Linear dynamic model for porous media saturated by two immiscible fluids

A linear isothermal dynamic model for a porous medium saturated by two immiscible fluids is developed in the paper. In contrast to the mixture theory, phase separation is avoided by introducing one energy for the porous medium. It is an important advantage of the model based on one energy approach that it can account for the couplings between the phases. The volume fraction of each phase is characterized by the porosity of the porous medium and the saturation of the wetting phase. The mass and momentum balance equations are constructed according to the generalized mixture theory. Constitutive relations for the stress, pore pressure are derived from the free energy function. A capillary pressure relaxation model characterizing one attenuation mechanism of the two-fluid saturated porous medium is introduced under the constraint of the entropy inequality. In order to describe the momentum interaction between the fluids and the solid, a frequency independent drag force model is introduced. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be calculated by the phenomenological parameters, which are measurable. The equations of motion in the frequency domain are obtained in terms of the Fourier transformation. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for three P waves and one S wave are calculated. The influences of the capillary pressure relaxation coefficient and the saturation of the wetting phase on the velocities and attenuation coefficients for the four wave modes are discussed in the numerical examples.