Mechanistic Model of Steady-State Two-Phase Flow in Porous Media Based on Ganglion Dynamics

Recent experimental work has shown that the pore-scale flow mechanism during steady-state two-phase flow in porous media is ganglion dynamics (GD) over a broad and practically significant range of the system parameters. This observation suggests that our conception and theoretical treatment of fractional flow in porous media need careful reconsideration. Here is proposed a mechanistic model of steady-state two-phase flow in those cases where the dominant flow regime is ganglion dynamics. The approach is based on the ganglion population balance equations in combination with a microflow network simulator. The fundamental information on the cooperative flow behavior of the two fluids at the scale of a few hundred pores is expressed through the system factors, which are functions of the system parameters and are calculated using the simulator. These system factors are utilized by the population balance equations to predict the macroscopic behavior of the process. The dependence of the conventional relative permeability coefficients not only on the wetting fluid saturation S_wbut also on the capillary number, Ca, the viscosity ratio the wettability (⁰ a, ⁰ r), the coalescence factor, Co, as well as the porous medium geometry and topology is explained and predicted on a mechanistic basis. Sample calculations have been performed for steady-state fully developed (SSFD) and steady-state nonfully developed (SSnonFD) flow conditions. The number distributions of the moving and the stranded ganglia, the mean ganglion size, the fraction of the nonwetting fluid in the form of mobile ganglia, the ratio of the conventional relative permeability coefficients and the fractional flows are studied as functions of the system parameters and are correlated with the flow phenomena at pore level and the system factors.     

Effects of Relative Permeability History Dependence on Two-Phase Flow in Porous Media

Hysteresis phenomena in multi-phase flow in porous media has been recognized by many researchers and widely believed to have significant effects on the flow. In an attempt to account for these effects, a theoretical model for history-dependent relative permeabilities is considered. This model is incorporated into 1-D two-phase nondiffusive flow system and the corresponding flow is predicted. Flow history is observed to have a notable impact on the saturation profile and fluids breakthrough.     

多孔介质渗透率的一种新分形模型

多孔介质的渗流特性是油气藏工程、地下水资源利用、高放废物深地质处置等实际工程领域的热门研究问题．基于分形理论及多孔介质由一束面积大小不等的椭圆形毛细管组成的假设,本文建立了流体在分形多孔介质中渗流时的绝对渗透率及相对渗透率的分形渗透率模型．结果表明,绝对渗透率是最大和最小孔隙面积、分形维数、形状因子ε的函数,且当ε =1时,本文模型可以简化成Yu与Cheng模型;而非饱和多孔介质的相对渗透率与饱和度和多孔介质微结构参数有关．将本文提出的渗透率分形模型预测与实验测量数据及其他模型结果进行对比,显示它们整体吻合很好．     

Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as N^1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator N^2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.     

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.     

A Two-Dimensional Network Simulator for Two-Phase Flow in Porous Media

We investigate a two-dimensional network simulator that model the dynamics of drainage dominated flow where film flow can be neglected. We present a new method for simulating the temporal evolution of the pressure due to capillary and viscous forces in the displacement process. To model the dynamics, we let the local capillary pressure change as if the menisci move in and out of hour-glass shaped tubes. Furthermore, a method has been developed to allow simultaneous flow of two liquids into one tube. The model is suitable to simulate different time dependencies in two-phase drainage displacements. In this paper, we simulate the temporal evolution of the fluid pressures and analyze the time dependence of the front between the two liquids. The front width was found to be consistent with a scaling relation w t h(t/t_s). The dynamical exponent, , describing the front width evolution as function of time, was estimated to = 1.0. The results are compared to experimental data of Frette and co-workers.     

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.     

The Discontinuous Solutions of the Transport Equations for Compositional Flow in Porous Media

In the present paper we consider a multicomponent multiphase isothermal flow in porous media with mass exchange between phases. The system of equations of multiphase multicomponent flow has discontinuous solutions, but is not hyperbolic, except some particular cases. For this general, non-hyperbolic system, we propose a free energy condition to select unique physically admissible discontinuous solutions. We also develop a geometrical procedure which provides a tool to analyze the free energy condition. For a two-component mixture, analytical formulae are obtained for the allowed discontinuities.     

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.     

An Improved IMPES Method for Two-Phase Flow in Porous Media

In this paper we develop and numerically study an improved IMPES method for solving a partial differential coupled system for two-phase flow in a three-dimensional porous medium. This improved method utilizes an adaptive control strategy on the choice of a time step for saturation and takes a much larger time step for pressure than for the saturation. Through a stability analysis and a comparison with a simultaneous solution method, we show that this improved IMPES method is effective and efficient for the numerical simulation of two-phase flow and it is capable of solving two-phase coning problems.     

A Dynamic Network Model for Two-Phase Flow in Porous Media

We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.     

Analytical Solutions for Multicomponent,Two-Phase Flow in Porous Media with Double Contact Discontinuities

This article presents the first instance of a double contact discontinuity in analytical solutions for multicomponent, two-phase flow in porous media. We use a three-component system with constant equilibrium ratios and fixed injection and initial conditions, to demonstrate this structure. This wave structure occurs for two-phase injection compositions. Such conditions were not considered previously in the development of analytical solutions for compositional flows. We demonstrate the stability of the double contact discontinuity in terms of the Liu entropy condition and also show that the resulting solution is continuously dependent on initial data. Extensions to four-component and systems with adsorption are presented, demonstrating the more widespread occurrence of this wave structure in multicomponent, two-phase flow systems. The developments in this article provide the building blocks for the development of a complete Riemann solver for general initial and injection conditions.     

A New Method for Calculating Two-Phase Relative Permeability from Resistivity Data in Porous Media

Many resistivity data from laboratory measurements and well logging are available. Papers on the relationship between resistivity and relative permeability have been few. To this end, a new method was developed to infer two-phase relative permeability from the resistivity data in a consolidated porous medium. It was found that the wetting phase relative permeability is inversely proportional to the resistivity index of a porous medium. The proposed model was verified using the experimental data in different rocks (Berea, Boise sandstone, and limestone) at different temperatures up to 300°F. The results demonstrated that the oil and water relative permeabilities calculated from the experimental resistivity data by using the model proposed in this article were close to those calculated from the capillary pressure data in the rock samples with different porosities and permeabilities. The results demonstrated that the proposed approach to calculating two-phase relative permeability from resistivity data works satisfactorily in the cases studied.     

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

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Effects of Snap-Off in Imbibition in Porous Media with Different Spatial Correlations

Quasi-static imbibition was simulated using random and correlated stochastic network models. Using the snap-off pore-scale displacement observed by Lernormand et al. (1983) the effects of many parameters on relative permeabilities and residual saturation reported in the literature were reproduced and explained. Increased relative permeabilities and decreased residual non-wetting phase saturation were the results of an increased contact angle (Li and Wardlaw, 1986b; Gauglitz and Radke, 1990; Blunt et al., 1992; Mogensen and Stenby, 1998) a decreased pore–throat aspect ratio, the presence of long-range pore-pore size correlations (Iaonnidis and Chatzis, 1993; Blunt, 1997a), or local pore–throat correlations (Jerauld and Salter, 1990; Iaonnidis and Chatzis, 1993). By modifying the level of snap-off, or its spatial distribution, these parameters varied the efficiency of the displacement patterns and ultimately affect relative permeabilities and residual saturations. Mani and Mohanty (1999) performed simulations on networks with infinite-ranged fractional Brownian motion (fBm) correlations and reported trends of relative permeabilities and residual saturations that were opposite to others’ results (Ioannidis and Chatzis, 1993; Blunt, 1997a). Applying a cut-off length to the fBm correlations reversed Mani and Mohanty’s trends to conform with the common observations.     

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.     

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.     

A Mechanistic Model of Gas-Condensate Flow in Pores

Recent experimental results reported in the literature indicate that the relative permeability of gas-condensate systems increases with rate (velocity) at some conditions. To gain a better understanding of the nature of the flow and the prevailing mechanisms resulting in such behaviour flow visualisation experiments have been performed, using high pressure micromodels. The observed flow behaviour at the pore level has been employed to develop a mechanistic model describing the coupled flow of gas and condensate phases. The results of the model simulating the observed simultaneous flow of gas and condensate phases have been compared with reported core experimental results. Most features of the reported rate effect are predictable by the developed single pore model, nevertheless, its extension to include multiple pore interaction is recommended.