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.     

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.     

Flow Through a Two-Scale Porosity,Oriented Fibre Porous Medium

A model is described for the meso- and micro-flow through an array of oriented fibre tows with meso-channels between the tows. Axial Stokes's flow was considered in the meso-channels and Darcy's law was applied within the porous fibre tows, taking into account injection pressure and capillary pressures in both types of flow. Transverse flow transfer was modelled from the leading flow front to the lagging flow and a partial-slip boundary condition was applied at the permeable boundaries of meso-channels. Flow visualisation experiments and microstructural characterisation of laminates provided appropriate experimental data for model validation. In this, the predictions for the progress of the leading meso-flow were in excellent agreement with the experimental data. Parametric studies followed including the effects of injection pressure and meso-channel size.     

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.     

Steady-state countercurrent flow in one dimension

Two phase countercurrent steady-state flow through permeable media in one dimension is discussed. For steady-state countercurrent flow in water wet porous media, a saturation profile is predicted with the water saturation decreasing in the direction that the water phase is flowing. The de la Cruz and Spanos equations predict that the Muskat relative permeability curves for countercurrent flow will be less than the Muskat relative permeability curves for steady-state cocurrent flow. This result has immediate implications regarding the use of external drive techniques to determine relative permeabilities based on the Buckley-Leverett theory and Muskat's equations. These equations and current experimental evidence involving countercurrent flow indicate that Muskat's equations do not adequately describe the multiphase flow of immiscible fluids.     

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.     

Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids. The theory is based on the Berryman–Thigpen–Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of “low frequency” underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g., seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.     

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.     

The exact analytical solutions to the new model of reservoir filtration problem

The present study has obtained the new model of the reservoir filtration problemby taking into account the effect of wellbore storage and skin and by making use ofthe coupled equations of doubled porous media filtration and consequently has got,through various forms of limits,the exact analytical solutions of the three commonreservoirs(fissure,homogeneous and the two-layered)pressure distribution under theconditions of three boundaries,i.e.,infinite boundary,sealed finite boundary and thefinite boundary at constant pressures.     

Unsteady Two-Dimensional Blood Flow in Porous Artery with Multi-Irregular Stenoses

The flow characteristics of an unsteady axisymmetric two-dimensional (2D) blood flow in a diseased porous arterial segment with flexible walls are investigated. The arterial walls mimic the irregular constrictions whereas the lumen containing the thrombus, cholesterol, and fatty plaques represents the porous medium. The governing equations with appropriate initial and boundary conditions are solved numerically using MAC method. The discretization is done on staggered grid with non-uniform grid size and pressure-poisson equation is solved following SOR method. The pressure and velocity corrections are made cyclically until the steady state is achieved. It is observed that for decreasing permeability, flow is highly decelerated while pressure drop and wall shear stress increases. The separation zones and re-circulation regions are found for severe stenoses. Flow separation and re-circulation diminishes for decreasing permeability of the porous medium. Comparisons are provided with published experimental and numerical results.     

Numerical Investigation of Multi-Jet Channel Flows

The results of a numerical simulation of the three-dimensional outflow of a system of circular supersonic turbulent jets into a cocurrent supersonic (or subsonic) air flow in a partially bounded region are given. Solutions are obtained by the splitting method using a matrix sweep of the parabolized Navier-Stokes equations. Assuming that the flow is nonseparated in the boundary layer, features of the three-dimensional structure of the jet system are investigated as functions of the pressure ratio number and the jet and cocurrent flow Mach numbers.     

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:     

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.     

Relative permeabilities from two- and three-dimensional pore-scale network modelling

We present a computer study of two-phase flow in a porous medium. The porous medium is represented by an isotropic network of up to 80 000 randomly placed nodes connected by thin tubes. We then simulate two-fluid displacements in this network and are able to demonstrate the effects of viscous and capillary forces. We use the local average flow rates and pressures to calculate effective saturation dependent relative pemeabilities, fractional flows and capillary pressures. Using a radial Buckley-Leverett theory, the mean saturation profile can be inferred from the solution of the fractional flow equation, which is consistent with the computed saturation. We show that the relative permeability may be a function of both viscosity ratio and capillary number.     

A unified approach to simulation and upscaling of single‐phase flow through vuggy carbonates

Numerical modeling of flow through vuggy porous media, mainly vuggy carbonates, is a challenging endeavor. Firstly, because the presence of vugs can significantly alter the effective porosity and permeability of the medium. Secondly, because of the co‐existence of porous and free flow regions within the medium and the uncertainties in defining the exact boundary between them. Traditionally, such heterogeneous systems are modeled by the coupled Darcy–Stokes equations. However, numerical modeling of flow through vuggy porous media using coupled Darcy–Stokes equations poses several numerical challenges particularly with respect to specification of correct interface condition between the porous and free‐flow regions. Hence, an alternative method, a more unified approach for modeling flows through vuggy porous media, the Stokes–Brinkman model, is proposed here. It is a single equation model with variable coefficients, which can be used for both porous and free‐flow regions. This also makes the requirement for interface condition redundant. Thus, there is an obvious benefit of using the Stokes–Brinkman equation, which can be reduced to Stokes or Darcy equation by the appropriate choice of parameters. At the same time, the Stokes–Brinkman equation provides a smooth transition between porous and free‐flow region, thereby taking care of the associated uncertainties. A numerical treatment for upscaling Stokes–Brinkman model is presented as an approach to use Stokes–Brinkman model for multi‐phase flow. Numerical upscaling methodology is validated by analyzing the error norm for numerical pressure convergence. Stokes–Brinkman single equation model is tested on a series of realistic data sets, and the results are compared with traditional coupled Darcy–Stokes model. Copyright © 2011 John Wiley & Sons, Ltd.     

Semi-Analytic Axisymmetric Pressure Distribution in a Consolidating Porous Medium

A semi-analytic solution of the consolidation problem in a finite hollow axisymmetric elastic porous medium is given. According to Biot's theory, we have rigorously derived the consolidation equations and demonstrated that in the axisymmetric problems, the pore pressure diffusion equation can be uncoupled. In the problem of infinite domain, the uncoupled pressure diffusion equation is homogeneous and only the diffusion coefficient is changed. In the problem of finite domain, the uncoupled pressure diffusion equation is nonhomogeneous. In fact, it is a linear differential-integral equation. We solve it by the variables separation method in the time domain.     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

Effective equations for two-phase flow in porous media: the effect of trapping on the microscale

In this article, we consider a two-phase flow model in a heterogeneous porous column. The medium consists of many homogeneous layers that are perpendicular to the flow direction and have a periodic structure resulting in a one-dimensional flow. Trapping may occur at the interface between a coarse and a fine layer. Assuming that capillary effects caused by the surface tension are in balance with the viscous effects, we apply the homogenization approach to derive an effective (upscaled) model. Numerical experiments show a good agreement between the effective solution and the averaged solution taking into account the detailed microstructure.     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Theoretical analysis on hydraulic transient resulted by sudden increase of inlet pressure for laminar pipeline flow

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.