Steady-state countercurrent flow in one dimension

Two phase countercurrent steady-state flow through permeable media in one dimension is discussed. For steady-state countercurrent flow in water wet porous media, a saturation profile is predicted with the water saturation decreasing in the direction that the water phase is flowing. The de la Cruz and Spanos equations predict that the Muskat relative permeability curves for countercurrent flow will be less than the Muskat relative permeability curves for steady-state cocurrent flow. This result has immediate implications regarding the use of external drive techniques to determine relative permeabilities based on the Buckley-Leverett theory and Muskat's equations. These equations and current experimental evidence involving countercurrent flow indicate that Muskat's equations do not adequately describe the multiphase flow of immiscible fluids.     

Numerical Investigations of the Steady State Relative Permeability of a Simplified Porous Medium

The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio. The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M=₁/₂; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993).(Now at AS Norske Shell, Norway.) e-mail:     

On the Construction of an Experimentally Based Set of Equations to Describe Cocurrent and Countercurrent, Two-Phase Flow of Immiscible Fluids Through Porous Media

Generalized flow equations developed for two-phase flow through porous media contain a second term that enables proper account to be taken of capillary coupling between the two flowing phases. In this study, a partition concept, together with a novel capillary pressure equation for countercurrent flow, have been introduced into Kalaydjian’s generalized flow equations to construct modified flow equations which enable a better understanding of the role of capillary coupling in horizontal, two-phase flow. With the help of these equations it is demonstrated that the reduced flux observed in countercurrent flow, as compared to cocurrent flow, can be explained by the reduction in the driving force per unit volume which comes about because of capillary coupling. Also, it is shown experimentally that, because fluids flow through a void space reduced in magnitude due to the presence of immobile irreducible and residual saturations, the capillary coupling parameter should be defined in terms of a reduced porosity, rather than in terms of porosity. Moreover, it is shown statistically that the countercurrent relative permeability curve is proportional to the cocurrent relative permeability curve, the constant of proportionality being the capillary coupling parameter. Finally it is suggested that one can eliminate the need to determine experimentally countercurrent relative permeability curves by making use of an equation constructed for predicting the magnitude of the capillary coupling parameter.     

Generalized relative permeability coefficients during steady-state two-phase flow in porous media,and correlation with the flow mechanisms

A parametric experimental investigation of the coupling effects during steady-state two-phase flow in porous media was carried out using a large model pore network of the chamber-and-throat type, etched in glass. The wetting phase saturation,S ₁, the capillary number,Ca, and the viscosity ratio,k, were changed systematically, whereas the wettability (contact angleθ _e ), the coalescence factorCo, and the geometrical and topological parameters were kept constant. The fluid flow rate and the pressure drop were measured independently for each fluid. During each experiment, the pore-scale flow mechanisms were observed and videorecorded, and the mean water saturation was determined with image analysis. Conventional relative permeability, as well as generalized relative permeability coefficients (with the viscous coupling terms taken explicitly into account) were determined with a new method that is based on a B-spline functional representation combined with standard constrained optimization techniques. A simple relationship between the conventional relative permeabilities and the generalized relative permeability coefficients is established based on several experimental sets. The viscous coupling (off-diagonal) coefficients are found to be comparable in magnitude to the direct (diagonal) coefficients over board ranges of the flow parameter values. The off-diagonal coefficients (k _rij /Μ _j ) are found to be unequal, and this is explained by the fact that, in the class of flows under consideration, microscopic reversibility does not hold and thus the Onsager-Casimir reciprocal relation does not apply. Thecoupling indices are introduced here; they are defined so that the magnitude of each coupling index is the measure of the contribution of the coupling effects to the flow rate of the corresponding fluid. A correlation of the coupling indices with the underlying flow mechanisms and the pertinent flow parameters is established.     

The Role of Capillarity in Two-Phase Flow through Porous Media

When determining experimentally relative permeability and capillary pressure as a function of saturation, a self-consistent system of macroscopic equations, that includes Leverett's equation for capillary pressure, is required. In this technical note, such a system of equations, together with the conditions under which the equations apply, is formulated. With the aid of this system of equations, it is shown that, at the inlet boundary of a vertically oriented porous medium, static conditions pertain, and that potentials, because of the definition of potential, are equal in magnitude to pressures. Consequently, Leverett's equation is valid at the inlet boundary of the porous medium, provided cocurrent flow, or gravity-driven, countercurrent flow is taking place, and provided the porous medium is homogeneous. Moreover, it is demonstrated that Leverett's equation is valid for flow along the length of a vertically oriented porous medium, provided cocurrent flow, or gravity-driven, countercurrent flow is taking place, and provided the porous medium is homogeneous and there are no hydrodynamic effects. However, Leverett's equation is invalid for horizontal, steady-state, forced, countercurrent flow. When such flow is taking place, it is the sum of the pressures, and not the difference in pressures, which is related to capillary pressure.     

A Simple Model for the Variation of Permeability due to Partial Saturation in Dual Scale Porous Media

The main focus of this work is to model macroscopically the effects of partial saturation upon the permeability of dual scale fibrous media made of fiber bundles when a Newtonian viscous fluid impregnates it. A new phenomenological model is proposed to explain the discrepancies between experimental pressure results and analytical predictions based on Darcy's law. This model incorporates the essential features of relative permeability but without the necessity of measuring saturation of the liquid for its prediction. The model is very relevant for the small scale industrial systems where a liquid is forced to flow through a fibrous porous medium. It requires four parameters. Two of them are the two permeability values based on the two length scales. One length scale is of the order of magnitude of the individual fiber radius and corresponds to the permeability of the completely staurated medium, the other is of the order of magnitude of the distance between the fiber bundles and corresponds to the permeability of the partially saturated medium. The other two parameters are the lengths of the two partially saturated regions of the flow domain. The two lengths of the partially saturated region and the permeability of the fully saturated flow domain can be directly measured from the experiments. The excellent agreement between the model and the experimental results of inlet pressure profile with respect to time suggests that this model may be used to describe the variation of the permeability behind a moving front in such porous media for correct pressure prediction. It may also be used to characterize the fibrous medium by determining the two different permeabilities and the relative importance of the unsaturated portion of the flow domain for a given architecture.     

Origin and quantification of coupling between relative permeabilities for two-phase flows in porous media

An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate. This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.     

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.     

Determining equations of two-phase flows through anisotropic porous media

A new interpretation of the concept of relative phase permeability is given. Relative phase permeabilities are represented in the form of fourth-rank tensors. It is shown that in the case of anisotropic porous media functions depending not only on the saturation but also on the anisotropy parameters represented in the form of ratios of the principal values of the absolute permeability coefficient tensor correspond to the classical representation of the relative phase permeabilities. For a two-phase flow in anisotropic porous media with orthotropic and transversely-isotropic symmetry a generalized two-term Darcy’s law is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–94, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00623).     

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.     

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.     

Investigation of Generalized Relative Permeability Coefficients for Electrically Assisted Oil Recovery in Oil Formations

Evaluation of relative permeability coefficients is one of the key steps in reliable simulation of two-phase flow in porous media. An extensive body of work exists on evaluation of these coefficients for two-phase flow under pressure gradient. Oil transport under an applied electrical gradient in porous media is also governed by the principles of two-phase flow, but is less understood. In this paper, relative permeability coefficients under applied electric field are evaluated for a specific case of two- phase fluid flow in water-wet porous media, where the second fluid phase is oil. It is postulated that the viscous drag on the oil phase, exerted by the electro-osmotic flow of the water phase, is responsible for the transport of oil in the absence of a pressure gradient. Reliable prediction of the flow patterns necessitates accurate representation and determination of the relative permeability coefficients under the electrical gradient. The contribution of each phase to the flow is represented mathematically, and the relative permeability coefficients are evaluated through electro-osmotic flow measurements conducted on oil bearing rock cores.     

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 Kalaydjian's 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, . Finally, it is argued that the capillary coupling parameter, , 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.     

Investigating the Temperature Dependency of Oil and Water Relative Permeabilities for Heavy Oil Systems

A look into the literature on the temperature dependency of oil and water relative permeabilities reveals contradictory reports. There are some publications reporting shifts in the water saturation range as well as variations in the relative permeability curves by temperature. On the other hand, some authors have blamed the experimental artifacts, viscous instabilities and fingering issues for these variations. We have performed core flooding experiments to further investigate this issue. Glass bead packs and sand packs were used as the porous media, and Athabasca bitumen with varying viscosities was displaced by hot water at differing temperatures. The unsteady-state method of relative permeability measurement was applied and the experimental data were history matched by a simulator that is tailor made to predict the relative permeabilities. The matches were obtained by varying the relative permeability correlation parameters. The results indicated that the initial water saturation has a direct relation with temperature, while residual oil saturation generally drops at higher temperatures. Although the water saturation range shifts, no direct and unique trend for either oil or water relative permeability is justified. The spread in relative permeabilities especially in the case of higher permeable cores suggests that viscous instabilities are present. As the same saturation shift happens by only changing the oil viscosity, the relative permeability variations with temperature can be attributed to oil to water viscosity ratio changes with temperature. Temperature dependency of relative permeabilities is more related to experimental artifacts, viscous fingering and viscosity changes than fundamental flow properties.     

Simulation of Three-Phase Displacement Mechanisms Using a 2D Lattice-Boltzmann Model

Using a numerical technique, known as the lattice-Boltzmann method, we study immiscible three-phase flow at the pore scale. An important phenomenon at this scale is the spreading of oil onto the gas–water interface. In this paper, we recognize from first principles how injected gas remobilizes initially trapped oil blobs. The two main flow mechanisms which account for this type of remobilization are simulated. These are the double-drainage mechanism and (countercurrent) film flow of oil. The simulations agree qualitatively with experimental findings in the literature. We also simulate steady-state three-phase flow (fixed and equal saturations) in a small segment of a waterwet porous medium under both spreading and nonspreading conditions. The difference between the two conditions with respect to the coefficients in the generalized law of Darcy (which also includes viscous coupling) is investigated.     

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

An Analysis of Three-Phase Pore Occupancies and Relative Permeabilities in Porous Media with Variable Wettability

In three-phase flow, the macroscopic constitutive relations of capillary pressure and relative permeability as functions of saturation depend in a complex manner on the underlying pore occupancies. These three-phase pore occupancies depend in turn on the interfacial tensions, the pore sizes and the degree of wettability of the pores, as characterised by the cosines of the oil–water contact angles. In this work, a quasi-probabilistic approach is developed to determine three-phase pore occupancies in media where the degree of wettability varies from pore to pore. Given a set of fluid and rock properties, a simple but novel graphical representation is given of the sizes and oil–water contact angles underlying three-phase occupancies for every allowed combination of capillary pressures. The actual phase occupancies are then computed using the contact angle probability density function. Since a completely accessible porous medium is studied, saturations, capillary pressures, and relative permeabilities are uniquely related to the pore occupancies. In empirical models of three-phase relative permeability it is of central importance whether a phase relative permeability depends only on its own saturation and how this relates to the corresponding two-phase relative permeability (if at all). The new graphical representation of pore sizes and wettabilities clearly distinguishes all three-phase pore occupancies with respect to these saturation-dependencies. Different types of saturation-dependencies may occur, which are shown to appear in ternary saturation diagrams of iso-relative permeability curves as well, thus guiding empirical approaches. However, for many saturation combinations three-phase and two-phase relative permeabilities can not be linked. In view of the latter, the present model has been used to demonstrate an approach for three-phase flow modelling on the basis of the underlying pore-scale processes, in which three-phase relative permeabilities are computed only along the actual flow paths. This process-based approach is used to predict an efficient strategy for oil recovery by simultaneous water-alternating-gas (SWAG) injection.