Plane longitudinal waves in a fluid-saturated porous medium with a nonlinear relationship between deformations and displacements of the liquid phase

A mathematical model is presented to describe the propagation of a plane longitudinal wave in a fluid saturated porous medium, taking into account the geometric nonlinearity of the liquid component of the medium. The nonlinear relationship between deformations and displacements refines the classical Biot’s theory, within the framework of which a fluid saturated porous medium is considered. Evolutionary equations for the displacements of the skeleton of the medium and fluid in the pores are obtained. It is shown that, if the liquid is confined in the pores, then the wave propagation is described by an equation that generalizes the well-known Burgers equation and has a solution in the form of a stationary shock wave resulting from mutual compensation of the effects of nonlinearity and dissipation. The dependence of the width of the shock wave front on the viscosity of the fluid saturating the pores and the shock wave amplitude is determined. As the viscosity coefficient increases, the wave profile becomes steeper, i.e., the wave front width decreases. With an increase in the wave amplitude, the front width can either increase or decrease, depending on the other parameters of the original system. The limiting cases of the obtained generalized Burgers equation are analyzed with respect to the fluid viscosity parameter. If the liquid flows freely in the pores, then the system of evolutionary equations is reduced to a single equation of a simple wave, i.e., the propagation of a plane longitudinal wave in a porous medium can be described by the well-known equation of nonlinear wave dynamics - the Riemann equation. The equation describes the nonlinear waves, which are characterized by a steepening of the leading edge with subsequent overturning resulting from the growth of nonlinear effects in the absence of compensating factors such as dispersion and dissipation.