Capillary Hysteresis and Gravity Segregation in Two Phase Flow Through Porous Media

TL;DR

This paper investigates how hysteresis in capillary pressure influences two-phase flow in porous media, revealing complex shock behaviors and extending classical models with new admissibility criteria.

Contribution

It introduces a new framework for understanding hysteretic effects in two-phase flow, including stationary shocks and their admissibility, extending classical hyperbolic models.

Findings

Hysteresis leads to non-convergent solutions in the hyperbolic limit.

Stationary shocks can occur due to hysteresis, unlike classical solutions.

Numerical results confirm theoretical predictions.

Abstract

We study the gravity-driven flow of two fluid phases in a one-dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves, and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function, and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.