Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks

Marios Valavanides,Tryfon Daras

doi:10.3390/e18020054

Abstract

Steady-state two-phase flow in porous media is a process whereby a wetting phase displaces a non-wetting phase within a pore network. It is an off-equilibrium stationary process—in the sense that it is maintained in dynamic equilibrium at the expense of energy supplied to the system. The efficiency of the process depends on its spontaneity, measurable by the rate of global entropy production. The latter has been proposed to comprise two components: the rate of mechanical energy dissipation at constant temperature (a thermal entropy component, Q/T, in the continuum mechanics scale) and the configurational entropy (a Boltzmann–Gibbs entropy component, klnW), due to the existence of a canonical ensemble of flow configurations, physically admissible to the externally imposed macrostate conditions. Here, we propose an analytical model to account the number of microstates, lnW, in two-phase flows in pore networks. Combinatorial analysis is implemented to evaluate the number of identified microstates per physically admissible internal flow arrangement, compatible with the imposed steady-state flow conditions. Then, Stirling’s approximation is applied to downscale the large factorial numbers. Finally, the number of microstates is estimated by contriving an appropriate mixing scheme over the canonical ensemble of the physically admissible flow configurations. Indicative computations are furnished.

Highlights

Introduction and Scope of WorkTwo-phase flow in porous media is a physical process whereby two phases simultaneously flow within a porous medium
The flow configurations have been chosen from the ensemble of physically admissible solutions obtained from DeProF model simulations for macroscopic flow conditions, (Ca, r), pertaining to combinations of typical capillary number, Ca = 1.0 ˆ 10 ́6, and flowrate ratio values, r = 0.1, 1.0 and 10.0, within a 3D model pore network and for fluid systems with a typical oil/water viscosity ratio, κ = 1.45
The constituent “micro” in “microstates” refers to the scale size of the fluidic elements, at the order of a few tens to hundred microns.The scope is to evaluate the global entropy production in this kind of processes and, in particular, in those maintained at off-equilibrium steady-state

Summary

Introduction

Introduction and Scope of WorkTwo-phase flow in porous media is a physical process whereby two phases simultaneously flow within a porous medium. Two-phase flow in porous media occupies a central position in physically important processes with practical applications of industrial and environmental interest, such as: enhanced oil recovery [1,2], carbon dioxide sequestration [3], groundwater and soil contamination and subsurface remediation [4], the operation of multiphase trickle-bed reactors [5], the operation of proton exchange membrane fuel cells [6], etc. The majority of those applications are based on inherently transient processes whereby one phase displaces the other. Flow Regimes and Relative Permeabilities during Steady-State Two-Phase

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