The Buckley-Leverett Equation with Dynamic Capillary Pressure

TL;DR

This paper extends the Buckley-Leverett equation by incorporating dynamic capillary pressure effects, resulting in a nonlinear pseudo-parabolic PDE, analyzed through phase plane methods and numerical simulations to understand shock structures.

Contribution

It introduces a modified model with rate-dependent capillary pressure and analyzes its wave solutions using phase plane analysis and numerical methods.

Findings

Supports traveling wave solutions with undercompressive shocks.

Provides a detailed phase plane analysis of the modified equation.

Numerical simulations confirm the theoretical shock structures.

Abstract

The Buckley-Leverett equation for two phase flow in a porous medium is modified by including a dependence of capillary pressure on the rate of change of saturation. This model, due to Gray and Hassanizadeh, results in a nonlinear pseudo-parabolic partial differential equation. Phase plane analysis, including a separation function to measure the distance between invariant manifolds, is used to determine when the equation supports traveling waves corresponding to undercompressive shocks. The Riemann problem for the underlying conservation law is solved and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation.