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.
Highlights
Different modeling approaches have been applied in order to increase understanding of immiscible two-phase flow in porous media
Pore network models have proven to be useful in order to reduce the computational cost [5], or enable the study of larger systems, while still retaining some pore-level detail
We present three numerical methods that can be utilized to perform stable simulations of two-phase flow in porous media with pore network models of the Aker type
Summary
Different modeling approaches have been applied in order to increase understanding of immiscible two-phase flow in porous media. One of the advantages of the Aker-type model is that a detailed picture of the fluid configuration is provided at any time during a simulation Dynamic phenomena, such as the retraction of the invasion front after a Haines jump [13,14,15,16], are resolved. We present three numerical methods that can be utilized to perform stable simulations of two-phase flow in porous media with pore network models of the Aker type.
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