Waves in Partially Saturated Porous Media

E orts to extend the theory of poroelasticity to semilinear and nonlinear elastic response, to partially saturated pores, to inhomogeneous solid frame materials, and to viscous losses due to localized ow e ects are summarized. The prospects for a comprehensive theory of wave propagation in partially saturated porous media and the reasons for needing such a theory are also discussed. The main results are these: (a) Using the physically reasonable assumption of negligible capillary pressure change during passage of an acoustic signal through the medium, equations of poroelasticity for partially saturated materials have been derived and boundary conditions assuring the uniqueness of the solutions have been found. (b) Coe cients for scattering from a spherical inclusion in a poroelastic medium have been calculated. These coe cients may then be used to estimate e ective constants in poroelastic wave equations when the medium is inhomogeneous; three common single-scattering approximations yield expressions that satisfy all known constraints on these constants and therefore provide generalized Gassmann's equations for inhomogeneous porous media. (c) The observed anomalously high attenuation of sound in partially saturated porous media can be explained in part by accounting for the e ects of inhomogeneous porosity and uid permeability. Regions of high permeability allow more uid motion than regions of low permeability and therefore may be expected to play the dominant role in sound attenuation. University of California, Lawrence Livermore National Laboratory, P. O. Box 808 L-202, Livermore, CA 94550. yWork performed under the auspices of the U. S. Department of Energy by LLNL under contract No. W-7405-ENG-48 and supported speci cally by the Geosciences Research Program of the DOE O ce of Energy Research within the O ce of Basic Energy Sciences, Division of Engineering and Geosciences. 1. Introduction. A limited theory of poroelasticity was formulated by Biot [1,2]. He assumed linear, isotropic elastic response on the macroscopic scale for porous media composed of homogeneous frame materials and fully uid-saturated pores. The principal attenuation mechanism of this theory was viscous attenuation due to shear induced during macroscopic ow of the single-phase uid lling the pores. Even with these simpli cations, the resulting theory has remained a scienti c oddity for over 30 years: (a) It is relatively hard to analyze the predictions of this theory [3{6] because it involves two coupled wave equations forming a system somewhat more complex than the equations of viscoelasticity [7,8] { which are nontrivial to analyze themselves! (b) The most startling predictions of the theory { such as the existence of a slow bulk compressional wave [1] or slow surface [9] and extensional waves [10] { are often very hard to verify in the laboratory [11{21]. (c) Even the validity of the form of the equations and the physical interpretation of many of the coe cients in the equations remained unclear for 25 years [12,22{27], and in some cases are still in dispute today [28,29]. It is therefore understandable that signi cant progress towards eliminating the many simplifying assumptions contained in the original work had not been made prior to the 1980s. Indeed, why complicate a subject which is already so di cult? Often we try to argue that an elementary theory should su ce to explain the gross behavior of such complex materials, justifying our approximations with the comparative simplicity and elegance of the resulting analysis. If the theory is really successful at explaining the preponderance of experimental data, then of course our arguments are justi ed and it appears to be of only academic interest to expend such e ort as would be required to construct a truly comprehensive theory. On the other hand, the theory to date has been unable to explain some of the most elementary experimental results for waves in geological materials, so it is essential to produce a more sophisticated theory capable of treating most of the complications encountered in practice. The need for a more realistic theory drives us to remove the simplifying assumptions. And this need for a more realistic theory becomes most apparent when we try to analyze sound wave data for real porous materials. For many of the geophysical applications of greatest interest, the pertinent geological materials are anisotropic and very heterogeneous, composed of multiple solid frame materials and multiple pore uids. In some applications, the exciting waves are of large amplitude so that linear equations of motion are simply inadequate to describe the phenomena we want to study. Various extensions of the elementary theory have been introduced. Biot himself had generalized the theory to include anisotropic e ects for dynamic problems [30] and nonlinear e ects for quasistatic problems [31]. When the saturating uid is air [32{34], connections between Biot's theory and earlier work on rigid frame porous media [35] have also been explored. Various other authors have treated the generalization to partial saturation at very low frequencies in an intuitively appealing manner [36{40], but without having any clear procedure for generalizing their results for higher frequencies. The most general form of the equations for the elastic coe cients when the solid frame material is composed of two or more constituents has been known for some time [37], but no method for obtaining the required data had been suggested. One goal of our research is a comprehensive theory of dynamic poroelasticity. Irreversible pore collapse [41] is important in some of our applications, but we have neglected such e ects initially in order to construct what is otherwise a quite general Lagrangian variational principle [42] for nonlinear and semilinear (reversible) deformations of dry and uid saturated porous solids. This approach is very closely related to an Eulerian variational formulation of Drumheller and Bedford [43] for ow of complex mixtures of uids and solids. We have shown that our theory reduces correctly to Biot's equations of poroelasticity [1] for small amplitude wave propagation and that it also reduces correctly to Biot's theory of nonlinear and semilinear rheology for porous solids [31] when the deformations are su ciently slow. The resulting theory is a nontrivial generalization of Biot's ideas including explicit equations of motion for changes of solid and uid density. Furthermore, if capillary pressure change may be neglected, the linear theory also shows that calculations on problems with only partially saturated pores may be reduced to computations of the same level of di culty as those for fully saturated pores [44]. Appropriate boundary conditions have been found to guarantee that solutions of these equations are unique [44,45]. We expect the general theory to give a very good account of the behavior of wet porous materials during elastic deformations. In the presentation that follows, we concentrate on three extensions of the theory of poroelasticity that tend to make the theory more realistic for applications to rocks. First, we show how the theory may be generalized to partially saturated porous media. Then, we use an e ective medium method to nd estimates of the coe cients in the equations when the frame material surrounding the pores is inhomogeneous. Finally, we analyze the attenuation of the fast compressional wave in heterogeneous media and show that the physically correct damping coe cient depends not on the global uidow permeability, but on a spatial average (a line integral) of the local permeability. 2. Wave Equations for Multiple Fluid Saturation. When the mechanical and thermodynamical processes set in motion by a deformation are reversible, an energy functional which includes all the important e ects involved in the motion may be constructed. Equations of motion may then be found by an application of Hamilton's principle. Such variational methods based on energy functionals are well-known in continuum mechanics [46]. Thus, the only really new feature in the present context is the degree of complexity; porous earth may be composed of many types of solid constituents and the pore space may be lled with a mixture of water and air. Some irreversible e ects may also be included in the variational method (e.g., losses of energy due to drag between constituents) when they may be analyzed in terms of a dissipation functional. Other irreversible e ects such as those associated with collapse of the pore space lie outside the scope of the traditional variational approaches; the forms normally used for the energy functionals are quadratic with constant coe cients in the linear problems or simply positive de nite polynomials with constant coe cients for nonlinear problems. During pore collapse, the usual assumptions about the form of the energy functionals are violated, so the usefulness of the variational method is questionable. However, if we restrict discussion to linear or semi-linear processes, the variational methods are entirely adequate. Using these variational methods, Berryman and Thigpen [44] have shown that the general equations of motion for linear elastic wave propagation through an isotropic porous medium containing both liquid and gas (or, more generally, any two uids) in the pores are given by (1) ( )0 ( ) = @E( ) @ ( ) 2( )0 ; (2) (s)0 u(s)i + X =g;l (s )0( u(s)i  u( )i) = (s)0 @E(s) @u(s)i;j + (s)0 ij ;j + d(s)i + (s)0b(s)i; and (3) ( )0 u( )i +X 6= ( )0( u( )i  u( )i) = ( )0( );i + d( )i + ( )0b( )i where = g or l and = g; l; or s. The generalization to multiple pore uids is immediate: let the index range over all uids in the pores, and the index range over all the uids and the solid frame. The displacements are u( )i. The local densities (mass per unit volume of constituent) are ( ). The partial densities (mass per unit total volume) are ( ) = ( ) ( ). The internal energies of these immiscible constituents are E( ). The induced mass coe cients are (s )0. The body forces are given by b( )i and the drag forces by d( )i. Thigpen and Berryman [47] have shown that the drag forces may be written in the form d( )i = P D( )( _ u( )i _ u(s)i) where D( ) is a symmetric, positive semide nite matrix whose matrix elements satisfy P D( ) = 0 for = g or l. For the present discussion, we will ignore the e ects of contact line motion that can be an added source of dissipation in partially saturated porous media [48,49]. We will also ignore possible e ects of interaction torque that may lead to shear wave coupling between the pore uids and the solid matrix [50]. One major simpli cation that occurs in the equations for partial saturation follows from (1) and the approximation ( )0 = 0. We nd that (4) = 2( )0 @E( ) @ ( ) p( ) where p( ) is the pressure for constituent . Eq. (4) implies that all the pressures are equal { which is consistent with an assumption that capillary pressure e ects are negligible for acoustics (also see References [51,52]). Without this approximation, the number of compressional waves through a porous medium will generally be one more than the number of uids in its pores. This result is however dependent on the spatial arrangement of the uids. If one uid dominates and the others are mixed into the dominant one, then only two compressional waves are expected. When (4) is valid, only two compressional waves will be found regardless of the spatial arrangement of the uids. The subscript may subsequently be dropped from p. If 2e( )ij u( )i;j+u( )j;i, then the rst two strain invariants are de ned by I( )1 = e( )ii and I( )2 = 1 2 [I2 ( )1 e( )ije( )ji]. The changes in density are de ned by ( ) = ( ) ( )0. In terms of these invariants, the standard de nitions of the internal energies for an isotropic medium are (5) (s)0E(s) = 1 2aI2 (s)1 + bI(s)2 + cI(s)1 (s) + 1 2d 2(s) and (6) ( )0E( ) = 12h( ) 2( ) for = g; l. Applying (4) to (5) and (6), we nd p = (s)0 (s)0 (cI(s)1 + d (s)) = (l)0 (l)0 h(l) (l) (7) = (g)0 (g)0 h(g) (g): The coe cients in (5) have been shown elsewhere [42] to be related to measurable quantities: a = (s)0K =( (f)0) + 4 3 , b = 2 , c = (s)0K = (s)0( (f)0), and d = [ (s)0= (s)0]2K(s)=( (f)0) where = 1 K =K(s). The bulk and shear moduli of the drained porous solid frame are K and . The bulk modulus of the (assumed) single constituent composing the microscopically homogeneous frame is K(s). If the solid frame is composed of two or more constituents, then these formulas must be modi ed. The coe cient h( ) is related to the bulk modulus K( ) of the -th uid constituent by (8) h( ) = ( )0 2( )0K( ): These equations are all based in essential ways on Gassmann's equations [22, 37, 53]. Methods for generalizing these relations for isotropic porous materials will be presented in Section 5. The methods presented here could easily be generalized for anisotropic porous media, but at the present time little work has been done to identify appropriate measurements to determine the coe cients needed in the resulting equations so we will not pursue this line of research here. Now we de ne the linearized increment of uid content for partial saturation to be (9) I(s)1 +X ( )0 ( )0 ( ): If only one uid phase is present, (9) reduces to the exact result obtained previously [42]. If more than one uid phase is present, then we observe that by de ning an e ective total uid density change according to (10) (f)0 (f)0 (f) (g)0 (g)0 (g) + (l)0 (l)0 (l) with (f)0 P ( )0 and we nd that (9) reduces again to the exact result. Furthermore, applying (8), it is straight forward to show that (4) implies that (11) ( ) ( )0 = p K( ) for = g or l. Substituting (11) into both sides of (10) shows that the e ective bulk modulus of the multiphase uid is given by (12) (f)0 K(f) X ( )0 K( ) which is just the harmonic mean or Reuss average of the constituents' bulk moduli. To check the consistency of our de nition of , we can show easily that (13) =X ( )0[I(s)1 I( )1]: If we de ne the average displacement of a uid relative to the solid frame by (14) w( )i = ( )0[u( )i u(s)i] for = g or l and the total relative uid displacement by (15) wi =X w( )i; then (13) becomes (16) = wi;i: Equation (16) reduces to the standard de nition for full saturation when only one uid saturates the pore space and is a natural generalization of this de nition for partially saturated materials. The total relative uid displacement wi de ned by (15) is important in partial saturation problems not only because of the analogy just developed with the fully saturated problems, but also for convenience in applying boundary conditions in practical problems. Berryman and Thigpen [44] have shown previously that uniqueness of the solutions to the equations (1)-(3) demands the speci cation of either p or the normal component of this same wi on the boundaries of the porous material. Therefore, it proves most convenient to combine these equations so that u(s)i and wi are the dependent variables. We will subsequently drop the subscript (s) on ui since no confusion will arise and also de ne e I(s)1. In addition, the zero subscripts on density and volume fraction may also be dropped in the remainder of the analysis. To determine the relations among p; ; and e, substitute (11) and the rst equation of (4) into (8) to eliminate ( ) for all . Using known identities and rearranging terms, we nd easily that (17) p = M Ce where the coe cients C and M are given by (18) C = ( (g) (l))=K(s) + (g)=K(g) + (l)=K(l) = 1; and (19) M = C= with (20) = 1 K =K(s): Substituting (12) into (18) gives (21) C = ( (f)0)=K(s) + (f)0=K(f) = 1 which is the standard result for single-phase saturation [53]. Next we suppose the body forces vanish and sum the equations (2) and (3) to obtain (22)  ui + (g)  w(g)i + (l)  w(l)i = (s) @E(s) @ui;j p ij ;j where =P ( ). Dividing (3) through by (f)0 and rearranging terms, we nd (23) (g) ui + (g) (g) (g)  w(g)i + D(gg) 2(g) _ w(g)i (gl) 2(g)  w(l)i = p;i and (24) (l) ui + (l) (l) (l)  w(l)i + D(ll) 2(l) _ w(l)i (lg) 2(l)  w(g)i = p;i: If an electrical tortuosity for a porous material is given by = F where is the e ective porosity and F is the e ective electrical formation factor (ratio of the conductivity of a conducting uid to that of the insulating porous material when it contains the conducting uid), then in (23) and (24) (g) is the electrical tortuosity of the pore space occupied only by the gas, while (l) is the electrical tortuosity of the pore space occupied only by the liquid. Introducing a Fourier time dependence of the form exp( i!t) into (23) and (24), combining, rearranging terms, and keeping the same names for the transformed and untransformed variables, we have (25) !2 q(g) r(g) r(l) q(l) w(g)i w(l)i = p;i + !2 (g)0ui p;i + !2 (l)0ui where (26) 2( )0q( ) ( ) ( )0 + iD( )=! and (27) 2( )0r( ) (gl)0: In (26), 6= so = l or g as = g or l. Inverting the matrix in (25) and summing the results gives (28) !2[q(g)q(l) r(g)r(l)]wi = (s(g) + s(l))p;i + !2[s(g) (l) + s(l) (g)]ui where (29) s( ) = q( ) + r( ): Using the expressions for w( )i from (25) again, we nd X ( )w( )i = [ (g)(q(l) + r(g)) + (l)(q(g) + r(l))]p;i !2[q(l) 2(g) + (r(g) + r(l)) (g) (l) + q(g) 2(l)]ui =!2[q(g)q(l) r(g)r(l)] (30) = (g)(q(l) + r(g)) + (l)(q(g) + r(l)) s(g) + s(l) wi ( (g) (l))2 s(g) + s(l) ui where p;i has been eliminated in the second step of (30) using (28). The nal form of these equations is found by substituting (30) into (22), using (18) in the result and also in (28), and nally rearranging terms. The equations then take the familiar form [54,55] (31) r2~u+ (H )re Cr + !2( uu~u+ uw ~ w) = 0; (32) Cre Mr + !2( wu~u+ ww ~ w) = 0; where the inertial coe cients are given by (33) uu = ( (g) (l))2 s(g) + s(l) ; (34) wu = (g)s(l) + (l)s(g) s(g) + s(l) = uw + (r(l) r(g))( (g) (l)) s(g) + s(l) ; and (35) ww = q(g)q(l) r(g)r(l) s(g) + s(l) : The coe cient H is given by (36) H = K + 43 + C while C and M are given by (18) and (19). Thus, we nd the remarkable result that the form of the equations of motion for partial saturation and for full saturation are the same { the only di erence being that the inertial coe cients, as well as the C and M coe cients, are more complicated when the porous solid is only partially saturated. 3. Biot's Theory for Porous Materials with Inclusions. Now we will change notation somewhat and consider two isotropic porous media (i.e., host and inclusion) each of whose connected pore space is saturated with a single-phase viscous uid. The fraction of the total volume occupied by the uid is the void volume fraction or porosity , which is assumed to be uniform within a constituent but which may vary between the the host and inclusion. The bulk modulus and density of the uid are Kf and f , respectively, in the host. The bulk and shear moduli of the drained porous frame for the host are K and . For now we assume the frame of the host is composed of a single constituent whose bulk and shear moduli and density are Km, m, and m. Corresponding parameters for the inclusion will be distinguished by adding a prime superscript. The frame moduli may be measured directly [55-57] or they may be estimated using one of the many methods developed to estimate elastic constants of composites [58,59]. For long-wavelength disturbances ( > h, where h is a typical pore size) propagating through such a porous medium, we de ne average values of the (local) displacements in the solid and also in the saturating uid. The average displacement vector for the solid frame is ~u while that for the pore uids is ~uf . The average displacement of the uid relative to the frame is ~ w = (~uf ~u). For small strains, the frame dilatation is (37) e = ex + ey + ez = r ~u; where ex; ey ; ez are the Cartesian strain components. Similarly, the average uid dilatation is (38) ef = r ~uf (ef also includes ow terms as well as dilatation) and the increment of uid content is de ned by (39) = r ~ w = (e ef): With these de nitions, Biot [1,2,30] shows that the strain-energy functional for an isotropic, linear medium is a quadratic function of the strain invariants [60] I1 = e; I2, and of having the form (40) 2E = He2 2Ce +M 2 4 I2; where (41) I2 = eyez + ezex + exey 1 4( 2 x + 2 y + 2 z ); and x; y ; z are the shear strain components. Our earlier de nitions (5) and (6) for partial saturation are completely consistent [42,61] with these de nitions. With time dependence of the form exp( i!t), the Fourier transformed version of the coupled wave equations of poroelasticity in the presence of dissipation take the form (42) r2~u+ (H )re Cr + !2( ~u+ f ~ w) = 0; (43) Cre Mr + !2( f~u+ q ~ w) = 0; where (44) = f + (1 ) m and (45) q = f [ = + iY ( ) = !]: The kinematic viscosity of the liquid is , the permeability of the porous frame is , and the dynamic viscosity factor [2,62] is given (for our present choice of sign for the frequency dependence) by (46) Y ( ) = 14 T ( )=[1 + 2T ( )=i ]; where (47) T ( ) = ber0( ) ibei0( ) ber( ) ibei( ) and (48) = (!h2= ) 1 2 : The functions ber( ) and bei( ) are the real and imaginary parts of the Kelvin function. The dynamic parameter h is a characteristic length generally associated with (and comparable in magnitude to) the steadyow hydraulic radius. The electrical tortuosity is a pure number related to the frame inertia which has been measured [27] for porous glass bead samples and has also been estimated theoretically [12,24]. The electrical tortuosity and the uid ow tortuosity are related by = 2 = F , where F is the electrical formation factor. The coe cients H;C; and M are given by [37,53] (49) H = K + 43 + C; (50) C = [( )=Km + =Kf ]= 1; (51) M = C= ; where (52) = 1 K=Km: The wave equations (42) and (43) decouple into Helmholtz equations for three modes of propagation if we note that the displacements ~u and ~ w can be decomposed as (53) ~u = r +r ~ ; ~ w = r +r ~ ; where , are scalar potentials and ~ ,~ are vector potentials. Substituting (53) into Biot's equations (42) and (43), we nd they are satis ed if two pairs of equations hold: (54) (r2 + k2 s)~ = 0; ~ = s~ ; where s = f=q and (55) (r2 + k2 )A = 0: In this notation, the subscripts +, , and s refer respectively to the fast and slow compressional waves and the shear wave. The wave vectors in (54) and (55) are de ned by (56) k2 s = !2( f s) and (57) k2 = (!2=2 )(b+ f [(b f)2 + 4cd] 1 2 ); where (58) b = M fC; c = fM qC; d = fH C; f = qH fC; with (59) = MH C2: The linear combination of scalar potentials has been chosen to be (60) A = + ; where (61) = d=[(k =!2)2 b] = [(k =!2)2 f ]=c: With the identi cation (61), the decoupling is complete. Since (54) and (55) are valid for any choice of coordinate system, they may be applied to boundary value problems with arbitrary symmetry. Biot's theory has therefore been applied to the scattering of elastic waves from a spherical inhomogeneity [4]. The results of that calculation will be summarized in the next section. 4. Scattering from a Poroelastic Spherical Inclusion. The full analysis of scattering from a spherical inhomogeneity in a uid-saturated isotropic porous medium is quite tedious. Fortunately, much of this work has already been done [4] and we may therefore merely quote the pertinent results here. Let the spherical inhomogeneity (see Figure 1) have radius a. For the present, we place no restrictions on the properties of the inhomogeneous region. Thus frame bulk and shear moduli, grain bulk modulus, density, porosity, and permeability of a solid inclusion may all di er from those of the host. Furthermore, bulk modulus, Figure 1. A spherical inclusion in a porous medium could be the result of local variations in uid content, grain composition, porosity, permeability, etc. density, and viscosity of the uid in an inhomogeneous region may also all di er from those of the host uid. Suppose now that a plane fast compressional wave is generated at a free surface far from the inclusion. Then, if the incident fast compressional wave has the form (62) ~u = ẑ A0 ik+ exp i(k+z !t); the radial component of the scattered compressional wave contains both fast and slow parts in the far eld and is given by (63) u1r = (ik+) 1exp i(k+r !t)=k+r[B(+) 0 B(+) 1 cos B(+) 2 (3 cos2 + 1)=4] (ik ) 1exp i(k r !t)=k r[B( ) 0 B( ) 1 cos B( ) 2 (3 cos 2 + 1)=4]: Then, with the de nitions = k a and s = ksa and with no restrictions on the materials, we nd that (64) B( ) 0 = i 3 A0 3M 0( + )(K 0 + 4 3 )h(C M )(K 0 + 43 ) (C 0 M 0 )(K + 43 ) + (C M )(C 0 M 0 ) C 0 M 0 C M i; and (65) B(+) 0 = 3+A0 3i [K 0 K + (C M )(C 0=M 0 C=M)] K 0 + 4 3 + ( += )3B( ) 0 : Expansions of the other coe cients in the small parameter = C=K have been given in [4]. However, for the present application, only the rst two coe cients are needed and these happen to be the ones known exactly at present. Of course, the full scattered wave also contains transverse components of the compressional wave, relative uid/solid displacement, and mode converted shear waves. However, the scattering coe cients for these contributions are linearly dependent on the the coe cients in (63) and therefore contain no new information. It is su cient then to base our discussion on the expression (63). As an elementary check on our analysis, we may rst consider the limit in which the porosity vanishes. Then the uid e ects disappear from the equations and only the rst line of (63) survives. Furthermore, it is not di cult to check [4] that the coe cients B(+) n for n = 0; 1; 2 reduce to the well-known results for scattering from a spherical elastic inclusion in an in nite elastic medium [58]. For example, (66) B(+) 0 = i 3+A0(K 0 K)=(3K 0 + 4 ) in this limit as expected. These results have a multitude of potential uses. One straightforward application is the calculation of energy losses from elastic wave scattering by randomly distributed particles. A second important application is to use these results as the basis for an e ective medium approximation for the e ective constants of complex porous media. The second application is the one we address in the next section. 5. Generalization of Gassmann's Equations. As noted previously, the equations of poroelasticity have several signi cant limitations. For example, these equations were derived with an explicit long-wavelength (low-frequency) assumption and also with strong implicit assumptions of homogeneity and isotropy on the macroscopic scale. Another restriction arises from the assumption that the pore uid is uniform and that it fully saturates the pore space. For the present application, we assume that a single uid saturates all the pore space for host as well as inclusion and scattering is caused by microscopic heterogeneity only in the solid properties. Before deriving the main result, consider the problem of the porous frame without a saturating uid (or with a highly compressible saturating gas). Then, since we take C =M = f = 0 in this limit, each term of Eq. (50) vanishes identically and the uid dependent terms of Eq. (49) also vanish, leaving only the terms for the elastic behavior of the porous frame remaining. Since no slow wave can propagate under these circumstances, the second line of Eq. (63) disappears and only the fast wave terms contribute to the scattering. This limit is formally equivalent to the problem of elastic wave scattering from a spherical inclusion that has been treated in detail previously (see [58] and other references therein). One e ective medium approximation (that we call the coherent potential approximation or CPA) requires the volume weighted average of the single-scattering results to vanish. This method simulates the physical requirement that the forward scattering should vanish at innity if the impedance of the \e ective medium" has been well matched to that of the composite. The resulting condition is that the volume weighted average of each of the B(+) n s for n = 0 2 must vanish. Using the convention that the e ective constants for the composite porous medium are distinguished by an asterisk, the formulas for the e ective bulk (K ) and shear( ) moduli for the drained porous frame of a microscopically heterogeneous medium are (67) 1 K + 43 = D 1 K(~x) + 4 3 E and (68) 1 + F = D 1 (~x) + F E where (69) F = ( =6)(9K + 8 )=(K + 2 ): The spatial(~x) average is denoted by h i. The remaining constant to be determined is the e ective density which is just the average density [4]. Equation (67) follows easily from the volume average of (66), while (68) follows similarly from the volume average of B+ 2 . Note that the equations for K and are coupled and therefore must be solved iteratively (i.e., self-consistently). Although the form of the equations (67) and (68) is identical to that obtained for elastic composites, the results can be quite di erent since the local constants K(~x) and (~x) appearing in the formulas are frame moduli of the constituent spheres of drained porous material, not (necessarily) the moduli of the individual material grains. Of course, since the formula reduces correctly in the absence of porosity to the corresponding result for the purely elastic limit, the user of Eqs. (67) and (68) has some discretion about conceptually lumping grains together to form a porous frame or treating them as isolated elastic inclusions. Now we will restrict discussion to the very low frequency limit where (70) + = H=C and (71) = 0: With these restrictions, the relevant scattering coe cients reduce to (72) B( ) 0 = i 3 CA0 3HM 0(K 0 + 4 3 )hC(K 0 + 43 + 0C 0) C 0(K + 4 3 + C)i; and (73) B(+) 0 = 3+A0 3i [K 0 K + ( 0 )C] K 0 + 4 3 + ( += )3B( ) 0 : The resulting conditions on the e ective constants are (74) DC K(~x) + 43 + (~x)C(~x) C(~x) K + 4 3 + C M(~x) K(~x) + 4 3 E = 0 and (75) DK(~x) K + (~x) C K(~x) + 4 3 E = 0: Recall that the averages in (74) and (75), as elsewhere in this paper, refer to spatial averages over (possibly) porous constituents of the overall porous aggregate. The limitations on the assumed geometry of the resulting aggregate have been discussed previously [63]. Note that (74) and (75) depend on the e ective medium frame moduli K and determined by (67) and (68). The new constants determined by (74) and (75) are C and . The expressions for C and are coupled as written but may be decoupled after some algebra. The nal expressions for these constants are (76) C = .hD 1 M(~x)E+ D 2(~x) ( )2 K(~x) + 4 3 Ei and (77) = D (~x) K(~x) + 4 3 E.D 1 K(~x) + 4 3 E: Notice that both constants are determined explicitly by the formulas, in contrast to the frame moduli K and which are determined only implicitly by (67) and (68). The author has also shown [63,64] that (76) and (77) are completely consistent with all known constraints [37,53] on the form of these coe cients. Furthermore, the coherent potential approximation (CPA) treated here is just one of three approximations | including the average T-matrix approximation (ATA) and the di erential e ective medium (DEM) | all of which satisfy the known constraints on coe cients [64]. The same idea used to derive (76) and (77) has also been used to show [65] that the speed of waves propagating through a mixture of liquid and gas in the low frequency limit is given by Wood's formula [66] as expected [37,40]. Recently Berryman and Milton [67] have also derived exact results for generalized Gassmann's equations in composite porous media with two constituents. Some of these exact results were rst obtained [64] using the single-scattering approach summarized here. 6. Anomalous Dissipation Caused by Inhomogeneous Fluid Permeability. A convincing demonstration has been given by Mochizuki [68] that, if we assume global uidow e ects dominate the viscous dissipation, Biot's theory of poroelasticity cannot explain the observed magnitude of wave attenuation in partially saturated rocks. Since the same theory explains the wave speeds quite well, it is reasonable to suppose that a small change in the theory may be adequate to repair this aw. Many explanations are possible of course, but within the context of Biot's theory the simplest postulate is to suppose that local { rather than global { uidow e ects dominate the dissipation [28,29]. We will distinguish two related issues in this section which are summarized in the following questions: (a) Does the physics of wave propagation require that the value of the permeability appearing in Biot's equations should be that for global ow or for local ow? Then, if we can show that the value should be that for local ow, (b) does this change in the interpretation make enough di erence so the theory can explain the correct magnitude for the attenuation? To address the rst question, we explore the consequences of assuming that Biot's theory should be applied at the local ow level rather than at the global ow level. This assessment is easily done by examining the dispersion relations. When the Fourier time dependence is e i!t with angular frequency ! su ciently low, Biot's theory predicts [see Eq. (57)] the dispersion relations for the fast (+) and slow ( ) compressional modes in any homogeneous porous material to be (78) k2 + ' !2 v2 + h1 + i! 2f q0 (1 v2 0=v2 +)2i and (79) k2 ' i!q0H MH C2 where (80) v2 + = H= ; v2 0 = C= f ; q0 = f = : The fraction of the total volume occupied by the uid is the void volume fraction or porosity , which is assumed to be uniform. The bulk modulus and density of the uid are Kf and f . The bulk and shear moduli of the drained porous frame are K and . For simplicity we assume the frame is composed of a single constituent whose bulk and shear moduli and density are Km, m, and m. Then the coe cients H;C; and M are given by (49){(52). The overall density is (81) = f + (1 ) m: The kinematic viscosity of the uid is and the permeability of the porous frame is .De ning the quality factor for the fast compressional wave Q+ by (82) k2 + = !2 v2 + [1 + i=Q+]; we nd [69] that Q+ is given by (83) 1=Q+ = ! f (1 v2 0=v2 +)2: Since 1=Q+ is proportional to the permeability, the attenuation is therefore greatest in regions of high permeability. Thus, we might say that the regions of high permeability control the attenuation. In the very low frequency limit, the slow compressional mode is known to reduce to Darcy ow with slowly changing magnitude and direction as the driving potential gradient oscillates sinusoidally [70{74]. Now consider a layered porous material (whose constants depend only on depth z) with constituents having identical physical constants except for the permeability which varies widely from layer to layer but which has a constant value n within the n-th layer (lying in the range zn 1 z zn with z0 = 0). Thus, the permeability is a piecewise constant function of z. The thickness of the n-th layer is given by ln = zn zn 1. If we impose a potential gradient along the z-direction in such a layered material, it is well-known that the e ective permeability for uid ow is found by taking the harmonic mean of the constituent permeabilities, i.e., (84) 1= f = 1 LXn ln n ; where the total sample length L is given by the sum of the layer thicknesses (85) L =Xn ln: From (84), we can conclude that the regions of lowest permeability dominate the e ective overall permeability for uid ow through a porous layered medium. Thus, we might say that the regions of low permeability control the uid ow { at least for this special choice of geometry. The apparent attenuation of a fast compressional mode at normal incidence on such a structure has two distinct components: (a) Re ection and mode conversion at layer interfaces will have a tendency to degrade the fast wave, but this e ect will be quite small at low frequencies for the model structure we are considering. (b) The attenuation within a layer is determined by the quality factor for that layer, as shown by Eq. (82). Assuming the attenuation is small enough, we may approximate (82) within any layer by k+(z) = (!=v+)[1 + i=2Q+(z)], where the functions k+(z) and Q+(z) take the piecewise constant values appropriate for the depth argument z. Neglecting the small e ects of re ection and mode conversion, the behavior of the fast compressional wave at normal incidence is then easily seen to be given by (86) A+exphi z Z0 dzk+(z) i!ti ' A+exphi ! v+ z i!t ! v+ z Z0 dz 1 2Q+(z)i; where A+ is the amplitude of the wave at z = 0. In writing (86), we have used the piecewise constant property of the functions. The integral in the exponent is given by (87) z Z0 dz 1 2Q+(z) = ! f 2 (1 v2 0=v2 +)2 z Z0 dz (z): At the z = L boundary of the material, we have (88) L Z0 dz (z) =Xn ln n: If the layering is periodic with period much less than either z or L or if it is statistically homogeneous on this length scale, then we may approximate the integral in the exponent of (86) using (87) and (89) z Z0 dz (z) ' az; where the e ective permeability for attenuation measurements is given by the mean (90) a = 1 LXn ln n: It is well-known that the mean is always greater than or equal to the harmonic mean of any function; thus, (91) f a: In answer to our rst query: the physics of wave propagation does dictate that localow e ects dominate the attenuation of the fast compressional wave. The necessity of this conclusion is nicely illustrated in Figure 2. Suppose that a fast compressional wave is incident on a layered material with alternating permeable and impermeable layers. If the impermeable layers are very thin and have an acoustic impedance closely matching that of the permeable layers, their presence has a negligible e ect on the propagating fast wave. The viscous attenuation of the fast wave occurs solely in the permeable layers and magnitude of that attenuation is completely determined by the permeability of these layers. By contrast, the global permeability of this material in the direction normal to the layering vanishes identically. If this null value were used in our predictions, the magnitude of the attenuation would be grossly underestimated. Although this choice of geometry is extreme, it clearly shows that errors in estimates of attenuation will arise if the value of permeability for global ow is used. Now, can the theory predict the correct magnitude for the attenuation even with this change in the interpretation of the permeability factor? To predict the wave attenuation from measurements of permeability, we need some independent means of measuring the local permeability distribution. Normal laboratory ow experiments will not su ce, because they necessarily measure the global permeability. One promising method of estimating the local permeability uses image processing techniques to measure pertinent statistical properties of rock topology from pictures of cross sections [75,76]. This approach is still under development and we will not attempt to describe it in detail here. Another approach, which is ultimately much less satisfactory than the image processing method but much easier to use at present, is to suppose that we can obtain reasonable estimates of the local permeability L from the known values of the global permeability G, the tortuosity = ( F ) 1 2 , and the porosity . To do so requires some formula, so we will use a form of the Kozeny-Carman relation derived by Walsh and Brace [77]. For tubes of arbitrary ellipsoidal (major and minor axes a,b) cross-section the e ective permeability of straight sections of such tubes is given by = ( =4A)[a3b3=(a2 + b2)]. The porosity for an ellipsoidal tube is Figure 2. Illustration of a simple experiment to prove that the attenuation of the fast wave depends on the local { not the global { value of permeability . A fast wave incident normal to the impermeable partitions will experience a small but nite attenuation even though the global permeability in this direction vanishes identically. = ab=A and the speci c surface area is well approximated by s ' 2 [(a2+ b2)=2] 1 2 =A. Then, a Kozeny-Carman relation satis ed by , , and s can be shown to be (92) = 1 2 3 s2 for the e ective permeability of a single tube oriented along the pressure gradient. If the tube is at an angle to this gradient, then Walsh and Brace [1984] show that (93) = 12 3 s2 2 ; where = 1=cos . If we suppose that (92) and (93) are fairly representative of the material of interest, then (92) describes the maximum local permeability L and (93) the e ective global permeability G. We then conclude that (94) L = 2 G = F G: The tortuosity has been measured for many sandstones; the values for samples studied by Simmons, Wilkens, Caruso, Wissler, and Miller [78{80] lie in the range 25, with most values ' 2. To obtain estimates of attenuation close toexperiment [39], we need to increase the value of permeability used in Mochizuki'scalculations [68] by a factor of 2 ' 10. This requirement implies a tortuosity of ' 3,which is clearly well within the established experimental bounds. A more detailedanalysis leading to the same qualitative conclusions has also been presented [69].These arguments provide strong evidence for the plausibility of a localow explana-tion of the observed discrepancies. However, a completely satisfying demonstrationmust await the collection of the required data on localow permeability.One unfortunate consequence of the observation that local permeability controlsattenuation is that measured attenuation in wet rocks cannot be used directly asa diagnostic of the global uidow permeability. Since the mean of the localpermeabilities will always be greater than the true uidow permeability regardlessof the actual spatial distribution of the constituent 's, the e ective permeabilitycomputed from attenuation measurements can nevertheless be used to provide anupper bound on the desired global permeability.Although the subject is really beyond the scope of this paper, in the contextof the present volume on Wave Propagation and Inversion we should mention thatindirect means of measuring the global permeability may still be viable. Seismicattenuation tomography [81] may be used to backproject e ective local attenuationin a region from global attenuation data (line integrals). From estimates of localattenuation, we may deduce estimates of local permeability. Then, from a regionalmap of local permeability estimates, we can in principle compute the global per-meability. Thus, although our results show that a simple direct measurement ofglobal permeability is impossible, it certainly does not prevent us from obtainingthe desired information from regional attenuation data.7. Discussion. Why should we care about poroelasticity in general and slowwaves in particular? Both the theory and the preponderance of experimental resultshave shown that, for earth materials containing some uid at normal temperatures,the attenuation of slow compressional waves is so strong in the relevant frequencyrange (10{1000 Hz) that it is extremely unlikely that propagating slow waves willever be directly observed in a eld experiment. If we can ignore the slow waves, thenthe theory reduces to elasticity or viscoelasticity { which is clearly advantageousboth conceptually and computationally. However, we miss something important ifwe try to compute wave propagation e ects in the earth without using the equa-tions of poroelasticity. The indisputable experimental evidence for the existence ofthe slow compressional mode in real materials [11{13,15,18,21,27,34] implies thatmode conversions occur at every interface in a complex medium; a fast compres-sional wave striking any boundary (even at normal incidence) is partially re ected,partially transmitted, and partially converted into transmitted and re ected slowcompressional modes [16]. Even if we never see a propagating slow wave in theeld, the fast wave feels its presence as an additional attenuation mechanism thatoperates at every interface. The slow wave therefore gradually bleeds energy out ofthe propagating fast compressional wave into a highly damped viscous motion ofuid in the pores. Thus, the slow wave itself acts as an additional source of unac-counted for (and therefore anomalous) attenuation for those easily measured wavesthat do propagate. This attenuation mechanism is not predicted by the theories of elasticity or viscoelasticity. It may be possible to incorporate such e ects intothese simpler theories, but it seems more natural to use the theory that predicts thephenomenon. This is one practical reason why we should care about poroelasticityand why it is important to develop a comprehensive theory.What then are the prospects for a comprehensive theory of poroelasticity? Itappears likely that we will soon have a completely satisfactory linear theory of bulkwaves including e ects of partial saturation and inhomogeneous frame materials. Asatisfactory nonlinear theory of bulk waves including e ects of fracture, plastic ow,and pore collapse is at a more elementary stage, but is still likely to be achieved bythe turn of the century. At present it appears that the most troublesome problemsare those involving surface waves rather than the bulk waves. Surface waves de-pend critically on the nature of the equations of motion near interfaces. 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