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.     

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.     

Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Evaporation and Condensation Fronts in Porous Media

Flow of a fluid through a porous medium is considered with allowance for heat conduction. Both fronts at which the liquid is transformed into steam or a liquid-steam mixture and fronts with inverse transformations are studied. The evolutionarity conditions of these fronts are considered and a model of their structure is proposed.     

A Class of Viscosity Profiles for Oil Displacement in Porous Media or Hele-Shaw Cell

This paper is devoted to the study of the stability of oil displacement in porous media. Results are applied to the secondary oil recovery process: the oil contained in a porous medium is obtained by pushing it with a second fluid (usually water). As in Saffman and Taylor (1958) and Gorell and Homsy (1983) the porous medium will be modelized by a Hele-Shaw cell. If the second fluid is less viscous, the fingering phenomenon appears, first studied by Saffman and Taylor (1959). In order to minimize this instability, we consider, as in Gorell and Homsy (1983), an intermediate polymer-solute region (i.r.), with a variable viscosity \mu , between water and oil. This viscosity increases from water to oil. The linear stability of the interfaces is governed by a Sturm–Liouville problem which contains eigenvalues in the boundary conditions. Its characteristic values are the growth constants of the perturbations. The stability can be improved by choosing a minimizing viscosity profile \mu which gives us an arbitrary small positive growth constant. In this paper, we suggest a class of minimizing profiles. This main result is obtained by considering the Rayleigh quotient to estimate – without any discretization – the characteristic values of the above Sturm–Liouville problem. A finite-difference procedure and Gerschgorins localization theorem were used by Carasso and Paa (1998) to solve the above problem. A formula of an exponential viscosity profile in (i.r.) was obtained. The new class of minimizing viscosity profiles described in this paper includes linear and exponential profiles. The corresponding total amount of polymer and the (i.r.) length are estimated in terms of the limit value of \mu on the (i.r.) – oil interface. Our results are compared with the previous theoretical viscosity profiles. We show that the linear case is more favorable compared with the exponential profile. We give lower estimates of the total amount of polymer and of the (i.r.) length for a given improvement of the stability , compared with the Saffman–Taylor case.     

Phenomenological Meniscus Model for Two-Phase Flows in Porous Media

A new macroscale model of a two-phase flow in porous media is suggested. It takes into consideration a typical configuration of phase distribution within pores in the form of a repetitive field of mobile menisci. These phase interfaces give rise to the appearance of a new term in the momentum balance equation, which describes a vectorial field of capillary forces. To derive the model, a phenomenological approach is developed, based on introducing a special continuum called the Meniscus-continuum. Its properties, such as a unique flow velocity, an averaged viscosity, a compensation mechanism and a duplication mechanism, are derived from a microscale analysis. The closure relations to the phenomenological model are obtained from a theoretical model of stochastic meniscus stream and from numerical simulations based on network models of porous media. The obtained transport equation remains hyperbolic even if the capillary forces are dominated, in contrast to the classic model which is parabolic. For the case of one space dimension, the analytical solutions are obtained, which manifest non-classical effects as double displacement fronts or counter-current fronts.     

Macroscopic Two-phase Flow in Porous Media Assuming the Diffuse-interface Model at Pore Level

The paper presents a model for two-phase flow, where liquid and gas are treated as one fluid with variable density. A one-component fluid and the diffuse-interface model for two-phase flow are assumed at pore level. The wetting properties of the fluid are described by the Cahn theory. Macroscopic equations are deduced in the framework of the Marle formalism. It is shown that two-phase flow in porous media can be described by the Cahn–Hilliard equation for the mass density. The concept of relative permeability is not needed. For non-neutral wetting, it is shown that a capillary pressure exists but that it is not a function of state. Two numerical illustrations are presented, one of them showing that the model is, at least in a simple steady-state situation, compatible with the generalized two-continuum model.     

Nonlinear Two-Phase Mixing in Heterogeneous Porous Media

For a two-phase immiscible flow through a heterogeneous porous medium in gravity field but with neglected capillary pressure, a macroscale model of first order is derived by a two-scale homogenization method while capturing the effect of fluid mixing. The mixing is manifested in the form of a nonlinear hydrodynamic dispersion and a transport velocity shift. The dispersion tensor is shown to be a nonlinear function of saturation. In the case offlow without gravity this function is proportional to the fractional flow derivative and depends on the viscosity ratio. For a flow which is one dimensional at the macroscale, the dispersion operator remains three dimensional and can be calculated in an analytical way. In the case of gravity induced flow, the longitudinal dispersion as the function of saturation is negative within some interval of saturation values. Numerical simulations of the microscale problemjustify the theoretical results of homogenization.     

Representation of the Functions of the Relative Phase Permeabilities for Anisotropic Porous Media

The most general representation of the functions of the relative phase permeabilities for porous media is written explicitly. The relations proposed for the relative phase permeabilities generalize those obtained earlier for media with transversely-isotropic and orthotropic percolation properties [1, 2] which can now be obtained as a particular case. A laboratory measurement technique for finding the percolation properties and determining the absolute and phase permeabilities for media with different types of anisotropy is discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 118–125.Original Russian Text Copyright © 2005 by Dmitriev, Dmitriev, and Maksimov.     

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.     

油水两相渗流压力脉动的实验及数值分析

本文用高精度的压差传感器和高速度的数据采集设备同时测量出油水两相流体在多孔介质流动时产生的较大压力降和微小压力脉动,应用自行研制的大型数据处理软件对恒流速水驱油的两相渗流压力脉动实验数据进行了分析,发现不同阶段压力脉动具有明显的频谱特性和时间相关特性的不同,在第三阶段（油为主,水增加阶段）谱能增加最大,时间正相关程度最强。     

流体饱和两相多孔介质拟静态问题的有限元解法

给出基于混合物理论的流体饱和两相多孔介质模型,该模型由一可变形固体 一流体相组成。采用Ｇａｌｅｒｋｉｎ加权残值法导出求解拟静态问题的有限元公式,并编制了二维有限元程序。用程序分析了一维和二维问题,得到合理的结果。     

Macro-Scale Dynamic Effects in Homogeneous and Heterogeneous Porous Media

It is known that the classical capillary pressure-saturation relationship may be deficient under non-equilibrium conditions when large saturation changes may occur. An extended relationship has been proposed in the literature which correlates the rate of change of saturation to the difference between the phase pressures and the equilibrium capillary pressure. This linear relationship contains a damping coefficient, \tau, that may be a function of saturation. The extended relationship is examined at the macro-scale through simulations using the two-phase simulator MUFTE-UG. In these simulations, it is assumed that the traditional equilibrium relationship between the water saturation and the difference in fluid pressures holds locally. Steady-state and dynamic numerical experiments are performed where a non-wetting phase displaces a wetting phase in homogeneous and heterogeneous domains with varying boundary conditions, domain size, and soil parameters. From these simulations the damping coefficient can be identified as a (non-linear) function of the water saturation. It is shown that the value of increases with an increased domain size and/or with decreased intrinsic permeability. Also, the value of for a domain with a spatially correlated random distribution of intrinsic permeability is compared to a homogeneous domain with equivalent permeability; they are shown to be almost equal.     

关于渗流中流线不封闭的特性和条件

本文对于流体在多孔介质中流动的特性进行理论研究和数值计算，提出两个关于渗流中流线不封闭的特性和条件，得到了在一般工程实际情况中的多孔介质区域内部不存在封闭流线的结论。本文以突变截面圆管中不可压缩渗流为算例，利用半人工瞬变方法进行数值计算，得到流体在充满多孔介质的突扩截面圆管和突缩截面圆管中流动时关于速度分布和压力分布的结果。由此表明，在突变截面附近的渗流区域中不存在回流和分离流，也不存在封闭的流线。渗流的这些流动特性不同于在无多孔介质的空间区域中的流动特性。     

Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by

非饱和多孔介质有限元分析的基本控制方程与变分原理

本文在对问题研究现状进行阐述的基础上较系统地给出了骨架可变形非饱和多孔介质的全耦合分析模型，模型中考虑了孔隙气体，水（油）流动对介质力学性能的影响，多孔介质的饱和度，渗透系数与毛吸压力的关系，由实验给出，所导出的控制方程以固体骨架的位移与孔隙流体压力为基本未知量，由于问题的非自共轭特征，文中构造了非饱和介质动力问题的参数变分形式，并在此基础上给出有限元离散方程。     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

Effects of Precursor Wetting Films in Immiscible Displacement Through Porous Media

A computer-aided simulator of immiscible displacement in strongly water-wet consolidated porous media that takes into account the effects of the wetting films is developed. The porous medium is modeled as a three-dimensional network of randomly sized unit cells of the constricted-tube type. Precursor wetting films are assumed to advance through the microroughness of the pore walls. Two types of pore wall microroughness are considered. In the first type of microroughness, the film advances quickly, driven by capillary pressure. In the second type, the meniscus moves relatively slowly, driven by local bulk pressure differences. In the latter case, the wetting film often forms a collar that squeezes the thread of oil causing oil disconnection. Each pore is assumed to have either one of the aforementioned microroughness types, or both. The type of microroughness in each pore is assigned randomly. The simulator is used to predict the residual oil saturation as a function of the pertinent parameters (capillary number, viscosity ratio, fraction of pores with each type of wall microroughness). These results are compared with those obtained in the absence of wetting films. It is found that wetting films cause substantial increase of the residual oil saturation. Furthermore, the action of the wetting films causes an increase of the mean volume of the residual oil ganglia.