A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure

Abstract

In this paper we investigate the pseudoparabolic equation

$${\partial }_{t}c=\mathrm{div}\left\{\frac{k(c)}{\mu }\right(-P_C^{\prime }(c)\mathrm{\nabla }c+\tau \mathrm{\nabla }{\partial }_{t}c+g\rho {\mathbf{e}}_n\left)\right\},$$

where τ is a positive constant, c is the moisture content, k is the hydraulic conductivity and $P_C$ is the static capillary pressure. This equation describes unsaturated flows in porous media with dynamic capillary pressure–saturation relationship. In general, such models arise in a number of cases when non-equilibrium thermodynamics or extended non-equilibrium thermodynamics are used to compute the flux.

For this equation existence of the traveling wave type solutions was extensively studied. Nevertheless, the existence seems to be known only for the non-degenerate case, when k is strictly positive. We use the approach from statistical hydrodynamics and construct the corresponding entropy functional for the regularized problem. Such approach permits to get existence, for any time interval, of an appropriate weak solution with square integrable first derivatives in x and in t and square integrable time derivative of the gradient. Negative part of such weak solution is small in $L^2$-norm with respect to x, uniformly in time, as the square root of the relative permeability at the value of the regularization parameter. Next we control the regularized entropy. A fine balance between the regularized entropy and the degeneracy of the capillary pressure permits to get an $L^q$ uniform bound for the time derivative of the gradient. These estimates permit passing to the limit when the regularization parameter tends to zero and obtaining the existence of global nonnegative weak solution.