Double-Porosity Governing Equations

In the double-porosity model, the goal is to determine the average fluid response in each of the porous phases in addition to the average displacement of the solid grains (Berryman and Wang, 1995; 2000). The averages are taken over regions large enough to significantly represent both porous phases, but smaller than wavelengths. Assuming an $e^{-i\omega t}$ time dependence, Pride and Berryman (2003a) have found the volume averaged local laws (1)-(4) in order to obtain the macroscopic ``double-porosity'' governing equations in the form

$$\begin{eqnarray} \phantom{\frac{1}{2}} \nabla \cdot \mbox{\boldmath$\tau$}^D &\!... ...bf v})^T \! - \frac{2}{3} \nabla \cdot {\bf v} \, {\bf I} \right].\end{eqnarray}$$ (6) (7) (8) (9) (10)

The macroscopic fields are: ${\bf v}$, the average particle velocity of the solid grains throughout an averaging volume of the composite; ${\bf q}_i$, the average Darcy flux across phase i; P_c, the average total pressure in the averaging volume; $\mbox{\boldmath$\tau$}^D$, the average deviatoric stress tensor; $\overline{p}_{fi}$, the average fluid pressure within phase i; and $-i\omega \zeta_{\rm int}$, the average rate at which fluid volume is being transferred from phase 1 into phase 2 as normalized by the total volume of the averaging region. The dimensionless increment $\zeta_{\rm int}$ represents the ``mesoscopic flow.''

Equation (7) is the generalized Darcy law allowing for fluid cross-coupling between the phases [c.f., Pride and Berryman (2003b)], equation (8) is the generalized compressibility law where $\nabla \cdot {\bf q}_i$ corresponds to fluid that has been depleted from phase i due to transfer across the external surface of an averaging volume, and equation (9) is the transport law for internal mesoscopic flow (fluid transfer between the two porous phases).

The coefficients in these equations have been modeled by Pride and Berryman (2003a,b). Before presenting these results, the nature of the waves implicitly contained in these laws is briefly commented upon. If plane-wave solutions for ${\bf v}$,${\bf q}_1$ and ${\bf q}_2$ are introduced, there is found to be a single transverse wave, and three longtitudinal responses: a fast wave and two slow waves (Berryman and Wang, 2000). The fast wave is the usual P-wave identified on seismograms, while the two slow waves correspond to fluid-pressure diffusion in phases 1 and 2. The only problem with analyzing the fast compressional wave in this manner is that the characteristic equation for the longtitudinal slowness s is cubic in s² and, therefore, analytically inconvenient.