Ramon G. Bentsen

Interfacial viscous coupling: a myth or reality?

This paper is a review of interfacial viscous coupling in multiphase porous media flow which has been a matter of debate for the last two decades. Several researchers have viewed the occurrence of interfacial viscous coupling phenomena as a reality and strongly recommended its incorporation into the existing Darcy formulation for multiphase flow through porous media. Other investigators, however, have argued that the effect(s) of mutual transfer of momentum between flowing fluids is extremely small; hence, it can be ignored and there is no real need to modify the conventional Darcy formulation. Others, however, disagree, even with the existence of such phenomena. This paper is an attempt to resolve this controversy. To accomplish this task, and to present a fair judgment about the controversy, three theoretical approaches are reviewed. These are (1) the volume averaging methods, (2) the irreversible thermodynamic methods and (3) the use of analogous models, which were used to develop a theoretical understanding of viscous coupling phenomena. On the basis of this review, it was found that the phenomena of interfacial viscous coupling is closer to a reality rather than a myth. Therefore, it is strongly recommended that the conventional two-phase flow formulation of Darcys law should be modified. Some of the important problems, related to validation of interfacial viscous coupling theory, and possible means for resolving them, are identified.

On the use of conventional cocurrent and countercurrent effective permeabilities to estimate the four generalized permeability coefficients which arise in coupled, two-phase flow

In the case of coupled, two-phase flow of fluids in porous media, the governing equations show that there are four independent generalized permeability coefficients which have to be measured separately. In order to specify these four coefficients at a specific saturation, it is necessary to conduct two types of flow experiments. The two types of flow experiments used in this study are cocurrent and countercurrent, steady-state permeability experiments. It is shown that, by taking this approach, it is possible to define the four generalized permeability coefficients in terms of the conventional cocurrent and countercurrent effective permeabilities for each phase. It is demonstrated that a given generalized phase permeability falls about midway between the conventional, cocurrent effective permeability for that phase, and that for the countercurrent flow of the same phase. Moreover, it is suggested that the conventional effective permeability for a given phase can be interpreted as arising out of the effects of two types of viscous drag: that due to the flow of a given phase over the solid surfaces in the porous medium and that due to momentum transfer across the phase 1-phase 2 interfaces in the porous medium. The magnitude of the viscous coupling is significant, contributing at least 15% to the total conventional cocurrent effective permeability for both phases. Finally, it is shown that the nontraditional generalized permeabilities which arise out of viscous coupling effects cannot equal one another, even when the viscosity ratio is unity and the surface tension is zero.

Effect of momentum transfer between fluid phases on effective mobility

Abstract Conventional methods for determining effective mobilities are based on the assumption that interfacial momentum transfer may be neglected. The effect of such neglect is investigated by using fractional flow theory, in conjunction with Kalaydjians transport equations and Liang and Lohrenzs method for calculating generalized mobilities, to construct new equations for calculating effective mobilities. With the help of these equations, it is shown that, for the sand–fluid system used in this study, neglect of viscous coupling between the two flowing phases can result in the introduction of relative errors as large as 30% into the calculated values of the effective mobilities. Because the largest errors occur only in the frontal region, and because the actual shape of the frontal region is unimportant in most practical problems, it is thought that the errors arising because of such neglect should be tolerable, provided that the displacement is stabilized. Finally, the results of this investigation are used as a basis for arguing that the cross coefficients, λ 12 and λ 21 , which control momentum transfer between fluid phases in porous media, cannot be equal. The fact that interfacial momentum transfer should not be neglected is important to a clear understanding of multiphase flow through porous media. Thus, the results of this study should be of interest to theoreticians, people who measure effective mobility, and the people who use such information in reservoir simulation.

The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media

Recently developed transport equations for two-phase flow through porous media usually have a second term that has been included to account properly for interfacial coupling between the two flowing phases. The source and magnitude of such coupling is not well understood. In this study, a partition concept has been introduced into Kalaydjians transport equations to construct modified transport equations that enable a better understanding of the role of interfacial coupling in two-phase flow through natural porous media. Using these equations, it is demonstrated that, in natural porous media, the physical origin of interfacial coupling is the capillarity of the porous medium, and not interfacial momentum transfer, as is usually assumed. The new equations are also used to show that, under conditions of steady-state flow, the magnitude of mobilities measured in a countercurrent flow experiment is the same as that measured in a cocurrent flow experiment, contrary to what has been reported previously. Moreover, the new equations are used to explicate the mechanism by which a saturation front steepens in an unstabilized displacement, and to show that the rate at which a wetting fluid is imbibed into a porous medium is controlled by the capillary coupling parameter, α. Finally, it is argued that the capillary coupling parameter, α, is dependent, at least in part, on porosity. Because a clear understanding of the role played by interfacial coupling is important to an improved understanding of two-phase flow through porous media, the new transport equations should prove to be effective tools for the study of such flow.

An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media

Many enhanced oil recovery schemes involve the displacement of oil by a miscible fluid. Whether a displacement is stable or unstable has a profound effect on how efficiently a solvent displaces oil within a reservoir. That is, if viscous fingers are present, the displacement efficiency and, hence, the economic return of the recovery scheme is seriously impaired bacause of macroscopic bypassing of the oil. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable.This paper presents a dimensionless scaling group for predicting the onset of hydrodynamic instability of a miscible displacement in porous media. An existing linear perturbation analysis was extended in order to obtain the scaling group. The new scaling group differs from those obtained in previous studies because it takes into account a variable unperturbed concentration profile, both transverse dimensions of the porous medium, and both the longitudinal and the transverse dispersion coefficient.It has been shown that stability criteria derived in the literature are special cases of the general condition given here. Therefore, the stability criterion obtained in this study should be used for a displacement conducted under arbitrary conditions. The stability criterion is verified by comparing it with miscible displacement experiments carried out in a Hele-Shaw cell. Moreover, a comparison of the theory with some porous medium experiments from the literature also supports the validity of the theory.

Influence of hydrodynamic forces and interfacial momentum transfer on the flow of two immiscible phases

Abstract Conventional theory regarding immiscible two-phase flow neglects the effect that hydrodynamic forces and interfacial momentum transfer have on flow through porous media. The significance of such neglect is explored by estimating the relative error incurred when these effects are neglected. It is shown, by investigating several one-dimensional problems, that relative errors of about 1% are incurred by neglecting the effect of the hydrodynamic forces, whereas errors of about 40% are introduced when viscous coupling across fluid–fluid interfaces is neglected.

An investigation into whether the nondiagonal mobility coefficients which arise in coupled, two phase flow are equal

Recent developments in two-phase flow through porous media show that four mobilities are required to define completely the flow characteristics of a particular porous medium. Because, in idealized porous media, it has been shown that two of these mobilities, the nondiagonal mobilities which represent the viscous coupling exerted between fluid phases, are equal, it has been suggested that they may be equal, as well, in real porous media. It is shown in this paper that these two interactive mobilities cannot be equal in real porous media. Moreover, it is demonstrated that the relative permeabilities which pertain to pure countercurrent flow differ from those which pertain to steady-state, countercurrent flow, and that the pure countercurrent-flow relative permeabilities depend strongly on viscosity ratio. Finally, it is suggested that, because three different experiments give rise to three different sets of relative permeability curves, the conventional description of two-phase flow is inadequate inasmuch as it does not account properly for the viscous coupling exerted between fluid phases.

Interfacial Coupling in Vertical, Two-Phase Flow Through Porous Media

Abstract: Vertical, two-phase flow through natural porous media is affected by interfacial coupling. Such coupling may be of two types: viscous and capillary. In this study, macroscopic defining equations for the viscous and capillary coupling parameters have been constructed. Moreover, these defining equations, together with modified forms of Kalaydjians transport equations, have been used to construct equations that can be used to analyze the effect of interfacial coupling on vertical, two-phase flow in natural porous media. These equations reveal that viscous coupling is dependent explicitly on mobility ratio, and that the effect of viscous coupling on vertical flow is bounded. Moreover, these equations show that capillary coupling depends explicitly on porosity. Furthermore, on the basis of the analysis carried out, it is argued that the effect of viscous coupling on vertical flow is small, and that the effect of capillary coupling on horizontal flow is an order of magnitude larger than that of viscous coupling. The data from two sets of gravity-driven, steady-state, cocurrent and countercurrent flow experiments were used to test the interfacial coupling theory. It was found that the experimentally determined and predicted values of the capillary coupling parameters were consistent for both the wetting and the nonwetting phase, provided proper account was taken of the contribution of the net buoyant force to the driving force for each phase. Because of a lack of sufficient experimental evidence, further experimental testing of the theory must be undertaken before it can be accepted with confidence. Even so, the new equations should be, because they show explicitly the role of mobility ratio and porosity, useful tools for improving our understanding of the effect of interfacial coupling on vertical, two-phase flow through natural porous media.

Effect of hydrodynamic forces on the pressure-difference equation

Because of the influence of hydrodynamic forces, the difference in macroscopic pressure which exists, at static equilibrium, between two immiscible phases located in a porous medium may be different from that which pertains during flow. In this paper, the concept of relative pressure difference, together with a new pressure-difference equation, is used to investigate the impact that the hydrodynamic forces have on the difference in macroscopic pressure which pertains when two immiscible fluids flow simultaneously through a homogeneous, water-wet porous medium. This investigation reveals that, in general, the equation defining the difference in pressure between two flowing phases must include a term which takes proper account of the hydrodynamic effects. Moreover, it is pointed out that, while neglect of the hydrodynamic effects introduces only a small amount of error when the two fluids are flowing cocurrently, such neglect is not permissible during steady-state, countercurrent flow. This is because failure to include the impact of the hydrodynamic effects in the latter case makes it impossible to explain the pressure behaviour observed in steady-state, countercurrent flow. Finally, the results of this investigation are used as a basis for arguing that, during steady-state, countercurrent flow, saturation is uniform, as is the case of steady-state, cocurrent flow.

Effect of Hydrodynamic Forces on Capillary Pressure and Relative Permeability

Because of the influence of hydrodynamic forces, the capillary pressure measured at static equilibrium may be different from that which pertains during flow. If such is the case, it may not be permissible to use steady-state relative permeabilities to predict unsteady-state flow. In this paper, the idea that the total flux of a given phase may be partitioned into several individual fluxes, together with a new pressure difference equation, is used to explore the possible impact that the hydrodynamic forces might have on capillary pressure and, as a consequence, relative permeability. This exploration reveals that, provided the pressure difference equation is implemented properly, capillarity has no impact on the relative permeability curves for the homogeneous, water-wet porous media considered. Moreover, it is demonstrated that, if the hydrodynamic effects are neglected, very little error is introduced into the analysis.