Effect of viscous cross coupling between two immiscible fluids on elastic wave propagation and attenuation in unsaturated porous media

A central issue in the theoretical treatment of a multiphase system is the proper mathematical description of momentum transfer across fluid-solid and fluid-fluid interfaces. Although recent studies have advanced our knowledge on modeling the coupling behavior between a porous framework and the fluids permeating it, the effect of viscous resistance caused by two-fluid flow on elastic wave behavior in unsaturated porous media still remains elusive. In the present study, the theoretical model developed for describing immiscible two-phase fluid flows in a deformable porous medium related to harmonic wave perturbation is generalized to account for viscous cross coupling due to relative velocity between two adjacent fluids. The corresponding dispersion relations whose coefficients feature all elasticity, inertial-drag, and viscous-drag parameters are then precisely formulated, in a physical context characterizing three compressional waves and one shear wave.To evaluate quantitatively this as-yet unknown effect, numerical calculations are conducted to solve the dispersion relations for Columbia fine sandy loam bearing an oil-water mixture as a function of water saturation and excitation frequency. Our results show that the phase speed and attenuation coefficient of the P3 wave which has the smallest speed is strongly sensitive to the presence of viscous cross coupling, as expected since this wave is attributed primarily to the out-of-phase motion of the two pore fluids. Viscous cross coupling also exerts an impact on the attenuation coefficient of the shear wave and the P1 wave whose speed is greatest, which exhibits two opposite trends at different ranges of low and high water contents. Relative differences in these wave attributes are principally independent of excitation frequency. A sensitivity analysis is carried out to assess how changes in viscous cross coupling affect these differences, revealing that some of them become more significant as viscous cross coupling is stronger.