Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D _e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

Monte Carlo Simulations of Observation Time-Dependent Self-Diffusion in Porous Media Models

Observation time-dependent self-diffusion coefficients can be used to obtain microstructural information of porous media. This paper presents two different kinds of Monte Carlo simulations of the self diffusion process of fluids like water in porous systems, a lattice-free method and a lattice-based method. The results for simple porous media model geometries agree well with each other and with published analytical as well as semi-analytical equations. The use of these equations, which are important for the interpretation of Pulsed Field Gradient-Nuclear Magnetic Resonance (PFG-NMR) time-dependent diffusion data with respect to properties of porous media, is discussed.     

An Alternative Description of Viscous Coupling in Two-Phase Flow through Porous Media

A new formalism is developed to describe the viscous coupling phenomena between two immiscible, flowing fluids in porous media. The formulation is based on the notation of ‘two-phase mixture’ in which the relative motion between an individual phase and the mixture in porous media can be described by a diffusion equation. The present formulation is derived from Darcy's law with cross-terms without making further approximations. However, the new formulation requires fewer effective parameters to be determined experimentally, thus offering a more viable tool for the study of two-phase flow dynamics with viscous coupling in porous media. Moreover, it is found that no new term appears in the present model in cases with and without viscous coupling; instead, the incorporation of viscous coupling only modifies the effective parameters. It can thus be concluded that viscous coupling does not represent a fundamentally new phenomenon within the framework of the present formulation.     

Numerical and experimental investigation of convective drying in unsaturated porous media with bound water

The heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles was investigated theoretically for relatively low convective drying rates. Local thermodynamic equilibrium was assumed in the mathematical model describing the multi-phase flow in the unsaturated porous media using the energy and mass conservation equations to describe the heat and mass transfer during the drying. The drying model included convection and capillary transport of the free water, diffusion of bound water, and convection and diffusion of the gas. The numerical results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing drying rate period. The numerical results agreed well with the experimental data verifying that the mathematical model can evaluate the drying performance of porous media for low drying rates. The effects of drying conditions such as the ambient temperature, the relative humidity, and the velocity of the drying air, on the drying process were evaluated by numerical solution.     

Knudsen’s Permeability Correction for Tight Porous Media

Various flow regimes including Knudsen, transition, slip and viscous flows (Darcy’s law), as applied to flow of natural gas through porous conventional rocks, tight formations and shale systems, are investigated. Data from the Mesaverde formation in the United States are used to demonstrate that the permeability correction factors range generally between 1 and 10. However, there are instances where the corrections can be between 10 and 100 for gas flow with high Knudsen number in the transition flow regime, and especially in the Knudsen’s flow regime. The results are of practical interest as gas permeability in porous media can be more complex than that of liquid. The gas permeability is influenced by slippage of gas, which is a pressure-dependent parameter, commonly referred to as Klinkenberg’s effect. This phenomenon plays a substantial role in gas flow through porous media, especially in unconventional reservoirs with low permeability, such as tight sands, coal seams, and shale formations. A higher-order permeability correlation for gas flow called Knudsen’s permeability is studied. As opposed to Klinkenberg’s correlation, which is a first-order equation, Knudsen’s correlation is a second-order approximation. Even higher-order equations can be derived based on the concept used in developing this model. A plot of permeability correction factor versus Knudsen number gives a typecurve. This typecurve can be used to generalize the permeability correction in tight porous media. We conclude that Knudsen’s permeability correlation is more accurate than Klinkenberg’s model especially for extremely tight porous media with transition and free molecular flow regimes. The results from this study indicate that Klinkenberg’s model and various extensions developed throughout the past years underestimate the permeability correction especially for the case of fluid flow with the high Knudsen number.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 1: The bundle-of-tubes model

A bundle-of-tubes construct is used as a model system to study ensemble averaged equations for multiphase flow in a porous material. Momentum equations for the fluid phases obtained from the method are similar to Darcy’s law, but with additional terms. We study properties of the additional terms, and the conditions under which the averaged equations can be approximated by the diffusion model or the extended Darcy’s law as often used in models for multiphase flows in porous media. Although the bundle-of-tubes model is perhaps the simplest model for a porous material, the ensemble averaged equation technique developed in this paper assumes the very same form in more general treatments described in Part 2 of the present work (Zhang, D.Z., 2009. Ensemble Phase Averaged Equations for Multiphase Flows in Porous Media, Part 2: A General Theory. Int. J. Multiphase Flow 35, 640–649). Any model equation system intended for the more general cases must be understood and tested first using simple models. The concept of ensemble phase averaging is dissected here in physical terms, without involved mathematics through its application to the idealized bundle-of-tubes model for multiphase flow in porous media.     

饱和-非饱和土壤中污染物运移过程的数值模拟

本文提出了一个模拟饱和 非饱和土壤中溶和污染物运移过程的数值模型．模拟的控制污染物运移的物理 化学现象包括：对流，机械逸散，分子弥散，吸附，蜕变，不动水效应．发展了一个修正的特征线Ｇａｌｅｒｋｉｎ方法以离散污染物运移过程的控制方程并导出了一个用于有限元方程求解的显式算法．数值例题结果表明所提出模型和算法的功能     

基于CFD模拟的翅片管束多孔介质模型建立方法及流场模拟

为了利用多孔介质模型对翅片管束外流场进行计算,首先利用实体模型计算了局部管束的流场特性以获得多孔介质参数,在验证了多孔介质模型的准确性后,使用该模型对大尺度管束的流场特性进行了计算。结果表明:可以采用多孔介质模型对翅片管束进行简化计算,局部计算得到的多孔介质参数可以应用到增大长度、高度和宽度后的大尺度同型管束区域计算上,多孔介质模型与实体模型的误差基本在10%以内。     

WAG Displacements of Oil-Condensates Accounting for Hydrocarbon Ganglia

During two-phase flow in porous media, non-wetting phase is present simultaneously in states of mobile connected continuum and of trapped isolated ganglia. Mass exchange between these two parts of non-wetting phase is going on by dissolution and diffusion of component in the wetting phase, so, compositions of non-wetting phase in both parts are different. Nevertheless, the traditional mathematical model for two-phase multicomponent transport in porous media assumes the homogeneous distribution of each component in the overall non-wetting phase. New governing equations honouring ganglia of non-wetting phase are derived. They are successfully verified by a number of laboratory tests. Analytical model is developed for miscible water-alternate-gas (WAG) displacement of oil-condensates. The modelling shows that the significant amount of oil-condensate is left in porous media after miscible WAG, while the traditional model predicts that the miscible displacement results in the total sweep.     

An Analysis of the Movement of Wetting and Nonwetting Fluids in Homogeneous Porous Media

The movement of wetting and nonwetting fluid flow in columns packed with glass beads is used to understand the more complicated flows in homogeneous porous media. The motion of two immiscible liquids (oil and water) is observed with different surfactants. Through dimensional analyses, fluid velocity is well correlated with interfacial tension and less dependent on particle size. In water–oil (W/O) experiments, finger pattern flows are observed if water is the displacing fluid that flows in an oil-filled porous media, whereas oil ganglia tend to form if oil is the displacing fluid in the water-wetted porous media. The results are well described by a simple model based on an earlier theory of flow in a tube.     

Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media

A smoothed particle hydrodynamics (SPH) model is presented for the study of diffusion in spatially periodic porous media. The method of SPH is formulated to solve the convection–diffusion equation for tracer diffusion under steady state and transient conditions. Solutions obtained using SPH are compared with other available solutions and the model is used to calculate diffusion coefficients of spatially periodic porous media for the steady state diffusion problem. Diffusion coefficients are then used to calculate nondimensional diffusivities of the media. The effects of media properties on the values of nondimensional diffusivity are also presented.     

Jots - a mathematical model of microscopic fluid flow in porous media

The model described in this paper is an approach to simulating flow through porous media on a microscopic scale. It is based on a variation of diffusion limited aggregation. The model is shown to match coreflood average saturation profiles and production histories as predicted by Darcy's equations while generating saturation distributions resembling viscous fingering. The model also is shown to simulate the limiting cases of infinite mobility ratio and zero flow rates as previously modeled by diffusion limited aggregation and percolation theory. With some simplifying assumptions, differential equations very similar to Darcy's equations are derived from the microscopic interpretation of fluid behavior in porous media used in this model.     

Constructal Design of Vascular Porous Materials and Electrokinetic Mass Transfer

According to constructal theory, the “generation of flow configuration” is a universal phenomenon in physics. This phenomenon is covered by the constructal law: “For a finite-size flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it.” This paper shows how the constructal law can be used to (1) predict and explain features of “design” in nature, and (2) design effective strategies and configurations for engineering. Many natural flow designs rely on two flow mechanisms: channels with relatively low resistivity, interwoven with diffusion across the interstices. The “design” is the balance between the two mechanisms. The flow from line to line (or plane to plane) through a sufficiently fine porous medium encounters less resistance than the flow through parallel channels when it is configured as trees that alternate with upside down trees: from this follows the prediction that natural porous media (e.g., hill slope) should be multiscale (bidisperse) and non-uniformly distributed. A porous medium contaminated with ionic species is decontaminated the fastest when the ionic flow is configured as two flow mechanisms in balance: “channeling” driven by potential differences between optimally positioned electrodes, and diffusion driven by concentration differences across the interstices between the channels.     

Interfaces of Phase Transition and Disappearance and Method of Negative Saturation for Compositional Flow with Diffusion and Capillarity in Porous Media

The specific case of interfaces separating a single-phase fluid and a two-phase continuum appears in the theory of compositional flow through porous media. They are usually called the interfaces of phase transition (PT-interfaces) or the interfaces of phase disappearing (PD-interfaces). The principle of equivalence is proved which shows that a single-phase multi-component fluid may be replaced by an equivalent fictitious two-phase fluid having specific properties. The equivalent properties are such that the extended saturation of a fictitious phase is negative. This principle enables us to develop the uniform system of two-phase equations in the overall domain in terms of the extended saturation (the NegSat model), and to apply the direct numerical simulation. In the case with diffusion, the uniform NegSat model contains a new term proportional to the gradient of saturation in the relation for flow velocity. The canonical NegSat model represents a transport equation with discontinuous nonlinearities. The qualitative analysis of this model shows that the PT-interfaces represent the shocks of the extended saturation, or, in some cases, can transform into weak shocks. The diffusion and capillarity do not destroy necessarily the shocks, but change their velocity. The analytical technique is developed which allows capturing PT-shocks. The method is illustrated by several examples of miscible gas injection in oil reservoir. In two-dimensional case, the effects of multiple shock collisions in heterogeneous media are automatically modeled. In the case of immiscible fluids and a classic interface, the suggested method converges to the VOF-method.     

Dynamic Network Model of Mobility Control in Emulsion Flow Through Porous Media

Modeling the flow of emulsion in porous media is extremely challenging due to the complex nature of the associated flows and multiscale phenomena. At the pore scale, the dispersed phase size can be of the same order of magnitude of the pore length scale and therefore effective viscosity models do not apply. A physically meaningful macroscopic flow model must incorporate the transport of the dispersed phase through the porous material and the changes on flow resistance due to drop deformation as it flows through pore throats. In this work, we present a dynamic capillary network model that uses experimentally determined pore-level constitutive relationships between flow rate and pressure drop in constricted capillaries to obtain representative transient macroscopic flow behavior emerging from microscopic emulsion flow at the pore level. A parametric analysis is conducted to study the effect of dispersed phase droplet size and capillary number on the flow response to both emulsion and alternating water/emulsion flooding in porous media. The results clearly show that emulsion flooding changes the continuous-phase mobility and consequently flow paths through the porous media, and how the intensity of mobility control can be tuned by the emulsion characteristics.     

A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—part I. Theoretical development and asymptotic analyses

In this paper a rigorous dual-porosity model is formulated, which accurately represents the coupling between large-scale fractures and the micropores within dual porosity media. The overall structure of the porous medium is conceptualized as being blocks of diffusion dominated micropores separated by natural fractures (e.g. cleats for coal) through which Darcy’s flow occurs. In the developed model, diffusion in the matrix blocks is fully coupled to the pressure distribution within the fracture system. Specific assumptions on the pressure behaviour at the matrix boundary, such as step-time function employed in some earlier studies, are not invoked. The model involves introducing an analytical solution for diffusion within a matrix block, and the resultant combined flow equation is a nonlinear integro-(partial) differential equation. Analyses to the equation in this text, in addition to the theoretical development of the proposed model, include: (1) discussion on the “fading memory” of the model; (2); one-dimensional perturbation solution subject to a specific condition; and (3) asymptotic analyses of the “long-time” and “short-time” responses of the flow. Two previous models, the Warren-Root and the modified Vermeulen models, are compared with the proposed model. The advantages of the new model are demonstrated, particularly for early time prediction where the approximations of these other models can lead to significant error.     

Impact of local diffusion on macroscopic dispersion in three-dimensional porous media

While macroscopic longitudinal and transverse dispersion in three-dimensional porous media has been simulated previously mostly under purely advective conditions, the impact of diffusion on macroscopic dispersion in 3D remains an open question. Furthermore, both in 2D and 3D, recurring difficulties have been encountered due to computer limitation or analytical approximation. In this work, we use the Lagrangian velocity covariance function and the temporal derivative of second-order moments to study the influence of diffusion on dispersion in highly heterogeneous 2D and 3D porous media. The first approach characterizes the correlation between the values of Eulerian velocity components sampled by particles undergoing diffusion at two times. The second approach allows the estimation of dispersion coefficients and the analysis of their behaviours as functions of diffusion. These two approaches allowed us to reach new results. The influence of diffusion on dispersion seems to be globally similar between highly heterogeneous 2D and 3D porous media. Diffusion induces a decrease in the dispersion in the direction parallel to the flow direction and an increase in the dispersion in the direction perpendicular to the flow direction. However, the amplification of these two effects with the permeability variance is clearly different between 2D and 3D. For the direction parallel to the flow direction, the amplification is more important in 3D than in 2D. It is reversed in the direction perpendicular to the flow direction.