Another important source of mesoscopic-scale heterogeneity is patchy fluid saturation. All natural hydrological processes by which one fluid non-miscibly invades a region initially occupied by another result in a patchy distribution of the two fluids. The patch sizes are distributed across the entire range of mesoscopic length scales and for many invasion scenarios are expected to be fractal. As a compressional wave squeezes such a material, the patches occupied by the less-compressible fluid will respond with a greater fluid-pressure change than the patches occupied by the more-compressible fluid. The two fluids will then equilibrate by the same type of mesoscopic flow already modeled in the double-porosity model.
An analysis almost identical to that of Pride and Berryman (2003a,b) can be carried out that leads to the same effective poroelastic moduli given by equations (11)-(13) but with different definitions of the aij constants and internal transport coefficient . In the model, a single uniform porous frame is saturated by mesoscopic-scale patches of fluid 1 and fluid 2. We define porous phase 1 to be those regions (patches) occupied by the less mobile fluid and phase 2 the patches saturated by the more mobile fluid; i.e., by definition . This most often (but not necessarily) corresponds to Kf1 > Kf2 and, therefore, to B1> B2.
Johnson (2001) approached this problem using a different
coarse-graining argument while starting from the same local physics
(assuming, however, that the porous medium is a Gassmann mono-mineral
material). The final undrained bulk modulus obtained by Johnson (2001)
is identical to our model in the limits of high and low
frequency and differs only negligibly in the transition range of frequencies
where the flow in either model is not explicitly treated.