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.     

Investigation of the porous drag and permeability at the porous-fluid interface: Influence of the filtering parameters on Darcy closure

The porous-fluid interface encompasses a region bridging the flow inside a porous medium and a free-flowing fluid. In the context of volume-averaged simulations, it can be described by a set of gradually changing parameters defining the porous medium, mainly porosity and permeability. In this paper, both the permeability and the porous-induced drag force are evaluated a-priori, by explicitly filtering a set of Particle-Resolved Simulations (PRS) of the flow in the channel partially occupied by the porous medium. Different porous matrices are considered and the influence of the geometry and filtering parameters on the macroscopic quantities is studied. Especially, the focus is placed on the requirements for the kernel type and size to perform filtering accurately, and their impact on the distribution of permeability at the interface. The performance of the typically used models for the permeability is compared to the explicitly filtered results. Lastly, a new model for permeability and the drag force is introduced, taking into account the information about the filtering size and non-uniformity of the velocity field. The model greatly improves the prediction of velocity at the porous-fluid interface and serves as a proof of concept that a successful porous drag model should strive to include information about both parameters.     

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.     

Vertical Permeability Distribution of Reservoirs with Impermeable Barriers

The presence of impermeable horizontal barriers (such as shales) in a reservoir is known to have a significant effect on its vertical permeability. Since calculation of an effective vertical permeability of such a reservoir is important, approximation of the distribution of vertical permeability may also be useful for analysis of the two-phase vertical flow of buoyant fluid, such as may occur in the subsurface injection of carbon dioxide into saline formations. In this situation, the maximum likely vertical permeability of a reservoir with impermeable barriers, which could be estimated from the probability distribution of the vertical permeability, is a more useful metric than an overall effective value for the vertical permeability due to its presumed relationship to breakthrough time. In this article, we derive expressions for the mean and variance of the vertical permeability of both two and three-dimensional reservoirs using the statistical streamline method of Begg and King (Paper No. 13529, 1985), and calculate the probability distribution of the vertical permeability of a reservoir with impermeable barriers. In addition, we also provide a simple statistical analysis of the presence of high vertical permeability regions in the reservoir, which may be of importance in coarse-scale simulations of vertical migration.     

Generalised Couette flow of two immiscible viscous fluids with heat transfer using Brinkman model

The present study deals with generalised Couette flow of two viscous, incompressible, immiscible fluids with heat transfer in presence of heat source through two straight parallel horizontal walls. The lower wall is bounded below, by a naturally permeable material of high porosity and the flow inside the porous medium is assumed to be of moderate permeability, modelled by Brinkman equation. The flow domain is divided into three zones to obtain analytical solutions of the momentum and energy equations. To link various flow regions, appropriate matching conditions have been used. The effects of permeability parameter, Reynolds number and viscous parameter on velocity field and the effects of Reynolds number, viscous parameter, permeability parameter, constant heat source and Brinkman number on temperature distribution in different zones are discussed graphically. The mass flow rate, skin-friction factor and rates of heat transfer at the upper boundary and porous interface are discussed with the help of tables.     

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

Separated two-phase flow in a nozzle

This work was performed to extend and further test the method of handling separated two-phase flow by studying each phase separately and, particularly, by placing emphasis on the study of the gas phase with interface transport expressions showing the influence of the liquid phase on it. A one-dimensional flow model for accelerating flows was used in conjunction with experimental data to obtain the pressure distribution and velocity distribution in a converging nozzle for several values of flow quality and nozzle inlet stagnation pressure. The results tend to support the use of the model (which includes the assumption that the gas is in critical flow when the two-phase mixture is in critical flow) and give some insight regarding the nature of the liquid distribution near the nozzle throat.     

基于WENO-THINC/WLIC模型的水气二相流数值模拟

水气二相流与诸多领域的实际工程问题密切相关. 对二相流运动进行高精度的数值模拟是计算流体力学研究的难点和热点. 针对开敞水域的自由表面流运动问题, 将水和空气均视为不可压缩流体, 采用五阶加权基本无震荡(weighted essentially non-oscillatory, WENO)格式求解描述流体运动的纳维斯托克斯(Navier-Stokes, NS)方程, 利用以加权线性界面算法改进的多维双曲正切函数界面捕捉法(tangent of hyperbola for interface capturing with weighed line interface calculation, THINC/WLIC)追踪水气界面, 建立WENO-THINC/WLIC水气二相流运动数值模型. 模型采用分步计算法离散求解控制方程, 通过压力投影法求解压强场, 并利用三阶总变差递减(total variation diminishing, TVD)龙格库塔(Runge-Kutta, RK)法对时间项进行离散求解. 通过对环境速度场下Zalesak's disk和shearing vortex界面运动问题, 线性液舱晃荡问题以及溃坝问题的模拟结果与理论分析或试验结果的比较, 对所建立的水气二相流数值模型的适用性及模拟精度进行了验证. 结果表明, 本模型的模拟结果与物理模型或理论分析结果吻合良好, 能较为准确地再现不可压缩水气二相流运动现象. 鉴于WENO格式和THINC法本身在算法及应用等方面仍在不断改进, 本研究提出的WENO-THINC耦合模型为后续更高精度的二相流计算模型开发提供了一种研究思路.      

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

Mixed convection flow over a vertical wedge embedded in a highly porous medium

 The steady mixed convection flow over a vertical wedge with a magnetic field embedded in a porous medium has been investigated. The effects of the permeability of the medium, surface mass transfer and viscous dissipation on the flow and temperature fields have been included in the analysis. The coupled nonlinear partial differential equations governing the flow field have been solved numerically using the Keller box method. The skin friction and heat transfer are found to increase with the parameters characterizing the permeability of the medium, buoyancy force, magnetic field and pressure gradient. However the effect of the permeability and magnetic field on the heat transfer is very small. The heat transfer increases with the Prandtl number, but the skin friction decreases. The buoyancy force which assists the forced convection flow causes an overshoot in the velocity profiles. Both the skin friction and heat transfer increase with suction and the effect of injection is just the reverse. Received on 21 May 1999     

Experimental study on the performance of a novel structure for two-phase flow distribution in parallel vertical channels

Uniform flow distribution is critical to obtain high thermal performance in many heat and mass transfer devices. It especially plays an important role in a compact heat exchanger. In this paper, a two-phase flow distributor is proposed for the evaporator unit of the plate-fin heat exchanger to alleviate the phase maldistribution in the multiphase flow. Air and water mixture was adopted as two-phase medium and distributions into ten parallel channels were measured in detail. The results show that the proposed distributor can improve the two-phase flow distribution of the plate-fin heat exchanger.     

Simultaneous heat and mass transfer under unsteady mixed convection along a vertical slender cylinder embedded in a porous medium

Unsteady laminar mixed convection flow (combined free and forced convection flow) along a vertical slender cylinder embedded in a porous medium under the combined buoyancy effect of thermal and species diffusion has been studied. The effect of the permeability of the medium as well as the magnetic field has been included in the analysis. The partial differential equations with three independent variables governing the flow have been solved numerically using a implicit finite difference scheme in combination with the quasilinearization technique. Computations have been carried out for accelerating, decelerating and oscillatory free stream velocity distributions. The effects of the permeability of the medium, buoyancy forces, transverse curvature and magnetic field on skin friction, heat transfer and mass transfer have been studied. It is found that the effect of free stream velocity distribution is more pronounced on the skin friction than on the heat and mass transfer. The permeability and magnetic parameters increase the skin friction, but reduce the heat and mass transfer. The skin friction, heat transfer and mass transfer are enhanced due to the buoyancy forces and curvature parameter. The heat transfer is strongly dependent on the viscous dissipation parameter and the Prandtl number, and the mass transfer on the Schmidt number.     

Contrôle passif d'écoulements incompressibles autour d'obstacles à l'aide de milieux poreux

Passive control of the flow behind a bluff-body is obtained by integrating porous area on the body. The penalisation method is used to modelize the flow in three different media. In fact each medium can be considered as a porous medium. The fluid is identified as a porous medium of infinite permeability and the solid is identified as a porous medium of zero permeability. This way, it is easy to compute the flow in each medium using the same parameter. Some benefical effects are due to the porous interface: the flow is smoothed, and the enstrophy and drag are significantly reduced.     

Analysis of forced-convection boiling flow instabilities in a single-channel upflow system

A numerical model is developed to predict the steady-state and transient behaviour of forced-convection boiling two-phase flow in a single channel. The model is based on the assumption of homogeneous two-phase flow and thermodynamic equilibrium of the phases. Compressibility effects in the two-phase region, motion of the bulk boiling interface and the thermal capacity of the heater wall have been included in the analysis. The model is used to study the effects of heat input, inlet subcooling and flow rate on the system behaviour. For comparison purposes, an experimental investigation was conducted using a single-channel, electrically heated, forced-convection upflow system. Steady-state operating characteristics, and stable and unstable regions, are determined as a function of heat flux, inlet subcooling and mass flow rate. Different modes of oscillation and their characteristics have been investigated. The model's predictions are in good agreement with the experimental results.     

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

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.     

Electrically charged two-phase flow past a sphere with a high electrostatic surface potential

A two-phase medium with a carrier phase in the form of an incompressible electrically neutral fluid and a dispersed phase in the form of inertial charged particles flows past an electrically charged sphere. It is assumed that the electrohydrodynamic interaction parameter is insignificant and that the flow conditions correspond to potential unseparated flow of the carrier medium over the sphere. The motion of the dispersed phase is described by continuum dynamic equations incorporating the electric field, which is the sum of the external field created by the sphere and the field induced by the dispersed particles. The electric field is determined by means of the equations of electrodynamics, which must be considered together with the dynamic equations. In the case considered a large electrostatic potential is applied to the sphere. This prevents the particles striking the surface of the sphere and leads to the intersection of the particle trajectories. In order to solve this problem within the framework of the two-velocity continuum we introduce a surface of discontinuity of the parameters to replace the zone of multiphase flow. The location of the surface of discontinuity, the distribution of the velocity and density of the dispersed phase and the distribution of electrostatic potential are found as a result of solving a system of elliptic and hyperbolic equations in two regions separated by the surface of discontinuity. The results of numerically integrating the system of equations formulated are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–95, March–April, 1987.     

Modeling the acidizing of a horizontal well opening a carbon-bearing oil reservoir

The model of two-phase seepage flow occurring when oil is displaced by an acid solution in the neighborhood of a horizontal well is considered with account for flow in the well bore. The effect of acid activation and acid fracturing is studied. The influence of the phase permeability variations accompanying the rock break-up on the treatment result is estimated. The nonuniformity of the inflow distribution along the horizontal bore is shown to increase as a result of the acidizing. Numerical calculations are carried out for actual situations.     

Unsteady flow through a porous medium in the presence of chemical reactions

The underground leaching of uranium ores and nonferrous and precious metals under natural conditions is one of the latest methods of mineral extraction [1]. It consists of pumping into isolated formations through reaction wells an acid solution which upon reacting with the rock yields a readily soluble salt that can be brought to the surface with water through extraction wells. Together with the acid solution, it is also possible to pump in other reactants to participate in the chemical reaction, for example, gases such as oxygen. Moreover, secondary gases may be formed as a result of the chemical reaction. Thus, the chemical reaction proceeds in the presence of a one or two-phase flow in the porous medium. The mathematical modeling of these processes is usually based on the approximation of one-phase flow without allowance for the changes in the porosity and permeability of the medium as a result of the reaction [2]. In this paper we solve the problem of unsteady flow in the presence of a chemical reaction for a two-phase system taking into account the changes in the flow parameters of the porous medium. The condition of stability of the plane reaction front is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–87, January–February, 1987.The author is grateful to R. I. Nigmatulin for his useful comments and interest in the work.