Lattice Boltzmann and analytical modeling of flow processes in anisotropic and heterogeneous stratified aquifers

Abstract We present analytical and Lattice Boltzmann (LB) solutions for steady-state saturated flows in 2D and 3D anisotropic heterogeneous aquifers. The analytical solution is easy to use and extends the known ones for ground-water whirls to more general combinations of the anisotropic properties of two-layered systems. The Bakker and Hemker’s “multi-layered” semi-analytical solution and the LB results are compared to the analytical solution for a broad range of anisotropic heterogeneous diffusion tensors. The main components of the LB scheme, the eigenvalues of the linear collision operator and/or the equilibrium functions, become discontinuous when the anisotropy changes between the layers. It is shown that the evolution equation of the LB method needs to be modified at the interfaces in order to satisfy the continuity conditions for the diffusion function and/or its tangential derivatives. The existing LB schemes for anisotropic advection–dispersion equations are formulated in a more general framework in which the leading-order interface corrections are constructed and analyzed for linear and highly nonlinear exact solutions. We also present some stability aspects of these schemes, introduce specified normal gradient boundary conditions and discuss the computation of total and local fluxes. The interface analysis developed here applies to generic LB schemes with discontinuous collision operators.

[1]  C. Pan,et al.  Lattice‐Boltzmann simulation of two‐phase flow in porous media , 2004 .

[2]  Cass T. Miller,et al.  An evaluation of lattice Boltzmann schemes for porous medium flow simulation , 2006 .

[3]  Irina Ginzburg,et al.  Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations , 2007 .

[4]  P. Lallemand,et al.  Momentum transfer of a Boltzmann-lattice fluid with boundaries , 2001 .

[5]  John P. Dempsey,et al.  On the numerical evaluation of Legendre's chi-function , 1991 .

[6]  David F. Griffiths,et al.  The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation , 1984 .

[7]  D. d'Humières,et al.  Multiple–relaxation–time lattice Boltzmann models in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  M. Bakker,et al.  Analytic solutions for groundwater whirls in box-shaped, layered anisotropic aquifers , 2004 .

[9]  W. G. Gray,et al.  Computational methods in water resources X , 1994 .

[10]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  John W. Crawford,et al.  A novel three‐dimensional lattice Boltzmann model for solute transport in variably saturated porous media , 2002 .

[12]  D. d'Humières,et al.  Thirteen-velocity three-dimensional lattice Boltzmann model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Irina Ginzburg,et al.  Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations , 2005 .

[14]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[15]  J. Jiménez,et al.  Boltzmann Approach to Lattice Gas Simulations , 1989 .

[16]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .

[17]  Dominique d'Humières,et al.  Multireflection boundary conditions for lattice Boltzmann models. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Konrad Steiner,et al.  Lattice Boltzmann model for free-surface flow and its application to filling process in casting , 2003 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  C. J. Hemker,et al.  Steady groundwater flow in leaky multiple-aquifer systems , 1984 .

[21]  Hans-Jörg Vogel,et al.  Comparison of a Lattice‐Boltzmann Model, a Full‐Morphology Model, and a Pore Network Model for Determining Capillary Pressure–Saturation Relationships , 2005 .

[22]  John W. Crawford,et al.  A lattice BGK model for advection and anisotropic dispersion equation , 2002 .

[23]  Louis Theodore,et al.  Developments In Water Science , 1980 .

[24]  Irina Ginzburg,et al.  Lattice Boltzmann approach to Richards' equation , 2004 .

[25]  U. Rüde,et al.  Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming , 2005 .

[26]  M. Bakker,et al.  Ground Water Whirls , 2004, Ground water.

[27]  G. Bruggeman.  Analytical solutions of geohydrological problems , 1999 .

[28]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[29]  Irina Ginzburg,et al.  Variably saturated flow described with the anisotropic Lattice Boltzmann methods , 2006 .