Effective Pore Fluid Bulk Modulus at Patchy Saturation: An Analytic Study

A patchy saturated two-layered porous rock, with each layer filled with a different fluid, is examined. We assume that this system is probed by an elastic wave having a wavelength much larger than the layers' thicknesses. We also assume that the diffusion length is smaller than the thickness of an individual layer, which implies hydraulic disconnection between layers. In the context of Gassmann's fluid substitution theory, we analytically derived an exact expression for the effective fluid bulk modulus assuming that both layers have the same porosity, dry frame, and mineral matrix properties. In addition we derived an approximate solution that works well at relatively high porosities. Both solutions are expressed as a weighted average of the arithmetic and harmonic averages of individual bulk moduli of the pore fluid. These weights are explicitly given as functions of the porosity, the fractional thicknesses of both layers, and the elastic moduli of the constituents. For the approximate solution, one does not require explicit knowledge of the shear modulus of the rock. The comparison with laboratory data showed that, in the case where a porous, isotropic rock is filled with water and gas, the approximate solution can be used to model the measured data for high values of water saturation.