Mechanics of stratified anisotropic poroelastic media

Inverting poroelastic compliance

Being in compliance form, the matrix in (1) has extremely simple poroelastic behavior in the sense that all the fluid mechanical effects appear only in the single coefficient $\gamma$ . I can simplify the notation a little more by lumping some coefficients together, combining the $3\times3$ submatrix in the upper left corner of the matrix in (1) as ${\bf S}$ , and defining the column vector ${\bf b}$ by

$${\bf b}^T \equiv (\beta_1, \beta_2, \beta_3).$$ (24)

The resulting $4\times4$ matrix and its inverse are now related by:

$$\left(\begin{array}{cc} {\bf S} & -{\bf b} \cr -{\bf b}^T & \gamm... ...(\begin{array}{cc} {\bf A} & {\bf q} \cr {\bf q}^T & z\end{array} \right)^{-1},$$ (25)

where the elements of the inverse matrix can be shown to be written in terms of drained stiffness matrix ${\bf C}^d = {\bf C} = {\bf S}^{-1}$ by introducing three components: (a) scalar $z = \left[\gamma - {\bf b}^T{\bf C}{\bf b}\right]^{-1}$ , (b) column vector ${\bf q} = z{\bf C}{\bf b}$ , and (c) undrained $3\times3$ stiffness matrix (i.e., the pertinent one connecting the principal strains to principal stresses) is given by ${\bf A} = {\bf C} + z{\bf C}{\bf bb}^T{\bf C} = {\bf C}^d + z^{-1}{\bf qq}^T \equiv {\bf C}^u$ , since ${\bf C}^d$ is drained stiffness and ${\bf A} = {\bf C}^u$ is clearly undrained stiffness by construction. This result is the same as that of Gassmann (1951) for anisotropic porous media, although his results were presented in a form somewhat harder to scan than the form shown explicitly here.

Also, note the important fact that the observed decoupling of the fluid effects occurs only in the compliance form (1) of the equations, and never in the stiffness (inverse) form for the poroelasticity equations.

From these results, it is not hard to show that

$${\bf S}^d = {\bf S}^u + \gamma^{-1}{\bf bb}^T.$$ (26)

This result emphasizes the remarkably simple fact that the drained compliance matrix can be found directly from knowledge of the inverse of undrained stiffness, and the still unknown, but sometimes relatively easy to estimate, values of $\gamma$ , together with the three distinct orthotropic $\beta_i$ coefficients, for $i = 1,2,3$ .

There are clearly many measurements required to determine all these various poroelastic coefficients. Furthermore, the strategy for finding the coefficients depends on available data sets, and whether the porous media of interest are constructed from a homogeneous or heterogeneous set of solid materials, and whether the individiual grains are isotropic or anisotropic. It also makes some difference if the pores are approximately rounded (for granular media) or flat (for fractured media). All these issues have been discussed previously at length, and this discussion will not be repeated here.

The remainder of the paper will concentrate on making use of the general poroelastic equations in situations where at least two and possibly many distinct layers of porous materials obeying these equations are under stress (either quasi-static or dynamic as would occur in a wave propagation scenario). As will be shown, the layered poroelastic equations behave somewhat differently from layered elastic equations because there are two distinct additional boundary conditions (drained and undrained) that can occur depending on the details of the excitation itself.

Mechanics of stratified anisotropic poroelastic media