[Paper Review] GeoChemFoam: Operator Splitting based time-stepping for efficient Volume-Of-Fluid simulation of capillary-dominated two-phase flow

Julien Maes, Hannah Menke|arXiv (Cornell University)|May 21, 2021

Lattice Boltzmann Simulation Studies17 references4 citations

TL;DR

This paper introduces OSCAR, a novel operator splitting-based time-stepping method for Volume-of-Fluid simulations of capillary-dominated two-phase flow. By decoupling viscous drag and capillary forces into separate time steps—using larger steps for viscous drag and adaptive, convergent steps for capillary relaxation—it achieves up to 250× speedup at low capillary numbers (Ca ≤ 10⁻⁵) while maintaining high accuracy and convergence.

ABSTRACT

We present a novel time-stepping method, called Operator Splitting with Capillary Relaxation (OSCAR), for efficient Volume-Of-Fluid simulations of capillary-dominated two-phase flow. OSCAR uses operator splitting methods to separate the viscous drag and the surface tension forces. Different time-steps are used for the viscous drag steps, controlled by the injection velocity, and for the capillary relaxation steps, controlled by the velocity of capillary waves. Although OSCAR induces an additional numerical error of order 0 in time resulting from the splitting, it is well suited for simulations at low capillary number. First, the splitting error decreases with the capillary number and at low capillary number, the relaxation steps converge before reaching their last iteration, resulting in a large speed-up (here up to 250x) compared to standard time-stepping methods. The method is implemented in GeoChemFoam, our OpenFOAM-based CFD solver. Convergence, accuracy and efficiency are demonstrated on three benchmark cases: (1) the steady motion of an air bubble in a straight 2D microchannel, (2) injection of supercritical CO2 in a 3D constricted channel leading to a snap-off, and (3) water drainage in a 2D oil-wet micromodel representing a porous media.

Motivation & Objective

• Address the high computational cost of Direct Numerical Simulation (DNS) of capillary-dominated two-phase flow at low capillary numbers.
• Overcome stability constraints from capillary wave velocities that force excessively small time steps in standard methods.
• Reduce computational cost while preserving accuracy in interfacial dynamics and capillary equilibrium.
• Enable long-duration simulations at Ca ≤ 10⁻⁵, which are infeasible with conventional time-stepping due to linear CPU cost scaling.
• Develop a robust, efficient time-stepping framework compatible with algebraic VOF and extendable to other multiphase methods.

Proposed method

• Apply operator splitting to decouple viscous drag (driven by injection velocity) and surface tension forces (driven by capillary waves).
• Use different time steps for each operator: CFL-based for viscous drag, and combined CFL-Brackbill-based for capillary relaxation.
• Perform multiple capillary relaxation steps per viscous drag step until convergence or maximum iteration count is reached.
• Reconstruct velocity and pressure fields using additive operator splitting, and track interface displacement via sequential splitting.
• Implement the method in GeoChemFoam, an OpenFOAM®-based CFD solver, using algebraic VOF for interface tracking.
• Apply convergence checks based on residual reduction and use topology-based or energy-based criteria to detect capillary equilibrium.

Experimental results

Research questions

• RQ1 Can operator splitting significantly reduce computational cost in low-capillary-number two-phase flow simulations without sacrificing accuracy?
• RQ2 How does the splitting error in OSCAR scale with capillary number, and can it be minimized in practice?
• RQ3 To what extent can capillary relaxation steps converge early, enabling large time-step savings?
• RQ4 How does OSCAR compare to standard PISO-based time-stepping in terms of accuracy and efficiency across benchmark cases?
• RQ5 Can OSCAR enable feasible simulations at Ca ≤ 10⁻⁵, where standard methods become prohibitively expensive?

Key findings

• OSCAR achieves up to 250× speedup over standard PISO-based time-stepping at Ca = 10⁻⁵ in a 2D microchannel bubble flow.
• For Ca ≤ 10⁻⁵, OSCAR simulations complete in ~60 hours for a 2D micromodel, compared to over 165 hours with PISO, which scales linearly with decreasing Ca.
• Splitting error is less than 0.001 for Ca ≤ 10⁻⁴ and decreases with decreasing capillary number, showing favorable convergence behavior.
• Snap-off dynamics in a 3D constricted channel are accurately captured by OSCAR at Ca = 10⁻⁷, with results matching PISO in bubble volume and snap-off time.
• Residual saturation in a 2D oil-wet micromodel differs by only 2% between OSCAR and PISO at Ca = 10⁻³, and is nearly identical at Ca ≤ 10⁻⁴.
• Capillary relaxation steps converge rapidly at low Ca, allowing early termination and significant computational savings, especially below Ca = 10⁻⁶.