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.     

Stochastic Effects of 2D Random Arrays of Cylinders on Rarefied Gas Permeability Using the Lattice Boltzmann Method

A large set of 2D random arrays of circular cylinders is generated to perform a statistical study on rarefied gas flow through micro-porous media. The flow regimes in this work lie for Knudsen numbers (Kn) ranging from the continuum to the transition regimes. Arrays are built by randomly placing cylinders with constant diameter with a uniform distribution without overlapping, and are generated for three target porosities. Fluid flow is assumed to be incompressible and isothermal. A modified lattice Boltzmann model is adopted to account for discrete effects, with slip-velocity boundary conditions and a Kn-dependent multi-relaxation time collision operator. The apparent permeability is modeled with Darcy’s law with a Klinkenberg-type relationship and compared with existing correlations. Velocity fields highlight the increasing contribution of fluid flow through small pores with increasing Kn. Numerical results show that porous media randomness leads to an uncertainty on rarefied gas permeability calculation despite the same structural characteristics and may not strictly follow a specific correlation. The influence of a local collision operator based on a local Kn instead of a global one in the numerical model is also studied. Results show that the permeability in rarefied regimes undergoes significant deviation when applying the local collision operator compared to the global one. These differences could result from a more accurate capture of the pore-scale behavior with a local Kn. Thus, it emphasizes the sensitivity of the model and the apparent permeability calculation to the appropriate definition of Kn.

     

利用格子Boltzmann方法数值模拟Rayleigh-Benard对流

采用格子Boltzmann方法对较大Rayleigh数范围下的二维Rayleigh-Benard对流进行了模拟研究.引入能量分布函数,利用该能量分布函数与粒子速度分布函数耦合来求解一个热流场,能量分布函数与粒子速度分布函数和Boltzmann方程构成了一个新的双分布格子Boltzmann模型.在考虑密度随温度变化的情况下,进行数值模拟,得到了Rayleigh-Benard对流速度、温度随时间的变化规律、系统的流线和等温线分布及平均Nusselt数与Rayleigh数的之间的关系,与相关文献数据进行了对比,模拟结果非常吻合,证明了改进的双分布格子Boltzmann模型的有效性.     

Ferrofluid Permeation into Three-Dimensional Random Porous Media: A Numerical Study Using the Lattice Boltzmann Method

Colloidal suspensions containing magnetic nanoparticles placed in appropriate carrier liquids present strong magnetic dipoles. These suspensions, in general, exhibit normal liquid behaviour coupled with super paramagnetic properties. This leads to the possibility of remotely controlling the flow of such liquids with a moderate-strength external magnetic field. In this study, we numerically investigate the viability of controlling and steering a base-fluid with magnetic-sensitive nanoparticles into randomly structured fibrous porous media. Three dimensional flow simulations are performed using the lattice Boltzmann method. The simulation results for the flow front are presented, and the effect of the magnetic field strength on the rate of ferrofluid penetration is discussed. It is shown that the porosity of the porous medium and the size of the fibres have a strong effect on the ferrofluid penetration rate.     

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.     

Stochastic Modeling of the Permeability of Randomly Generated Porous Media via the Lattice Boltzmann Method and Probabilistic Collocation Method

Transport in Porous Media - The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such...     

黏弹性流体扩展及收缩流动的格子Boltzmann模拟分析

本文应用格子Boltzmann 方法(LBM)并结合Oldroyd-B 模型,讨论了不可压缩的 Navier-Stokes 方程和平流扩散本构方程的解耦及各自求解方法,以及两类问题的边界处理格式,实现了黏弹性流体在二维1:3 扩展流道以及3:1 收缩流道中的流动的数值模拟．获得了不同雷诺数Re 和维森伯格数Wi 以及黏度vs 下流动的流线分布,计算给出了漩涡的涡心位置和大小,并分析了参数Re、Wi 和vs 对流动特点的影响．模拟结果表明本文所采用模型和边界处理方法具有良好的精度和稳定性．     

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...     

粘弹流体轴对称流动的格子Boltzmann方法

基于BGK碰撞模型，通过在迁移方程中引入作用力项，建立了粘弹流体的轴对称格子Boltzmann模型．通过Chapman-Enskog展开，获得了准确的柱坐标下轴对称宏观流动方程．采用双分布函数对运动方程和本构方程进行迭代求解，模拟分析了粘弹流体管道流动，获得了流场中的速度和构型张量的分布，通过与解析解进行比较，验证了模型的准确性．研究了作为粘弹流体流动基准问题的收敛流动，对涡旋位置进行了定量分析，将回转长度的计算结果与有限体积法进行了比较，两种数值结果十分吻合．研究结果表明，模型能够准确表征粘弹流体的轴对称流动，具有较广阔的应用前景．     

Dynamics of Isotropic Homogeneous Turbulence with Linear Forcing Using a Lattice Boltzmann Method

The Lattice Boltzmann Method (LBM) has proved to be a promising approach to solve the Navier–Stokes equations, especially for incompressible and isothermal cases. For turbulent flows, the quality of the predictions has been previously studied considering standard spectral forced (ten Cate et al., Comput Fluids 35:1239–1251, 2006) statistically homogeneous isotropic turbulence. In the present contribution, a recently proposed linear forcing scheme working in physical space (Lundgren 2003; Rosales and Meneveau, Phys Fluids 17(9):095106–1,8, 2005) has been integrated in a three-dimensional fifteen-velocity LBM formulation. Results have been analyzed, with special attention to the dynamics of the flow through the invariants of the velocity tensor. This topic had not been studied yet for the linear forcing, regardless of the nature (spectral or LBM) of the numerical method. Results fully agree with standard pseudo-spectral direct numerical simulations, results proving the validity of the LBM with linear forcing in real space to study this kind of turbulent flows.     

格子Boltzmann方法处理复杂边界新进展

格子玻尔兹曼方法是相对较新的计算流体力学方法,具有其独特的优点,如自然并行计算以及复杂边界的处理．在日益重要的高性能计算上,将发挥越来越大的作用．在已经较为成熟的计算模型基础上,为计算复杂几何边界对流场的作用,大量学者提出了各类针对格子玻尔兹曼方法的边界处理格式．边界处理方式对计算的影响主要有三个方面：计算精度,算法稳定性以及并行性．本文简单介绍格子玻尔兹曼方法在边界处理上的一些最新进展．并通过对实际算例的计算比较各类边界处理方式的优劣．     

Lattice Boltzmann model for nanofluids

A nanofluid is a particle suspension that consists of base liquids and nanoparticles and has great potential for heat transfer enhancement. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles, a lattice Boltzmann model is proposed for simulating flow and energy transport processes inside the nanofluids. First, we briefly introduce the conventional lattice Boltzmann model for multicomponent systems. Then, we discuss the irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids and describe a lattice Boltzmann model for simulating nanofluids. Finally, we conduct some calculations for the distribution of the suspended nanoparticles.     

Analysis of Thermal Flow in a Rotating Porous U-Turn Duct Using Lattice Boltzmann Method

The lattice Boltzmann method is carried out to investigate the heat transfer enhancement in a U-turn duct which is partially filled with a porous media. The porous layer is inserted at the core of the duct and is modeled using the Brinkman–Forchheimer assumptions. In order to validate the results, first a channel flow problem without any porous layer is compared with available data. Second, the porous Couette flow and partially porous channel flow are successfully compared with the studies of other researchers. Then, fluid flow in a clear U-turn duct is studied looking carefully at the velocity, curvature and rotation effects. Finally, the effects of porous layer thickness on the rate of heat transfer and pressure drop are investigated. Parametric studies are conducted to evaluate the effects of various parameters (i.e., Reynolds number, Darcy number, rotation number), highlighting their influences on the thermo-hydrodynamics behavior of the flow. The optimum values of porous layer thickness are presented for specific flow parameters.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Study of Slip Flow in Unconventional Shale Rocks Using Lattice Boltzmann Method: Effects of Boundary Conditions and TMAC

Steady, laminar, fully developed flows of a Newtonian fluid driven by a constant pressure gradient in (1) a curvilinear constant cross section triangle bounded by two straight no-slip segments and a circular meniscus and (2) a wedge bounded by two rays and an adjacent film bulging near the corner are studied analytically by the theory of holomorphic functions and numerically by finite elements. The analytical solution of the first problem is obtained by reducing the Poisson equation for the longitudinal flow velocity to the Laplace equation, conformal mapping of the corresponding transformed physical domain onto an auxiliary half-plane and solving there the Signorini mixed boundary value problem (BVP). The numerical solution is obtained by meshing the circular sector and solving a system of linear equations ensuing from the Poisson equation. Comparisons are made with known solutions for flows in a rectangular conduit, circular annulus and Philip’s circular duct with a no-shear sector. Problem (2) is treated by the Saint-Venant semi-inverse method: the free surface (quasi-meniscus) is reconstructed by a one-parametric family, which specifies a holomorphic function of the first derivative of the physical coordinate with respect to an auxiliary variable. The latter maps the flow domain onto a quarter of a unit disc where a mixed BVP for a characteristic function is solved by the Zhukovsky–Chaplygin method. Velocity distributions in a cross section perpendicular to the flow direction are obtained. It is shown that the change of the type of the boundary condition from no slip to perfect slip (along the meniscus) causes a dramatic increase of the total flow rate (conductance). For example, the classical Saint-Venant formulae for a sector, with all three boundaries being no-slip segments, predict up to four times smaller rate as compared to a free surface meniscus. Mathematically equivalent problems of unconfined flows in aquifers recharged by a constant-intensity infiltration are also addressed.     

Pore-Scale Simulation of Fluid Flow Through Deformable Porous Media Using Immersed Boundary Coupled Lattice Boltzmann Method

Transport in Porous Media - The interaction between the fluid flow and the deformable porous media is crucial in the applications of adsorption/absorption. The immersed boundary coupled lattice...     

Lattice Boltzmann method for polymer kinetic theory

The purpose of this work is to extend the applicability of the lattice Boltzmann method (LBM) to the field of polymer kinetic theory or more generally suspensions that could be described in the Fokker–Planck formalism. This method has been, in a first time, used for gas kinetic theory, where the resolution space corresponds to the physical space coordinate. In a second time is has been generalized to be applied to fluid flow involving different behaviours: turbulence, porous media, multiphase flow, etc. However this powerful, parallel, and efficient algorithm has not been applied for solving Fokker–Planck equations widely used to describe suspension kinetic theory. In this scale, molecular models involve a high computational costs because of the multidimensionality of the fully coupled micro–macro complex flow. The originality of this work consists to apply the lattice Boltzmann technique for solving Fokker–Planck equation based on a discretization of the configuration space where the resolution coordinates correspond to the microscopic configuration space (and not the physical coordinates). The result of this work emphasizes the optimality of the used technique that, in addition to its parallel ability, gathers the simplicity of the stochastic simulation and the robustness of the traditional fixed mesh support (such as the finite element method). Accuracy and convergence of the LBM will be compared to the stochastic and the finite element techniques for homogeneous shear flow.     

基于格子Boltzmann方法的挤出胀大的数值模拟

将单相格子Boltzmann方法(lattice Boltzmann method, LBM)引入到粘弹流体的瞬态挤出胀大的数值模拟中,建立了基于双分布函数的自由面粘弹性流动格子Boltzmann模型．分析得到的流道中流动速度分布和构型张量结果与理论解十分吻合．对粘弹流体瞬态挤出胀大过程进行了模拟,并分析了运动粘度比和剪切速率对挤出胀大率的影响,得到的胀大率结果与理论分析和其它模拟结果基本一致．表明给出的LBM可以捕捉挤出胀大的瞬态效应．     

颗粒在黏弹性流体扩张和收缩流中沉降的格子Boltzmann模拟分析

对于Oldroyd-B型黏弹性流体,本文应用格子Boltzmann方法(LBM),实现了流体在二维1:3扩张流道及3:1收缩流道中流动的数值模拟,获得了黏弹性流体在扩张和收缩流道中的流场分布．结合颗粒的受力和运动规则,基于点源颗粒模型,数值分析了颗粒在扩张流和收缩流中的沉降过程和特征,讨论了颗粒相对质量和起始位置以及雷诺数Re和威森伯格数Wi对颗粒沉降特征的影响．结果表明,颗粒相对质量和起始位置以及Re对颗粒沉降轨迹和落点影响较大,而Wi的影响则较小．