A mixture-theory-based dynamic model for a porous medium saturated by two immiscible fluids

In terms of the mixture theory, a linear isothermal dynamic model for a porous medium saturated by two immiscible fluids is developed in the paper. The volume fraction of each phase is characterized by the saturation of the wetting phase and the porosity of the porous medium. The mass and momentum balance equations are obtained according to the generalized mixture theory. The isothermal constitutive relations for the stress, the pore pressure are derived from the entropy inequality of the porous medium. Three kinds of attenuation mechanisms are introduced in terms of the entropy inequality. The drag force model is introduced to account for the attenuation due to global fluid flow between the fluids and the solid skeleton, while the capillary pressure relaxation and the porosity relaxation mechanism are used to describe the relaxation process related to the variation of the saturation and the porosity. The capillary pressure relaxation mechanism is related to relaxation of the interface between the two fluids, while the porosity relaxation mechanism is related to the local fluid flow of the porous medium. In terms of the proposed model, a linear model for the two-fluid saturated porous medium is presented in the paper. The physical meaning and the evaluation of the material parameters in the linear model are discussed in the paper. In terms of the established two-fluid linear model, the velocities and the attenuations for different wave modes are calculated. It follows from numerical results that the capillary pressure relaxation and the porosity relaxation have a significant influence on the attenuations of the P wave modes.