Simulation of incompressible two-phase flow in porous media with large timesteps

TL;DR

This paper introduces a phase-field based homotopy method that significantly increases timestep size in simulations of incompressible two-phase flow in porous media, improving computational efficiency while maintaining convergence.

Contribution

It presents a novel phase-field and convex energy splitting approach to enable large timesteps in Buckley-Leverett equations, addressing nonlinear solver convergence issues.

Findings

Timestep size increased by over six orders of magnitude.

Method maintains convergence in highly heterogeneous media.

Applicable to petroleum engineering and CO2 sequestration simulations.

Abstract

Multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering, petroleum systems' analysis, and CO$_2$ sequestration. While simulations often use a fully implicit discretization to increase the time step size, restrictions on the time step often exist due to non-convergence of the nonlinear solver (e.g. Newton's method). Here this problem is addressed for the Buckley-Leverett equations, which model incompressible, immiscible, two-phase flow with no capillary potential. The equations are recast as a gradient flow using the phase-field method, and a convex energy splitting scheme is applied to enable large timesteps, even for high degrees of heterogeneity in permeability and viscosity. By using the phase-field formulation as a homotopy map, the underlying hyperbolic flow equations can be solved with large timesteps. For a heterogeneous test problem, the new homotopy method allows the timestep to be increased by more than six orders of magnitude relative to the unmodified equations while maintaining convergence.