sign convention differs from the one employed in De Vries (1988).) With the aid of the boundary condition (2.34). Eq. (2.41) can be written as

i - f Akkmr^v'r dA = Kf'dt ^ j - K''d, o . (2.42)

Vg Jx'eZelx) 3

where the left-hand side is a contracted version of the surface integral occurring in Eq. (2.38). The latter surface integral represents the time rate of the net deformation transfer via the interface D€ through which the solid and the fluid

phases are coupled. The integrand consists of a diagonal part (time rate of cubic dilatation) and a deviatoric part. From the boundary condition (2.34) we note that the diagonal part is continuous while the deviatoric part may be discontinuous across the interface; therefore only the diagonal part can be linked to the fluid (see Eq. (2.42)). In this respect we observe t h a t the diagonal part represents an increase or decrease of the volume in the solid or fluid, while the deviatoric part is associated with a change in shape of the interface without accompanying volume increase or decrease. It is now assumed that the total surface term in Eq. (2.38) is linearly related to the volume averaged stresses of the fluid and solid phases in the following way

4 - ƒ , „ , , \imymKdA = Kl]pqdt'-pq - » e " ae< c r ) ^ , (2.43)

Ve ■'x'eEgtx) 3

with

Kn„ = « " i , (2.44)

where

Kvpq = mutually-induced tensorial compliance ( P a- 1) .

Upon substituting Eqs. (2.39) - (2.43) in Eqs. (2.35) - (2.38) we end up with

6kmdm[o) - rn[idt{vi) - m£fit{*i) = - { ƒ / • (2.45) 6mrdm{v{) - KS'dt(o\- KftdA] = hf . (2.46)

and

< Wwdm( rM) - mïMO- mlidt{vfr) =-{ft), (2.47)

(30)

2 . 4 ENERGY CONSIDERATIONS 21

w h e r e

mil = p{

r

- m\[ (2.49)

is t h e self-induced tensorial fluid volume density of mass ( k g / m3) ,

K! I = Kf - K>f (2.50)

is the self-induced scalar fluid compressibility ( P a- 1) ,

™°k'r = Pi ~ ™{'r (2.51)

is t h e self-induced tensorial solid volume density of mass ( k g / m3) , and

KI;P9 = S'JPI ~ **>3P<i (2.52)

is the self-induced tensorial solid compliance ( P a- 1) . In deriving Eqs. (2.45)

-(2.48) we have assumed that by taking the volume average of the fluid and solid constitutive equations (2.21) - (2.24), pkr and K.1', and p'kr and s,jpq, respectively,

can be taken to be constant over the representative elementary domain D€,

i.e., over D{ and D's, respectively. Further, the properties (2.11) and (2.12)

have been used. Eqs. (2.45) - (2.48) constitute the basic partial differential equations t h a t describe the propagation of acoustic waves in the anisotropic, inhomogeneous. fluid/solid composite. Note that the, in general inhomogeneous, volume fractions ^ ( x ) and <f>'(x) do not explicitly occur in Eqs. (2.45) - (2.48).

The macroscopic picture that is associated with Eqs. (2.45) - (2.48) is t h a t the fluid and solid phases are fully mixed and simultaneously present in some domain in space, while their interaction (on the basis of a linear theory) is incorporated in the mutually-induced coefficients {m'/r, mfk'} and {«*', «/'}■

2.4 Energy considerations

In this section we investigate the exchange of acoustic energy between a cer tain portion of a porous medium and its surroundings. Let the portion of the medium under consideration be located in some bounded domain D interior to the piecewise smooth, closed, surface 3D and let the unit vector along the nor mal to 3D, pointing away from D, be denoted by um. A subdomain DSRC °f D

(31)

Figure 2.2: Domain D for which the energy relation is obtained, if dis tributed sources are present in

DSRC-is occupied by the elastodvnamic sources. Further, the complement of D U 3D in the total domain occupied by the porous medium is denoted by D' (see Fig ure 2.2). Multiplying Eqs. (2.45) - (2.48) by the macroscopic field quantities -{vjO, -(c), ~(v'k), and —{%), respectively, and adding the resulting equations

we obtain

ESRC = -£KIN + £ D E F + dmSm. (2.53)

in which

(32)

2 . 4 ENERGY CONSIDERATIONS ₂₃ denotes the volume density of the time rate at which the sources deliver me chanical work to the acoustic disturbance (Si-unit: J / m3s ) ,

SKIN = m[i{v[)dt{vi) + m£(v{)dt{v'r) + " £ < * * > * < • ƒ ) + ™ £ R > 3 , < < ) (2-55)

the volume density of the time rate of kinetic energy (Si-unit: J / m3s ) ,

W = KfI(a)dt{0) - Kt'Wttf) + K'l£f)dM + <;M{ni>Öt(rM) (2.56)

the volume density of the time rate of deformation (potential) energy (Si-unit: J / m3s ) , and

Sm = -6mr(v'r)(o) - Am r p, « ) ( rM) (2.57)

denotes the area density of acoustic power flow (acoustic Poynting vector; Si-unit: W / m2) . Equation (2.53) is the local form of the acoustic power balance

in a porous medium.

Next, we investigate the conditions under which the volume density of the time rate of kinetic energy (cf. Eq. (2.55)) can be written as the time derivative of the kinetic energy density, where the latter is a state quantity, i.e., a quantity t h a t only depends on the instantaneous values of the averaged particle velocities of the fluid and the solid. The relevant conditions are:

m{i = m{l (2-58)

m{> = m'/k, (2.59)

m g = m'/k. (2.60)

On the microscopic level the same argument holds, where it has the consequence t h a t the volume density of mass tensors p[T and p'kr are symmetric. With this,

and the symmetry of mkr and m"T, it follows from Eqs. (2.49) and (2.51) t h a t

mk'r and mkr are symmetrical tensors, and therefore Eq. (2.59) can be extended

to

m{'r = mi'k - < = m%. (2.61)

Similarly, we investigate the conditions under which the volume density of the time rate of deformation energy (cf. Eq. (2.56)) can be written as the time derivative of the deformation energy density, where the latter, too, is a state

(33)

quantity, i.e.. a quantity that only depends on the instantaneous values of the averaged fluid traction and the averaged solid stress. The relevant conditions are:

Kf' = K,f, (2.62)

* & , = *£»• (2.63)

On the microscopic level the same argument holds, where it has the consequence that the compliance tensor s,]pq is symmetric. With this, and the symmetry of

/c"p 9, it follows from Eq. (2.52) that the mutually-induced compliance K,}PQ (cf.

Eq. (2.43)) is a symmetrical tensor, i.e.,

Kim = KMij: (2.64)

Eqs. (2.58) - (2.64) are the symmetry relations pertaining to a so-called recip rocal. or self-adjoint, porous medium. With the aid of Eqs. (2.58) - (2.64). Eqs. (2.55) and (2.56) can be rewritten as

£KIN = dtEK\K, (2.65)

£ D E F = dtEDEF. (2.66)

in which

EKW = km^(vM) + lmZ(vlHv^+mt(vi)(v'rh (2.67)

£DEF = iK^ioUa^^^r^lr^-^Uo^f;. (2.68) where £ K I N ar>d £ D E F denote the volume densities of the kinetic and the deforma

tion energy, respectively (Si-unit: J. m3) . Further, it is assumed that the kinetic

and deformation volume densities of energy are non-negative definite quadratic forms, i.e., the right-hand sides of Eqs. (2.67) and (2.68) are non-negative for any choice of {v{), (v}), {a), and (rp ?).

Integration of Eq. (2.53) over the bounded domain D and the application of Gauss' divergence theorem to the last term on the resulting right-hand side leads t o

(34)

2 . 5 MACROSCOPIC BOUNDARY CONDITIONS 25

in which

^ S R C = / ÈSRCdV (2.70)

denotes the total time rate at which the sources in D deliver mechanical work to the acoustic wave motion in D (Sl-unit: J / s ) ,

W K I N = / EKlsdV (2.71)

the work t h a t has gone into the motion of the acoustic wave field in the porous medium in D. i.e., t h e total kinetic energy stored in the acoustic wave field in

D (Si-unit: J ) ,

W-W = f EDEF dV (2.72)

JxeD

the work that has gone into t h e building up of stress of the acoustic wave field in the porous medium in D. i.e., the total deformation (potential) energy stored in the acoustic wave field in D (Si-unit: J ) , and

Pi = f unSmdA (2.73)

JxzdD

RAD

JxedD

the amount of acoustic power t h a t is radiated away, across the boundary dD of

D, towards D' (Si-unit: J / s ) .

Equation (2.69) is t h e global form, for t h e domain D, of the acoustic power balance in a porous medium. It expresses t h a t t h e power delivered by the sources in D that generate t h e acoustic wave field, is partly used to build up the kinetic and the deformation (potential) energies that are stored in t h e wave motion in

D, while the remainder is radiated, across 3D, away t o the surroundings of D.

(Note t h a t the porous medium is assumed to be lossless in its acoustic behavior.) When there are no sources located at t h e boundary dD of D, it is obvious that PRAD (cf. Eq. (2.73)) is continuous across this boundary. Therfore, t h e ''normal component" of the Poynting vector ((cf. Eq. (2.57)), i.e., vmSm, is con

tinuous across a sourcefree macroscopic interface of two different porous media. This will be shown in t h e next section.

2.5 Macroscopic b o u n d a r y conditions

In this section, boundary conditions are derived for configurations where two po rous media with different acoustic properties are in contact with each other, as

(35)

well as for the special cases where one of the two porous media degenerates into either a pure fluid, a pure solid, or a vacuum. They follow from the consideration t h a t , due to the non-zero volume of the representative elementary domain, all volume averaged quantities change, in the macroscopic space, continuously with position. The macroscopic field quantities (volume averaged fluid traction, vol ume averaged fluid particle velocity, volume averaged solid stress, and volume averaged solid particle velocity) are then continuously differentiable functions of position and satisfy the partial differential equations (2.45) - (2.48). Across interfaces where the volume averaged quantities change rapidly due to rapid changes (or even jumps) in the microscopic configurational quantities, they can mathematically be modeled as j u m p discontinuities. On physical grounds we assume that these jumps will remain bounded; hence, across the discontinuity surface, or interface, they can at most j u m p by finite amounts. In the limit of such discontinuous behavior, the macroscopic field quantities are no longer continuously differentiable in a domain that contains (part of) an interface, and Eqs. (2.45) - (2.48) cease to hold. As a consequence, they have to be supple mented by boundary conditions that interconnect, in a certain manner, (parts of) the relevant macroscopic field values at either side of the interface under con sideration. To arrive at these boundary conditions, we shall asume t h a t along the interface the solid phases of the two porous media are in rigid contact with each other. Then all components of the volume averaged solid particle velocity are continuous across the interface. Further, because the fluid motion is consid ered at a high Reynolds number, i.e., the effects of viscosity are neglected with respect to the inertia effects, slip may occur along the interface between the particle velocities of the fluid phases of the two porous media. Therefore, the components of the fluid volume averaged particle velocity parallel to the inter face are not interrelated, while their component perpendicular to the interface is continuous. To arrive at the boundary conditions we proceed as follows.

Let I denote a macroscopic interface and assume that I has everywhere a unique tangent plane. Further let n (with components nm) denote the unit

vector along the normal to / such t h a t upon traversing I in the direction of n , we pass from the domain Di to the domain D\, Dx and D* being located at

(36)

Figure 2.3: Configuration employed for t h e derivation of macroscopic boundary conditions at a porous-medium/porous-medium interface I.

on / . Suppose now t h a t some (or all) acoustic wave quantities j u m p across

I. In the direction parallel to I, all acoustic wave quantities still vary in a

continuously differentiable manner, and hence the spatial derivatives parallel to

I give no problem in Eqs. (2.45) - (2.48). The spatial derivatives perpendicular

to I, on the contrary, meet functions t h a t show a j u m p discontinuity across I: these give rise to surface Dirac delta distributions (surface impulses) located on I. Distributions of this kind would, however, physically be representative of surface sources located on I. In the absence of such surface sources, the absence of surface impulses in the spatial derivatives should be enforced. The latter is done by requiring that these spatial derivatives only meet functions t h a t are continuous across I.

We shall apply this reasoning to derive boundary conditions t h a t inter-connnect, in a consistent manner, (parts of) the relevant macroscopic field values

(37)

at either side of an interface in a porous-medium porous-medium configuration. a porous-medium/vacuum configuration, a porous-medium 'fluid configuration. and a porous-medium 'solid configuration for the acoustic wave motion t h a t is governed by our macroscopic system of equations (2.45) - (2.48).

B o u n d a r y c o n d i t i o n s a t a p o r o u s - m e d i u m / p o r o u s - m e d i u m i n t e r f a c e The vectorial spatial derivative dm that shows up in Eqs. (2.45) - (2.48) can be

separated in the following way (see Figure (2.3))

dm = nm(n,d,) - dm - nm( n , a . ) ; . (2.74)

T h e first term on the right-hand side of Eq. (2.74) contains the component of dm parallel to nm (i.e., perpendicular to ƒ); the second term contains the

components of dm perpendicular to nm (i.e., parallel to / ) . This separation of

dm is now used in Eqs. (2.45) - (2.48) to investigate which functions are to be

continuous across / . As Eq. (2.45) shows, the occurrence of a surface Dirac distribution is avoided if <5*mnm(a) is continuous across the interface, or

(a) is continuous across I. (2.75)

The use of Eq. (2.74) in Eq. (2.46) leads to

nT(v{) is continuous across I. (2.76)

Equation (2.76) had already been conjectured on physical grounds, but is shown here to be consistent with the volume averaged fluid deformation rate equation (2.46). T h e use of Eq. (2.74) in Eq. (2.47) shows t h a t the occurrence of a surface Dirac distribution is avoided if Afcmp(Jnm<Ypfl} is continuous across the interface:

this implies t h a t the volume averaged solid traction

(**) = At n wnm( rM) = \nm{{rmk) + 'jkm) is continuous across / . (2.77)

T h e use of Eq. (2.74) in Eq. (2.48) leads to the condition that A ,; m rn „ , v')

is continuous across the interface. Multiplying this condition by n, and rc.n,, respectively, and using the relations nmnm = 1, n , nmA ,j m r = ±(<5;r - M;nr), and

(38)

n,rijAijmr = nmnr, the conditions

5_{vj) + nj(nr(v'))\ is continuous a c r o s s / , (2.78)

nr(v') is continuous across / , (2.79)

are obtained. Upon combining Eqs. (2.78) and (2.79), the final condition be comes

(vt) is continuous across I. (2.80)

Equation (2.80) had already been anticipated on physical grounds, but is shown here to be consistent with the volume averaged solid deformation rate equation (2.48).

The requirement of consistency with the partial differential equations (2.45) - (2.48) with rapidly varying coefficients has led to the boundary conditions (2.75) - (2.77) and (2.80) at a sourcefree interface between two porous media with different acoustic properties whose solid phases are in rigid contact. Our conditions, t h a t have not been used in the earlier literature, automatically guar antee continuity of the normal component of the acoustic Poynting vector (cf. Eq. (2.57)) upon crossing an interface, which is a prerequisite for a consistent macroscopic theory of energy transfer; i.e.,

nmSm = nm(-6mr(v;[){<?) - Amrpq(v'}(TPg}) is continuous across / . (2.81)

Our macroscopic boundary conditions automatically guarantee the uniqueness of a solution of the partial differential equations (2.45) - (2.48), with given source terms, as far as this uniqueness is based on energy considerations.

In the discussions on the Biot theory and the boundary conditions as they have been suggested by Deresiewickz and Skalak (1963), invariably the micro scopic structure of the interface between two porous media turns up, with the possibility of closed, open, or partly open pore connnections. Even if, with the aid of an area-averaging procedure carried out over a two-dimensional represen tative elementary disk, some kind of interrelation of the thus averaged quantities at either side of a macroscopic interface could be arrived at, these results would in general not be compatible with the volume averaged differential equations

(39)

Figure 2.4: Configuration employed for the derivation of macroscopic boundary conditions at a porous-medium, vacuum interface I.

(2.45) - (2.48) and, hence, no consistent theory would result. Since the devel opment of a consistent macroscopic theory is the purpose of our analysis, the question of how our boundary conditions (2.75) - (2.77) and (2.80) are related to the microscopic structure of this interface is further left aside, just as the detailed structure of the microscopic interface between the fluid and the solid phase does not explicitly turn up in the corresponding macroscopic interaction terms.

B o u n d a r y c o n d i t i o n s a t a p o r o u s - m e d i u m / v a c u u r n i n t e r f a c e

In this subsection, we discuss the boundary conditions that interconnect the macroscopic acoustic field values at either side of an interface in a porous-medium/vacuum configuration. A medium is denoted as a vacuum if in it the traction or stress is negligibly small compared to the traction or stress in the medium on the other side of the interface. For example, in many applications the

(40)

pressure of the acoustic wave motion in the atmosphere is negligible with respect to the stress of the acoustic wave motion in an adjacent porous medium. This situation applies, for example, to the air/porous-earth interface in geophysical applications.

Suppose t h a t the domain D] (see Figure 2.4) is occupied by a porous medium and the domain D2 by a vacuum in which the traction and the stress are neg

ligibly small, while the conditions of continuity of the volume averaged fluid traction and the volume averaged solid traction across the boundary surface I are maintained. As a consequence, the boundary conditions upon approaching the boundary surface I via medium Dt become

l i m ( » ( x - kn,t) = 0 for any x € / , (2.82)

MO

\im(tk)(x- hn,t) - 0 for any x € / , (2.83)

where the volume averaged solid traction (i*) has been defined in Eq. (2.77), n denotes the unit vector along the normal to I, pointing away from D2, and h is

a scalar. We are not free to prescribe the volume averaged solid particle velocity and the normal component of the volume averaged fluid particle velocity in the porous-medium side of I in this case. In fact, the volume averaged solid particle velocity and the normal component of the volume averaged fluid particle velocity will, in general, have a non-zero value a t I, while they are not defined in D2.

T h e requirement of consistency with the partial differential equations (2.45) -(2.48) with rapidly varying coefficients has led to the boundary conditions (2.82) and (2.83) at a sourcefree interface between a porous medium and a vacuum with different acoustic properties. As far as the acoustic Poynting vector is concerned, we observe t h a t (cf. Eq (2.81)) l i m nm5m( x + hn,t) = 0 for any x 6 ƒ, (2.84) hlO or limn-n [-Smr(v()(x-rkn,t)(c)(x+hn,t) - Amrpq{v'r){x+hn,t){rM)(x + hn,t)j (2.85) = 0 for any x € / ,

(41)

Figure 2.5: Configuration employed for the derivation of macroscopic boundary conditions at a porous-medium/fluid interface / .

i.e., there is no acoustic powerflow across I. Our macroscopic boundary con ditions (2.82) and (2.83), too, are admissible ones for the uniqueness of the solution of the partial differential equations (2.45) - (2.48). with given source terms, as far as this uniqueness is based on energy considerations.

B o u n d a r y c o n d i t i o n s a t a p o r o u s - r n e d i u m / f l u i d i n t e r f a c e

In this subsection, we discuss the boundary conditions that interconnect the macroscopic acoustic field values at either side of an interface in a porous-medium/fluid configuration, i.e., the porous medium on one side of the boundary is degenerated into a pure fluid by the absence of the solid phase. As examples, we mention the sea-floor/sea interface in marine seismics and fluid-filled bore holes with a mudcake (that can be modeled as a porous medium) surrounding the borehole in land or subsurface seismics.

(42)

case of two porous media joining at an interface and has therefore to be treated separately. The basic equations of acoustic wave theory in an ideal fluid are given by Eqs. (2.13), (2.14), (2.21), and (2.22). The fundamental field quantities are the traction (opposite of the pressure) and the particle velocity which are denoted by the symbols a' and v7r , respectively. Since in the fluid only the fluid

phase is present, these quantities equal their corresponding volume averages. Suppose that, the domain D\ (see Figure 2.5) is occupied by the porous medium and the domain D2 by the fluid. Upon moving a representative elementary

domain from the porous medium in D\, via the interface / , to the medium in

D2 where only a fluid is present, we see a continuation of the fluid phase (with

possibly discontinuous acoustic properties) only. To derive boundary conditions for this particular configuration, we again use the type of reasoning t h a t has been set up in the main section. We substitute Eq. (2.74) in Eqs. (2.13) and (2.45) and observe t h a t the occurrence of a surface Dirac distribution in the acoustic fluid quantities is avoided by requiring t h a t 6kmnmo and èkmnmW) are

each other's continuations across the interface, i.e., by requiring the condition

lim(a)(x + hn,t) — lim<r'(x + hn,t) for any x € I, (2.86) where n is the unit vector along the normal to I. pointing away from D2, and h

is a scalar (see Figure 2.5). The use of Eq. (2.74) in Eqs. (2.14) and (2.46) leads to

limwr(f^)(x — hn,t) = \\mnrv*(x + hn,t) for any x € I. (2.87)

Mo MO

The solid phase of the porous medium in £>j is not continued into D2 and there

fore the volume averaged solid traction satisfies, on the grounds of consistency, the boundary condition

lim(ti)(x + hn,t) = 0 for any x € I, (2.88) MO

where the volume averaged solid traction {t/,) has been defined in Eq. (2.77). We are not free to prescribe the volume averaged solid particle velocity on the porous-medium side of I in this case. In fact, the volume averaged solid particle velocity will, in general, have a non-zero value on ƒ, while it is not defined in

(43)

The requirement of consistency with the partial differentia] equations (2.13). (2.14). (2.21), (2.22), and (2.45) - (2.48) with rapidly varying coefficients has led to the boundary conditions (2.86) - (2.88) at a sourcefree interface between a porous medium and an ideal fluid with different acoustic properties. As far as the acoustic Poynting vector is concerned, we observe t h a t (cf. Eq (2.81))

lim nmSm(x - fen. <) = I i m nmSm( x — fen. t) for anv x ~ I. (2.89)

hiO MO

or

\\mnm \-6mr(vh{x+hn,t)(a (x-hn.t) - Am r„ rr' ( x - f e n . r ) 'jM ( x - f e n . f ) l

(2.90) = l i m nm — <5mrr;'(x - fen,f)<r' (x + fen.i) for any x £ I.

h'0 I J

Our macroscopic boundary conditions (2.86) - (2.88). too, are admissible ones for the uniqueness of the solution of the partial differential equations (2.45)

-(2.48). with given source terms, as far as this uniqueness is based on energy considerations.

B o u n d a r y c o n d i t i o n s a t a p o r o u s - m e d i u m / s o l i d i n t e r f a c e

In this subsection, we discuss the boundary conditions t h a t interconnect the macroscopic acoustic field values at either side of an interface in a porous-medium 'solid configuration, i.e., the porous porous-medium on one side of the boundary is degenerated into a pure solid by the absence of the fluid phase. As an example. we mention the gas or oil-reservoir/solid interface and the borehole with a mud-cake (that can be modeled as a porous medium) surrounded by a non-porous

rock in geophysical applications.

The degenerate case under consideration is a singular perturbation of the case of two porous media joining at. an interface and has therefore to be treated separately. The basic equations of acoustic wave theory in an ideal solid are given by Eqs. (2.15), (2.16), (2.23). and (2.24). The fundamental field quanti ties are the stress and the particle velocity which are denoted by the symbols T-and v*, respectively. The traction AkmMnmTpq at the boundary I is denoted as

(44)

2 . 5 MACROSCOPIC BOUNDARY CONDITIONS

₃₅

Figure 2.6: Configuration employed for the derivation of macroscopic boundary conditions at a porous-medium/solid interface I.

corresponding volume averages. Suppose t h a t the domain D\ (see Figure 2.6) is occupied by the porous medium and the domain D% by the solid. Upon mov ing a representative elementary domain from the porous medium in D%, via the interface / , to the medium in D2 where only a solid is present, we see a contin

uation of the solid phase (with possibly discontinuous acoustic properties) only. To derive boundary conditions for this particular configuration, we use again the type of reasoning t h a t has been set up in the main section. We substitute Eq. (2.74) in Eqs. (2.15) and (2.47) and observe t h a t the occurrence of a surface Dirac distribution in the acoustic solid quantities is avoided by requiring t h a t

&kmpqnmTpq and Akmpgnm(Tpg) are each other's continuations across the interface,

i.e., by requiring the condition

lim(<t)(x -r hn,t) = \\mtsk(x + hn.t) for any x € I, (2.91)

where the volume averaged solid traction {£*) has been defined in Eq. (2.77), n denotes the unit vector along the normal to / , pointing away from Di, and h is

(45)

a scalar (see Figure 2.6). The use of Eq. (2.74) in Eqs. (2.16) and (2.48) leads to

linn>;')(x - hn,t) = l i m rr s( x - hn.t) for any x £ / . (2.92)

where the same type of reasoning has been used as in the derivation of Eq. (2.80). The fluid phase of the porous medium in Di is not continued into D-> and there fore the volume averaged fluid traction satisfies, on the grounds of consistency. the boundary condition:

Iim(<r)(x + kn,t) = 0 for any x€l. (2.93)

We are not free to prescribe on the porous-medium side of I the normal com

ponent of the volume averaged fluid particle velocity in this case. In fact, the normal component of the volume averaged fluid particle velocity will, in general. have a non-zero value on / , while it is not defined in D%.

The requirement of consistency with the partial differential equations (2.15). (2.16), (2.23), (2.24), and (2.45) - (2.48) with rapidly varying coefficients has led to the boundary conditions (2.91) - (2.93) at a sourcefree interface between a porous medium and a solid with different acoustic properties: the solid phases are assumed to b e in rigid contact. As far as the acoustic Povnting vector is concerned, we observe t h a t (cf. Eq (2.81))

l i m r cmSm( x + hn,t) = lim nmSm( x -f hn,t) for any x € / . (2.94)

or

l i m nm \—Smr(v{)(x+hn,t)(a)(x+hn.t) - Am r„ v' (x+tm,t)(TM ( x - / m . < ) j

(2.95)

= limn

m

[-A

mrp

,u

rs

(x - hn,t)r^{x - An.)] f°*

r

any x e I.

Our macroscopic boundary conditions (2.91) - (2.93), too, are admissible ones for the uniqueness of a solution of the partial differential equations (2.45) -(2.48), with given source terms, as far as this uniqueness is based on energy considerations.

Propagation of transient acoustic waves in porous media

ACOUSTIC WAVES IN

POROUS MEDIA

Sytze M. de Vries

TR diss

PROPAGATION OF TRANSIENT

ACOUSTIC WAVES IN

POROUS MEDIA

P R O E F S C H R I F T

ter verkrijging van de graad van doctor aan

de Technische Universiteit Delft, op gezag van

de Rector Magnificus, prof. drs. P.A. Schenck,

in het openbaar te verdedigen ten overstaan van

een commissie aangewezen door het College van Dekanen

op dinsdag 30 mei 1989 te 16.00 uur

door

Sytze Marten de Vries

geboren te IJlst

elektrotechnisch ingenieur

TR diss

1 7 3 0

Contents

Chapter 1

I N T R O D U C T I O N

Summary

1.1 Introduction

1.2 Outline of t h e different chapters

C h a p t e r 2

B A S I C R E L A T I O N S F O R

T R A N S I E N T A C O U S T I C

WAVES I N A P O R O U S

M E D I U M

Summary

2.1 I n t r o d u c t i o n

2.2 Spatial averaging considerations

(0')(x,O = i - / 1>'{x',t)dV.

(r>(x,z) = i - / y,*{x',t)dV.

<^)'(x,t) = - i - ƒ" ^(x'.tJdV, (2.5)

(P)[x>t) = ^(xKVTM) (2.8)

W)(x,0 = (W)(x,«) - i - / vfy

(x!,t)dA, (2.9)

diW)(x,t) = ( W M ) - i - / »;p{x',t)dA, (2.io)

2.3 Volume averaging of t h e basic equations of

acoustic wave theory

fi

H

vi

è

H =

(2.21)

H =

Ki =

mil = p{

- m\[ (2.49)

2.4 Energy considerations

2.5 Macroscopic b o u n d a r y conditions

35

Iim(<r)(x + kn,t) = 0 for any x€l. (2.93)

= limn

[-A

,u

(x - hn,t)r^{x - An.*)] f°

any x e I.

(r>(x,z) = i - / y,{x',t)dV.*

₃₅

(x - hn,t)r^{x - An.)] f°*