Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media

Subhadeep Roy,Alex Hansen,Santanu Sinha

doi:10.3389/fphy.2020.00004

Abstract

We present a theoretical framework for immiscible incompressible two-phase flow in homogeneous porous media that connects the distribution of local fluid velocities to the average seepage velocities. By dividing the pore area along a cross-section transversal to the average flow direction up into differential areas associated with the local flow velocities, we construct a distribution function that allows us not only to re-establish existing relationships between the seepage velocities of the immiscible fluids but also to find new relations between their higher moments. We support and demonstrate the formalism through numerical simulations using a dynamic pore-network model for immiscible two-phase flow with two- and three-dimensional pore networks. Our numerical results are in agreement with the theoretical considerations.

Highlights

When two immiscible fluids compete for the same pore space, we are dealing with immiscible two-phase flow in porous media [1]
There is a second goal behind this numerical work: the dynamic network model is a model at pore level and by its use, we show how the formalism developed here connect to the flow patterns at the pore level
We may Fourier transform ap, aw, and an, shows that the combined differential transversal area based upon the thermodynamic velocity distributions is the same as that based upon the distributions giving the seepage velocities

Summary

INTRODUCTION

When two immiscible fluids compete for the same pore space, we are dealing with immiscible two-phase flow in porous media [1]. A recent work [21] has explored a new approach to immiscible two-phase flow in porous media based on elements borrowed from thermodynamics That is, it is using the framework of thermodynamics, but without connecting it to processes involving heat. Statistical mechanics is the theory that makes the connection between thermodynamics and the underlying atomistic picture It is the aim of this paper to formulate a description of immiscible two-phase flow in porous media that may form a link between the continuum-level approach of Hansen et al [21] and the pore-level description of the problem—a sort of “statistical mechanics” from which the pseudo-thermodynamics may be derived, but which describes the flow problem at the pore level.

SYSTEM DEFINITION

PSEUDO-THERMODYNAMIC RELATIONS

DIFFERENTIAL TRANSVERSAL AREA DISTRIBUTIONS

NUMERICAL OBSERVATIONS

SUMMARY