[Paper Review] Stable and efficient time integration at low capillary numbers of a dynamic pore network model for immiscible two-phase flow in porous media

Magnus Aa. Gjennestad, Morten Vassvik+2 more|arXiv (Cornell University)|Feb 2, 2018

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TL;DR

This paper proposes a novel semi-implicit time integration method for pore network models of immiscible two-phase flow in porous media, significantly improving stability and efficiency at low capillary numbers (Ca). Unlike explicit methods that require prohibitively small time steps at low Ca, the semi-implicit method enables stable simulations with up to 1,000x reduction in computational cost at Ca ~ 10⁻⁸, extending the practical applicability of pore network models to previously intractable low-capillary-number regimes.

ABSTRACT

We study three different time integration methods for a 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, and numerical experiments are performed demonstrating that stabilization is achieved. 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 $ ext{Ca}$, and at $ ext{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.

Motivation & Objective

• To address numerical instabilities in explicit time integration methods when simulating immiscible two-phase flow at low capillary numbers in pore network models.
• To develop a more efficient and stable time integration scheme that enables simulations in the low-capillary-number regime where explicit methods become computationally prohibitive.
• To extend the range of capillary numbers over which pore network models remain a tractable and practical modeling tool.
• To quantify the computational efficiency gain of the new method compared to established explicit schemes across varying capillary numbers.

Proposed method

• A new semi-implicit time integration method is developed, which treats capillary forces implicitly while handling viscous and inertial forces explicitly.
• The method is derived from a modified form of the pore-scale momentum balance, ensuring stability under large time steps at low Ca.
• A new time-step criterion is proposed for explicit methods (forward Euler and midpoint) to stabilize them at low capillary numbers, based on capillary force dynamics.
• Numerical experiments are conducted to compare stability and computational cost between the explicit and semi-implicit methods across a range of capillary numbers.
• Performance analysis evaluates computational time and stability, focusing on the scaling of computational effort with decreasing capillary number.

Experimental results

Research questions

• RQ1 Can explicit time integration methods be stabilized at low capillary numbers through a modified time-step criterion?
• RQ2 How does the computational cost of explicit methods scale with decreasing capillary number, and at what point does it become infeasible?
• RQ3 To what extent does the proposed semi-implicit method reduce computational effort compared to explicit methods at low Ca?
• RQ4 What is the maximum capillary number range for which pore network models remain computationally tractable using the new semi-implicit method?

Key findings

• The proposed time-step criterion successfully stabilizes forward Euler and midpoint methods at low capillary numbers, enabling otherwise unstable simulations.
• The semi-implicit method achieves stable simulations at capillary numbers as low as Ca ~ 10⁻⁸, where explicit methods fail due to instability.
• At Ca ~ 10⁻⁸, the semi-implicit method reduces computational time by three orders of magnitude compared to stabilized explicit methods.
• The efficiency gain of the semi-implicit method increases with decreasing capillary number, making it increasingly advantageous in the low-Ca regime.
• The semi-implicit method extends the practical range of capillary numbers for which pore network models are computationally feasible, significantly broadening their applicability.