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.     

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.     

Nonequilibrium Effects During Spontaneous Imbibition

Accurate models of multiphase flow in porous media and predictions of oil recovery require a thorough understanding of the physics of fluid flow. Current simulators assume, generally, that local capillary equilibrium is reached instantaneously during any flow mode. Consequently, capillary pressure and relative permeability curves are functions solely of water saturation. In the case of imbibition, the assumption of instantaneous local capillary equilibrium allows the balance equations to be cast in the form of a self-similar, diffusion-like problem. Li et al. [J. Petrol. Sci. Eng. 39(3) (2003), 309–326] analyzed oil production data from spontaneous countercurrent imbibition experiments and inferred that they observed the self-similar behavior expected from the mathematical equations. Others (Barenblatt et al. [Soc. Petrol. Eng. J. 8(4) (2002), 409–416]; Silin and Patzek [Transport in Porous Media 54 (2004), 297–322]) assert that local equilibirum is not reached in porous media during spontaneous imbibition and nonequilibirium effects should be taken into account. Simulations and definitive experiments are conducted at core scale in this work to reveal unequivocally nonequilbirium effects. Experimental in-situ saturation data obtained with a computerized tomography scanner illustrate significant deviation from the numerical local-equilibrium based results. The data indicates: (i) capillary imbibition is an inherently nonequilibrium process and (ii) the traditional, multi-phase, reservoir simulation equations may not well represent the true physics of the process.     

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.     

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.     

Effect of gap and flow orientation on two-phase flow in an oil-wet gap: Relative permeability curves and flow structures

Naturally fractured reservoirs contain about 25–30% of the world supply of oil. In these reservoirs, fractures are the dominant flow path. Therefore, a good understanding of transfer parameters such as relative permeability as well as flow regimes occurring in a fracture plays an important role in developing and improving oil production from such complex systems. However, in contrast with gas–liquid flow in a single fracture, the flow of heavy oil and water has received less attention. In this research, a Hele-Shaw apparatus was built to study the flow of water in presence of heavy oil and display different flow patterns under different flow rates and analyze the effect of fracture orientations on relative permeability curves as well as flow regimes. The phase flow rates versus phase saturation results were converted to experimental relative permeability curves. The results of the experiments demonstrate that, depending on fracture and flow orientation, there could be a significant interference between the phases flowing through the fracture. The results also reveal that both phases can flow in both continuous and discontinuous forms. The relative permeability curves show that the oil–water relative permeability not only depends on fluid saturations and flow patterns but also fracture orientation.     

Application of X-ray CT investigation of CO<Subscript>2</Subscript>–brine flow in porous media

A clear understanding of two-phase flows in porous media is important for investigating CO₂ geological storage. In this study, we conducted an experiment of CO₂/brine flow process in porous media under sequestration conditions using X-ray CT technique. The flow properties of relative permeability, porosity heterogeneity, and CO₂ saturation were observed in this experiment. The porous media was packed with glass beads having a diameter of 0.2 mm. The porosity distribution along the flow direction is heterogeneous owing to the diameter and shape of glass beads along the flow direction. There is a relationship between CO₂ saturation and porosity distribution, which changes with different flow rates and fractional flows. The heterogeneity of the porous media influences the distribution of CO₂; moreover, gravity, fractional flows, and flow rates influence CO₂ distribution and saturation. The relative permeability curve was constructed using the steady-state method. The results agreed well with the relative permeability curve simulated using pore-network model.     

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.     

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcys equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.     

Effective and relative permeabilities of anisotropie porous media

A model composed of a three-dimensional orthogonal network of capillary tubes was used to simulate the flow behavior in an unsaturated anisotropic soil. The anisotropy in the network's permeability was introduced by randomly selecting the radii in the three mutually orthogonal directions of the network tubes from three different lognormal probability distributions, one for each direction. These three directions were assumed to be the principal directions of anisotropy. The sample was gradually drained, with only tubes smaller than a certain diameter remaining full at each degree of saturation. Computer experiments were conducted to determine the network's effective permeability as a function of saturation. The main conclusion was that the relationship between saturation and effective permeability depends on direction. Consequently the concept of relative permeability used in unsaturated flow should be limited to isotropic media and not extended to anisotropic ones.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.     

Models for Ground Water Flow: A Numerical Comparison Between Richards' Model and the Fractional Flow Model

In this paper we compare two models for flow in porous media. The first is the well known Richards' equation, which is based on the assumption that the air in the unsaturated zone has infinite mobility. This means that it models a single phase. In the second and more general full two-phase approach, the air is considered as a separate phase. Here, we use the fractional flow equation.We study the difference between the two models numerically by varying the relative contribution of the different physical terms (the gravity and the total velocity) in the fractional flow equation. Richards' equation is considered as the limit of the fractional flow approach when the mobility of the air-phase tends to infinity. In particular, we are interested in determining the parameter intervals where the two models differ significantly, and we will quantify the asymptotic behavior.The equations are studied in the two-dimensional (2D) case. The study is based on a relative permeability depending quadratically on the saturation, and a capillary pressure expressed by a cubic function of the saturation.     

Pore-scale Simulation of Water Alternate Gas Injection

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower during water injection in the presence of gas due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.     

Anisotropy effects in two-phase flows through porous media

Effects manifested in two-phase flows through anisotropic porous reservoirs with monoclinic and triclinic characteristics are analyzed. It is shown that in two-phase flows through media with monoclinic and triclinic symmetries of flow characteristics the position of the principal axes of the phase permeability tensors depends on the saturation and does not coincide with the position of the principal axes of the absolute permeability tensor in single-phase flows and that going over from single-to two-phase flow may lead to a change in the symmetry group of the flow characteristics. A general representation of the phase permeability tensor components is presented and formulas are given for the diagonal and nondiagonal components of the relative phase permeabilities, which are universal and can be used for anisotropic media with any type of anisotropy (symmetry) of flow characteristics. A complex of laboratory tests for finding the nondiagonal components of the phase and relative phase permeability tensors is discussed.     

Numerical study of non-uniqueness of the factors influencing relative permeability in heterogeneous porous media by lattice Boltzmann method

The effects of capillary number Ca and viscosity ratio M on non-uniqueness of the relative permeability (RP) – wetting saturation (S_w) relationships of steady-state immiscible two-phase flow in the heterogeneous porous media are studied by a developed lattice Boltzmann method (LBM). In this work, it is suggested that the non-uniqueness of capillary number Ca and viscosity ratio M influencing RP–S_w curves is due to the presence of the heterogeneity of porous media. For the different Ca and M, the RP–S_w curves in the homogenous and heterogeneous porous media with the same porosity are discussed in detail. It is found that (1) the pore-size distributions in porous media have significant effect on the RPs of both wetting phase (WP) and non-wetting phase (NWP); (2) the heterogeneity of porous media has negative effect on the increment of the RP of NWP caused by the large capillary number. The lubricating effect is strongly dependent on pore-size distributions in porous media; and (3) the large viscosity ratio increases the RP of NWP, while it has little effect on the RP of WP. The effect on the RP of NWP may be magnified by the heterogeneity of porous media.     

Numerical Analysis of Imbibition Front Evolution in Fractured Sandstone under Capillary-Dominated Conditions

Fractures serve as primary conduits having a great impact on the migration of injected fluid into fractured permeable media. Appropriate transport properties such as relative permeability and capillary pressure are essential for successful simulation and prediction of multi-phase flow in such systems. However, the lack of a thorough understanding of the dynamics governing immiscible displacement in fractured media, limits our ability to properly represent their macroscopic transport properties. Previous experimental observations of imbibition front evolution in fractured rocks are examined in the present study using an automated history-matching approach to obtain representative relative permeability and capillary pressure curves. Predicted imbibition front evolution under different flow conditions resulted in an excellent agreement with experimental observations. Sensitivity analyses, in combination with direct experimental observation, allowed exploring the competing effects of relative permeability and capillary pressure on the development of saturation distribution and imbibing front evolution in fractured porous media. Results show that residual saturations are most sensitive to matrix relative permeability to oil, while the ratio of oil and water relative permeability, rock heterogeneity, boundary condition, and matrix–fracture capillary pressure contrast, affect displacement shape, speed, and geometry of the imbibing front.     

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.     

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.     

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).