# The unsaturated flow in porous media with dynamic capillary pressure

## TL;DR

This paper proves the existence of weak solutions for a degenerate pseudoparabolic PDE modeling unsaturated flow in porous media with dynamic capillary pressure, using Galerkin approximation and entropy methods.

## Contribution

It introduces a rigorous mathematical framework for a complex PDE with saturation-dependent relaxation, extending previous models with new existence results.

## Key findings

- Existence of weak solutions established
- A priori estimates derived using entropy-inspired test functions
- Handling of mixed derivatives in the PDE achieved

## Abstract

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [12] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed derivative term by combining the information from the capillary pressure with obtained a priori estimates on the saturation.