格子Boltzmann方法模拟多孔介质惯性流的边界条件改进
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摘要
格子Boltzmann方法可以有效地模拟水动力学问题,边界处理方法的选择对于可靠的模拟计算至关重要.本文基于多松弛时间格子Boltzmann模型开展了不同边界条件下,周期对称性结构和不规则结构中流体流动模拟,阐述了不同边界条件的精度和适用范围.此外,引入一种混合式边界处理方法来模拟多孔介质惯性流,结果表明:对于周期性对称结构流动模拟,体力格式边界条件和压力边界处理方法是等效的,两者都能精确地捕捉流体流动特点;而对于非周期性不规则结构,两种边界处理方法并不等价,体力格式边界条件只适用于周期性结构;由于广义化周期性边界条件忽略了垂直主流方向上流体与固体格点的碰撞作用,同样不适合处理不规则模型;体力–压力混合式边界格式能够用来模拟周期性或非周期性结构流体流动,在模拟多孔介质流体惯性流时,比压力边界条件有更大的应用优势,可以获得更大的雷诺数且能保证计算的准确性.
The lattice Boltzmann method has been considered as an effective method for the simulation of hydrodynamic flows. Handling the boundary condition accurately in simulation is extremely essential for a reliable study. In this paper,a multiple relaxation time lattice Boltzmann model with different boundary conditions was applied to mimic the flows in periodically symmetric and irregular structures. The scope of application and accuracy for different boundary conditions in various geometries was investigated. In addition, a hybrid boundary treatment method was introduced to simulate the non-Darcy flow in porous media, the simulation results of which were also compared to the results obtained using pressure boundary condition. The results show that for the symmetric and periodic flow simulation, both the body force and the pressure driven boundary treatments are perfectly equivalent and both can accurately capture the flow characteristics.While for the fluid flow in irregular structures, the body force and pressure boundary conditions are not equivalent, and the body force one has limited use and can only be applied to periodic structures. This implies that one must be cautious of the reliability of modeling when conducting model validation with simple structures. It seems that the regular structures could be inadequate to validate the modeling, which depends on the research issues, i.e., the flow patterns in what kinds of structures. Furthermore, the generalized periodic boundary condition proposed by previous authors combines periodic density momentum with a pressure gradient in one dimension is also not appropriate to conduct flow simulation in irregular models since this method ignores the effect of asymmetric obstacles in the direction perpendicular to the main streamlines.Moreover, the hybrid boundary condition can be used to perform flow simulations not only in periodic structures but also the irregular ones. In particular, for the inertial flow of fluids in porous media, the relatively high Reynolds number can be achieved readily with the hybrid boundary condition. For the pressure driven boundary condition, the pressure gradient comes from the density difference between the inlet and outlet. To provide a higher Reynolds number, it is necessary to implement a great density contrast in inlet and outlet nodes. However, this approach is inconsistent with physical situation and causes undesirable errors in simulation. All in all, the hybrid boundary condition has greater advantages over the pressure boundary condition.
The lattice Boltzmann method has been considered as an effective method for the simulation of hydrodynamic flows. Handling the boundary condition accurately in simulation is extremely essential for a reliable study. In this paper,a multiple relaxation time lattice Boltzmann model with different boundary conditions was applied to mimic the flows in periodically symmetric and irregular structures. The scope of application and accuracy for different boundary conditions in various geometries was investigated. In addition, a hybrid boundary treatment method was introduced to simulate the non-Darcy flow in porous media, the simulation results of which were also compared to the results obtained using pressure boundary condition. The results show that for the symmetric and periodic flow simulation, both the body force and the pressure driven boundary treatments are perfectly equivalent and both can accurately capture the flow characteristics.While for the fluid flow in irregular structures, the body force and pressure boundary conditions are not equivalent, and the body force one has limited use and can only be applied to periodic structures. This implies that one must be cautious of the reliability of modeling when conducting model validation with simple structures. It seems that the regular structures could be inadequate to validate the modeling, which depends on the research issues, i.e., the flow patterns in what kinds of structures. Furthermore, the generalized periodic boundary condition proposed by previous authors combines periodic density momentum with a pressure gradient in one dimension is also not appropriate to conduct flow simulation in irregular models since this method ignores the effect of asymmetric obstacles in the direction perpendicular to the main streamlines.Moreover, the hybrid boundary condition can be used to perform flow simulations not only in periodic structures but also the irregular ones. In particular, for the inertial flow of fluids in porous media, the relatively high Reynolds number can be achieved readily with the hybrid boundary condition. For the pressure driven boundary condition, the pressure gradient comes from the density difference between the inlet and outlet. To provide a higher Reynolds number, it is necessary to implement a great density contrast in inlet and outlet nodes. However, this approach is inconsistent with physical situation and causes undesirable errors in simulation. All in all, the hybrid boundary condition has greater advantages over the pressure boundary condition.
引文
1柳占立,庄茁,孟庆国等.页岩气高效开采的力学问题与挑战.力学学报,2017,49(3):507-516(Liu Zhanli,Zhuang Zhuo,Meng Qingguo,et al.Problems and challenges of mechanics in shale gas efficient exploitation.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):507-516(in Chinese))
2刘文超,刘曰武.低渗透煤层气藏中气-水两相不稳定渗流动态分析.力学学报,2017,49(4):828-835(Liu Wenchao,Liu Yuewu.Dynamic analysis on gas-water two-phase unsteady seepage flow in low-permeable coalbed gas reservoirs.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):828-835(in Chinese))
3郑艺君,李庆祥,潘明等.多孔介质壁面剪切湍流速度时空关联的研究.力学学报,2016,48(6):1308-1318(Zheng Yijun,Li Qingxiang,Pan Ming,et al.Space-time correlations of fluctuating veloctuating in porous wall-bounded turbulent shear flows.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1308-1318(in Chinese))
4 Sukop MC,Thorne DT.Lattice Boltzmann Modeling:An Introduction for Geoscientists and Engineers.Springer Publishing Company,Incorporated,2007
5 Krüger T,Kusumaatmaja H,Kuzmin A,et al.The Lattice Boltzmann Method:Principles and Practice.Springer Publishing Company,2016
6 Chen L,Zhang L,Kang Q,et al.Nanoscale simulation of shale transport properties using the lattice Boltzmann method:Permeability and diffusivity.Scientific Reports,2015,5:8089
7 Chukwudozie C,Tyagi M.Pore scale inertial flow simulations in 3-D smooth and rough sphere packs using lattice Boltzmann method.AIChE Journal,2013,59(12):4858-4870
8 Arabjamaloei R,Ruth D.Numerical study of inertial effects on permeability of porous media utilizing the lattice Boltzmann method.Journal of Natural Gas Science and Engineering,2017,44:22-36
9 Kakouei A,Vatani A,Rasaei M,et al.Cessation of Darcy regime in gas flow through porous media using LBM:Comparison of pressure gradient approaches.Journal of Natural Gas Science and Engineering,2017,45:693-705
10 Zhao H,Ning Z,Kang Q,et al.Relative permeability of two immiscible fluids flowing through porous media determined by lattice Boltzmann method.International Communications in Heat and Mass Transfer,2017,85:53-61
11 Zhao T,Zhao H,Li X,et al.Pore scale characteristics of gas flow in shale matrix determined by the regularized lattice Boltzmann method.Chemical Engineering Science,2018,187:245-255
12 Kandhai D,Koponen A,Hoekstra A,et al.Implementation aspects of 3D lattice-BGK:Boundaries,accuracy,and a new fast relaxation method.Journal of Computational Physics,1999,150(2):482-501
13 Zhang J,Kwok DY.Pressure boundary condition of the lattice Boltzmann method for fully developed periodic flows.Physical Review E,2006,73(4):047702
14 Grser O,Grimm A.Adaptive generalized periodic boundary conditions for lattice Boltzmann simulations of pressure-driven flows through confined repetitive geometries.Physical Review E,2010,82(1):016702
15 Newman MS,Yin X.Lattice Boltzmann simulation of non-Darcy flow in stochastically generated 2D porous media geometries.SPEJournal,2013,18(1):12-26
16 Chai Z,Shi B,Lu J,et al.Non-Darcy flow in disordered porous media:A lattice Boltzmann study.Computers&Fluids,2010,39(10):2069-2077
17 Sukop MC,Huang H,Alvarez PF,et al.Evaluation of permeability and non-Darcy flow in vuggy macroporous limestone aquifer samples with lattice Boltzmann methods.Water Resources Research,2013,49(1):216-230
18 Bhatnagar PL,Gross EP,Krook M.A model for collision processes in gases.I.Small amplitude processes in charged and neutral onecomponent systems.Physical Review,1954,94(3):511-525
19 Humières D.Multiple-relaxation-time lattice Boltzmann models in three dimensions.Philosophical Transactions of the Royal Society of London Series A:Mathematical,Physical and Engineering Sciences,2002,360(1792):437
20 Lallemand P,Luo L-S.Theory of the lattice Boltzmann method:Dispersion,dissipation,isotropy,Galilean invariance,and stability.Physical Review E,2000,61(6):6546-6562
21 Qian YH,Humières DD,Lallemand P.Lattice BGK models for Navier-Stokes equation.EPL(Europhysics Letters),1992,17(6):479
22 Guo Z,Zheng C,Shi B.Discrete lattice effects on the forcing term in the lattice Boltzmann method.Physical Review E,2002,65(4):046308
23 Yu Z,Fan LS.Multirelaxation-time interaction-potential-based lattice Boltzmann model for two-phase flow.Physical Review E,2010,82(4):046708
24 Huang H,Huang JJ,Lu XY.Study of immiscible displacements in porous media using a color-gradient-based multiphase lattice Boltzmann method.Computers&Fluids,2014,93:164-172
25 Guo ZL,Zheng CG,Shi BC.Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method.Chinese Physics,2002,11(4):366
26 Kim S H,Pitsch H.A generalized periodic boundary condition for lattice Boltzmann method simulation of a pressure driven flow in a periodic geometry.Physics of Fluids,2007,19(10):108101
27陶实,王亮,郭照立.微尺度振荡Couette流的格子Boltzmann模拟.物理学报,2014,63(21):239-247(Tao Shi,Wang Liang,Guo ZhaoLi.Lattice Boltzmann modeling of microscale oscillating Couette flow.Acta Physica Sinica,2014,63(21):239-247(in Chinese))
28姚军,赵建林,张敏等.基于格子Boltzmann方法的页岩气微观流动模拟.石油学报,2015,36(10):1280-1289(Yao Jun,Zhao Jianlin,Zhang Min,et al.Microscale shale gas flow simulation based on lattice Boltzmann method.Acta Petrolei Sinica,2015,36(10):1280-1289(in Chinese))
29 Zeng Y,Ning Z,Wang Q,et al.Gas transport in self-affine rough microchannels of shale gas reservoir.Journal of Petroleum Science and Engineering,2018,167:716-728
30 Cheng Z,Ning Z,Wang Q,et al.The effect of pore structure on nonDarcy flow in porous media using the lattice Boltzmann method.Journal of Petroleum Science and Engineering,2019,172:391-400
31 Guo WB,Wang NC,Shi BC,et al.Lattice-BGK simulation of a two-dimensional channel flow around a square cylinder.Chinese Physics,2003,12(1):67
32 Breuer M,Bernsdorf J,Zeiser T,et al.Accurate computations of the laminar flow past a square cylinder based on two different methods:lattice-Boltzmann and finite-volume.International Journal of Heat and Fluid Flow,2000,21(2):186-196
33 Mei R,Yu D,Shyy W,et al.Force evaluation in the lattice Boltzmann method involving curved geometry.Physical Review E,2002,65(4):041203
34 Lu J,Guo Z,Chai Z,et al.Numerical study on the tortuosity of porous media via lattice Boltzmann method.Communications in Computational Physics,2009,6(2):354-366
35 Chukwudozie C,Tyagi M,Sears S,et al.Prediction of non-Darcy coefficients for inertial flows through the castlegate sandstone using image-based modeling.Transport in Porous Media,2012,95(3):563-580
36 Guo Z,Zhao T.Lattice Boltzmann model for incompressible flows through porous media.Physical Review E,2002,66(3):036304
37 Forchheimer P.Wasserbewegung durch boden,Z.Ver.Deutsch,Ing.,1901,45:1782-1788
38 Ruth D,Ma H.On the derivation of the Forchheimer equation by means of the averaging theorem.Transport in Porous Media,1992,7(3):255-264
39 Zeng Z,Grigg R.A criterion for non-Darcy flow in porous media.Transport in Porous Media,2006,63(1):57-69
40 Andrade Jr J,Costa U,Almeida M,et al.Inertial effects on fluid flow through disordered porous media.Physical Review Letters,1999,82(26):5249
2刘文超,刘曰武.低渗透煤层气藏中气-水两相不稳定渗流动态分析.力学学报,2017,49(4):828-835(Liu Wenchao,Liu Yuewu.Dynamic analysis on gas-water two-phase unsteady seepage flow in low-permeable coalbed gas reservoirs.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):828-835(in Chinese))
3郑艺君,李庆祥,潘明等.多孔介质壁面剪切湍流速度时空关联的研究.力学学报,2016,48(6):1308-1318(Zheng Yijun,Li Qingxiang,Pan Ming,et al.Space-time correlations of fluctuating veloctuating in porous wall-bounded turbulent shear flows.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1308-1318(in Chinese))
4 Sukop MC,Thorne DT.Lattice Boltzmann Modeling:An Introduction for Geoscientists and Engineers.Springer Publishing Company,Incorporated,2007
5 Krüger T,Kusumaatmaja H,Kuzmin A,et al.The Lattice Boltzmann Method:Principles and Practice.Springer Publishing Company,2016
6 Chen L,Zhang L,Kang Q,et al.Nanoscale simulation of shale transport properties using the lattice Boltzmann method:Permeability and diffusivity.Scientific Reports,2015,5:8089
7 Chukwudozie C,Tyagi M.Pore scale inertial flow simulations in 3-D smooth and rough sphere packs using lattice Boltzmann method.AIChE Journal,2013,59(12):4858-4870
8 Arabjamaloei R,Ruth D.Numerical study of inertial effects on permeability of porous media utilizing the lattice Boltzmann method.Journal of Natural Gas Science and Engineering,2017,44:22-36
9 Kakouei A,Vatani A,Rasaei M,et al.Cessation of Darcy regime in gas flow through porous media using LBM:Comparison of pressure gradient approaches.Journal of Natural Gas Science and Engineering,2017,45:693-705
10 Zhao H,Ning Z,Kang Q,et al.Relative permeability of two immiscible fluids flowing through porous media determined by lattice Boltzmann method.International Communications in Heat and Mass Transfer,2017,85:53-61
11 Zhao T,Zhao H,Li X,et al.Pore scale characteristics of gas flow in shale matrix determined by the regularized lattice Boltzmann method.Chemical Engineering Science,2018,187:245-255
12 Kandhai D,Koponen A,Hoekstra A,et al.Implementation aspects of 3D lattice-BGK:Boundaries,accuracy,and a new fast relaxation method.Journal of Computational Physics,1999,150(2):482-501
13 Zhang J,Kwok DY.Pressure boundary condition of the lattice Boltzmann method for fully developed periodic flows.Physical Review E,2006,73(4):047702
14 Grser O,Grimm A.Adaptive generalized periodic boundary conditions for lattice Boltzmann simulations of pressure-driven flows through confined repetitive geometries.Physical Review E,2010,82(1):016702
15 Newman MS,Yin X.Lattice Boltzmann simulation of non-Darcy flow in stochastically generated 2D porous media geometries.SPEJournal,2013,18(1):12-26
16 Chai Z,Shi B,Lu J,et al.Non-Darcy flow in disordered porous media:A lattice Boltzmann study.Computers&Fluids,2010,39(10):2069-2077
17 Sukop MC,Huang H,Alvarez PF,et al.Evaluation of permeability and non-Darcy flow in vuggy macroporous limestone aquifer samples with lattice Boltzmann methods.Water Resources Research,2013,49(1):216-230
18 Bhatnagar PL,Gross EP,Krook M.A model for collision processes in gases.I.Small amplitude processes in charged and neutral onecomponent systems.Physical Review,1954,94(3):511-525
19 Humières D.Multiple-relaxation-time lattice Boltzmann models in three dimensions.Philosophical Transactions of the Royal Society of London Series A:Mathematical,Physical and Engineering Sciences,2002,360(1792):437
20 Lallemand P,Luo L-S.Theory of the lattice Boltzmann method:Dispersion,dissipation,isotropy,Galilean invariance,and stability.Physical Review E,2000,61(6):6546-6562
21 Qian YH,Humières DD,Lallemand P.Lattice BGK models for Navier-Stokes equation.EPL(Europhysics Letters),1992,17(6):479
22 Guo Z,Zheng C,Shi B.Discrete lattice effects on the forcing term in the lattice Boltzmann method.Physical Review E,2002,65(4):046308
23 Yu Z,Fan LS.Multirelaxation-time interaction-potential-based lattice Boltzmann model for two-phase flow.Physical Review E,2010,82(4):046708
24 Huang H,Huang JJ,Lu XY.Study of immiscible displacements in porous media using a color-gradient-based multiphase lattice Boltzmann method.Computers&Fluids,2014,93:164-172
25 Guo ZL,Zheng CG,Shi BC.Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method.Chinese Physics,2002,11(4):366
26 Kim S H,Pitsch H.A generalized periodic boundary condition for lattice Boltzmann method simulation of a pressure driven flow in a periodic geometry.Physics of Fluids,2007,19(10):108101
27陶实,王亮,郭照立.微尺度振荡Couette流的格子Boltzmann模拟.物理学报,2014,63(21):239-247(Tao Shi,Wang Liang,Guo ZhaoLi.Lattice Boltzmann modeling of microscale oscillating Couette flow.Acta Physica Sinica,2014,63(21):239-247(in Chinese))
28姚军,赵建林,张敏等.基于格子Boltzmann方法的页岩气微观流动模拟.石油学报,2015,36(10):1280-1289(Yao Jun,Zhao Jianlin,Zhang Min,et al.Microscale shale gas flow simulation based on lattice Boltzmann method.Acta Petrolei Sinica,2015,36(10):1280-1289(in Chinese))
29 Zeng Y,Ning Z,Wang Q,et al.Gas transport in self-affine rough microchannels of shale gas reservoir.Journal of Petroleum Science and Engineering,2018,167:716-728
30 Cheng Z,Ning Z,Wang Q,et al.The effect of pore structure on nonDarcy flow in porous media using the lattice Boltzmann method.Journal of Petroleum Science and Engineering,2019,172:391-400
31 Guo WB,Wang NC,Shi BC,et al.Lattice-BGK simulation of a two-dimensional channel flow around a square cylinder.Chinese Physics,2003,12(1):67
32 Breuer M,Bernsdorf J,Zeiser T,et al.Accurate computations of the laminar flow past a square cylinder based on two different methods:lattice-Boltzmann and finite-volume.International Journal of Heat and Fluid Flow,2000,21(2):186-196
33 Mei R,Yu D,Shyy W,et al.Force evaluation in the lattice Boltzmann method involving curved geometry.Physical Review E,2002,65(4):041203
34 Lu J,Guo Z,Chai Z,et al.Numerical study on the tortuosity of porous media via lattice Boltzmann method.Communications in Computational Physics,2009,6(2):354-366
35 Chukwudozie C,Tyagi M,Sears S,et al.Prediction of non-Darcy coefficients for inertial flows through the castlegate sandstone using image-based modeling.Transport in Porous Media,2012,95(3):563-580
36 Guo Z,Zhao T.Lattice Boltzmann model for incompressible flows through porous media.Physical Review E,2002,66(3):036304
37 Forchheimer P.Wasserbewegung durch boden,Z.Ver.Deutsch,Ing.,1901,45:1782-1788
38 Ruth D,Ma H.On the derivation of the Forchheimer equation by means of the averaging theorem.Transport in Porous Media,1992,7(3):255-264
39 Zeng Z,Grigg R.A criterion for non-Darcy flow in porous media.Transport in Porous Media,2006,63(1):57-69
40 Andrade Jr J,Costa U,Almeida M,et al.Inertial effects on fluid flow through disordered porous media.Physical Review Letters,1999,82(26):5249