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

Modeling of Two-Phase Flow in Fractured Porous Media on Unstructured Non-Uniformly Coarsened Grids

In this article, we investigate two strategies for coarsening fractured geological models. The first approach, which generates grids that resolve the fractures, is referred to as explicit fracture-matrix separation (EFMS). The second approach is based on a non-uniform coarsening strategy introduced in Aarnes et al. (Adv Water Resour 30(11):2177–2193, 2007a). A series of two-phase flow simulations where the saturation is modeled on the respective coarse grids are performed. The accuracy of the resulting solutions is examined, and the robustness of the two strategies is assessed with respect to number of fractures, degree of coarsening, well locations, phase viscosities, and fracture permeability. The numerical results show that saturation solutions obtained on the non-uniform coarse grids are consistently more accurate than the corresponding saturation solutions obtained on the EFMS grids. The numerical results also reveal that it is much easier to tune the upscaling factor with the non-uniform coarsening approach.     

Theoretical Analysis of Viscous Coupling in Two-phase Flow through Porous Media

Theoretical analysis is presented to quantify the viscous coupling effect in two-phase flow through porous media. The analysis is based on the principle of potential difference equations as well as on the interfacial contact area and partition concept. The analysis shows that viscous coupling effect is negligible throughout the normalized saturation range. The expression, Xϕ ², was developed for the quantification of the parameter that controls the amount of viscous coupling, where X was theoretically found to have a maximum value of 2.     

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 Dynamic Model of Capillary Hysteresis in Immiscible Fluid Displacement

The capillary hysteresis in a dynamic and quasi-static two-fluid flow in a porous medium is discussed. Thermodynamic background is presented. It is shown that physically acceptable constitutive relations satisfying the thermodynamic conditions can be constructed in terms of Preisach hysteresis operators.     

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.     

Modeling the Formation of Fluid Banks During Counter-Current Flow in Porous Media

Fluid banks sometimes form during gravity-driven counter-current flow in certain natural reservoir processes. Prediction of flow performance in such systems depends on our understanding of the bank-formation process. Traditional modeling methods using a single capillary pressure curve based on a final saturation distribution have successfully simulated counter-current flow without fluid banks. However, it has been difficult to simulate counter-current flow with fluid banks. In this paper, we describe the successful saturation-history-dependent modeling of counter-current flow experiments that result in fluid banks. The method used to simulate the experiments takes into account hysteresis in capillary pressure and relative permeabilities. Each spatial element in the model follows a distinct trajectory on the capillary pressure versus saturation map, which consists of the capillary hysteresis loop and the associated capillary pressure scanning curves. The new modeling method successfully captured the formation of the fluid banks observed in the experiments, including their development with time. Results show that bank formation is favored where the p_c-versus-saturation slope is low. Experiments documented in the literature that exhibited formation of fluid banks were also successfully simulated.     

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.     

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.     

Boiling in Porous Media: Model and Simulations

We present a modelization of the heat and mass transfers within a porous medium, which takes into account phase transitions. Classical equations are derived for the mass conservation equation, whereas the equation of energy relies on an entropy balance adapted to the case of a rigid porous medium. The approximation of the solution is obtained using a finite volume scheme coupled with the management of phase transitions. This model is shown to apply in the case of an experiment of heat generation in a porous medium. The vapor phase appearance is well reproduced by the simulations, and the size of the two-phase region is correctly predicted. A result of this study is the evidence of the discrepancy between the air – water capillary and relative permeability curves and water – water vapor ones.     

Surfactant Concentration and End Effects on Foam Flow in Porous Media

Foaming injected gas is a useful and promising technique for achieving mobility control in porous media. Typically, such foams are aqueous. In the presence of foam, gas and liquid flow behavior is determined by bubble size or foam texture. The thin-liquid films that separate foam into bubbles must be relatively stable for a foam to be finely textured and thereby be effective as a displacing or blocking agent. Film stability is a strong function of surfactant concentration and type. This work studies foam flow behavior at a variety of surfactant concentrations using experiments and a numerical model. Thus, the foam behavior examined spans from strong to weak.Specifically, a suite of foam displacements over a range of surfactant concentrations in a roughly 7m², one-dimensional sandpack are monitored using X-ray computed tomography (CT). Sequential pressure taps are employed to measure flow resistance. Nitrogen is the gas and an alpha olefin sulfonate (AOS 1416) in brine is the foamer. Surfactant concentrations studied vary from 0.005 to 1wt%. Because foam mobility depends strongly upon its texture, a bubble population balance model is both useful and necessary to describe the experimental results thoroughly and self consistently. Excellent agreement is found between experiment and theory.     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

A Lattice Gas Automaton Simulation of the Nonlinear Diffusion Equation: A Model for Moisture Flow in Unsaturated Porous Media

We investigate a two-dimensional lattice gas automaton (LGA) for simulating the nonlinear diffusion equation in a random heterogeneous structure. The utilility of the LGA for computation of nonlinear diffusion arises from the fact that, the diffusion coefficient in the LGA depends on the local density of fluid particles which statistically determines the collision rate and thus, the mean free path of the particles at the microscopic scale. The LGA may therefore be used as a physical analogue to simulate moisture flow in unsaturated porous media. The capability of the LGA to account for unsaturated flow is tested through a set of numerical experiments simulating one-dimensional infiltration in a simplified semi-infinite homogenous isotropic porous material. Different mechanisms of interactions are used between the fluid and the solid phase to simulate various fluid–solid interfaces. The heterogeneous medium, initially at low density is submitted to a steep density gradient by continuously injecting fluid particles at high concentration and zero velocity along one face of the model. The propagation of the infiltration front is visualized at different time steps through concentration profiles parallel to the applied concentration gradient and the infiltration rate is measured continuously until steady-state flow is reached. The numerical results show close agreement with the classical theory of flow in unsaturated porous media. The cumulative absorption exhibits the expected t ^1/2 dependence. The evolution of the effective diffusion coefficient with the particle concentration is estimated from the measured density profiles for the various porous materials. Depending on the applied fluid–solid interactions, the macroscopic effective diffusivity may vary by more than two orders of magnitude with density.     

A Markov Chain Model for Effective Relative Permeabilities and Capillary Pressure

We present a new method for calculating the effective two-phase parameters of one-dimensional randomly heterogeneous porous media, which avoids the timeconsuming use of simulations on explicit realizations. The procedure is based on the steady state saturation distribution. The idea is to model the local variation of saturation and saturation dependent parameters as Markov chains, in such a way that the effective parameters are given by the asymptotic expectations of the chains. We derive the exact asymptotic moment equations and solve them numerically, based on their second order approximation. The method determines the effective parameters to a high degree of accuracy, even with large variations in rock properties. In particular, the capillary limit and viscous limit effective parameters are recovered exactly. The applicability of the effective parameters in the unsteady state case is studied by comparing the displacement production profiles in heterogeneous media and their homogenized counterpart.     

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。     

可压缩气体定常非Darcy渗流的流动分析及其应用

气体通过多孔介质的非Darcy流动具有广泛的工程应用背景，因此对多孔介质中的气体非Darcy流动进行流动分析有着非常重要的意义。然而，在通常的研究中，一般都将气体考虑为不可压缩流体，很少考虑气体的压缩性。对于高压气体以较高的速度通过多孔介质的情况，在进行流动分析时，不仅要考虑非Darcy效应，还必须考虑气体的压缩性。在本文中，对可压缩气体通过多孔介质的定常非Darcy流动进行了一维流动分析，得出了多孔介质中气体的压力分布和速度分布。还进一步给出了在高压差和高流速情况下，测定多孔介质材料渗透率和惯性系数的方法，以及多孔介质材料前后压力差与材料厚度的比Δp/L和材料有气流速度u1的解析关系。     

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。     

可变形多孔介质中的一维非定常耦合渗流

在Ｂｉｏｔ理论的基础上,考虑到可变形多孔介质的渗透系数依赖于孔隙变形的特点,建立了耦合渗流问题的基本方程;用初始层校正法求出了一维非定常耦合渗流问题的摄动解;实例计算表明,耦合分析与非耦合分析之间的判别较大,因此耦合效应不能忽略。     

Electrical Resistivity Index in Multiphase Flow through Porous Media

The simultaneous flow of two phases through a three-dimensional porous medium is calculated by means of a Lattice-Boltzmann algorithm. The time-dependent phase configurations can be derived and also macroscopic quantities such as the relative permeabilities. When one phase only is supposed to be conductive, the Laplace equation which governs electrical conduction can be solved in each phase configuration; an instantaneous value of the macroscopic conductivity is obtained and it is averaged over many configurations. The influence of saturation on the resistivity index is studied for six different samples and two viscosity ratios. The saturation exponent is systematically determined. The numerical results are also compared to other possible models and also to experimental results; finally, they are discussed and criticized.