Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 18578 Continuum models that describe two-phase flow of immiscible fluids in porous media often treat momentum exchange between the two phases by simply generalizing the single-phase Darcy law and introducing saturation-dependent permeabilities. Here we study models of creeping flows that include an explicit coupling between both phases via the addition of cross terms in the generalized Darcy law. Using an extension of the Buckley-Leverett theory, we analyze the impact of these cross terms on saturation profiles and pressure drops for different couples of fluids and closure relations of the effective parameters. We show that these cross terms in the macroscale models may significantly impact the flow compared to results obtained with the generalized Darcy laws without cross terms. Analytical solutions, validated against experimental data, suggest that the effect of this coupling on the dynamics of saturation fronts and the steady-state profiles is very sensitive to gravitational effects, the ratio of viscosity between the two phases, and the permeability. Our results indicate that the effects of momentum exchange on two-phase flow may increase with the permeability of the porous medium when the influence of the fluid-fluid interfaces become similar to that of the solid-fluid interfaces.

Bounds to Air‐Flow Patterns during Cyclic Air Injection into Partially Saturated Soils Inferred from Extremum States

“…Using analytical solutions, the models proposed here delineate the plausible range of air contents and pressures under the assumption of instantaneous water relaxation. This assumption enables a description of single-phase air flow with analytical solutions and avoids the complexity of the numerical solutions for two-phase flow, such as those of van Dijke et al (1995), Silin et al (2009), and Pasquier et al (2017). The main suppositions of the proposed approach were corroborated using a numerical, one-dimensional, two-phase-flow model.…”

Relative Permeability Scaling From Pore‐Scale Flow Regimes

“…Only recently, Karadimitriou et al (2014) proposed a method to estimate such state variables from experimental measurements, while Schluter et al (2016) and McClure et al (2018) provided alternative approaches to identify the missing variables in terms of integral geometry (i.e., Euler characteristics). Over the years, more practical and ad hoc formulations of the relative permeabilities have been proposed (Chierici, 1984;Clavier et al, 2017;Dehghanpour et al, 2011;Gunstensen & Rothman, 1993;Li et al, 2018;Oliveira & Demond, 2003;Pasquier et al, 2017;Yiotis et al, 2007;Zhang et al, 2018), including the study of the impact of spatial heterogeneity on relative permeabilities (Jackson et al, 2018). To the best of our knowledge, a general formulation that accounts for the phase topology at the pore-scale of a 3-D porous medium has not been proposed yet.…”

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

“…As particular case, the authors have investigated the flow of water and mercury under action of a sinusoidal time-dependent pressure gradient. Pasquier et al 9 have developed a mathematical model of creeping flows that includes an explicit coupling term between both phases by adding of cross terms in the generalized Darcy law. The impact of the added terms on saturation profiles and pressure drops has been investigated by using an extension of the Buckley-Leverett theory.…”