Comparison of Lattice Boltzmann and Finite Difference Methods in the Solution of Coupled Groundwater and Contaminant Transport Equations

Document Type : Research Article


1 M.Sc. Graduate of Water and Hydraulic Structures, Civil and Environmental Engineering Department, Shiraz University of Technology, Shiraz, Iran

2 - Assistant professor, Civil and Environmental Engineering Department, Shiraz University of Technology, Shiraz, Iran

3 Professor, Civil and Environmental Engineering Department, Shiraz University, Shiraz, Iran


Lattice Boltzmann method is a powerful numerical method for the simulation of fluid flow. There are two approaches for the description of groundwater flow in porous media. In the first approach, Navier-Stokes equations are used, while diffusion equation is applied to describe groundwater flow in the second approach. In this research, groundwater flow is modeled using the second approach. Furthermore, the governing equation of contaminant transport is called advection-dispersion equation. Here for the first time, the coupled solution of groundwater flow and contaminant transport using Lattice Boltzmann method is performed. The results indicated that Lattice Boltzmann method is capable of solving groundwater and contaminant transport equations simultaneously with high precision. Moreover, it was found that although the accuracy of Lattice Boltzmann and Crank-Nicolson method are the same, the speed of Lattice Boltzmann method is much higher than finite difference methods. In addition, Lattice Boltzmann method has higher range of stability and consistency in comparison with explicit finite difference method. Regarding this issue, grid Peclet number smaller than 7 is recommended for D1Q2 scheme of LBM.


مشهدگرمه، ن.، ولی‌سامانی، ج. م. و مظاهری، م. (1392). "حل تحلیلی معادله انتقال آلودگی به ازای الگوی زمانی دلخواه منابع آلاینده نقطه‌ای توسط روش تابع گرین"، مجله هیدرولیک، دوره 8، شماره 4، ص.ص. 13-25.
نصرتی، ک.، زهتابیان، غ. ر.، مرادی، ا. و شهبازی، ا. (1386). "ارزیابی روش شبیه‌سازی تصادفی برای تولید داده‌های هواشناسی"، پژوهش‌های جغرافیایی، شماره 62، ص.ص. 1-9.
Anwar, S. and Sukop, M.C. (2009a). ''Lattice Boltzmann models for flow and transport in saturated karst''. Ground Water. 47, pp. 401-413.
Anwar, S. and Sukop, M.C. (2009b). ''Regional scale transient groundwater flow modeling using Lattice Boltzmann methods''. Computers & Mathematics with Applications. 58, pp. 1015-1023.
Bin, D., Bao-Chang, S. and Guang-Chao, W. (2005). ''A new lattice Bhatnagar–Gross–Krook Model for the convection–diffusion equation with a source term''. Chinese Physics Letters. 22, pp. 267-270.
Budinski, L., Fabian, J. and Stipić, M. (2015). ''Lattice Boltzmann method for groundwater flow in non-orthogonal structured lattices''. Computers & Mathematics with Applications. 70. pp. 2601-2615.
Charbeneau Randall, J. (2000). Groundwater hydraulics and pollutant transport. Upper Saddle River. Prentice Hall, XIII.
Chen, S. and Doolen, G.D. (1998). ''Lattice Boltzmann method for fluid flows''. Annual Review of Fluid Mechanics. 30, pp. 329-364.
Gao, D., Chen, Z. and Chen, L. (2014). ''A thermal lattice Boltzmann model for natural convection in porous media under local thermal non-equilibrium conditions''. International Journal of Heat and Mass Transfer. 70, pp. 979-989.
Graille, B. (2014). ''Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method''. Journal of Computational Physics. 266, pp. 74-88.
Grucelski, A. and Pozorski, J. (2012). ''Lattice Boltzmann simulation of fluid flow in porous media of temperature-affected geometry''. Journal of Theoretical and Applied Mechanics. 50, pp. 193-214.
Guo, Z. and Zhao, T. (2002). ''Lattice Boltzmann model for incompressible flows through porous media''. Physical Review. E 66, 036304.
Guo, Z. and Zhao, T. (2005). ''A lattice Boltzmann model for convection heat transfer in porous media''. Numerical Heat Transfe. Part B 47, pp. 157-177.
Li, L., Mei, R. and Klausner, J.F. (2013). ''Multiple-relaxation-time lattice Boltzmann model for the axisymmetric convection diffusion equation''. International Journal of Heat and Mass Transfer. 67, pp. 338-351.
Li, Y. and Huang, P. (2008). ''A coupled lattice Boltzmann model for advection and anisotropic dispersion problem in shallow water''. Advances in Water Resources. 31, pp. 1719-1730.
Liu H, Zhou JG, Li M, and Zhao, Y. (2013) ''Multi-block lattice Boltzmann simulations of solute transport in shallow water flows''. Advances in Water Resources. 58, pp. 24-40.
Liu, Q., He, Y.L., Li, Q. and Tao W.Q. (2014). ''A multiple-relaxation-time lattice Boltzmann model for convection heat transfer in porous media''. International Journal of Heat and Mass transfer. 7, pp. 761-775.
Mohamad, A. and Kuzmin, A. (2012). ''The soret effect with the D1Q2 and D2Q4 lattice Boltzmann model''. International Journal of Nonlinear Sciences and Numerical Simulation. 13. pp. 289-293.
Mohamad A., Tao Q., He. Y., and Bawazeer, S. (2015) Treatment of transport at the interface between multilayers via the lattice Boltzmann method. Numerical Heat Transfer, Part B: Fundamentals 67, pp. 124-134.
Mohamad, A.A. (2011). Lattice Boltzmann method: fundamentals and engineering applications with computer codes. Springer Science & Business Media.
Perko, J. and Patel, R.A. (2014). ''Single-relaxation-time lattice Boltzmann scheme for advection-diffusion problems with large diffusion-coefficient heterogeneities and high-advection transport''. Physical Review E 89, pp. 1-9.
Rong, F., Guo, Z., Chai, Z., et al. (2010). ''A lattice Boltzmann model for axisymmetric thermal flows through porous media''. International Journal of Heat and Mass Transfer. 53, pp. 5519-5527.
Seta, T., Takegoshi, E. and Okui K. (2006). ''Lattice Boltzmann simulation of natural convection in porous media''. Mathematics and Computers in Simulation. 72, pp. 195-200.
Wang HF. and Anderson MP. (1982). Introduction to groundwater modeling: finite difference and finite element methods. Academic Press.
Zheng C and Bennett GD. (2002). Applied contaminant transport modeling. Wiley-Interscience, New York.
Zhou JG. (2007a) A lattice Boltzmann model for groundwater flows. International Journal of Modern Physics C 18: 973-991.
Zhou JG. (2007b) A rectangular lattice Boltzmann method for groundwater flows. Modern Physics Letters B 21: 531-542.
Zhou JG. (2009) A lattice Boltzmann method for solute transport. International Journal for Numerical Methods in Fluids 61: 848-863.