A macroscopic model for immiscible two-phase flow in porous media

Abstract

This work provides the derivation of a closed macroscopic model for immiscible two-phase, incompressible, Newtonian and isothermal creeping steady flow in a rigid and homogeneous porous medium without considering three-phase contact. The mass and momentum upscaled equations are obtained from the pore-scale Stokes equations, adopting a two-domain approach where the two fluid phases are separated by an interface. The average mass equations result from using the classical volume averaging method. A Green's formula and the adjoint Green's function velocity pair problems are used to obtain the pore-scale velocity solutions that are averaged to obtain the upscaled momentum balance equations. The macroscopic model is based on the assumptions of scale separation and the existence of a periodic representative elementary volume allowing a local description as usually postulated for upscaling. The macroscopic momentum equation in each phase includes the generalized Darcy-like dominant and viscous coupling terms and, importantly, an additional compensation term that accounts for surface tension effects to momentum transfer that is, otherwise, incompletely captured by the Darcy terms. This interfacial term, as well as the dominant and viscous coupling permeability tensors, can be predicted from the solutions of two associated closure problems that coincide with those reported in the literature. The relevance of the compensation term and the upscaled model validity are assessed by comparisons with direct numerical simulations in a model two-dimensional periodic structure. Upscaled model predictions are found to be in excellent agreement with direct numerical simulations.

Lattice Boltzmann modelling of isothermal two-component evaporation in porous media

“…2020) and thermodynamically constrained averaging theory (Jackson, Miller & Gray 2009). Besides, statistical information from pore-scale data can be used to validate the upscaled models (Lasseux & Valdés-Parada 2022). Therefore, there is an increasing need for reliable numerical models able to allow pore-scale investigations of flows in porous media coupled with liquid–vapour phase change (Ackermann, Bringedal & Helmig 2021; Vorhauer-Huget & Shokri 2022).…”

“…Pore-scale investigation is essential since it not only helps to elucidate the underlying mechanisms of behaviour at the macroscale, but also provides guidelines for constructing macroscopic models using various upscaling techniques, such as the homogenization technique (Whitaker 1977), the volume averaging method (Ahmad et al 2020) and thermodynamically constrained averaging theory (Jackson, Miller & Gray 2009). Besides, statistical information from pore-scale data can be used to validate the upscaled models (Lasseux & Valdés-Parada 2022). Therefore, there is an increasing need for reliable numerical models able to allow pore-scale investigations of flows in porous media coupled with liquid-vapour phase change (Ackermann, Bringedal & Helmig 2021;Vorhauer-Huget & Shokri 2022).…”

Upscaled dynamic capillary pressure for two-phase flow in porous media

“…The decomposition defined in (2.3) can be applied to the pore-scale pressure in both phases, so that (2.1 b ), (2.1 d ), (2.1 g ) and (2.1 h ), respectively, can be replaced by () Here, and , yielding the formulation used in Lasseux & Valdés-Parada (2022) in which was taken as the gravitational acceleration. In (2.5 b ), is used to locate points at .…”

“…Note that the average pressure difference can also be expressed at the centroid, , of the unit cell by using the relationship , with the definition . Indeed, equality of the two pressure gradients in both phases is a compatibility requirement with the assumption of a periodic unit cell representative of the process (see details in Appendix B in Lasseux & Valdés-Parada (2022)). The effective-medium coefficient associated with the pressure gradient in that case can be shown to be .…”

“…In a recent development proposed by Lasseux & Valdés-Parada (2022), a formal derivation of the macroscopic momentum and mass conservation equations for isothermal, creeping Newtonian flow of two immiscible and incompressible phases, and , in homogeneous porous media, was obtained, which reads (see the definitions of the different quantities in § 2) with and . In the above momentum equation, the (dominant and coupling) permeability tensors are defined as , where are closure variables that solve two ancillary closure problems (see (5.14) and (5.15) in Lasseux & Valdés-Parada (2022)).…”