Does Klinkenberg's Law Survive Upscaling?

This work deals with the large-scale mathematical modelling of flow of gas at low pressure in porous media. At the pore scale, this type of flow is characterised by a wall-slip effect, which at the sample scale entails a dependence of permeability upon gas pressure. This latter property is described by Klinkenberg's law. The goal of the present work is to examine the robustness of this law, by determining whether it is still verified on a large-scale: upscaling is thus applied, starting with Klinkenberg's law at the local scale. A Klinkenberg's flow of gas in a two-constituent composite porous medium is considered, and the constituents are firstly assumed to be homogeneous. The cases of low and of high permeability contrast are successively examined. Upscaling is performed using the homogenisation method of multiple scale expansions. In both cases, the large-scale permeability tensor differs from its liquid counterpart. Except in the particular case of equal Klinkenberg factors, Klinkenberg's law is not verified at low permeability contrast. At high permeability contrast, the large-scale gas permeability verifies Klinkenberg's law. The case of heterogeneous constituents is then examined. It is shown that the large-scale permeability differs from its liquid counterpart, but it does not verify Klinkenberg's law.     

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.     

Gas Flow in Porous Media With Klinkenberg Effects

Gas flow in porous media differs from liquid flow because of the large gas compressibility and pressure-dependent effective permeability. The latter effect, named after Klinkenberg, may have significant impact on gas flow behavior, especially in low permeability media, but it has been ignored in most of the previous studies because of the mathematical difficulty in handling the additional nonlinear term in the gas flow governing equation. This paper presents a set of new analytical solutions developed for analyzing steady-state and transient gas flow through porous media including Klinkenberg effects. The analytical solutions are obtained using a new form of gas flow governing equation that incorporates the Klinkenberg effect. Additional analytical solutions for one-, two- and three-dimensional gas flow in porous media could be readily derived by the following solution procedures in this paper. Furthermore, the validity of the conventional assumption used for linearizing the gas flow equation has been examined. A generally applicable procedure has been developed for accurate evaluation of the analytical solutions which use a linearized diffusivity for transient gas flow. As application examples, the new analytical solutions have been used to verify numerical solutions, and to design new laboratory and field testing techniques to determine the Klinkenberg parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs from The Geysers geothermal field. We show that this new approach and the traditional method of Klinkenberg yield similar results of Klinkenberg constants for the laboratory tests; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Mathematical Model of Coalbed Gas Flow with Klinkenberg Effects in Multi-Physical Fields and its Analytic Solution

The deep-mining coal seam impacted by high in situ stress, where Klinkenberg effects for gas flow were very obvious due to low gas permeability, could be regarded as a porous and tight gas-bearing media. Moreover, the Klinkenberg effects had a significant effect on gas flow behavior of deep-mining coal seam. Based on the gas flow properties of deep-mining coal seams affected by in situ stress field, geothermal temperature field and geo-electric field, a new mathematical model of coalbed gas flow, which reflected the impact of Klinkenberg effects on coalbed gas flow properties in multi-physical fields, was developed by establishing the flow equation, state equation, and continuity equation and content equation of coalbed gas. The analytic solution was derived for the model of one-dimensional steady coalbed gas flow with Klinkenberg effects affected by in situ stress field and geothermal temperature field, and a sensitivity analysis of its physical parameters was carried out by comparing available analytic solutions and the measured values. The results show that the analytic solutions of this model of coalbed gas flow with Klinkenberg effects are closer to the measured values compared to those without Klinkenberg effects, and this model can reflect more accurately gas flow of deep-mining coal seams. Moreover, the analytic solution of this model is more sensitive to the change of Klinkenberg factor b and temperature grad G than depth h.     

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 the Permeability of Very Low Porosity Sandstones

The permeability predictions of two geometric pore-scale models, one being predominantly granular and the other consolidated with tube-like pores, are compared with experimental results for Fontainebleau sandstones and the results interpreted. Percolation thresholds are determined from experimental data and applied in the modelling exercise by means of cut-off asymptotes on porosity. It is found that, although both granular and foamlike models yield plausible results, the granular model appears to be superior, at least for the sets of data considered. The Klinkenberg correction is analytically derived and incorporated into the models to relate gas and liquid permeabilities and an analytical expression for the Klinkenberg factor is proposed for each model. The permeability predictions are promising and yield an effective manner to correlate sandstone percolation data.     

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.     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

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.     

Flow of Maxwell fluids in porous media

In this paper we analyze the flow of a Maxwell fluid in a rigid porous medium using the method of volume averaging. We first present the local volume averaged momentum equation which contains Darcy-scale elastic effects and undetermined integrals of the spatial deviations of the pressure and velocity. A closure problem is developed in order to determine the spatial deviations and thus obtain a closed form of the momentum equation that contains a time-dependent permeability tensor. To gain some insight into the effects of elasticity on the dynamics of flow in porous media, the entire problem is transformed to the frequency domain through a temporal Fourier transform. This leads to a dynamic generalization of Darcy's law. Analytical results are provided for the case in which the porous medium is modeled as a bundle of capillary tubes, and a scheme is presented to solve the transformed closure problem for a general microstructure.     

Contribution of the Dusty Gas Model to Permeability/Diffusion Tests on Partially Saturated Clay Rocks

Gas transfer experiments on claystone and numerical simulations have been conducted to enhance the knowledge of gas transport in nuclear waste repositories in the Callovo-Oxfordian clay formation in Bure, France. Laboratory Gas transfer experiments were performed with a specific device dedicated to very low permeability measurement (10^?23 to 10^?20 m²). Experiments were performed on both dry and close to saturation claystone. The Dusty Gas Model, based on multi-component gas transfer equations with Knudsen diffusion, was used to describe the experimental results. The parameters obtained are the effective permeability, the Knudsen diffusion (Klinkenberg effect) and molecular diffusion coefficients and the porosity accessible to gas. Numerical simulations were carried with various boundary conditions and for different gases (helium vs hydrogen) and were compared with experiments to test the reliability of the model parameters and to better understand the mechanisms involved in clays close to saturation. The numerical simulation fitted the experimental data well whereas simpler models cannot describe the complexity of the Knudsen/Klinkenberg effects. Permeabilities lie between 10^?22 and 10^?20 m². Claystones close to saturation have an accessible porosity to gas transfer that is lower than 0.1?C1% of the porosity. Analysis of the Klinkenberg effect suggests that this accessible pore network should be made of 50?C200?nm diameter pores. It represents pore networks accessible at capillary pressure lower than 4?MPa.     

Numerical simulation of the fountain flow instability in injection molding

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numerical simulation. A two-component viscoelastic numerical model is developed which can predict fountain flow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used. The levelset method is used for modeling the two-component flow of polymer and gas. The difficulties arising from the three-phase contact point modeling are addressed, and solved by treating the wall as an interface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solved in a decoupled way using a finite element formulation. Since the model for the polymer does not contain a solvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressure at the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetric fountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found with a spatial period of three times the channel height, which corresponds to experimental observations.     

Two Types of Transient Phenomena and Full Relaxation Macroscale Model for Single Phase Flow through Double Porosity Media

Darcy's flow of a weakly compressible fluid through double porosity media is studied in the framework of the homogenization theory. In previous papers, various classes of single-phase flow have been detected with various determination of the effective permeability tensor for each class. In this paper, the full model including transient phenomena is developed, where the macroscale momentum balance equation represents a modification of Darcy's law with a nonequilibrium term. The effective permeability tensor appears to be nonstationary and is changing during the system evolution in time. Three relaxation times characterize the transient transformations of each component of the macroscale flow velocity.This effect is superposed with the second relaxation phenomenon caused by the exchange flow between dense blocks and the highly conductive matrix. The relaxation times for the effective permeability and for the exchange flow are shown to have different orders.All relaxation parameters are explicitly determined through solutions of cell problems.     

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.     

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.     

涡识别技术及其在船用螺旋桨伴流场中的应用

涡识别是很重要的流体问题, 为了在船用螺旋桨伴流场中寻找一种合理的涡识别方法, 本文结合实践, 研究了六种涡识别技术理论, 其中使用Burgers涡流和Lamb-Oseen涡流作了必要的解释, 讨论了各种识别方法的优缺点. 局部低压标准比较直观, 但深究其黏性和非定常影响后, 明显不足; 迹线或流线显然不能满足伽利略不变性, 会使辨别变得混乱; 涡度大小需要规定其阈值, 具有一定不确定性, 且也会识别不是涡的涡片; 速度梯度张量的复特征值也会有识别不出的区域; 速度梯度张量的第二不变量标准和对称张量的第二特征值标准能更好地识别涡核, 这两种标准有时等效. 螺旋桨伴流场的数值模拟是在开源软件OpenFOAM平台上实现的, 湍流大涡模型由一种局部动态方程建模, 此模型优于动态Smagorinsky模型. 最终的结果显示: 对于船用螺旋桨伴流场中的涡, 采用速度梯度张量的第二不变量和对称张量的第二特征值的结果基本一致, 而最小压力标准、流线或迹线标准、涡度值标准和速度张量的复特征值标准都存在一定的缺陷, 不适用于船用螺旋桨伴流场中的涡识别.      

The influence of thermal expansion flow on droplet evaporation

It has been demonstrated recently that it follows from conservation of mass that unsteady temperature fields create flow in an incompressible fluid with a temperature-dependent density even in the absence of gravity. The paper studies the influence of thermal expansion flow on spherically symmetric evaporation of an isolated droplet. A model problem of a droplet evaporating at a constant rate is first considered. In this idealized situation one can use the assumption of a thin thermal boundary layer to solve analytically the unsteady moving-boundary heat conduction problem to find the temperature field inside the droplet both with and without the thermal expansion flow. Next evaporation of a fuel droplet in a diesel engine is studied numerically. The heat diffusion equation is solved in the liquid phase while the standard quasi-steady model is used for the gas phase. The results of the calculation show that for high ambient temperatures the influence of the thermal expansion flow on the droplet lifetime can be considerable.     

水平均流中细管排放气泡的三维数值模拟

在液体为无粘不可压，流动有势和气体遵循完全气体绝热关系的假定下，本文应用边界积分方程方法数值模拟了水平均流中垂直细管排放气泡的三维动力学问题，边界采用高阶有限元表达。文中介绍了有关泡面法向矢量、切向速度、曲率和接触线等的计算技术。与已知解的比较，表明了这一数值方法的高精度和优越性。算例显示了水平均流对于气泡形状和体积的影响     

Permeability identification of a weakly permeable partially saturated porous rock

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.