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Multiphase Flow Modelling in Multiscale Porous Media: An Open-Sourced Micro-Continuum Approach
Francisco J. Carrillo, Ian C. Bourg, Cyprien Soulaine
TL;DR
Multiphase flow in inherently multiscale structures remains difficult to predict and characterize. The paper derives, implements, tests, and verifies an open-source multiscale model that simulates multiphase Darcy and Navier–Stokes flow across two length scales.
Problem
The unresolved problem is predicting and characterizing multiphase flow physics within inherently multiscale structures.
Method
The paper derives and implements a micro-continuum framework using volume averaging to model multiphase flow simultaneously at Darcy and other length scales.
Results
The model was successfully derived, implemented, tested, and verified, simulating multiphase Darcy and Navier–Stokes flow to the same standard as conventional single-scale solvers.
Takeaways & Limitations
The framework and its open-source hybridPorousInterFoam implementation extend multiscale multiphase modelling across Darcy and Navier–Stokes flow regimes.
Takeaways & Limitations
Modelling multiphase flow at rough porous interfaces remains an open question, and no complete boundary description currently exists.
Abstract
from arXiv · showhide
An open-sourced multiphase Darcy-Brinkman approach is proposed to simulate two-phase flow in hybrid systems containing both solid-free regions and porous matrices. This micro-continuum model is rooted in elementary physics and volume averaging principles, where a unique set of partial differential equations is used to represent flow in both regions and scales. The crux of the proposed model is that it tends asymptotically towards the Navier-Stokes volume-of-fluid approach in solid-free regions and towards the multiphase Darcy equations in porous regions. Unlike existing multiscale multiphase solvers, it can match analytical predictions of capillary, relative permeability, and gravitational effects at both the pore and Darcy scales. Through its open-source implementation, hybridPorousInterFoam, the proposed approach marks the extension of computational fluid dynamics (CFD) simulation packages into porous multiscale, multiphase systems. The versatility of the solver is illustrated using applications to two-phase flow in a fractured porous matrix and wave interaction with a porous coastal barrier.
1. Introduction
Multiphase flow in heterogeneous porous media remains difficult to model across scales because inertial, viscous, capillary, and interfacial effects are strongly coupled. The paper proposes a volume-averaged micro-continuum framework using one equation set across pore, hybrid, and continuum representations.
- Motivation: The paper addresses the unresolved challenge of describing multiphase flow across inherently multiscale porous structures.The relevant physics spans microscopic interfaces, pore networks, and larger continuum scales.
- Related approaches: Existing multiscale strategies separately use homogenization, multiscale algorithms, or domain-decomposed multiphysics equations.These approaches may rely on scale separation, differently refined grids, or independently governed sub-domains.
- Approach: The proposed micro-continuum approach uses a single equation framework for systems containing solid-free and porous domains.Its Darcy-Brinkman formulation includes porosity weighting and fluid-solid drag while remaining valid in solid-free regions.
- Approach: The Darcy-Brinkman equation becomes Navier-Stokes-like in solid-free regions and tends toward Darcy’s law in porous regions.The drag term vanishes in solid-free regions, whereas porous-region drag dominates viscous dissipation.
- Results: The two-phase framework rigorously derives gravity and capillary effects and is verified against reference solutions across pore and continuum scales.Reported tests include Buckley-Leverett, capillary-gravity equilibrium, heterogeneous-reservoir drainage, droplets, capillary rise, film deposition, and complex porous flow.
- Results: The solver converges to two-phase Darcy solutions at the continuum scale and two-phase Navier-Stokes solutions at the pore scale.The implementation is released as the open-source hybridPorousInterFoam solver, with applications to fractured matrices and coastal barriers.
2. Mathematical model
The model derives two-phase micro-continuum equations by volume-averaging pore-scale conservation laws into a single formulation for solid-free and porous regions. Its parameters are matched to VOF behavior in solid-free domains and multiphase Darcy behavior in porous domains.
- Volume averaging and single-field formulation: Volume averaging transforms continuous conservation laws into finite-volume equations while representing unresolved interfaces and microstructure through sub-grid descriptions.The derivation averages equations over Eulerian control volumes and closes information below the grid-cell scale.
- Volume averaging and single-field formulation: Porosity distinguishes solid-free regions, where φ = 1, from porous regions, where 0 < φ < 1, and varies with simulation resolution.The porosity field represents unresolved microstructure, including image-based sub-voxel features.
- Volume averaging and single-field formulation: The model uses single-field pressure and velocity variables defined throughout the grid, with phase contributions combined through volume- and porosity-weighted averages.The resulting single-field velocity equals the sum of the phase filtration velocities.
- Volume averaging and single-field formulation: A single equation set applies across cells containing different combinations of fluids and solids, covering solid-free, porous, pore-resolved, and unresolved regions.The same multiphase flow and transport equations are used regardless of cell content, avoiding separate multiphysics solvers with mortars.
- Closure and multi-scale parameters: The formulation asymptotically approaches two-phase Navier–Stokes at the pore scale and two-phase Darcy equations at the continuum scale.This behavior is obtained by deriving multiscale parameters through asymptotic matching.
- Closure and multi-scale parameters: Interface compression is controlled by Cα: higher values sharpen the interface, but excessive values introduce parasitic velocities and unphysical solutions.The cited implementation typically selects Cα within a practical range, though the supplied passage does not include its numerical bounds.
3. Numerical implementation
The solver implements the micro-continuum equations in OpenFOAM using finite-volume discretization and segregated pressure–velocity coupling. Its algorithm advances saturation explicitly, predicts velocity, corrects pressure and velocity through PISO, and provides an open-source toolbox with verification cases and supporting libraries.
- Open-source toolbox: OpenFOAM supplies finite-volume solution of coupled equations on structured and unstructured three-dimensional grids with parallel-computing scalability.The implementation targets OpenFOAM version 7.0.
- Discretization of the equations: The momentum equation is discretized with an implicit Euler scheme, combining convective, diffusive, and Darcy momentum-exchange contributions.Pressure gradient, buoyancy, and capillary forces are handled separately from other explicit source terms at this stage.
- Solver algorithm: The implementation is built on OpenFOAM’s interFoam VOF solver and uses a segregated predictor–corrector algorithm for pressure–velocity coupling.The pressure equation is solved with GAMG, and the PISO loop may be repeated to enforce mass conservation.
- Solver algorithm: Saturation is solved explicitly with MULES, while relative permeability, density, and viscosity are updated before the momentum solve.The saturation equation uses a Van Leer limiter and forward Euler time discretization.
- Open-source toolbox: The open-source hybridPorousInterFoam toolbox contains the solver, tutorials with verification and example cases, and libraries for contact-angle and porous-media models.The toolbox also includes Brooks–Corey and van Genuchten models for relative permeability and capillary-pressure descriptions.
4. Verification
The solver is verified against reference solutions across its two asymptotic limits: Darcy-scale porous flow and pore-scale VOF flow.
- The verification uses analytical and semi-analytical reference solutions across the tested cases.
- The verification targets convergence to the two-phase Darcy model at continuum scale and the VOF formulation at pore scale.
4.1. Darcy scale validation
Darcy-scale tests show agreement with analytical and semi-analytical solutions for Buckley–Leverett displacement, gravity, capillary equilibrium, and heterogeneous oil drainage.
- Buckley–Leverett validation: Numerical saturation profiles agree well with Buckley–Leverett semi-analytical solutions for shock position, shock velocity, and trailing saturation gradients.
- Gravity dominated Buckley-Leverett: Numerical solutions agree with semi-analytical solutions when gravity dominates water injection, using both Brooks-Corey and Van Genuchten relative permeability models.
- Gravity-capillarity equilibrium: The model accurately reproduces the analytical steady-state saturation profile and equilibrium gradient for both implemented capillary pressure models.
- Oil drainage in a heterogeneous reservoir: HybridPorousInterFoam and impesFoam produce very similar saturation profiles in heterogeneous oil drainage, including viscosity-driven fingering instabilities.
- Oil drainage in a heterogeneous reservoir: Despite different governing equations and numerical algorithms, the solver simulates complex porous systems traditionally handled by single-scale Darcy solvers.
4.2. Pore scale validation
Pore-scale validation recovers prescribed contact angles and capillary-flow benchmarks, while complex drainage reproduces interFoam flow structures with minor contact-angle-related differences.
- Contact-angle validation: Prescribed equilibrium contact angles are reproduced within 5° for droplets on porous boundaries.
- Capillary rise: Capillary-rise simulations produce an 8.8 mm meniscus height, equivalent to an imposed contact angle of 52°.
- Thin-film drainage: The predicted ethanol film thickness is 4.50 µm versus an analytical value of 4.35 µm, a relative error of about 3% or 0.15 µm.
- Oil drainage in a complex pore network: Pore-scale oil-drainage results from hybridPorousInterFoam and interFoam are practically identical, including preferential paths and droplet snap-off at 5 ms.
- Oil drainage in a complex pore network: Minor interface-rate differences are attributed to differing implementations of the contact angle at solid boundaries.
5. Hybrid Scale Applications
Hybrid-scale applications demonstrate coupled Navier–Stokes and Darcy dynamics across porous interfaces, including wave interaction with a coastal barrier and fracture imbibition or drainage.
- Coastal barrier simulation: Coastal-barrier simulations simultaneously model coupled wave and Darcy dynamics, with porous structure producing repeated wave absorption and dissipation.
- The framework models hybrid-scale multiphase flow in complex porous media without different meshes, solvers, or complex interfacial conditions.
- Coastal barrier simulation: The coastal-barrier model retains capillary effects and is intended for barriers with arbitrarily small pores, including sand or gravel structures.
- Fractured porous matrix: In imbibition, capillary pressure drives water invasion into the surrounding microporous matrix, producing near-complete system saturation by 500 ms.
- Fractured porous matrix: In drainage, capillary effects resist air invasion, trap water droplets, and make the invading interface progress about three times more slowly in the microporous matrix.
6. Conclusion
The paper presents a verified micro-continuum model for two-phase flow across Darcy and pore scales using one governing framework. Its implementation supports hybrid applications while leaving rough porous-interface modelling and additional multiphysics extensions for future work.
- The open-source framework hybridPorousInterFoam provides an accurate and straightforward route for introducing porous-media two-phase-flow physics into CFD software.The implemented interface has been shown to predict single-phase flow into porous media, impose static contact angles, and approximate multiphase porous flow.
- The model simultaneously represents Darcy-scale porous flow and pore-scale Navier–Stokes flow using one momentum equation without separate meshes, solvers, or complex interfacial conditions.Darcy-scale interactions use relative permeability and capillary-pressure models, while pore-scale interactions use a continuum Young–Laplace representation.
- The framework was derived from fundamental principles using volume averaging and asymptotic matching, with parameters matched to VOF in solid-free regions and multiphase Darcy flow in porous regions.The same equations govern multiphase flow and transport in both solid-free and porous regions.
- The model successfully simulates multiphase Darcy and Navier–Stokes flow to the standard of conventional single-scale solvers and couples porous interfaces through mass and stress continuity plus constant-contact-angle wettability.These interface conditions support applications involving wave interaction with a porous coastal barrier and drainage or imbibition in a fractured porous matrix.
- Further study is required to rigorously model multiscale flow at rough porous interfaces and to incorporate erosion, chemical reactions, and solid mechanics.The paper identifies rough-interface boundary conditions as an open question and lists these extensions as future work.
Appendix A.1. Relative Permeability Models
The appendix describes two implemented relative-permeability formulations based on effective wetting-fluid saturation. It presents Brooks–Corey and Van Genuchten alternatives for relating phase mobility to saturation.
- The implemented relative-permeability models define effective wetting-fluid saturation using each fluid’s irreducible saturation.
- The Brooks–Corey model relates each phase’s relative permeability to effective saturation through a porous-media coefficient m.
- The Van Genuchten model provides an alternative relation between phase relative permeabilities and the wetting fluid’s effective saturation.
Appendix A.2. Capillary Pressure Models
The appendix presents alternative capillary-pressure models based on effective wetting-fluid saturation. Both formulations use porous-medium parameters including entry pressure and pore-size-distribution information.
- The implemented capillary-pressure models use different formulations for effective wetting-fluid saturation.
- The Brooks–Corey formulation calculates porous-medium capillary pressure using the wetting fluid’s saturation, entry capillary pressure, and a pore-size-distribution parameter.The entry pressure is denoted pc,0, while 1/β can be calculated from the pore-size distribution.
- The Van Genuchten model supplies an alternative relation for calculating capillary pressure.
Appendix B. Alternative formulation of multi-scale parameters
The alternative formulation checks the capillary-force representation against equilibrium multiphase Darcy behavior. It requires the equilibrium capillary term to remain independent of fluid mobilities.
- The alternative formulation is obtained by recasting the preceding equation and redefining density identically in clear-fluid and porous regions.
- At equilibrium, multiphase Darcy’s law gives (ρ_l − ρ_g)g + ∇p_c = 0.
- The derived equation is consistent with an equilibrium capillary term independent of fluid mobilities.
- The overall momentum equation reduces at equilibrium to 0 = −∇p + ρg − p_c∇α_l.