Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions

Vugs and fractures are common features of carbonate formations. The presence of vugs and fractures in porous media can significantly affect pressure and flow behavior of a fluid. A vug is a cavity (usually a void space, occasionally filled with sediments), and its pore volume is much larger than the intergranular pore volume. Fractures occur in almost all geological formations to some extent. The fluid flow in vugs and fractures at the microscopic level does not obey Darcy’s law; rather, it is governed by Stokes flow (sometimes is also called Stokes’ law). In this paper, analytical solutions are derived for the fluid flow in porous media with spherical- and spheroidal-shaped vug and/or fracture inclusions. The coupling of Stokes flow and Darcy’s law is implemented through a no-jump condition on normal velocities, a jump condition on pressures, and generalized Beavers–Joseph–Saffman condition on the interface of the matrix and vug or fracture. The spheroidal geometry is used because of its flexibility to represent many different geometrical shapes. A spheroid reduces to a sphere when the focal length of the spheroid approaches zero. A prolate spheroid degenerates to a long rod to represent the connected vug geometry (a tunnel geometry) when the focal length of the spheroid approaches infinity. An oblate spheroid degenerates to a flat spheroidal disk to represent the fracture geometry. Once the pressure field in a single vug or fracture and in the matrix domains is obtained, the equivalent permeability of the vug with the matrix or the fracture with matrix can be determined. Using the effective medium theory, the effective permeability of the vug–matrix or fracture–matrix ensemble domain can be determined. The effect of the volume fraction and geometrical properties of vugs, such as the aspect ratio and spatial distribution, in the matrix is also investigated. It is shown that the higher volume fraction of the vugs or fractures enhances the effective permeability of the system. For a fixed-volume fraction, highly elongated vugs or fractures significantly increase the effective permeability compared with shorter vugs or fractures. A set of disconnected vugs or fractures yields lower effective permeability compared with a single vug or fracture of the same volume fraction.     

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 Tensor for Heterogeneous Porous Medium of Fibre Type

A theoretical model which allows us to determine the permeability of a fibrous porous medium is proposed. Fibres are assumed to be parallel and nonuniform in space and material with a low volume fraction of fibres is considered. The model includes two geometric parameters: the diameter of fibres and the diameter of caverns or fissures inside the bundle of fibres. The tensor of permeability of the porous medium is determined based upon a generalized cell model. The components of permeability tensor depend on two parameters which are determined using experimental data and least-squares approximation. The influence of the geometric parameters on components of permeability tensor is discussed.     

General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs

One of the techniques to calculate the effective property of a heterogeneous medium is the effective medium theory. The present paper presents a general mathematical formulation for the effective medium approximation using a self-consistent choice of the effective permeability, to apply it to the case of a general anisotropic 2D medium and to the case of a 3D isotropic medium with randomly oriented ellipsoidal inclusions. The 2D results are compared with analytical results and with a homogenization technique with good result. The 3D correlations are used to derive percolation thresholds in two-phase systems with a large permeability contrast, which are compared to numerical results from the literature, also with good results.     

Effect of Bedding Planes on the Permeability and Diffusivity Anisotropies of Berea Sandstone

Transport in Porous Media - Evaluating the anisotropy of transport parameters in rocks is important for various applications, such as reservoir engineering and rock mechanics. Owing to their...     

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.     

Effects of Medium Permeability Anisotropy on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

This paper deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including medium anisotropic effects. Since a general anisotropic medium can be described as an orthotropic medium in the corresponding principal directions, a two-dimensional orthotropic porous medium is considered to derive the analytical solution for the critical condition, which is used to judge whether or not the chemical dissolution front can become unstable during its propagation. In the case of the mineral dissolution ratio (that is defined as the ratio of the dissolved-mineral equilibrium concentration in the pore-fluid to the molar concentration of the dissolvable mineral in the solid matrix of the fluid-saturated porous medium) approaching zero, the corresponding critical condition has been mathematically derived when medium permeability anisotropic effects are considered. As a complementary tool, the computational simulation method is used to simulate the morphological evolution of chemical dissolution fronts in two-dimensional fluid-saturated porous media including medium anisotropic effects. The related theoretical and numerical results demonstrated that: (1) a decrease in the medium anisotropic permeability factor (or ratio), which is defined as the ratio of the principal permeability in the transversal direction to that in the longitudinal direction parallel to the pore-fluid inflow direction, can stabilize the chemical dissolution front so that it becomes more difficult for a planar chemical-dissolution front to evolve into different morphologies in the chemical dissolution system; (2) the medium anisotropic permeability ratio can have significant effects on the morphological evolution of the chemical dissolution front. When the Zhao number of the chemical dissolution system is greater than its critical value, the greater the medium anisotropic permeability ratio, the faster the irregular chemical-dissolution front grows.     

Characteristics of Wave Propagation in The Saturated Thermoelastic Porous Medium

Took into consideration the coupling effect of thermo, hydraulics and mechanics, a set of thermo–hydro-mechanical coupled wave equations for fluid–saturated soil are developed. In these wave equations, the $P_{3}$ -wave in solid phase and $P_{4}$ -wave in fluid phase are coupled into $T$ -wave in fluid–saturated soil by the assumption that the temperature of the solid phase is equal to the temperature of liquid phase at the same position. The dispersion equations for the thermo-elastic wave, which can be degraded to the equations for elastic wave in fluid–saturated soil, are derived from the above equations by introducing four potential functions. Then, these equations are solved numerically. The characteristics of wave phase velocity, attenuation and the effect of thermal expansion, initial temperature and porosity, etc., on phase velocities of $P_{1}$ -, $P_{2}$ -, and $T$ -wave are discussed. As a reference, the characteristics of the propagation of elastic waves in fluid–saturated soil are also studied. The computation results show that (1) the phase velocity of $P_{1}$ -wave obtained by the theory of thermoporoelascity (THM) is faster than that by the theory of poroelasticity (HM); (2) the attenuation of $P_{1}$ -wave obtained by either the theory of THM or HM are consistent; (3) the dissemination characteristics of $P_{2}$ -wave are almost consistent; (4) the phase velocity of $T$ -wave is the slowest among the three compressional waves; and (5) The attenuation versus frequency characteristic of $T$ -wave is similar to that of $P_{2}$ -wave.     

Effective Diffusivity Tensors of Point-Like Molecules in Anisotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of random walks in anisotropic structured media are performed to determine the dependence of effective diffusivities on geometrical properties. The anisotropic media used in this study are periodic systems, which are generated by extending primitive, face-centered, and body-centered unit cells indefinitely in all axial directions. Results of simulations compare well with published experimental data and the calculations by the volume averaging method. In addition, these results suggest that if the 2D media with percolation thresholds subtantially differ from those of 3D, 2D approximations of 3D media are not satisfactory. When percolation thresholds are the same, the effective diffusivity tensors depend solely on the porosity. This fact has been suggested for isotropic media and it seems to hold for anisotropic media.     

Effective Diffusivity of Porous Materials with Microcracks: Self-Similar Mean-Field Homogenization and Pixel Finite Element Simulations

We investigate the influence of distributed microcracks on the overall diffusion properties of a porous material using the self-similar cascade continuum micromechanics model within the framework of mean-field homogenization and computational homogenization of diffusion simulations using a high-resolution pixel finite element method. In addition to isotropic, also anisotropic crack distributions are considered. The comparison of the results from the cascade continuum micromechanics model and the numerical simulations provides a deeper insight into the qualitative transport characteristics such as the influence of the crack density on the complexity and connectivity of crack networks. The analysis shows that the effective diffusivity for a disordered microcrack distribution is independent of the absolute length scale of the cracks. It is observed that the overall effective diffusivity of a microcracked material with the microcracks oriented in the direction of transport is not necessarily higher than that of a material with a random orientation of microcracks, independent of the microcrack density.     

Numerical Investigations of the Steady State Relative Permeability of a Simplified Porous Medium

The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio. The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M=₁/₂; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993).(Now at AS Norske Shell, Norway.) e-mail:     

Chemical Reaction Effect on Convection in Bidispersive Porous Medium

Transport in Porous Media - The object of this study is to investigate the question of convective movement of a reacting solute in a viscous incompressible occupying a plane layer in a saturated...     

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.     

Influence of Variable Permeability on Combined Convection Along a Nonisothermal Wedge in a Saturated Porous Medium

Mixed convection along a vertical nonisothermal wedge embedded in a fluid-saturated porous media incorporating the variation of permeability and thermal conductivity is studied. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter and a pseudo-similarity variable are introduced to cast the governing boundary layer equations into a system of dimensionless equations which are solved numerically using finite difference method. The entire mixed convection regime is covered by the single nonsimilarity parameter =[1+(Ra _x /Pe _x )^1/2]^–1 from pure forced convection (=1) to pure free convection (=0). The problem is solved using nonsimilarity solution for the case of variable wall temperature. Velocity and temperature profiles as well as local Nusselt number are presented. The wedge angle geometry parameter is ranged from 0 to 1.     

The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium

The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is discussed. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability. This package is applied to the case of a cube partitioned into octants and to the cases where the permeability and thermal conductivity vary continuously across a cube in either a linear or a quadratic manner.     

The Effect of Pulsating Deformation on the Onset of Convection in a Porous Medium

A linear instability analysis has been performed for the onset of convection in a horizontal layer of a porous medium whose permeability pulsates with time as a result of vertical movement of one of the boundaries. It was found that, to a first-order approximation in the pulsation amplitude, the effect of deformation is destabilizing for disturbances whose period is of the order of the thermal diffusion time scale. The effects of the average porosity, pulsation amplitude, and pulsation frequency were investigated.     

Determination of the Effective Permeability of Doubly Porous Materials by a Two-Scale Homogenization Approach

Transport in Porous Media - The present work is dedicated to determining the effective permeability of doubly porous materials made of a solid phase comprising a network of interconnected pores at...