To use the patchy-saturation model, appropriate values for the two geometric terms L1 and V/S must be specified. Immiscible fluid distributions in the earth have very complicated geometries since they arise from slow flow that often produces fractal patch distributions. In particular, analytical solutions of the boundary-value problem (25)-(27) that defines L1 for such real-earth situations are impossible. Recall that L1 is a characteristic length of phase 1 (the phase having the smaller fluid mobility ) that defines the distance overwhich the fluid-pressure gradient is defined during the final stages of equilibration. For complicated geometries it may either be numerically determined, guessed at, or treated as a target parameter for a full-waveform inversion of seismic data. In the numerical examples that follow, we simply assume that the individual patches correspond to disconnected spheres for which simple analytical results are available for L1 and V/S.
If we consider phase 2 (porous continuum saturated by the less viscous fluid) to be in the form of spheres of radius a embedded within each radius R sphere of the two-phase composite, then v2 = (a/R)3, V/S = a v2/3, and L12 = 9 v2-2/3 a2/14 [1 - 7 v21/3/6]. This model is particularly appropriate when . Since the fluid 2 patches are disconnected, the definitions (11)-(13) of the effective poroelastic moduli again hold. Further, fluid 2 may be taken to be immobile relative to the framework of grains in the wavelength-scale Biot equilibration so that the inertial properties of equations (29) and (30) are identified as , and .
In situations where it is more appropriate to treat fluid 1 (the more viscous fluid) as occuping disconnected patches (e.g., when ), the effective poroelastic moduli are defined by replacing 2 with 3 (and 3 with 2) in the subscripts of equations (11)-(13). Again assuming the phase-1 patches to be spheres of radius a embedded within radius R sphere of the two-phase composite, we have that v1 = (a/R)3 and V/S = a v1/3. The elliptic boundary-value problem (25)-(27) can be solved in this case to give L12 = a2/15. Furthermore, the effective inertial coefficients in the Biot theory are defined , , and .
In situations where both phases form continuous paths across each averaging volume, it is best to determine the attenuation and phase velocity by seeking the plane longtitudinal-wave solution of non-reduced ``double-porosity'' governing equations of the form (6)-(10). However, this approach is not pursued here. We conclude by noting that if the embedded fluid is fractally distributed, the lengths L1 will remain finite while as the fractal surface area S becomes large (however, V/S never reaches zero because the fractality has a small-scale cutoff fixed by the grain size of the material).