Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

The quantitative description of elastic wave propagation in an elastic porous medium containing two immiscible fluids is one of the classic problems in the physics of flow through unsaturated porous materials. An analytical theory of the low-frequency behavior of dilatational waves propagating through such a porous medium is presented based on the Berryman-Thigpen-Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions.