Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.     

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.     

Microvisual Study of Multiphase Gas Condensate Flow in Porous Media

Gas condensate reservoirs constitute a significant portion of hydrocarbon reserves worldwide. The liquid drop-out in these reservoirs may lead to recovery problems such as near wellbore permeability impairment and uncertainty in the actual location of the target condensate. Such technical issues can be addressed through improved understanding of the formation of condensate and the multiphase flow of gas/condensate/water in the reservoir as characterized by relative permeability curves. The appropriate relative permeability curves in turn can be used in reservoir simulators to assist in optimization of field development. This paper reports results of experiments conducted in micromodels, in support of possible core flow tests, using reservoir fluids under reservoir conditions. In particular, visualizations of condensate formation with and without connate water are presented and the differences between the two cases as well as the possible implications for the relative permeability measurements are discussed. Furthermore, the flow of gas and condensate at different force ratios (capillary and Bond numbers) are presented. It is postulated that a single dimensionless number may not be sufficient to characterize the multiphase flow in gas condensate reservoirs. The physical mechanisms occurring under various field conditions are examined in the light of these observations.     

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.     

Gas Flow in Porous Media With Klinkenberg Effects

Gas flow in porous media differs from liquid flow because of the large gas compressibility and pressure-dependent effective permeability. The latter effect, named after Klinkenberg, may have significant impact on gas flow behavior, especially in low permeability media, but it has been ignored in most of the previous studies because of the mathematical difficulty in handling the additional nonlinear term in the gas flow governing equation. This paper presents a set of new analytical solutions developed for analyzing steady-state and transient gas flow through porous media including Klinkenberg effects. The analytical solutions are obtained using a new form of gas flow governing equation that incorporates the Klinkenberg effect. Additional analytical solutions for one-, two- and three-dimensional gas flow in porous media could be readily derived by the following solution procedures in this paper. Furthermore, the validity of the conventional assumption used for linearizing the gas flow equation has been examined. A generally applicable procedure has been developed for accurate evaluation of the analytical solutions which use a linearized diffusivity for transient gas flow. As application examples, the new analytical solutions have been used to verify numerical solutions, and to design new laboratory and field testing techniques to determine the Klinkenberg parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs from The Geysers geothermal field. We show that this new approach and the traditional method of Klinkenberg yield similar results of Klinkenberg constants for the laboratory tests; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Solution Construction to a Class of Riemann Problems of Multiphase Flow in Porous Media

Transport in Porous Media - Fluid displacement in porous media can usually be formulated as a Riemann problem. Finding the solution to such a problem helps shed light on the dynamics of flow and...     

A Model for Multiphase and Multicomponent Flow in Porous Media,Built on the Diffuse-Interface Assumption

This article presents a new theory for flow in porous media of a mixture of nonreacting chemical components. In the examples considered, these components are hydrocarbons and water. The model presented assumes that porosity is constant and uniform, and that the wetting properties of the medium are nearly neutral. The flow equations are obtained by starting with the balance equations (mass, momentum, and energy) at pore level, and averaging them over a large number of pores, using the diffuse interface assumption then the methods of irreversible thermodynamics, thus obtaining, among other things, the collective convective velocity and the component-wise diffusive velocities as functions of the component densities. When the simplification of uniform temperature is introduced, the flow equations are of the Cahn–Hilliard type (with an extra term accounting for gravitation) where the thermodynamic function is the Helmholtz free energy per unit volume of the mixture. There are no relative permeabilities. Also, the set of equations is complete in the sense that no flash calculations are necessary, phase segregation being part of the calculation. The numerical examples considered are: (i) phase segregation in a gravitational field and (ii) coning where the initial state is fully segregated.     

Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media

The paper deals with the effects of large scale permeability–heterogeneity on flows at high velocities through porous media. The media is made of a large number of homogeneous blocks where the flow is assumed to be governed by the Forchheimer equation with a constant inertial coefficient. By assuming the validity of the Forchheimer equation at the large scale, an effective inertial coefficient is deduced from numerical simulations. Different media are investigated: serial-layers, parallel-layers and correlated media. The numerical results show that: (i) for the serial-layers, the effective inertial coefficient is independent of the Reynolds number and decreases when the variance and the mean permeability ratio increases; (ii) for the parallel-layers and the correlated media, the effective inertial coefficient is function of the Reynolds number and increases when the variance and the mean permeability ratio increases. Theoretical relationships are proposed for the inertial coefficient as function of the Reynolds number and the characteristics of the media.     

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

Hyperelastic Multiphase Porous Media with Strain-Dependent Retention Laws

This article presents poroelastic laws accounting for a retention behavior dependent also on porosity, as suggested by experimental evidence. Motivated by the numerical formulation of the corresponding boundary-value problem presented in a companion article, these constitutive equations employ displacements and fluid pressures as primary variables. The thermodynamic admissibility of the proposed rate laws for stress and fluid contents is assessed by means of symmetry and Maxwell conditions obtained from the Biot theory. In the case of strain-dependent saturation, the two elasticity tensors describing the drained response in saturated and unsaturated conditions, respectively, are proven to be in general not coincident, with their difference depending on capillary pressure and porosity. Furthermore, it is shown that besides the stress decomposition proposed by Coussy, also the stress split proposed by Lewis and Schrefler is consistent with the Biot framework. The former decomposition is obtained for retention laws depending only on capillary pressure, as expected. The Lewis–Schrefler split is proven to be consistent with retention models depending also on porosity. In these developments, the compressibility of all the phases is taken into account, in order to assess the thermodynamic consistency of an extension of the Biot’s coefficient to partially saturated anisotropic porous media.     

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.     

Network Model of Flow,Transport and Biofilm Effects in Porous Media

In this paper, we develop a network model to determine porosity and permeability changes in a porous medium as a result of changes in the amount of biomass. The biomass is in the form of biofilms. Biofilms form when certain types of bacteria reproduce, bond to surfaces, and produce extracellular polymer (EPS) filaments that link together the bacteria. The pore spaces are modeled as a system of interconnected pipes in two and three dimensions. The radii of the pipes are given by a lognormal probability distribution. Volumetric flow rates through each of the pipes, and through the medium, are determined by solving a linear system of equations, with a symmetric and positive definite matrix. Transport through the medium is modeled by upwind, explicit finite difference approximations in the individual pipes. Methods for handling the boundary conditions between pipes and for visualizing the results of numerical simulations are developed. Increases in biomass, as a result of transport and reaction, decrease the pipe radii, which decreases the permeability of the medium. Relationships between biomass accumulation and permeability and porosity reduction are presented.     

High-Velocity Laminar and Turbulent Flow in Porous Media

We model high-velocity flow in porous media with the multiple scale homogenization technique and basic fluid mechanics. Momentum and mechanical energy theorems are derived. In idealized porous media inviscid irrotational flow in the pores and wall boundary layers give a pressure loss with a power of 3/2 in average velocity. This model has support from flow in simple model media. In complex media the flow separates from the solid surface. Pressure loss effects of flow separation, wall and free shear layers, pressure drag, flow tube velocity and developing flow are discussed by using phenomenological arguments. We propose that the square pressure loss in the laminar Forchheimer equation is caused by development of strong localized dissipation zones around flow separation, that is, in the viscous boundary layer in triple decks. For turbulent flow, the resulting pressure loss due to average dissipation is a power 2 term in velocity.     

Macroscopic Flow Potentials in Swelling Porous Media

In swelling porous media, the potential for flow is much more than pressure, and derivations for flow equations have yielded a variety of equations. In this article, we show that the macroscopic flow potentials are the electro-chemical potentials of the components of the fluid and that other forms of flow equations, such as those derived through mixture theory or homogenization, are a result of particular forms of the chemical potentials of the species. It is also shown that depending upon whether one is considering the pressure of a liquid in a reservoir in electro-chemical equilibrium with the swelling porous media, or the pressure of the vicinal liquid within the swelling porous media, a critical pressure gradient threshold exists or does not.     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

Modelling Foamy Oil Flow in Porous Media

This paper describes a dynamic model for the simulation of foamy oil flow in porous media. The model includes expressions for the rate processes of nucleation, bubble growth and disengagement of dispersed gas bubbles from the oil. The model is used to simulate experimental results pertaining to primary depletion tests conducted in a sand pack. Using the model to interpret experimental results indicated that, although the lifetimes of supersaturation and dispersed gas bubbles may be short, supersaturated conditions are likely to exist, and dispersed gas bubbles are likely to be present during the entire production period, as long as the pressure continues to decline at a high rate. The model developed in this paper gave better agreement with experimental data than other proposed models. The effect of foamy oil flow increases as the rate of pressure decline increases.