Stable and efficient time integration of a dynamic pore network model for two-phase flow in porous media

TL;DR

This paper compares explicit and semi-implicit time integration methods for a pore network model of two-phase flow in porous media, demonstrating that the semi-implicit method offers greater stability and efficiency at low capillary numbers.

Contribution

A new semi-implicit time integration method is developed and shown to outperform explicit methods in stability and computational efficiency for low capillary number flows.

Findings

Semi-implicit method stabilizes simulations at low capillary numbers.

Semi-implicit method reduces computational time by up to three orders of magnitude.

The method extends the feasible range of pore network simulations in porous media.

Abstract

We study three different time integration methods for a dynamic pore network model for immiscible two-phase flow in porous media. Considered are two explicit methods, the forward Euler and midpoint methods, and a new semi-implicit method developed herein. The explicit methods are known to suffer from numerical instabilities at low capillary numbers. A new time-step criterion is suggested in order to stabilize them. Numerical experiments, including a Haines jump case, are performed and these demonstrate that stabilization is achieved. Further, the results from the Haines jump case are consistent with experimental observations. A performance analysis reveals that the semi-implicit method is able to perform stable simulations with much less computational effort than the explicit methods at low capillary numbers. The relative benefit of using the semi-implicit method increases with decreasing capillary number $\mathrm{Ca}$, and at $\mathrm{Ca} \sim 10^{-8}$ the computational time needed is reduced by three orders of magnitude. This increased efficiency enables simulations in the low-capillary number regime that are unfeasible with explicit methods and the range of capillary numbers for which the pore network model is a tractable modeling alternative is thus greatly extended by the semi-implicit method.