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Journal ArticleDOI

Chapman–Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

TL;DR: The gray lattice Boltzmann equation (GLBE) method has been used to simulate fluid flow in porous media and it employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) to account for the linear drag of the medium.
Abstract: The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman–Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in $$ \epsilon $$ only if $$ \theta = {\rm O}\left( \epsilon \right) $$ in which $$ \epsilon $$ is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.
Citations
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Journal ArticleDOI
TL;DR: In this paper, a phase-field model formulated within the system of lattice Boltzmann (LB) equation for simulating solidification and dendritic growth with fully coupled melt flow and thermosolutal convection-diffusion is proposed.

10 citations

Journal ArticleDOI
Zhang Qi1, Yu Huibin1, Li Xiaofeng1, Liu Tiesheng1, Hu Junfeng1 
TL;DR: In this paper, a new upscaling method integrated gray lattice Boltzmann method (GLBM) and pore network model (PNM), accounting for the fluid flow in heterogeneous porous media.
Abstract: High heterogeneity and nonuniformly distributed multiscale pore systems are two characteristics of the unconventional reservoirs, which lead to very complex transport mechanisms. Limited by inadequate computational capability and imaging field of view, flow simulation cannot be directly performed on complex pore structures. The traditional methods usually coarsen the grid to reduce the computational load but will lead to the missing microstructure information and inaccurate simulation results. To develop a better understanding of flow properties in unconventional reservoirs, this study proposed a new upscaling method integrated gray lattice Boltzmann method (GLBM) and pore network model (PNM), accounting for the fluid flow in heterogeneous porous media. This method can reasonably reduce the computational loads while preserving certain micropore characteristics. Verifications are conducted by comparing the simulation and experimental results on tight sandstones, and good agreements are achieved. The proposed method is proven to be capable of estimating bulk properties in highly heterogenous unconventional reservoirs. This method could contribute to the development of multiscale pore structure characterizations and enhance the understandings of fluid flow mechanisms in unconventional reservoirs.

Cites result from "Chapman–Enskog Analyses on the Gray..."

  • ...The simulation results have a high consistency with the analytical solution [16, 17]....

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References
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Journal ArticleDOI
01 Feb 1992-EPL
TL;DR: In this article, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order approximation with a properly chosen equilibrium distribution, with a relaxation parameter that influences the stability of the new scheme.
Abstract: We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.

4,481 citations

Journal ArticleDOI
TL;DR: In this paper, the viscous force exerted by a flowing fluid on a dense swarm of particles is described by a modification of Darcy's equation, which was necessary in order to obtain consistent boundary conditions.
Abstract: A calculation is given of the viscous force, exerted by a flowing fluid on a dense swarm of particles. The model underlying these calculations is that of a spherical particle embedded in a porous mass. The flow through this porous mass is decribed by a modification of Darcy's equation. Such a modification was necessary in order to obtain consistent boundary conditions. A relation between permeability and particle size and density is obtained. Our results are compared with an experimental relation due to Carman.

2,519 citations

Journal ArticleDOI
TL;DR: In this paper, Chen et al. used the half-way wall bounceback boundary condition for the 2-D Poiseuille flow with forcing to obtain second-order accuracy for the 3-D square duct flow.
Abstract: Pressure (density) and velocity boundary conditions inside a flow domain are studied for 2-D and 3-D lattice Boltzmann BGK models (LBGK) and a new method to specify these conditions are proposed. These conditions are constructed in consistency of the wall boundary condition based on an idea of bounceback of non-equilibrium distribution. When these conditions are used together with the improved incompressible LBGK model by Zou et al., the simulation results recover the analytical solution of the plane Poiseuille flow driven by pressure (density) difference with machine accuracy. Since the half-way wall bounceback boundary condition is very easy to implement and was shown theoretically to give second-order accuracy for the 2-D Poiseuille flow with forcing, it is used with pressure (density) inlet/outlet conditions proposed in this paper and in Chen et al. to study the 2-D Poiseuille flow and the 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback is comparable with that using some other published boundary conditions. Besides, the bounceback condition has a much better stability behavior than that of other boundary conditions.

2,001 citations

Journal ArticleDOI
TL;DR: The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail and linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long-wavelength limit (wave vector k=0).
Abstract: The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.

1,859 citations

Journal ArticleDOI
TL;DR: The half-way wall bounceback boundary condition is also used with the pressure ~density! inlet/outlet conditions proposed in this article to study 2-D Poiseuille flow and 3-D square duct flow.
Abstract: Pressure ~density! and velocity boundary conditions are studied for 2-D and 3-D lattice Boltzmann BGK models ~LBGK! and a new method to specify these conditions is proposed. These conditions are constructed in consistency with the wall boundary condition, based on the idea of bounceback of the non-equilibrium distribution. When these conditions are used together with the incompressible LBGK model @J. Stat. Phys. 81 ,3 5 ~1995!# the simulation results recover the analytical solution of the plane Poiseuille flow driven by a pressure ~density! difference. The half-way wall bounceback boundary condition is also used with the pressure ~density! inlet/outlet conditions proposed in this paper and in Phys. Fluids 8, 2527 ~1996! to study 2-D Poiseuille flow and 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback boundary condition is comparable with that of other published boundary conditions and it has better stability behavior. © 1997 American Institute of Physics. @S1070-6631~97!03406-5#

1,854 citations