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 1996! 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 @MAB# 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 2!. 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: ~i! 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; ~ii! 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.; ~iii! 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 physicists!. In particu- I. NONLINEAR POROELASTICITY lar, the jumps of second derivatives across acceleration wavefronts satisfy31,33 A. Governing equations 2 1 ] ui All further analysis is based on the following governing @u #15h n n , 5h c2, ~4a! i, jk 2 i j k F ]t2 G i equations of the nonviscous, fluid-filled poroelastic medium: 2 2 i 2 i ] u ] w 2w 1 ij 1 ] i 2 r 2 1r f 2 5t, j , ~1a! @w # 5H n n , 5H c , ~4b! ]t ]t i, jk 2 i j k F ]t2 G i 2 2 i 2 ] w ] uj 1 p q 1 p q ij ij where hk5[ui,pq]2n n and Hk5[wi,pq]2n n are the r f 1r f K 52K p,j , ~1b! ]t2 ]t2 second-order amplitude vectors of discontinuity, n j is the where unit normal to the wavefront, and c is the velocity of the front. The compatibility conditions indicate that the two vec- ]W ]W ij tor functions hk and Hk completely define the jumps of 60 t 5 ~z,um,n!, p5 ~z,um,n!. ~2! ]ui,j ]z partial derivatives of the displacements. 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, tij are the stresses, p is the tion from the governing Eqs. ~3!. These equations are not fluid pressure, W(z,u ) is the poroelastic potential, r and m,n valid at the wavefront since the second derivatives are unde- r are the averaged density and the density of the fluid, re- f fined at the front. By definition, only one-sided second de- spectively, 2z5wi 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. ~3!. We first subtract, termwise, Eq. ~3a! 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 t50!. 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. ~1! and ~2!. By inserting the value of z in terms of wi we can rewrite 2 ik ijkl 2 ik ij k ~rc d 2W njnl!hk1~rfc d 1Wz njn !Hk50, the equations of motion as ~5a! 2 i 2 i 2 ik kl i 2 ik i k ] u ] w ij k r c d 1W n n !h 1 r c K 2W n n !H 50, r 1r 5Wijklu 2W w , ~3a! ~ f z l k ~ f inv zz k ]t2 f ]t2 k,jl z ,jk ~5b! 2 i 2 ij ij ] w ] uj where Kinv is the inverse of the permeability matrix K , r 1r Kij 52Kij~Wklu 2W wk !. ~3b! f ]t2 f ]t2 z k,lj zz ,kj which is assumed to be invertible. 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 #15qn .