Surface Permeability of Particulate Porous Media

The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by superfast nonlinear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace–Beltrami Dirichlet boundary value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques, we examine numerical solutions to the Laplace–Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to directly estimate tortuosity effects of the surface flows in the particle ensemble setting.

     

Permeability of the Fluid-Filled Inclusions in Porous Media

In this article, we propose an approach to obtain the equivalent permeability of the fluid-filled inclusions embedded into a porous host in which a fluid flow obeys Darcy’s law. The approach consists in the comparison of the solutions for one-particle problem describing the flow inside the inclusion, firstly, by the Stokes equations and then by using Darcy’s law. The results obtained for spheres (3D) and circles (2D) demonstrate that the inclusion equivalent permeability is a function of its radius and, additionally, depends on the host permeability. Based on this definition of inclusion permeability and using effective medium method, we have calculated the effective permeability of the double-porosity medium composed of the permeable matrix (with small scale pores) and large scale secondary spherical pores.     

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．     

The Effective Permeability of Cracks and Interfaces in Porous Media

The presence of interfaces in fluid/solid biphasic media is known to strongly influence their behavior both in terms of solid deformation and fluids flow. Mathematical models have traditionally represented these interfaces as lines of no-thickness and whose behavior is given in terms of effective permeabilities whose physical meaning is often disconnected to the microscopic nature of the interface. This article aims to reconcile macroscopic and microscopic interface representations by investigating how the nature of microscopic flows and pressures in the interface can be used to explain its macroscopic behavior. By invoking a proper thickness average operation, we derive an closed form expression that relates the effective interfaces permeabilities to its microscopic properties. In particular, we find that the effective interface permeabilities are strongly influenced by three factors: the ratio of bulk and interface permeabilities, the fluid viscosity, and the physical thickness of the interface.

     

Permeability of Porous Media from Simulated NMR Response

Nuclear Magnetic Resonance (NMR) is an increasingly popular well-logging tool in petroleum industry because it is the only tool that attempts to estimate formation permeability. In this paper, spatially correlated porous media are generated. Permeabilities of these media are computed by the lattice Boltzmann method. NMR relaxation responses are simulated by a random walk technique and formation factors are computed by solving a Laplacian equation. The testing of commonly used NMR permeability correlations shows that three conditions should be met for the validity of these correlations. The surface relaxivity should not vary significantly. The formation factor should depend only on porosity. And the characteristic pore body radius should be proportional to the characteristic throat radius. The correlations are improved by including surface relaxivity and formation factor.     

Effective Correlation of Apparent Gas Permeability in Tight Porous Media

Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accomplished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.     

Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals

Transport in Porous Media - Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport...     

Low Pressure Gas Percolation Characteristic in Ultra-low Permeability Porous Media

Low pressure gas percolation characteristic in ultra-low permeability porous media is investigated in this article through core flow experiments. The results show that the wall-slip layer covers more than 10% of the average porous channel radius on account of minimum pore size when the permeability is below 0.1 × 10^?3μ m ² order, and seepage behavior is contrasted to that in mid-high permeability pore media. When the gas pressure is not high enough, the flow regime turns into transitional flow instead of slip flow, and nonlinear relationship between the measured gas permeability and the reciprocal of average pressure exists. The gas measuring permeability experiment would be influenced by the non-linear relationship. If Klinkenberg-corrected method is applied to speculate the equivalent liquid permeability, the extrapolated value will become less or minus. Simultaneously, actual gas flow velocity at the outlet is beyond the calculated value with Klinkenberg formula. A new gas seepage model based on the general slip boundary condition is derived from the homogenization technique in this article. At last the flow model is examined to be suitable for representing the gas flow behavior in ultra-low permeability media and estimating the absolute permeability from single-point, steady-states measurements.     

Practical Considerations for Selection of Representative Elementary Volumes for Fluid Permeability in Fibrous Porous Media

We investigate the lower bound of the area of a square-shaped representative elementary volume (REV) for the permeability tensor for transverse Stokes flow through randomly packed, parallel, and monodisperse cylinders. The investigation is significant to flow models using small calculation regions for fibrous porous media, such as modeling defect formation during directional solidification in the mushy zone of dendritic alloys. Using 90 ensembles of 1,000 domains, where each ensemble comprises domains with the same number and size of cylinders, we develop correlations between the permeability tensor invariants and macroscopic features of the domain. We find that for ensembles of domains with fewer than 200 cylinders, the eigenvectors of the permeability tensors exhibit preferential alignment with the domain axes, demonstrating that the estimated permeability is significantly affected by the periodic boundary conditions for these cases. Our results also suggest that the anisotropy of the permeability tensor may not be insignificant even for large sampling volumes. These results provide a practical lower bound for the calculation volumes used in permeability simulations in fibrous porous media, and also suggest that modelers should consider using an anisotropic tensor for small calculation volumes if phenomena such as channeling are important.     

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.     

Sensitivity of Intrinsic Permeability to Electrokinetic Coupling in Shaly and Clayey Porous Media

Classical Darcy’s law assumes that the intrinsic permeability of porous media is only dependent on the micro-geometrical and structural properties of the inner geometry of the medium. There are, however, numerous experimental evidences that intrinsic permeability of shaly and clayey porous material is a function of the fluid phase used in the experiments. Several pore-scale processes have been proposed to explain the observed behavior. In this study, we conduct a detailed investigation of one such mechanism, namely the electrokinetic coupling. We have developed a numerical model to simulate this process at the pore-scale, incorporating a refined model of the electrical double layer. The model is used to conduct a detailed sensitivity analysis to elucidate the relative importance of several chemical–physical parameters on the intensity of the electrokinetic coupling. We found that permeability reduction due to this mechanism is likely to occur only if the effective pore-radius is smaller than 10⁻⁶ m. We also observed that electrokinetic coupling is strongly sensitive to electrophoretic mobility, which is normally reduced in clays compared to free-water conditions. Based on these findings, we set up a suite of stochastic pore-network simulations to quantify the extent of permeability reduction. We found that only if the effective pore-radius is ranging from 5 × 10⁻⁷ m to 5 × 10⁻⁸, electrokinetic coupling can be responsible for a 5–20% reduction of the intrinsic permeability, and, therefore, this mechanism has a minor impact on situations of practical environmental or mining interest.     

The Effect of Sudden Permeability Changes in Porous Media Filling Box Flows

We report upon experimental and analytical investigations of filling box flows in non-uniform porous media characterized by a sudden change in permeability. The porous medium consists of two layers separated by a horizontal permeability jump and is initially filled with light ambient fluid. A line source located at the top of the upper layer supplies dense contaminated fluid that falls toward the bottom of the domain. Two configurations are studied, i.e., a low-permeability layer on top of a high-permeability layer and vice versa. In the former scenario, the flow dynamics are qualitatively similar to the case of a uniform porous medium. Thus, the analytical formulation of Sahu and Flynn (J Fluid Mech 782:455–478, 2015) can be adopted to compute the parameters of interest, e.g., the plume volume flux. In the latter scenario, the flow dynamics are significantly different from those of the uniform porous medium case; after reaching the permeability jump, some fraction of the dense plume propagates horizontally as a pair of oppositely directed interfacial gravity currents. Meanwhile, the remaining fraction of the plume flows downward into the lower layer where it accumulates along the bottom boundary in the form of a deepening layer of discharged plume fluid. Depending on the permeability ratio of the upper and lower layers and the source conditions, the gravity currents may become temporarily arrested after traveling some finite horizontal length. An analytical prediction for this so-called run-out length is derived, motivated, in part, by the immiscible analysis of Goda and Sato (J Fluid Mech 673:60–79, 2011). Finally, a prediction of the filling box time, consisting of the time required to fill the control volume up to the point of contaminated fluid overflow, is made. These predictions are compared with analog experimental measurements. Generally positive agreement is found when the higher-permeability layer is located below the lower-permeability layer. In the opposite circumstance, the agreement is conditional. If the run-out length of the gravity current is less than the horizontal dimensions of the control volume (or tank in case of the experiments), the agreement is good. By contrast, when the run-out length is large, comparatively poor agreement may be realized: In spite of the higher density of the contaminated fluid, it may occupy the entirety of the upper layer before filling the lower layer.     

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.     

Flow and Transport Properties of Deforming Porous Media. I. Permeability

Transport in Porous Media - Estimating flow and transport properties of porous media that undergo deformation as a result of applying an external pressure or force is important to a wide variety of...     

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.     

A New Model for Viscous Dissipation in Porous Media Across a Range of Permeability Values

In this paper a unified mathematical theory for the viscous dissipation term in the governing Brinkman equation is derived. This term has, unlike other models, the correct asymptotic behaviour in both the fully Darcy and Newtonian fluid flow limits.