POROELASTICITY ESTIMATES AND BOUNDS

My main focus here will be the extension of earlier work in elasticity to the case of locally layered poroelastic media (Wang, 2000; Coussy, 2004), where the laminated grains (or crystals) are formed by sequential layering of N porous isotropic layers. Although these grains each have the same quasi-static anisotropic elastic behavior, they do not necessarily have the same shapes or the same orientations of their crystal symmetry axes. Specifically, we want to study the case of isotropic random polycrystals, wherein the individuals can and do take on all possible orientations of their symmetry axes (equiaxed, statistically isotropic polycrystals) so that the overall composite polycrystal has isotropic behavior at the macroscopic level. Furthermore, in some applications, the pores of some grain layers may be filled with different fluids (heterogeneous saturation conditions) than those in other layers. This model may or may not be a realistic one for any given fluid-bearing reservoir whose geomechanics we need to model. My first goal is arrive at a model for which many of the available modern tools of elastic and poroelastic analysis apply, including Hashin-Shtrikman bounds for a reservoir having isotropic constituents (Hashin and Shtrikman, 1962a,b,c; 1963a,b), Peselnick-Meister-Watt bounds for random polycrystals (Peselnick and Meister, 1965; Watt and Peselnick, 1980), certain exact relationships known for two-component poroelastic media (Berryman and Milton, 1991), and -- whenever appropriate -- self-consistent or other effective medium estimates of both elastic constants and conductivities (electrical, thermal, and hydraulic). By constructing such a model material, we expect to be able to make estimates of the behavior of the system and at the same time be able to predict the range of variation likely to be observed around these estimates, as well as identifying what material and microgeometry properties control those variations. My further goal is to be able to make fairly precise statements about this model that are then useful to our (both mine and the reader's) intuition and to quantify how much is really known about these complex systems. In particular, the hope is to identify assumptions currently and commonly used in the literature without much apparent justification and to provide a means of either verifying or falsifying these assumptions in the context of this model -- if that proves to be possible.

Two distinct results that will be required from poroelasticity theory are: (a) Gassmann's equations and (b) certain relationships that determine the overall effective stress coefficient of a composite poroelastic medium when it is composed of two porous materials satisfying Gassmann's assumptions. Gassmann's results (Gassmann, 1951; Berryman, 1999; Wang, 2000) for the undrained bulk (K) and shear ($\mu$) moduli of microhomogeneous (one solid constituent) porous media are:  

$$K_u = K_d + \frac{\alpha^2}{(\alpha-\phi)/K_m + \phi/K_f} = \frac{K_d}{1-\alpha B}$$ (17)

and  

$$\mu_u = \mu_d.$$ (18)

Here, K_u and $\mu_u$ are the undrained (pore fluid trapped) constants, while K_d and $\mu_d$ are the drained (pore fluid untrapped) constants. Porosity (void volume fraction) is $\phi$.Grain bulk and shear moduli of the sole mineral constituent are K_m and $\mu_m$. The bulk modulus of the pore fluid is K_f. The factor $\alpha$ is the Biot-Willis (Biot and Willis, 1957) or volume effective stress coefficient (Nur and Byerlee, 1971; Berryman, 1992; Gurevich, 2004), related to K_m and K_d within each layer by  

$$\alpha^{(n)} = 1 - K^{(n)}_d/K^{(n)}_m.$$ (19)

Skempton's coefficient (Skempton, 1954) is B in (17).

Although my presentation is based on quasi-static results, my ultimate interest is often applications to seismic wave propagation. In such circumstances a slightly different terminology is used by some authors (Mavko and Jizba, 1991). In particular, for high frequency wave propagation, fluid may be effectively trapped in the pores as it is unable to equilibrate through pore-pressure diffusion on the time scale of wave passage. In this case, the term ``unrelaxed'' is sometimes used instead of ``undrained.'' We will not make any further issue of this distinction here and stick to the single term ``undrained'' for both types of applications.

For a porous medium composed of only two constituent porous media, each of which is microhomogeneous and obeys Gassmann's equations, the exact relation (Berryman and Milton, 1991) that determines the overall effective stress coefficient $\alpha^*$ - assuming only that the constituents are in welded contact (volume fractions and spatial distribution of constituents do not directly affect the result) - is:  

$$\frac{\alpha^*-\alpha^{(1)}}{\alpha^{(2)}-\alpha^{(1)}} = \frac{K^*_d-K^{(1)}_d}{K^{(2)}_d-K^{(1)}_d}.$$ (20)

Here K^*_d is the overall drained bulk modulus of the composite system, and the superscripts (1) and (2) reference the two distinct components in the composite porous medium.