Local Governing Equations

Each porous phase is locally modeled as a porous continuum and obeys the laws of poroelasticty [e.g., Biot (1962)]

$$\begin{eqnarray} \nabla \cdot \mbox{\boldmath$\tau$}_i^D - \nabla p_{ci} &=& \rh... ... {\bf u}_i^T -\frac{2}{3} \nabla \cdot {\bf u}_i \, {\bf I}\right)\end{eqnarray}$$ (1) (2) (3) (4)

where the index i represents the two phases (i=1,2). The response fields in these equations are themselves local volume averages taken over a scale larger than the grain sizes but smaller than the mesoscopic extent of either phase. The local fields are: ${\bf u}_i$, the average displacement of the framework of grains; ${\bf Q}_i$, the Darcy filtration velocity; p_fi, the fluid pressure; p_ci, the confining pressure (total average pressure); and $\mbox{\boldmath$\tau$}_i^D$, the deviatoric (or shear) stress tensor. In the linear theory of interest here, the overdots on these fields denote a partial time derivative. In the local Darcy law (2), $\eta$ is the fluid viscosity and the permeability k_i is a linear time-convolution operator whose Fourier transform $k_i(\omega)$ is called the ``dynamic permeability'' and can be modeled using the theory of Johnson et al. (1987) (see the Appendix).

In the local compressibility law (3), K^d_i is the drained bulk modulus of phase i (confining pressure change divided by sample dilatation under conditions where the fluid pressure does not change), B_i is Skempton's (1954) coefficient of phase i (fluid pressure change divided by confining pressure change for a sealed sample), and $\alpha_i$is the Biot and Willis (1957) coefficient of phase i defined as

$$\alpha_i = (1 - K^d_i/K^u_i)/B_i,$$ (5)

where K^u_i is the undrained bulk modulus (confining pressure change divided by sample dilatation for a sealed sample). In the present work, no restrictions to single-mineral isotropic grains will be made. Finally, in the deviatoric constitutive law (4), G_i is the shear modulus of the framework of grains. At the local level, all these poroelastic constants are taken to be real constants. In the appendix we give the Gassmann (1951) fluid-substitution relations that allow B_i and $\alpha_i$ to be expressed in terms of the porosity $\phi_i$, the fluid and solid bulk moduli K_f and K_s, and the drained modulus K^d_i.