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.     

Direct Numerical Simulation of Pore-Scale Trapping Events During Capillary-Dominated Two-Phase Flow in Porous Media

Transport in Porous Media - This study focuses on direct numerical simulation of imbibition, displacement of the non-wetting phase by the wetting phase, through water-wet carbonate rocks. We...     

A Study of the Time Constant in Unsteady Porous Media Flow Using Direct Numerical Simulation

We consider unsteady flow in porous media and focus on the behavior of the coefficients in the unsteady form of Darcy’s equation. It can be obtained by consistent volume-averaging of the Navier–Stokes equations together with a closure for the interaction term. Two different closures can be found in the literature, a steady-state closure and a virtual mass approach taking unsteady effects into account. We contrast these approaches with an unsteady form of Darcy’s equation derived by volume-averaging the equation for the kinetic energy. A series of direct numerical simulations of transient flow in the pore space of porous media with various complexities are used to assess the applicability of the unsteady form of Darcy’s equation with constant coefficients. The results imply that velocity profile shapes change during flow acceleration. Nevertheless, we demonstrate that the new kinetic energy approach shows perfect agreement for transient flow in porous media. The time scale predicted by this approach represents the ratio between the integrated kinetic energy in the pore space and that of the intrinsic velocity. It can be significantly larger than that obtained by volume-averaging the Navier–Stokes equation using the steady-state closure for the flow resistance term.     

Finite-Difference Approximation for Fluid-Flow Simulation and Calculation of Permeability in Porous Media

We introduce a finite-difference method to simulate pore scale steady-state creeping fluid flow in porous media. First, a geometrical approximation is invoked to describe the interstitial space of grid-based images of porous media. Subsequently, a generalized Laplace equation is derived and solved to calculate fluid pressure and velocity distributions in the interstitial space domain. We use a previously validated lattice-Boltzmann method (LBM) as ground truth for modeling comparison purposes. Our method requires on average 17 % of the CPU time used by LBM to calculate permeability in the same pore-scale distributions. After grid refinement, calculations of permeability performed from velocity distributions converge with both methods, and our modeling results differ within 6 % from those yielded by LBM. However, without grid refinement, permeability calculations differ within 20 % from those yielded by LBM for the case of high-porosity rocks and by as much as 100 % in low-porosity and highly tortuous porous media. We confirm that grid refinement is essential to secure reliable results when modeling fluid flow in porous media. Without grid refinement, permeability results obtained with our modeling method are closer to converged results than those yielded by LBM in low-porosity and highly tortuous media. However, the accuracy of the presented model decreases in pores with elongated cross sections.     

Numerical Simulation of Two-Phase Inertial Flow in Heterogeneous Porous Media

In this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.     

Coupling of Foam Drainage and Viscous Fingering in Porous Media Revealed by X-ray Computed Tomography

We investigate the development foam in granular porous media and the subsequent flow of the surfactant solution, where the fluid fraction variations are visualized and quantified using X-ray computed tomography. It is found that foam flows in a front like manner leading to a residual liquid fraction of 0.18±0.01, far from the inlet surface of the porous sample. A desaturation backward wave is also observed during foam development. We provided direct evidence that the flow of surfactant solution in porous media containing foam gives rise to superposition of a drainage wave and a characteristic viscous fingering pattern. In the wave the liquid fraction ranges from the above residual value to nearly 0.25±0.01. The liquid fraction associated with the viscous fingering decays as a function of distance but the inlet value increases up to 0.06±0.01. Certain ideas about the physics of foam flow in porous media are revised in the light of our findings.     

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.     

Resolution Effect: An Error Correction Model for Intrinsic Permeability of Porous Media Estimated from Lattice Boltzmann Method

Transport in Porous Media - In digital rock physics, the intrinsic permeability of a porous rock sample can be evaluated from its micro-computed tomography ($$\upmu$$-CT) image through lattice...     

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.     

Coupled Processes in Charged Porous Media: From Theory to Applications

Charged porous media are pervasive, and modeling such systems is mathematically and computationally challenging due to the highly coupled hydrodynamic and electrochemical interactions caused by the presence of charged solid surfaces, ions in the fluid, and chemical reactions between the ions in the fluid and the solid surface. In addition to the microscopic physics, applied external potentials, such as hydrodynamic, electrical, and chemical potential gradients, control the macroscopic dynamics of the system. This paper aims to give fresh overview of modeling pore-scale and Darcy-scale coupled processes for different applications. At the microscale, fundamental microscopic concepts and corresponding mass and momentum balance equations for charged porous media are presented. Given the highly coupled nonlinear physiochemical processes in charged porous media as well as the huge discrepancy in length scales of these physiochemical phenomena versus the application, numerical simulation of these processes at the Darcy scale is even more challenging than the direct pore-scale simulation of multiphase flow in porous media. Thus, upscaling the microscopic processes up to the Darcy scale is essential and highly required for large-scale applications. Hence, we provide and discuss Darcy-scale porous medium theories obtained using the hybrid mixture theory and homogenization along with their corresponding assumptions. Then, application of these theoretical developments in clays, batteries, enhanced oil recovery, and biological systems is discussed.

     

裂缝介质渗流多尺度混合有限元数值模拟研究

针对裂缝介质具有多尺度特点,建立了Darcy/Stokes-Brinkman多尺度耦合模型,采用多尺度混合有限元方法,对裂缝介质渗流问题进行了研究.阐述了多尺度混合有限元方法的基本原理,并推导得到Darcy/Stokes-Brinkman方程的多尺度混合有限元计算格式.数值计算结果表明,大尺度Darcy模型能够捕捉到小尺度上裂缝网络渗流特征;与网格粗化、传统有限元方法相比,多尺度混合有限元方法的基函数具有能反映单元内参数变化的优点,在保证计算精度的同时能够减少计算量,对于裂缝油藏具有良好的适用性.     

Numerical Simulation of Hydraulic Fracturing Water Effects on Shale Gas Permeability Alteration

Hydraulic fracturing has been recognized as the necessary well completion technique to achieve economic production from shale gas formation. However, following the fracturing, fluid–wall interactions can form a damaged zone nearby the fracture characterized by strong capillarity and osmosis effects. Here, we present a new reservoir multi-phase flow model which includes these mechanisms to predict formation damage in the aftermath of the fracturing during shut-in and production periods. In the model, the shale matrix is treated as a multi-scale porosity medium including interconnected organic, inorganic slit-shaped, and clay porosity fields. Prior to the fracturing, the matrix holds gas in the organic and the inorganic slit-shaped pores, water with dissolved salt in the inorganic slit-shaped pores and the clay pores. During and after fracturing, imbibition causes water invasion into the matrix, and then, the injected water–clay interaction may lead to clay-swelling pressure development due to osmosis. The swelling pressure gives additional stress to slit-shaped pores and cause permeability reduction in the inorganic matrix. We develop a simulator describing a system of three pores, two phases (aqueous and gaseous phases), and three components (\(\hbox {H}_{2}\hbox {O}, \hbox {CH}_{4}\), and salt), including osmosis and clay-swelling effect on the permeability. The simulation of aqueous-phase transport through clay shows that high swelling pressure can occur in clays as function of salt type, salt concentration difference, and clay-membrane efficiency. The new model is used to demonstrate the damage zone characteristics. The simulation of two-phase flow through the shale formation shows that, although fracturing is a rapid process, fluid–wall interactions continue to occur after the fracturing due to imbibition mechanism, which allows water to penetrate into the inorganic pore network and displace the gas in-place near the fracture. This water invasion leads to osmosis effect in the formation, which cause clay swelling and the subsequent permeability reduction. Continuing shale–water interactions during the production period can expand the damage zone further.     

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.

     

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.     

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.     

Linking Morphology of Porous Media to Their Macroscopic Permeability by Deep Learning

Flow, transport, mechanical, and fracture properties of porous media depend on their morphology and are usually estimated by experimental and/or computational methods. The precision of the computational approaches depends on the accuracy of the model that represents the morphology. If high accuracy is required, the computations and even experiments can be quite time-consuming. At the same time, linking the morphology directly to the permeability, as well as other important flow and transport properties, has been a long-standing problem. In this paper, we develop a new network that utilizes a deep learning (DL) algorithm to link the morphology of porous media to their permeability. The network is neither a purely traditional artificial neural network (ANN), nor is it a purely DL algorithm, but, rather, it is a hybrid of both. The input data include three-dimensional images of sandstones, hundreds of their stochastic realizations generated by a reconstruction method, and synthetic unconsolidated porous media produced by a Boolean method. To develop the network, we first extract important features of the images using a DL algorithm and then feed them to an ANN to estimate the permeabilities. We demonstrate that the network is successfully trained, such that it can develop accurate correlations between the morphology of porous media and their effective permeability. The high accuracy of the network is demonstrated by its predictions for the permeability of a variety of porous media.

     

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．     

Intrinsic Permeability Evolution in High Temperature Concrete: An Experimental and Numerical Analysis

This paper presents an experimental and thermo-hydro-chemical-mechanical numerical analysis of concrete at high temperatures, aiming at the definition of a law describing the evolution of intrinsic permeability. The evaluation of heat and mass transfers, evolution of the phases composing the porous medium, and mechanical performances of concrete are taken into account in a full three phases coupled analysis. An experimental set-up and a numerical simulation are then presented. A hollow cylinder has been heated up to 523.15 K (250 °C) on the internal side and subjected to gas pressure/temperature measurements; the experience has been then simulated by means of a numerical code. The analysis has allowed for the definition of an original relationship describing intrinsic permeability evolution. Finally, the law has been validated measuring the actual value of intrinsic permeability on heated concrete.     

A High Resolution Scheme for the Numerical Simulation of Second Sound in Crystals

A high resolution essentially non-oscillatory (ENO) numerical scheme is applied to a new model of heat propagation in crystals for an accurate reconstruction of second sound waves. A splitting of the governing equations allows to approximate the differential terms and the forcing term by means of the ENO scheme and a Runge--Kutta third order scheme, respectively. Results obtained on data relative to a high purity crystal of NaF are in good agreement with the theoretical predictions.