Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole–Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pru?  equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.     

Effect of Vibrations on Pore Fluid Distribution in Porous Media

Understanding the role of shuttle vibrations in pore fluid distribution is an essential task in the exploration of plant growth in root modules aboard space flights. Results from experimental investigations are reported in this paper on the distribution of immiscible fluid phases in glass beads under vibrations. Hexadecane, a petroleum compound immiscible with and lighter than water, was used in the experiments. The higher freezing point of Hexadecane (18 °C) allowed the solidification of the entrapped blobs in the presence of water in porous media, so that their size distribution can be obtained. van Genuchten function, commonly used to express moisture retention curves, is found to be an adequate fit for blob size distribution at residual saturation. The effect of vibrations on the fate (mobilization, stranding, or breakup) of a solitary ganglion in porous media was studied using a network model. A mobility criterion considering viscous, gravity, and capillary forces was developed to determine the fate of a solitary ganglion in a porous medium. It is concluded that the effect of vibrations is to increase the likelihood of breakup and mobilization of blobs entrapped in porous media at residual saturation. The pore fluid distributions after vibrations are less uniform than those before vibrations.     

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.     

Quantifying the Representative Size in Porous Media

We present a new approach to quantify the representative quality of pore-scale samples of porous media. It is shown that the flow field uniformity serves as a reliable criterion to decide if the computed flow properties are representative at larger sample sizes. The proposed approach is computationally inexpensive and requires minimal effort to implement. We rely on the correlation matrix of flow field to quantify the representative quality of the computed flow properties at the pore scale. Using this approach, we have been able to study several pore-networks and a high-resolution image of a sandstone, and quickly answer if the computed flow properties from these pore-networks/images are representative of the actual media.     

Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow

For a non-isothermal reactive flow process, effective properties such as permeability and heat conductivity change as the underlying pore structure evolves. We investigate changes of the effective properties for a two-dimensional periodic porous medium as the grain geometry changes. We consider specific grain shapes and study the evolution by solving the cell problems numerically for an upscaled model derived in Bringedal et al. (Transp Porous Media 114(2):371–393, 2016. doi: 10.1007/s11242-015-0530-9). In particular, we focus on the limit behavior near clogging. The effective heat conductivities are compared to common porosity-weighted volume averaging approximations, and we find that geometric averages perform better than arithmetic and harmonic for isotropic media, while the optimal choice for anisotropic media depends on the degree and direction of the anisotropy. An approximate analytical expression is found to perform well for the isotropic effective heat conductivity. The permeability is compared to some commonly used approaches focusing on the limiting behavior near clogging, where a fitted power law is found to behave reasonably well. The resulting macroscale equations are tested on a case where the geochemical reactions cause pore clogging and a corresponding change in the flow and transport behavior at Darcy scale. As pores clog the flow paths shift away, while heat conduction increases in regions with lower porosity.     

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.     

双重孔隙介质非线性流固耦合渗流

本文给出了考虑双孔双涌介质生变形的流固耦合渗流模型。不仅考虑了固结对渗流的影响,同时也考虎了固体变形对渗流参数（孔隙度和渗透率）的影响。这样。渗流就成了双孔双渗介质中非线性流固耦合渗流。在此基础上,本文还推导了双重孔隙介质非线性流固耦合渗流计算。给出了算例并作了对比。结果表明,固体变形引起的介质参数变化对流体渗流早中期过程有重要的影响,对渗流后期影响并不大。这对于石油开采有重要的参考价值。     

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.     

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.     

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.     

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.     

Measurement of Void Size Correlation in Inhomogeneous Porous Media

In previous works, we have described a void space reconstruction method based on non-wetting fluid intrusion, wetting fluid drainage, and image analysis data. The method has been applied to a wide range of substances, including sandstone, compressed and sintered powders, paper substrates and coatings, soil and fibrous mats. We have also demonstrated in a previous work that the spatial correlation of similarly sized voids within inhomogeneous porous media has a huge effect on permeability. We therefore describe a method of measuring such correlation, suitable for use in our void space reconstructions. The method involves a cubic spline smoothing of a variogram of the void sizes in a binary image of the porous medium. It has been successfully tested on an artificially correlated void network, comprising two sintered glass discs of different void size ranges. Stereological effects, caused by the off-centre sectioning of voids, can interfere with the variogram features. Our method is sh own to be insensitive to artificially generated stereological interference. The method is also applied to sandstone samples.     

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.     

Investigation of Post-Darcy Flow in Thin Porous Media

Transport in Porous Media - We present numerical simulations of post-Darcy flow in thin porous medium: one consisting of staggered arrangements of circular cylinders and one random distribution of...     

Effect of Hydrogen Sulfide on Condensation and Flow of Gas Condensate Fluids in Porous Media

Condensation and flow experiments were conducted at subsurface conditions in a glass micromodel using reservoir fluids with and without the hydrogen sulfide component. It has been noted that the formation of the condensing phase as well as modes of condensate flow are similar for both fluids. Furthermore, an additional condensate transport mechanism, termed lamella flow, was observed with the sour fluid. It has been concluded that core flow experiments conducted with sweet reservoir fluid should reproduce the flow of sour fluid to a large extent.     

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.     

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.     

Nonequilibrium Effects and Multiphase Flow in Porous Media

In this paper we develop a more general formulation for transient multiphase flow in porous media based on physics observed in core-scale and micromodel experiments. We account for non-equilibrium effects by considering redistribution time and treat saturation by evolving locally moving time-averages of the saturation. Several families of models arise from approximations to the general formulation with various degrees of accuracy. The classical Buckley-Leverett and Barenblatt expressions are special cases of these families. We explore the behaviors of a number of special cases arising from the proposed general formulation using established and novel numerical schemes that provide nonlinear physics-based preconditioning. The agreement observed between numerical and experimental results demonstrates the consistency of the proposed abstraction.