Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?

We study the generation and flow of foam through rough-walled, fractured marble rocks that mimic natural fracture systems in carbonate reservoirs. Flow was isolated to the fracture network because of the very low rock permeability of the marble samples and foam generated in situ during co-injection of surfactant solution and gas. The foam apparent viscosities were calculated at steady pressure gradients for a range of gas fractions, and similar to foam flow in porous media, we identified two flow regimes for foam flow in fractures: a high-quality flow regime only dependent on liquid velocity and a low-quality flow regime determined by the gas and liquid velocities. Variations in local fluid saturation during co-injection were visualized and quantified using positron emission tomography combined with computed tomography.

     

不同滑移边界下的页岩渗透率修正模型

页岩中的孔隙直径通常为纳米量级,基于连续流的达西定律已不能描述纳米级孔隙内的气体流动规律,一般采用附加滑移边界条件的Navier-Stokes方程对其进行描述. 由此可导出与压力相关的渗透率公式(称为"视渗透率"),并用来修正达西定律.因而,渗透率修正方法研究成为页岩气流动研究的热点之一.首先,基于Hagen-Poiseuille 流推导出一般形式二阶滑移模型下的速度分布和流量公式,并推导出相应的渗透率修正公式.该渗透率修正公式基本能将现有的滑移速度模型统一表达为对渗透率的修正. 基于一般形式的渗透率修正公式,重点研究了Maxwell, Hsia, Beskok与Ng 滑移模型速度分布渗透率修正系数、及其对井底压力的影响;提出了基于Ng 滑移速度模型的渗透率修正公式. 基于页岩实际储层温压系统及孔隙分布,计算了Kn 范围及储层条件下页岩气的流动形态,表明页岩气流动存在滑移流、过渡流与分子自由流. 而Ng 模型能描述Kn<88 的滑移流、过渡流、自由分子流的流量规律,因此可以用于描述页岩实际储层中页岩气的流动特征. 计算表明,随着Kn 的增加,不同滑移模型下的渗透率修正系数差异增大.Maxwell与Hsia模型适用于滑移流与过渡流早期,Beskok与Ng 模型可描述自由分子流下的流动规律,但二者在虚拟的孔径均为10nm页岩中,井底压力的差别开始显现;在虚拟的孔径均为1nm页岩中,井底压力的差别开始明显.      

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.     

A LGA model for fluid flow in heterogeneous porous media

A lattice gas automaton (LGA) model is proposed to simulate fluid flow in heterogeneous porous media. Permeability fields are created by distributing scatterers (solids, grains) within the fluid flow field. These scatterers act as obstacles to flow. The loss in momentum of the fluid is directly related to the permeability of the lattice gas model. It is shown that by varying the probability of occurrence of solid nodes, the permeability of the porous medium can be changed over several orders of magnitude. To simulate fluid flow in heterogeneous permeability fields, isotropic, anisotropic, random, and correlated permeability fields are generated. The lattice gas model developed here is then used to obtain the effective permeability as well as the local fluid flow field. The method presented here can be used to simulate fluid flow in arbitrarily complex heterogeneous porous media.     

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.     

Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

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.     

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.     

Verification of the nature of the approximation of the Boltzmann integral by Krook's kinetic relaxation model

Up to now a significant number of aerodynamic problems have been solved with the aid of Krook's kinetic relaxation model. However, because of the absence of reliable solutions of boundary problems for the Boltzmann equation, the correctness of the assumed model of the collision integral remains unclarified. In the present paper, in order to verify the nature of the approximation of the collision operator by the given model, a machine experiment is undertaken. The Boltzmann collision operator is computed for a variety of test functions characteristic of the motion of a rarefied gas and the values obtained according to it are compared to the Krook model. Some physical hypotheses embedded in the relaxation model are also examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–7, July–August, 1970.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Effective Transport Properties from Homogenization of the Kinetic Equation

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This article develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e.g., by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to macroscopic transfer relations, and to closure problems in for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor, which is distinct from the permeability and Knudsen diffusion tensors.     

Numerical experiments on the gas flow between eccentric rotating cylinders

A numerical experiment was carried out on the gas flow field between two eccentric cylinders, one of which is rotating. Attention was paid to the presence of separated recirculating regions from the continuum to the rarefied regimes. The direct simulations were performed by means of a Monte Carlo (DSMC) method and bi‐polar co‐ordinates were adopted. The calculations were relative to isothermal walls at the same temperature. Streamlines and velocity profiles were evaluated as functions of the Knudsen number, of the Mach number and of the geometric parameters. The gas considered was argon. Copyright © 2000 John Wiley & Sons, Ltd.     

Investigation of slow monatomic gas flows past a circular cylinder

Slow low-Knudsen-number monatomic-gas flow past a circular cylinder is numerically investigated on the basis of a model kinetic equation. The gas flow is described by a new kinetic equation, from which the continuum equations for slow nonisothermal gas flows containing temperature stresses follow rigorously. It is shown that a closed convective-flow region arises near a nonuniformly heated cylinder in a slow gas flow if the flow impinges on the hot side of its surface. Using a new model of the Boltzmann equation makes it possible to study gas flows both in continuum and rarefied flow regimes.     

Case Method Employed for Solving the Problem of Thermal Creep of a Rarefied Gas along a Solid Cylindrical Surface

An analytical solution of a halfspace boundaryvalue problem is constructed for an inhomogeneous kinetic Boltzmann equation with the collision operator in the form of an operator of an ellipsoidal statistical model in the problem on thermal creep of a rarefied gas along a solid cylindrical surface. Corrections to the thermal creep coefficient are obtained for the cases of longitudinal and transverse flow past a straight circular cylinder in the approximation linear with respect to the Knudsen number, allowing for the interfacial curvature. The results are compared with available data.     

Exact solution of the kinetic equation in the problem of nonisothermal flow in the neighborhood of a weakly curved surface

An analytic method for solving the half-space boundary value problem for the inhomogeneous Boltzmann equation with the collision operator in the form of an elliptico-statistical model (the ES-model of the Boltzmann equation) is proposed for the problem of nonisothermal rarefied gas flow in the neighborhood of a curved surface. An exact analytic expression is derived for the thermal slip of a monatomic gas along the surface of a rigid spherical aerosol particle. A numerical value of the gas-kinetic coefficient which takes into account the effect of the curvature of the surface on the thermal slip coefficient is obtained. A comparison with published data is carried out. Moscow, Arkhangelsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–173, March–April, 1998.     

Direct numerical test of the B-G-K model equation by the DSMC method

In the present paper the rarefied gas flow caused by the sudden change of the wall temperature and the Rayleigh problem are simulated by the DSMC method which has been validated by experiments both in global flow field and velocity distribution function level. The comparison of the simulated results with the accurate numerical solution of the B-G-K model equation shows that near equilibrium the B-G-K equation with corrected collision frequency can give accurate result but as farther away from equilibrium the B-G-K equation is not accurate. This is for the first time that the error caused by the B-G-K model equation has been revealed. The project supported by the National Natural Science Foundation of China (19772059, 19889209)     

Invariant submodels of the model of thermal motion of gas in a rarefied space

A model describing the thermal motion of a gas in a rarefied space is investigated. This model can be used in the study of the motion of gas in outer space, and the processes occurring inside the tornado, and the state of the medium behind the shock front of the wave after a very intense explosion. For a given initial pressure distribution, a special choice of mass Lagrange variables leads to a reduced system of differential equations describing this motion, in which the number of independent variables is one less than the original system. This means that there is a stratification of a highly rarefied gas with respect to pressure. Namely, in a strongly rarefied space for each given initial pressure distribution, at each instant of time all gas particles are localized on a two-dimensional surface moving in this space. At each point of this surface, the acceleration vector is collinear with its normal vector. The resulting system admits an infinite Lie transformation group. All significantly various submodels that are invariant with respect to the subgroups of its eight-parameter subgroup generated by the transfer, extension, rotation, and hyperbolic rotation operators (the Lorentz operator) are found. For invariant submodels of rank 1, the basic mechanical characteristics of the gas flow described by them are obtained. Conditions for the existence of these submodels are given. For invariant submodels of rank 2, integral equations describing these submodels are obtained. For some submodels, the problem of describing the gas flow from the initial location of its particles and the distribution of their velocities has been investigated.     

Gas-kinetic unified algorithm for plane external force-driven flows covering all flow regimes by modeling of Boltzmann equation

The nonequilibrium steady gas flows under the external forces are essentially associated with some extremely complicated nonlinear dynamics, due to the acceleration or deceleration effects of the external forces on the gas molecules by the velocity distribution function. In this article, the gas-kinetic unified algorithm (GKUA) for rarefied transition to continuum flows under external forces is developed by solving the unified Boltzmann model equation. The computable modeling of the Boltzmann equation with the external force terms is presented at the first time by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The conservative discrete velocity ordinate (DVO) method is utilized to transform the governing equation into the hyperbolic conservation forms at each of the DVO points. The corresponding numerical schemes are constructed, especially the forward-backward MacCormack predictor-corrector method for the convection term in the molecular velocity space, which is unlike the original type. Some typical numerical examples are conducted to test the present new algorithm. The results obtained by the relevant direct simulation Monte Carlo method, Euler/Navier-Stokes solver, unified gas-kinetic scheme, and moment methods are compared with the numerical analysis solutions of the present GKUA, which are in good agreement, demonstrating the high accuracy of the present algorithm. Besides, some anomalous features in these flows are observed and analyzed in detail. The numerical experience indicates that the present GKUA can provide potential applications for the simulations of the nonequilibrium external-force driven flows, such as the gravity, the electric force, and the Lorentz force fields covering all flow regimes.     

转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1:5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.