Two-Phase Filtration in a Transversally Isotropic Porous Medium: Experiment and Theory

The results of an experimental determination of the relative phase permeabilities during flow of two immiscible fluids in stratified sandstone with transversally isotropic characteristics are presented. The measurements were performed on samples oriented in three directions: along, perpendicular to and at an angle of 45° to the stratification plane. An approximate solution of the problem of steady two-phase flow toward a finite gallery in an anisotropic porous medium for arbitrary relative orientation of the gallery and the principal axes of the absolute permeability tensor is given. This solution was tested against the experimental results. The good agreement between the theoretical and experimental results makes it possible to recommend for engineering calculations both the relations between the absolute and phase permeabilities for transversally isotropic and orthotropic porous media and the approximate solution obtained.     

Compressibility and anisotropy effects in the processes of two-phase flow through a porous medium

Functions of the relative phase permeabilities which take into account the tensor nature of the relations between the absolute and phase permeabilities are constructed on the basis of experimental data. Problems of the displacement of gas by water and of water by gas in an orthotropic pore space are studied. The relative phase permeabilities obtained are compared with functions that depend only on the local saturation. The effect of the phase compressibility and the anisotropy of the absolute permeability tensor is considered.     

Generalized Darcy's Law and the Structure of the Phase and Relative Phase Permeabilities for Two-Phase Flows through Anisotropic Porous Media

For two-phase immiscible fluid flows a generalized Darcy's law is written in invariant tensor form for crystallographic point symmetry groups and anisotropic textures. The representation of the phase permeability coefficient tensors and the structure of the expressions for the relative phase permeabilities are analyzed for all symmetry groups. The relation between the phase and absolute permeability coefficient tensors is specified by a fourth-rank tensor with the external symmetry coinciding with external symmetry of the phase permeability tensors. It is shown that the external symmetry of the phase permeability coefficient tensors can differ from the external symmetry of the absolute permeability tensor. For triclinic and monoclinic symmetry groups it is shown that the phase permeability coefficient tensors may not be coaxial with each other and with the absolute permeability tensor; moreover, the directions of the principal axes of the phase permeability coefficient tensors can depend on the saturation.     

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.     

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:     

Experimental investigation into the influence of polymer additives in water on the relative permeabilities of porous media

A method has been developed for investigating the relative permeabilities of porous media for oil and for aqueous solutions of polymers; experimental equipment has been developed for determining the phase permeabilities by a stationary method. Investigations were made of the influence of polyacrylamide additives on the change in the relative permeabilities for the simultaneous flow of water and a nonpolar hydrocarbon liquid. It was established that addition of the polymer can lead to a simultaneous reduction in the relative permeability for the wetting liquid and an increase for the nonwetting liquid. The phase permeabilities were obtained for oil and water moving behind a fringe of polymer substance. It was established that the phase permeability for the water phase is a function of the saturation and the amount of sorbate. A cycle of experimental investigations was made into the influence of the rate of pumping and the concentration of the dissolved polymer on the change in the relative permeabilities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 163–167, July–August, 1980.     

Pore Scale Modeling of Rate Effects in Imbibition

We use pore scale network modeling to study the effects of flow rate and contact angle on imbibition relative permeabilities. The model accounts for flow in wetting layers that occupy roughness or crevices in the pore space. Viscous forces are accounted for by solving for the wetting phase pressure and assuming a fixed conductance in wetting layers. Three-dimensional simulations model granular media, whereas two-dimensional runs represent fracture flow.We identify five generic types of displacement pattern as we vary capillary number, contact angle, and initial wetting phase saturation: flat frontal advance, dendritic frontal advance, bond percolation, compact cluster growth, and ramified cluster growth. Using phase diagrams we quantify the range of physical properties under which each regime is observed. The work explains apparently inconsistent experimental measurements of relative permeability in granular media and fractures.     

Application of Effective Flux Boundary Conditions to Two-Phase Upscaling in Porous Media

A new algorithm is introduced for upscaling relative permeabilities, and tested in simulations of two-dimensional reservoir displacement processes. The algorithm is similar to existing algorithms for computing upscaled relative permeabilities from subgrid simulations, but uses new boundary conditions for the pressure field. The new 'effective flux boundary conditions' were introduced in a previous paper and provide a more accurate estimate of flux through high permeability channels. The algorithm was tested in conjunction with uniform grid coarsening and upscaled absolute permeabilities for a broad range of coarsenings. The permeability fields were highly heteroge-neous and layered, and were obtained from synthetic data and from conditioned realizations of actual oil reservoirs. The algorithm was tested for a wide variety of grid aspect ratios, and for both viscous-and gravity-dominated flow. Typical fine grids were of the order of 100×100 cells; the coarsest scaled-up grids were on the order of 5×5 cells. The quality of scale up was evaluated by comparing oil cut curves for the fine and coarse grid simulations. We consistently obtained excellent agreement, even at the coarsest levels of scale up.     

Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

Percolation model of two-phase equilibrium flow in a porous medium with microheterogeneous wettability

The percolation model of two-phase flow described in [1, 2] is used as a basis for examining the problem of the behavior of the characteristics of two-phase equilibrium flow in a porous medium when the capillaries have a radius distribution and differ with respect to the wettability properties of their surfaces. Analytic expressions describing the dependence of the relative phase permeability coefficients on the saturation of the medium by the displacing phase and the microinhomogeneous wettability parameters are obtained. A qualitative comparison shows the theoretical results to be consistent with the data of a direct numerical computer calculation of a grid model [3]. The effect of the microinhomogeneity parameters and the form of the capillary radius distribution function on the phase permeabilities is analyzed within the framework of the approach developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–93, September–October, 1989.     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.     

Accounting for Fluid Rheology in the Hydrophobization Model of Relative Permeability Hysteresis

A complex mathematical model of relative permeability hysteresis is developed based on percolation theory. The model takes into account the change in the surface properties of the pore space and the rheology of percolating fluids during the transition from drainage to imbibition, which gives rise to hysteresis. It is shown that accounting for the change in the rheology of percolation fluids, along with accounting for the hydrophobization of the surface of the pore space, provides a better agreement between the calculated and experimental curves of relative permeabilities.     

Principles of the analytic description of fluid flow in network models of porous media

A method of calculating the conductivity of capillary networks which combines percolation concepts of conductivity in the neighborhood of the flow threshold with the results obtained using the effective medium model is proposed. The conductivity calculated theoretically is compared with experimental data. Water and oil phase permeabilities for combined flow through a capillary network are determined as an example of the use of the relations obtained. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 79–83, January–February, 2000.     

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).     

Estimation of three-phase relative permeabilities

We have extended the Alemán-Slattery model to provide a self-consistent prediction for the residual saturation of the intermediate-wetting phase. Previous experimental studies of three-phase relative permeabilities are critiqued. Only a portion of the data of Oaket al. (J. Petrol. Tech. 42 (1990) 1054) and Oak (SPE/DOE 20183, Society of Petroleum Engineers, 1990) is regarded as suitable for comparison with available models. While the extended Alemán-Slattery model appears to give the best representation of these data, a definitive conclusion is premature.