scholarly journals Fluctuation-Dissipation Theorems for Multiphase Flow in Porous Media

Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 46
Author(s):  
Dick Bedeaux ◽  
Signe Kjelstrup

A thermodynamic description of porous media must handle the size- and shape-dependence of media properties, in particular on the nano-scale. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course-grained scale, by applying Hill’s thermodynamics of small systems to the subsystems of the medium. We argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we then derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation-dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of the Onsager conductivity matrix for transport through porous media. Practical possibilities are discussed.

Author(s):  
Dick Bedeaux ◽  
Signe Kjelstrup

A thermodynamic description of nano-porous media must handle the size- and shape-dependence of the media properties. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course grained scale, applying Hill's thermodynamics for small systems to the subsystems. we argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to di¤erences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation- dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of porous media permeabilities. Practical possibilities are further discussed.


2015 ◽  
Vol 114 (2) ◽  
pp. 581-600 ◽  
Author(s):  
P. Kunz ◽  
I. M. Zarikos ◽  
N. K. Karadimitriou ◽  
M. Huber ◽  
U. Nieken ◽  
...  

2021 ◽  
Author(s):  
S.R. Zhu ◽  
L.Z. Wu ◽  
T. Ma ◽  
S.H. Li

Abstract The numerical solution of various systems of linear equations describing fluid infiltration uses the Picard iteration (PI). However, because many such systems are ill-conditioned, the solution process often has a poor convergence rate, making it very time-consuming. In this study, a control volume method based on non-uniform nodes is used to discretize the Richards equation, and adaptive relaxation is combined with a multistep preconditioner to improve the convergence rate of PI. The resulting adaptive relaxed PI with multistep preconditioner (MP(m)-ARPI) is used to simulate unsaturated flow in porous media. Three examples are used to verify the proposed schemes. The results show that MP(m)-ARPI can effectively reduce the condition number of the coefficient matrix for the system of linear equations. Compared with conventional PI, MP(m)-ARPI achieves faster convergence, higher computational efficiency, and enhanced robustness. These results demonstrate that improved scheme is an excellent prospect for simulating unsaturated flow in porous media.


Author(s):  
David Jon Furbish

The concept of conservation of mass holds a fundamental role in most problems in fluid physics. For a given problem this concept is cast in the form of an equation of continuity. Such an equation describes a condition—conservation of mass—that must be satisfied in any formal analysis of a problem. Thus an equation of continuity often is one of several complementary equations that are solved simultaneously to arrive at a solution to a flow problem, for example, the flow velocity as a function of coordinate position in a flow field. (Typically these complementary equations, as we will see in later chapters, involve conservation of momentum or energy, or both.) Although we did not explicitly use this idea in analyzing the one-dimensional flow problems at the end of Chapter 3, it turns out that continuity was implicitly satisfied in setting up each problem. We will return to these problems to illustrate this point. We will develop equations of continuity for three general cases: purely fluid flow, saturated single-phase flow in porous media, and unsaturated flow in porous media. The most general of the three equations is that for unsaturated flow, where pores are partially filled with the fluid phase of interest, such that the degree of saturation with respect to that phase is less than one. We will then show that this equation reduces, in the special case in which the degree of saturation equals one, to a simpler form appropriate for saturated single-phase flow. Then, this equation for saturated flow could be reduced further, in the special case in which the porosity equals one, to a form appropriate for purely fluid flow. For pedagogical reasons, however, we shall reverse this order and consider purely fluid flow first. In addition we will consider conservation of a solid or gas dissolved in a liquid, and take this opportunity to introduce Fick’s law for molecular diffusion. For simplicity we will consider only species that do not react chemically with the liquid, nor with the solid phases of a porous medium. Most of the derivations below are based on the idea of a small control volume of specified dimensions embedded within a fluid or porous medium.


1970 ◽  
Vol 10 (01) ◽  
pp. 51-55 ◽  
Author(s):  
Robert A. Albrecht ◽  
Sullivan S. Marsden

Abstract Although foam usually will flow in porous media, under certain controllable conditions it can also be used to block the flow of gas, both in unconsolidated sand packs and in sandstones. After steady gas or foam flow has been established at a certain injection pressure pi, the pressure is decreased until flow pressure pi, the pressure is decreased until flow ceases at a certain blocking pressure pb. When flow is then reestablished at a second, higher pi, blocking can again occur at another pb that will usually be greater than the first pi. The relationship between pi and Pb depends on the type of porous medium and the foamer solution saturation in the porous medium. A process is suggested whereby porous medium. A process is suggested whereby this phenomenon might be used to impede or block leakage in natural gas storage projects. Introduction The practice of storing natural gas in underground porous rocks has developed rapidly, and it now is porous rocks has developed rapidly, and it now is the major way of meeting peak demands in urban areas of the U. S. Many of these storage projects have been plagued with gas leakage problems that have, in some cases, presented safety hazards and resulted in sizeable economic losses. Usually these leaks are due to such natural factors as faults and fractures, or to such engineering factors as poor cement jobs and wells that were improperly abandoned. For the latter, various remedies such as spot cementing have been tried but not always with great success. In recent years several research groups have been studying the flow properties of aqueous foams and their application to various petroleum engineering problems. Most of this work has been done under problems. Most of this work has been done under experimental conditions such that the foam would flow in either tubes or porous media. However, under some extreme or unusual experimental conditions, flow in porous media becomes very difficult or even impossible. This factor also has suggested m us as well as to others that foam can be used as a gas flow impeder or as a sealant for leaks in gas storage reservoirs. In such a process, the natural ability of porous media to process, the natural ability of porous media to generate foam would be utilized by injecting a slug of foamer solution and following this with gas to form the foam in situ. This paper presents preliminary results of a sandy on the blockage of gas flow by foam in porous media. It also describes how this approach might be applied to a field process for sealing leaks in natural gas storage reservoirs. Throughout this report, we use the term "foam" to describe any dispersed gas-liquid system in which the liquid is the continuous phase, and the gas is the discontinuous phase. APPARATUS AND PROCEDURE A schematic drawing of the apparatus is shown in Fig. 1. At least 50 PV of filtered, deaerated foamer solution were forced through the porous medium to achieve liquid saturation greater than 80 percent. Afterwards air at controlled pressures was passed into the porous medium in order to generate foam in situ. Table 1 shows the properties and dimensions of the several porous media that were used. The beach sands were washed, graded and packed into a vibrating lucite tube containing a constant liquid level to avoid Stoke's law segregation over most of the porous medium. JPT P. 51


Sign in / Sign up

Export Citation Format

Share Document