Acoustoelasticity theory and applications for fluid-saturated porous media Michael A. Grinfeld and Andrew N. Norris Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08855-0909 ͑Received 17 April 1995; accepted for publication 6 May 1996͒ The general theory for small dynamic motion superimposed upon large static deformation, or acoustoelasticity, is developed for isotropic fluid-filled poroelastic solids. Formulas are obtained for the change in acoustic wave speeds for arbitrary loading, both on the frame and the pore fluid. Specific experiments are proposed to find the complete set of third-order elastic moduli for an isotropic poroelastic medium. Because of the larger number of third-order moduli involved, seven as compared with three for a simple elastic medium, experiments combining open-pore, closed-pore, jacketed, and unjacketed configurations are required. The details for each type of loading are presented, and a set of possible experiments is discussed. The present theory is applicable to fluid-saturated, biconnected porous solids, such as sandstones or consolidated granular media. © 1996 Acoustical Society of America. PACS numbers: 43.25.Dc, 43.25.Ba, 43.20.Jr ͓MAB͔
INTRODUCTION An isotropic poroelastic medium has four static moduli that may be measured by a combination of stress-strain experi- The linear theory of poroelastic fluid-saturated media is ments on ‘‘jacketed’’ and ‘‘unjacketed’’ samples. The role of a mature subject, with the classic studies of Frenkel1 and the jacketing is to constrain either the fluid pressure or its especially of Biot2 standing as the significant achievements mass. The nonlinear moduli, in contrast, should be deter- in its modern development. Several different methods now exist for arriving at governing equations, including the mined by measurement of essentially nonlinear effects. One theory of interpenetrating continua3 and the method of ‘‘ho- of the simplest nonlinear phenomena is the elastoacoustic mogenization’’ based on two-scale asymptotic expansions.4,5 effect, whereby the speeds of small amplitude waves are Despite the fact that these approaches do not always yield the changed by applying stress or strain. The theory of acous- toelasticity has been thoroughly discussed for purely elastic same governing equations and that further general theoretical 20–22 studies are required, they have certainly proved useful in materials, and is now commonly used in ultrasonics;23–25 it has also been adapted to multiphase several branches of civil and geo-engineering and the Earth 26 sciences,6–8 and also in unexpected fields, such as low tem- materials. perature physics9 and pattern formation in polymer gels.10 The subject of this article is the acoustoelastic effect However, there are definite limitations to the linear theory. within the context of poroelasticity. The theory is based on a For example, a proper analysis of the dynamics of large am- nonlinear generalization of the classic Biot theory. Biot him- plitude sound in sediments or of small amplitude sound in self discussed thermodynamic aspects of a nonlinear theory rocks under large confining stress is obviously impossible of poroelastic media, but did not specify any nonlinear within the framework of a linear theory. Even seismic waves stress–strain behavior ͑see papers 15, 16, 18, and 19 in Ref. of very small intensity at a large distance from their epicenter 2͒. There has been some work on nonlinear poroelasticity should be studied in the framework of a nonlinear theory within the framework of the theory of interpenetrating 3 because of the well-known effect of the accumulation of continua, and also using the method of two-scale 27,28 nonlinear distortions leading to the ‘‘gradient catastrophe’’ homogenization. Three sources of nonlinearity are tradi- phenomenon.11,12 tionally distinguished in elasticity: ͑i͒ the physical nonlinear- There is renewed interest in the nonlinear acoustics of ity, that is, the nonlinearity of the constitutive relations relat- rocks that is based partly on several observations of distinct ing the stress and the displacement gradient; ͑ii͒ nonlinearity nonlinear effects. These include the change of the in situ of the universal equations, such as the equations of conser- velocities of seismic waves13–15 and direct measurement of vation of mass, momentum, etc.; ͑iii͒ geometric nonlinearity, harmonic distortion.16,17 These experiments have clearly which results from the nonlinear relationship between the demonstrated that the in situ nonlinear elastic moduli of deformation gradient and the tensor of finite deformations. rocks, soil, and sediments are much greater than the linear In order to keep the analysis as simple as possible, in ones. The nonlinear behavior of fluid-filled poroelastic solids this article we ignore all dissipative effects and concentrate is not as well understood, and further progress depends upon on the first kind of nonlinearity only because it appears to be accurate experimental measurement of the effective nonlin- the most significant for geophysical materials. We begin with ear moduli of poroelastic media. a discussion of acceleration waves in nonlinear poroelastic The determination of the linear moduli has been the sub- media. These are exact solutions independent of the state of ject of numerous experimental and theoretical studies.7,18,19 prestress in the medium, and they reduce to the well-known
1368 J. Acoust. Soc. Am. 100 (3), September 1996 0001-4966/96/100(3)/1368/7/$10.00 © 1996 Acoustical Society of America 1368 Biot fast and slow waves for unstressed isotropic materials. poses geometric and kinematic constraints on possible jumps These general results serve as the basis for considering the of the second derivatives since the ‘‘tangential’’ components principal body waves propagating in a slightly prestressed of the second derivatives are continuous. These constraints isotropic nonlinear saturated poroelastic substance. Finally, are known as compatibility conditions and are, essentially, we discuss application of the general formulas for the stress geometric conditions originally developed by outstanding dependence of the wave speeds to several specifically de- geometers like Hadamard,29 Levi-Civita,32 and Thomas30 signed experimental configurations. ͑fortunately all three were outstanding mathematical physi- cists so their works are accessible to physicists͒. In particu- I. NONLINEAR POROELASTICITY lar, the jumps of second derivatives across acceleration wavefronts satisfy31,33 A. Governing equations 2 ϩ ui ץ All further analysis is based on the following governing ͓u ͔ϩϭh n n , ϭh c2, ͑4a͒ t2 ͬ iץ ͫ i, jk Ϫ i j k equations of the nonviscous, fluid-filled poroelastic medium: Ϫ 2 i 2 i w 2w ϩ ץ u ץ i 2 ץ ij ϩ 2 ϩ f 2 ϭ, j , ͑1a͒ ͓w ͔ ϭH n n , ϭH c , ͑4b͒ t2 ͬ iץ ͫ t i, jk Ϫ i j kץ tץ Ϫ 2 i 2 uj ϩ p q ϩ p q ץ w ץ ij ij where hkϭ[ui,pq]Ϫn n and Hkϭ[wi,pq]Ϫn n are the f ϩ f K ϭϪK p,j , ͑1b͒ t2 second-order amplitude vectors of discontinuity, n j is theץ t2ץ where unit normal to the wavefront, and c is the velocity of the front. The compatibility conditions indicate that the two vec- Wץ Wץ ij tor functions hk and Hk completely define the jumps of 60 ϭ ͑,um,n͒, pϭ ͑,um,n͒. ͑2͒ . partial derivatives of the displacementsץ ui,jץ In order to determine the acceleration vectors them- Here, ui is the displacement of the solid skeleton, wi is the selves one has to extract some additional dynamic informa- relative displacement of the fluid, ij are the stresses, p is the tion from the governing Eqs. ͑3͒. These equations are not fluid pressure, W(,u ) is the poroelastic potential, and m,n valid at the wavefront since the second derivatives are unde- are the averaged density and the density of the fluid, re- f fined at the front. By definition, only one-sided second de- spectively, Ϫϭwi is a divergence of the relative displace- ,i rivatives are defined at the front, and these one-sided limits ment of the fluid, and Kij is the ‘‘instantaneous’’ magnitude satisfy Eqs. ͑3͒. We first subtract, termwise, Eq. ͑3a͒ for the of a symmetric tensor of permeability ͑the value of the per- two one-sided limits taking into account continuity of the meability hereditary operator at tϭ0͒. Also, xi are the spatial first derivatives and then do the same operation with Eq. coordinates, the Latin indices take the values 1, 2, and 3; 3b . Then, evaluating the jumps of the second derivatives summation over repeated indices is implied, and a comma ͑ ͒ using the second order compatibility conditions of Eqs. 4 , followed by a Latin suffix symbolizes partial differentiation. ͑ ͒ we get the following linear algebraic system for the ampli- We refer to the Refs. 2, 4, 27, and particularly Ref. 28 and tude vectors: for the motivation behind Eqs. ͑1͒ and ͑2͒. By inserting the value of in terms of wi we can rewrite 2 ik ijkl 2 ik ij k ͑c ␦ ϪW njnl͒hkϩ͑fc ␦ ϩW njn ͒Hkϭ0, the equations of motion as ͑5a͒
2 i 2 i 2 ik kl i 2 ik i k ,w ij k c ␦ ϩW n n ͒h ϩ c K ϪW n n ͒H ϭ0 ץ u ץ ϩ ϭWijklu ϪW w , ͑3a͒ ͑ f l k ͑ f inv k t2 k,jl ,jk ͑5b͒ץ t2 fץ
2 i 2 ij ij , uj where Kinv is the inverse of the permeability matrix K ץ w ץ ϩ Kij ϭϪKij͑Wklu ϪW wk ͒. ͑3b͒ .t2 k,lj ,kj which is assumed to be invertibleץ t2 fץ f The pressure p is continuous across the wavefront, but In these and subsequent equations we use the following no- its gradient is not, specifically, tation for the derivatives of the potential energy function W: ij 2 2 ij 2 ͓p ͔ϩϭqn . ͑6͒ , ,i Ϫ iץ ui,jץ/W ץ , W ϭץ/W ץui,j , Wϭץ/WץW ϭ ijkl 2 uk,l , etc. At the same time, it follows from the Eqs. ͑2͒ and ͑4͒ thatץ ui,jץ/W ץW ϭ
ϩ kl k B. Acceleration waves ͓p,i͔ϪϭW ninlhkϪWnin Hk . ͑7͒ An acceleration wavefront is a propagating surface de- Combining the previous two equations allows us to eliminate k fined such that displacements and their first derivatives are the quantity n Hk in favor of the jump in the pressure gra- continuous across this surface, but the second and higher dient, q. Then Eq. ͑5b͒ provides an expression for Hi, derivatives and, in particular, the accelerations possess finite q jumps. Acceleration waves are the most convenient object of i ij H ϭϪK hjϩ 2 nj . ͑8͒ theoretical study for both linear and nonlinear dynamics. ͩ fc ͪ 29 They have been investigated thoroughly by Hadamard and After some manipulation, Eqs. ͑5͒ imply that by Thomas,30 while Chen31 provides a more recent review ik 2 ˜ijkl ˜ij for elastic materials. Continuity of the first derivatives im- ͑˜ c ϪW njnl͒hkϪK njqϭ0, ͑9a͒
1369 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1369 ij K ninj 1 When the material is isotropic then all directions are K˜ kln h ϩ Ϫ qϭ0, ͑9b͒ l k ͩ c2 W ͪ principal directions, i.e., all acceleration waves are either f longitudinal or transverse in nature. where ik ik ik ˜ ϭ␦ ϪfK , ͑10a͒ II. THE ACOUSTOELASTIC EFFECT ˜ ijkl ijkl ij kl W ϭW ϪW W /W , ͑10b͒ A. Third-order moduli for poroelasticity ˜ ik ik ik We now consider the effects of prestress on the longitu- K ϭK ϩW /W . ͑10c͒ dinal and transverse waves in an isotropic medium. The per- Equations ͑9͒ form a system of four linear algebraic equa- meability for an isotropic poroelastic medium is Kijϭ␦ij, tions with respect to the four unknowns hk and q. This sys- while the potential W is a function of and the principal tem has a nonzero solution when the determinant vanishes, invariants I of the symmetric strain ϭ(u ϩu )/2. 2 M ij i,j j,i leading to a fourth order polynomial equation in c . The Thus coefficients of the polynomial depend on uk,l , , and the WϭW͑I1 ,I2 ,I3 ,͒, ͑16͒ orientation of the unit normal nk , and thus the same is true for the velocities c. Each real root generates an associated where nonzero solution (hk ,q). Further discussion of the fourth- ij 1 qj rk rj qk order system can be found in Ref. 34. On the other hand, the I1ϭ␦ ij , I2ϭ2͑␦ ␦ Ϫ␦ ␦ ͒jqkr , system defined by Eqs. ͑5͒ is sixth order, because it involves 1 ijk pqr ͑17͒ I3ϭ 6⑀ ⑀ ipjqkr , the unknowns hi and H j . The fourth and sixth order systems ijk are connected by the fact that the latter possesses the root and ⑀ is the third-order alternating tensor. To within third- c2ϭ0 of multiplicity 2. Thus, when c2ϭ0, the system of Eqs. order terms in ij and the potential W can be approximated by the polynomial ͑5͒ has infinitely many solutions of the form hkϭ0, HkϭRk , where R is an arbitrary vector in the two-dimensional ͑2-D͒ 2 2 3 k Wϭ⌫11I1ϩ⌫2I2ϩ⌫1I1ϩ⌫ ϩ⌫111I1ϩ⌫12I1I2 space orthogonal to the wave normal nk . In summary, the 2 2 3 propagating waves and the nonzero wave speeds can be ϩ⌫3I3ϩ⌫1I1 ϩ⌫11I1ϩ⌫2I2ϩ⌫ . found from either the fourth-order or the sixth-order systems. ͑18͒ We consider the following field of perturbations with respect to the reference ͑unstressed͒ configuration: C. Principal directions j j A principal direction is defined as one in which an ei- ␦uiϭ ␣Nn jx niϩ ␣Ax eAj , ␦ϭ⌬, ͑19͒ A͚ϭ1,2 genvector, such as hi, is aligned with or orthogonal to the ͩ ͪ i wave normal n . For example, let us consider a longitudinal where e1i and e2i are two mutually orthogonal transverse i wave in the direction of ni , that is, a solution of the form vectors, i.e., both are orthogonal to the n direction. The perturbation magnitude is defined by , with ͉͉Ӷ1byas- hkϭhnk , HkϭHnk . ͑11͒ sumption, and the parameters ␣N , ␣A , Aϭ1,2, and ⌬ are Substituting from Eq. ͑11͒ into Eqs. ͑5͒ and contracting with O͑1͒ and independent. The prestrain is, from Eq. ͑19͒, ni gives 2 ijkl 2 il ijϭ ␣Nnjniϩ ͚ ␣AeAieAj . ͑20͒ ͑c ϪW ninjnknl͒hϩ͑fc ϩW ninl͒Hϭ0, ͑12a͒ ͩ Aϭ1,2 ͪ 2 il 2 ik ͑ f c ϩW ninl͒hϩ͑fc KinvninkϪW͒Hϭ0. ͑12b͒ Differentiating Eq. ͑18͒ in the vicinity of the reference configuration ͑ϭ0͒ we obtain the second order elasticity It follows immediately from Eqs. ͑12͒ that the corresponding tensors in terms of the elastic moduli: eigenvalues c2 are defined by a biquadratic characteristic ijkl ij kl il jk ik jl equation, whereas the amplitudes h and H are connected by W ϭc␦ ␦ ϩ͑␦ ␦ ϩ␦ ␦ ͒, the relation ij ij ͑21͒ W ϭϪ␣M␦ , WϭM, 2 ijkl 2 c ϪW ninjnknl H ϭ . ͑13͒ where , ␣, and M are the Biot parameters.35 In fact, Biot 2 ik h2 c fc KinvninkϪW defines the linearized stress–strain relations in terms of a
A transverse wave, on the other hand, satisfies hkϭhek , quadratic energy potential similar to Eq. ͑18͒. Comparison of i k 35 HkϭHek , where e is a polarization vector with n ekϭ0 and the quadratic terms in the latter with Eq. ͑3.4͒ of Biot im- k e ekϭ1. Under these circumstances, Eqs. ͑5͒ become mediately implies that 2 ik ijkl 2 i 1 ͑c ␦ ϪW njnl͒ekhϩfc e Hϭ0, ͑14a͒ cϭ2⌫11ϩ⌫2 , ϭϪ 2⌫2 , Mϭ2⌫ , ͑22͒ 2 ik kl i 2 ik ␣ϭϪ⌫ /͑2⌫ ͒. ͑ f c ␦ ϩW nln ͒ekhϩfc KinvekHϭ0, ͑14b͒ 1 implying that the transverse wave velocity is given by There are many different notations used for describing the linear response of saturated porous media, and even Biot’s 2 pq Ϫ1 Ϫ1 ijkl ct ϭ͓Ϫ f͑Kinvepeq͒ ͔ W eieknjnl . ͑15͒ own notation evolved from the time of his early work on
1370 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1370 consolidation ͑see paper 1 of Ref. 2͒ to his later work. His In particular, the split in the velocities of two transverse 1962 paper35 provides a good comparison of the notations waves with the same propagation direction but different po- used. larizations is, from Eq. ͑27͒, For the third-order quantities we get 2 2 ␦ctAϪ␦ctB ⌫3 ijkl ij kl 1 il jk ik jl W ϭ͑⌫2ϩ2⌫11͒␦ ␦ Ϫ 2⌫2͑␦ ␦ ϩ␦ ␦ ͒, ϭ . ͑28͒ ␣AϪ␣B 2͑Ϫf͒ ij ij ͑23͒ Wϭ2⌫1␦ , Wϭ6⌫ , In order to establish analogous formulae for longitudinal ijklmn 1 ikm jln principal waves we first write the system of Eqs. ͑5͒ as W ϭ͑6⌫111ϩ3⌫12͒␦ij␦kl␦mnϩ 4⌫3͑⑀ ⑀ 2 ϩ⑀jkm⑀ilnϩ⑀ilm⑀jknϩ⑀jlm⑀ikn͒ AXϭc CX, ͑29͒ 1 ij km ln kn lm kl im jn where Xϭ(h,H)T, and the symmetric matrices A, C are Ϫ 2⌫12„␦ ͑␦ ␦ ϩ␦ ␦ ͒ϩ␦ ͑␦ ␦ in jm mn ik jl il jk ijkl il ϩ␦ ␦ ͒ϩ␦ ͑␦ ␦ ϩ␦ ␦ ͒…. W ninjnknl ϪW ninl f Aϭ il , Cϭ Ϫ1. In the absence of poroelastic effects, the third-order elastic ͫ ϪW ninl W ͬ ͫffͬ moduli Wijklmn are equivalent to the standard third order ͑30͒ moduli C ,orC in the concise Voigt notation.23 In ijklmn IJK The inertial matrix C is again unchanged by the initial de- order to compare the moduli here with the more common formation, and the new wave speeds depend upon the varia- notation, we note the equivalences tion in A. By making use of Eqs. ͑18͒–͑23͒ we can approxi- 1 1 ⌫111ϭ 6C111 , ⌫12ϭ 2͑C112ϪC111͒, mate this to within first order in ,
1 ͑24͒ 0 1 ⌫3ϭ 2͑C111Ϫ3C112ϩ2C123͒. AϭA ϩA , ͑31͒
The connection with other notations for third-order moduli where can be inferred from the table in Green’s review23 that com- pares many different systems of notation. We note for future cϩ2␣M A0ϭ , reference the following formulas: ͫ ␣MMͬ Wijklmnn n n n ϭ6⌫ ␦mnϩ2⌫ ͑␦mnϪnmnn͒, ͑25a͒ ͑32͒ i j k l 111 12 6⌫ Ϫ2⌫ 1 111 11 Wijklmne n e n ϭϪ 1 mnϪ 1 mnϪnmnnϪemen . A ϭ␣N ϩ͑␣1ϩ␣2͒ i j k l 2⌫12␦ 2⌫3͑␦ ͒ ͫ Ϫ2⌫ 2⌫ ͬ ͑25b͒ 11 1 6⌫111ϩ2⌫12 Ϫ2⌫11Ϫ⌫2 ϫ B. Wave-speed dependence on the applied strain ͫϪ2⌫ Ϫ⌫ 2⌫ ͬ 11 2 1 For each deformed configuration there are three orthogo- 2⌫11 Ϫ2⌫1 nal directions of principal strain, and the small amplitude ϩ⌬ . waves that propagate in these directions are either longitudi- ͫϪ2⌫1 6⌫ ͬ nal or transverse. We will now derive approximative formu- The eigenvalue problem for the matrix CϪ1A0 was first stud- las for the incremental changes of the velocities of these ied by Biot in 1956 ͑see paper 8 of Ref. 2͒ when he estab- principal waves to leading order in . lished the existence of two different longitudinal modes: the First, the variation in the velocity of the transverse wave ‘‘fast’’ and ‘‘slow’’ waves with velocities c f and cs , respec- with the polarization vector eCi follows from Eqs. ͑15͒ and tively. The eigenvectors Xf and Xs associated with these ͑20͒ as velocities can be calculated by making use of the following 2 Ϫ1 ijkl equation implied by Eq. ͑13͒: ␦ctCϭ͑Ϫf͒ ␦W eCieCknjnl Ϫ1 ijklmn 2 2 ϭ͑Ϫf͒ ͑W eCieCknjnlmn c Ϫ͑cϩ2͒ H Ϫ1 2 ϭ 2 . ͑33͒ ijkl f c ϪM h ϩW eCieCknjnl⌬͒. ͑26͒ Corrections to the Biot fast and slow velocities due to the Note that the densities and f are unchanged because we are using the reference or Lagrangian description as opposed incremental deformations can now be calculated using stan- to the current or Eulerian description. Conservation of mass dard perturbation theory, leading to the formula requires that these densities are constant. The structure con- A1 X0iX0j 2 ij stant is also assumed to remain the same under the defor- ␦c ϭ 0i 0j . ͑34͒ mation. Combining Eqs. ͑20͒, ͑25͒, and ͑26͒ we arrive at the CijX X formula for the elastoacoustic effect for transverse waves, Equations ͑27͒, ͑32͒, and ͑34͒ show that by measurement of elastoacoustic effects we can, in principle, find the third or- 2 Ϫ ␦c ϭ ͑␣ ϩ␣ ϩ␣ ͒⌫ ϩ͑␣ ϩ␣ der elastic moduli by solving a system of linear algebraic tC 2͑Ϫ ͒ „ 1 2 N 12 1 2 f equations. In Sec. III we shall examine some possible experi- Ϫ␣C͒⌫3ϩ⌫2⌬…. ͑27͒ mental configurations with this purpose in mind.
1371 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1371 TABLE I. Different compressibilities of a porous medium, with definitions of Kc , K, and K M . Here, p and pc are the pore and confining pressure, respectively. Note that c , , ␣, and M are defined in Eqs. ͑22͒, and 2 ϭcϪ␣ M ͑Ref. 35͒. The relations between these and other moduli are explored in greater detail by Ku¨mpel ͑Ref. 36͒, and also by Brown and Korringa ͑Ref. 37͒.
In current Compressibility—definition notation Terminology, notation, and references
V 1 Closed compressibility ͑Ref. 35͒ץ Ϫ1 1 ϭ p ͯ 2 undrained compressibility, cu ͑Ref. 36͒ץ K V c c cϩ 3 V 1 Open or jacketed compressibility ͑Refs. 35 and 18͒ץ Ϫ1 1 ϭ p ͯ 2 drained or matrix compressibility, c ͑Ref. 36͒ץ K V c p ϩ 3 ,V 1Ϫ␣ Grain or pore compressibility ͑Ref. 37͒, cs ͑Ref. 36͒ץ Ϫ1 1 ϭ unjacketed compressibility, ␦ ͑Ref. 18͒ p ͯ 2ץ K V M c pϭpc ϩ 3
III. THE ELASTOACOUSTIC EFFECT IN mathematical description and their implications for elastoa- EXPERIMENTS coustic measurements. A. Experimental nomenclature and moduli In state ͑a͒ the closed-pore system corresponds to con- stancy of the fluid content or mass, and in the geometrically The application of the previous ideas to a poroelastic linear approximation this implies that ϭ0. The closed-pore sample requires the ability to impose various types of stress experiment is the simplest for theoretical analysis but is states if all seven third-order moduli are to be determined, rather difficult to realize in practice, at least for a high level i.e., ͕⌫111 ,⌫12 ,⌫3 ,⌫1,⌫11,⌫2, We first discuss the types of of wetting. Conceptually, it requires that the sample is cov- measurements employed in determining the four second- ered by an impervious closed deformable jacket; see Fig. order moduli ͕⌫11 ,⌫2 ,⌫1,⌫͖ or ͕c ,,␣,M͖. The second- 1͑a͒. ͑b͒ Alternatively, the open porous system, in the termi- order moduli are related to one another via Eq. ͑22͒ and yield nology of Biot and Willis,18 corresponds to the case of con- 35 the classic Biot theory for linear poroelasticity for which stant fluid pressure, or pϭ0. However, it is more consistent to refer to this as a ‘‘drained’’ test and to treat it as a special ϭ uk ␦ ϩ͑u ϩu ͒Ϫ␣M␦ , ͑35a͒ ij c ,k ij i,j j,i ij case of a more general open configuration with pϭconst. k Such an experiment is sketched in Fig. 1͑b͒ and is regarded pϭϪ␣Mu,kϩM. ͑35b͒ as the jacketed test. The amount of absorbed liquid is not Note that p is the pore pressure, and ij is the total or con- fixed in this experiment since it can leave the sample through fining stress. The effective stress, ijϩ␣p␦ij , is also com- the tube connected with the air. The unjacketed tests are monly used. For example, if a hydrostatic confining stress ijϭϪpc␦ij is applied, then the effective pressure is peϭpcϪ␣p. The main point is that there are two indepen- dent stress variables, or pressures, if the system is hydrostati- cally loaded. Table I lists and defines three distinct bulk moduli ͑inverse of compressibility͒, Kc , K, and KM , associ- ated with hydrostatic deformation of undrained, jacketed, and unjacketed samples, respectively. We also define the bulk modulus for the fluid alone, K f , and the ‘‘coefficient of fluid content’’ ␥,18,35
V f 1 1ץ Ϫ1 1 ϭ , ␥ϭ Ϫ , ͑36͒ ͪ p ͩ K Kץ K V f f f M in terms of which the Gassmann relation35 is 1 ␣ ␥ϭ Ϫ . ͑37͒ M KM Different types of experiments have been used to deter- mine the second-order elasticities of fluid-saturated poroelas- tic media. We shall discuss them briefly in order to avoid possible confusion in terminology. The terms ‘‘open’’ and ‘‘closed,’’ ‘‘jacketed’’ and ‘‘unjacketed,’’ and ‘‘drained’’ and ‘‘undrained’’ are all relevant to the problem, but have FIG. 1. Different schemes of loading: ͑a͒ closed-pore jacketed test, ͑b͒ different connotations. We shall next describe the four states open-pore jacketed test, ͑c͒ conventional unjacketed test, and ͑d͒ triaxial of deformation illustrated in Fig. 1, and then consider their unjacketed test.
1372 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1372 sketched in Fig. 1͑c͒ and Fig. 1͑d͒. Although the system as a B. Wave speeds for specific experimental whole is again contained within an impervious jacket, the configurations porous solid sample can now exchange fluid with the sur- The above formulas together imply specific relations for rounding free fluid in the reservoir. ͑c͒ A ‘‘conventional’’ the change in the velocities of the three wave types: trans- unjacketed test is shown in Fig. 1͑c͒. In this case the hydro- verse, and fast and slow longitudinal waves. For the first static stress on the solid is caused by pressurizing the sur- three cases below, the loading is hydrostatic, defined by the rounding fluid, and therefore both the skeleton and the fluid confining pressure pc . The change in the transverse speed is are hydrostatically stressed with the same pressure, therefore independent of its polarization direction. ijϭϪpc␦ij and pϭpc . This type of conventional unjack- ͑a͒ The ‘‘closed-pore jacketed’’ tests: for the transverse eted test does not permit triaxial deformation of the sample, wave, and is thus too restrictive for the acoustoelastic effect. ͑d͒ 2 1 The experiment sketched in Fig. 1͑d͒ might be helpful in dct ⌫12ϩ 3⌫3 ϭ , ͑42͒ avoiding this drawback, although it is presumably rather dif- dpc 2Kc͑Ϫ f ͒ ficult to realize practically. In what follows we call it a ‘‘tri- and for the fast and slow waves, axial unjacketed’’ test, and it is characterized by a constant 2 0i 0j pressure p in the pore fluid, and a triaxial state of stress in dc BijX X ϭ 0i 0j , ͑43͒ the skeleton, implying three independent elements for ij . dpc CijX X The four states outlined above are as follows. First, for where the ‘‘closed-pore’’ jacketed test we have ͑see Table I for 4 2 definitions͒ 1 Ϫ6⌫111Ϫ 3 ⌫12 2⌫11ϩ 3 ⌫2 Bϭ . ͑44͒ Kc 2 ␦ϭ0 pc /Kcϭpe /K. ͑38͒ ͫ 2⌫11ϩ 3 ⌫2 Ϫ2⌫1 ͬ ⇔ In the ‘‘open-pore’’ jacketed test, on the other hand, ͑b͒ The ‘‘open-pore jacketed’’ tests: 2 1 pϭ0 ␦ϭϪ␣pc /K. ͑39͒ dct ⌫12ϩ 3⌫3ϩ␣⌫2 ⇔ ϭ . ͑45͒ In the ‘‘conventional’’ unjacketed test the skeleton is in a dpc 2K͑Ϫ f ͒ state of hydrostatic stress, experiencing the same pressure as The variation in the fast and slow wave speeds is given by the fluid, i.e., pcϭp. In the case at hand the skeleton defor- Eq. ͑43͒, where now mation is purely dilatational, and Eqs. ͑35͒ then imply 4 2 1 Ϫ6⌫111Ϫ 3⌫12Ϫ2␣⌫11 2⌫11ϩ 3⌫2ϩ2␣⌫1 ␦ui, jϭϪ͑p/3KM ͒␦ij , ␦ϭ␥p, ͑40͒ Bϭ . K 2 ͫ 2⌫11ϩ 3⌫2ϩ2␣⌫1 Ϫ2⌫1Ϫ6␣⌫ ͬ where ␥ is defined in Eq. ͑36͒. Finally, in the triaxial unjack- ͑46͒ eted test both the fluid pressure and the deformation of the skeleton are simultaneously kept under control. In this case ͑c͒ The conventional ‘‘unjacketed’’ tests: we can express the fluid dilatation parameter in terms of 2 1 dct ⌫12ϩ 3⌫3Ϫ␥KM⌫2 these as ϭ . ͑47͒ dpc 2KM͑Ϫ f ͒ k ␦ϭ␣␦u,kϩp/M. ͑41͒ The fast and slow speed changes are given by Eq. ͑43͒ with
4 2 1 Ϫ6⌫111Ϫ 3⌫12Ϫ2␥KM⌫11 2⌫11ϩ 3⌫2ϩ2␥KM⌫1 Bϭ . ͑48͒ KM 2 ͫ 2⌫11ϩ 3⌫2ϩ2␥KM⌫1 Ϫ2⌫1Ϫ6␥KM⌫ ͬ
͑d͒ The ‘‘unjacketed’’ triaxial tests: The three principal To this end one should measure changes of velocities of strains ␣N , ␣1 , ␣2 , and the pore pressure pϵP are now acoustic waves induced by prestrain. Although it is of higher independent parameters, and, according to Eq. ͑41͒, order as compared to linear effects, acoustoelasticity can be reliably detected in sedimentary materials since the nonlinear ⌬ϭ␣͑␣ ϩ␣ ϩ␣ ͒ϩP/M. ͑49͒ N 1 2 elastic moduli are of several orders higher that their linear The changes in the three wave speeds are then given by the moduli.16 The experiments can be performed either on core general formulas in Eqs. ͑27͒, ͑32͒, and ͑34͒. samples or in situ as, for instance, by experimental measure- ment of strain-induced changes in seismic waves velocities due to the solid Earth tides.13 In fact, it is found that the in IV. DISCUSSION AND CONCLUSIONS situ nonlinearity is as large or larger than that observed in the The acoustoelastic effect is the simplest nonlinear elastic laboratory, and may be due to the large scale heterogeneity effect that can be used to determine nonlinear elastic moduli. in the Earth.13 Such measurements can provide valuable in-
1373 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1373 formation about the Earth’s interior, just as measurements of 10 A. Onuki, ‘‘Theory of phase transition in polymer gels,’’ Adv. Polym. Sci. the velocities themselves allow one to determine linear 109, 61–121 ͑1993͒. 11 moduli. J. Engelbrecht, Nonlinear Wave Processes of Deformation in Solids ͑Pit- man, Boston, 1983͒. In the present article we have established relations for 12 A. A. Gvozdev, M. A. Grinfeld, and N. V. Zvolinski, ‘‘Nonlinear effects the strain-induced changes of acoustic wave speeds in fluid- accompanying seismic waves propagation,’’ in Nonlinear Deformation filled poroelastic media. We have concentrated on the longi- Waves, edited by U. Nigul and J. Engelbrecht ͑Springer, New York, tudinal and transverse waves in isotropic poroelastic sub- 1983͒, pp. 391–396. 13 T. L. De Fazio, K. Aki, and J. 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1374 J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 M. A. Grinfeld and A. N. Norris: Fluid-saturated porous media 1374