Reduction to an Effective Biot Theory

The approach that we take instead is to first reduce these double-porosity laws (6)-(10) to an effective single-porosity Biot theory having complex frequency-dependent coefficients. The easiest way to do this is to assume that phase 2 is entirely embedded in phase 1 so that the average flux ${\bf q}_2$ into and out of the averaging volume across the external surface of phase 2 is zero. By placing $\nabla \cdot {\bf q}_2=0$ into the compressibility laws (8), the fluid pressure p_f2 can be entirely eliminated from the theory. In this case the double-porosity laws reduce to effective single-porosity poroelasticity governed by laws of the form (3) but with effective poroelastic moduli given by

$$\begin{eqnarray} &&\frac{1}{K_D} = a_{11} - \frac{a_{13}^2}{a_{33} - \gamma /i \... ...a_{13}(a_{23} + \gamma/i\omega)}{a_{33} - \gamma/i \omega}\right).\end{eqnarray}$$ (11) (12) (13)

Here, $K_D(\omega)$ is the effective drained bulk modulus of the double-porosity composite, $B(\omega)$ is the effective Skempton's coefficient, and $K_U(\omega)$ is the effective undrained bulk modulus. An effective Biot-Willis constant can then be defined using $\alpha(\omega) = [1- K_D(\omega)/K_U(\omega)]/B(\omega)$.

The complex frequency dependent ``drained'' modulus K_D again defines the total volumetric response when the average fluid pressure throughout the entire composite is unchanged; however, the local fluid pressure in each phase may be non-uniform even though the average is zero resulting in mesoscopic flow and in K_D being complex and frequency dependent. Similar interpretations hold for the undrained moduli K_U and B. An undrained response is when no fluid can escape or enter through the external surface of an averaging volume; however, there can be considerable internal exchange of fluid between the two phases resulting in the complex frequency-dependent nature of both K_U and B.