A Monte Carlo Algorithm for Immiscible Two-Phase Flow in Porous Media

We present a Markov Chain Monte Carlo algorithm based on the Metropolis algorithm for simulation of the flow of two immiscible fluids in a porous medium under macroscopic steady-state conditions using a dynamical pore network model that tracks the motion of the fluid interfaces. The Monte Carlo algorithm is based on the configuration probability, where a configuration is defined by the positions of all fluid interfaces. We show that the configuration probability is proportional to the inverse of the flow rate. Using a two-dimensional network, advancing the interfaces using time integration, the computational time scales as the linear system size to the fourth power, whereas the Monte Carlo computational time scales as the linear size to the second power. We discuss the strengths and the weaknesses of the algorithm.     

Network Modeling of Non-Darcy Flow Through Porous Media

Darcy's law is inadequate for describing high-velocity gas flow in porous media, which occurs in the near well-bore region of high capacity gas and condensate reservoirs. This study is directed at understanding the non-Darcy flow behavior. A pore-level network model has been developed to describe high velocity flow. The inputs to the model are pore size distributions and network coordination numbers. The outputs are permeability, non-Darcy coefficient, tortuousity and porosity. The additional pressure gradient term is found to be proportional to the square of the velocity in accordance with the Forchheimer's equation. The correlation between the non-Darcy coefficient and other flow properties (the permeability, the porosity and the tortuousity) is found to depend on the morphological parameters being changed. General correlations are derived between these flow properties.     

Homogenization of Wall-Slip Gas Flow Through Porous Media

The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy's law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media.     

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.     

Effective Flux Boundary Conditions for Upscaling Porous Media Equations

We introduce a new algorithm for setting pressure boundary conditions in subgrid simulations of porous media flow. The algorithm approximates the flux in the boundary cell as the flux through a homogeneous inclusion in a homogeneous background, where the permeability of the inclusion is given by the cell permeability and the permeability of the background is given by the ambient effective permeability. With this approximation, the flux in the boundary cell scales with the cell permeability when that permeability is small, and saturates at a constant value when that permeability is large. The flux conditions provide Neumann boundary conditions for the subgrid pressure. We call these boundary conditions effective flux boundary conditions (EFBCs). We give solutions for the flux through ellipsoidal inclusions in two and three dimensions, assuming symmetric tensor permeabilities whose principal axes align with the axes of the ellipse. We then discuss the considerations involved in applying these equations to scale up problems in geological porous media. The key complications are heterogeneity, fluctuations at all length scales, and boundary conditions at finite scales.     

Pore-Scale Simulation of Fluid Flow Through Deformable Porous Media Using Immersed Boundary Coupled Lattice Boltzmann Method

Transport in Porous Media - The interaction between the fluid flow and the deformable porous media is crucial in the applications of adsorption/absorption. The immersed boundary coupled lattice...     

About the Porous Media Flow Through Circular and Elliptical Anisotropic Inhomogeneities

In this article, we describe horizontal groundwater flow due to a uniform flow at infinity around a cylindrical or elliptical inhomogeneity, where the permeability inside the inhomogeneity is anisotropic and different from the isotropic domain outside the inhomogeneity. The orientation of the uniform flow with respect to the orientation of the ellipse is arbitrary as well as the orientation of the anisotropy inside the ellipse. We derive an expression for the ratio of the flow through the ellipse with respect to the flow in the homogeneous case.     

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.     

Tomography-Based Characterization and Optimization Of Fluid Flow Through Porous Media

Numerical simulations to characterize fluid flow through porous media have been carried out using tomography-derived real geometry data that has been manipulated using digital image processing techniques to obtain a wide range of porosities. Two kinds of porous media have been analyzed: (a) a reticulated porous ceramic (RPC) foam and (b) a packed bed of CaCO₃ particles. The porosity of the media is varied via morphological operations between 0.727 and 0.913 in case of the RPC and between 0.329 and 0.824 in case of the packed bed. A mesh generator based on the pore space indicator function is then used to generate unstructured tetrahedral grids from the processed tomography data. Fluid flow simulations are carried out for Reynolds numbers ranging from 0.1 to 200 and the results are used to determine the permeability and the Dupuit?CForchheimer coefficient in each case. The results are then compared with existing analytical models and the applicability of the models is examined. In the RPC case, the Happel?CBrenner (parallel-flow) model predicts the permeability with a normalized root mean square error (NRMSE) of 11.8 % across the porosity range and Modified Ergun (Macdonald et. al) model predicts the Dupuit?CForchheimer coefficient within a NRMSE of 13.5 %. In the packed-bed case, the Brinkman drag model predicts the permeability within a NRMSE of 8.26 % across the porosity range and the Modified Ergun model predicts the Dupuit?CForchheimer coefficient within an NRMSE of 5.94 %. For each material, an adjusted Kozeny constant is determined. For the RPC, the Kozeny constant is evaluated at 7.73 and for the CaCO₃ packed bed, it is found to be 6.10, leading to predictions of the permeability with an NRMSE of 4.16 and 3.37 %, respectively.     

关于渗流中流线不封闭的特性和条件

本文对于流体在多孔介质中流动的特性进行理论研究和数值计算，提出两个关于渗流中流线不封闭的特性和条件，得到了在一般工程实际情况中的多孔介质区域内部不存在封闭流线的结论。本文以突变截面圆管中不可压缩渗流为算例，利用半人工瞬变方法进行数值计算，得到流体在充满多孔介质的突扩截面圆管和突缩截面圆管中流动时关于速度分布和压力分布的结果。由此表明，在突变截面附近的渗流区域中不存在回流和分离流，也不存在封闭的流线。渗流的这些流动特性不同于在无多孔介质的空间区域中的流动特性。     

Dynamics of Flow Through Porous Media with Unsteady Phase Permeabilities

The influence of nonequilibrium effects developing due to the formation of an emulsion of each phase (gas bubbles in the water and water dust in the gas) on the flow dynamics is investigated with reference to the displacement of water by a gas. The nonequilibrium effects manifest themselves in a change in the shape of the phase permeability curves (they become convex) and the threshold phase saturations in the course of flow through the porous medium. A kinetic equation in which the relaxation time is proportional to the seepage rate is used to describe such effects. The case in which the liquid displaced by the gas is itself gassed and the volume concentration of the gas bubbles is constant is considered.     

膨胀性非牛顿流体多孔介质流动的模拟分析

在表征体元尺度采用格子Boltzmann方法分析膨胀性非牛顿流体在多孔介质中的流动,基于二阶矩模型在演化方程中引入表征介质阻力的作用力项,求解描述渗流模型的广义Navier-Stokes方程．采用局部法计算形变速率张量,通过循环迭代得到非牛顿粘度和松弛时间．对多孔介质的Poiseuille流动进行分析,通过比较发现结果与孔隙尺度的解析解十分吻合,并且收敛较快,表明方法合理有效．分析了渗透率和幂律指数对速度和压力降的影响,研究结果表明,膨胀性流体的多孔介质流动不符合达西规律,压力降的增加幅度小于渗透率的减小幅度．当无量纲渗透率Da小于10-5时,流道中的速度呈现均匀分布,并且速度分布随着幂律指数的减小趋于平滑．压力降随着幂律指数的增加而增加,Da越大幂律指数对压力降的影响越明显．     

Shock Propagation in a One-Dimensional Flow Through Deformable Porous Media

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution possibly in the case of onset of singularities, which can be interpreted as a local collapse of the medium. In particular we analyze the case in which the boundary pressure has a piecewise constant derivative.     

New General Maximum Entropy Model for Flow Through Porous Media

Transport in Porous Media - New experimental and numerical techniques constitute the major recent advancements in the study of flow through porous media. However, a model that duly links the micro-...     

Transient Heat Transport in Gas Flow Through Granular Porous Media

Models are developed to describe the time-evolution of gas and solid temperature profiles in a class of granular porous media; this time-dependence being the result of a heat source supplying energy uniformly to the gas as it flows into the medium. The solid-phase is treated as a fixed, axi-symmetric bed of randomly packed spheres of uniform size and material properties. An analytic expression for the locally averaged surface temperature of the solid phase is formulated in terms of the time history of the corresponding local gas temperature. This approach avoids the need to assume locally uniform temperatures within the solid phase, and provides a coupled model for the thermal energy transport in the system. A numerical scheme for treating the resulting transport equations is outlined, and results presented. A quasi-steady approximation is proposed, and this approximation is assessed by reference to numerical results obtained from the numerical scheme. One application of the work is to fixed-bed catalytic reactors and absorbers, and results are presented indicating how the regeneration times of such systems depend on operating parameters.     

Effective Diffusivities of Point-Like Molecules in Isotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of point-like molecules in random and structured media are used to determine and characterize the effective diffusion coefficients of the molecules in the media. Simulations were carried out in 2D and 3D media. Monte Carlo simulation results in 2D and 3D media are compared with those obtained by analytical techniques. Simulation results indicate that for the structured, isotropic media the effective diffusivities can be characterized according to percolation thresholds in addition to porosity. The effective diffusivities in two isotropic media with the same porosity but different percolation thresholds can differ significantly. The effects of dimensionality on the effective diffusivities can also be significant. It is shown that in general the effective diffusion coefficients obtained from 2D simulation are not a good approximation to those of 3D, especially when the percolation thresholds of the 2D media and the 3D media are very different.     

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.