Localized lattice Boltzmann equation model for simulating miscible viscous displacement in porous media

Abstract A localized lattice Boltzmann equation (LBE) model for simulating the miscible viscous displacement in porous media is proposed. The Darcy’s law for flow and the convection–diffusion equation (CDE) describing the transport of solute are solved numerically by the present model. To ensure the local implementation of the collision process in this model, the pressure and concentration gradients are computed from the moments of the nonequilibrium distribution functions, which are of second-order accuracy. Consequently, the advantages of the lattice Boltzmann method (LBM) are retained. The model is validated with a stable displacement problem, and is employed to study the viscous fingering instability that occurs in the process of the miscible viscous displacement. The results agree well with previous studies. Furthermore, although the present model is an explicit scheme, it is interesting to find that it is capable of simulating the viscous displacement over a wide range of Peclet (Pe) numbers, indicating the superior stability of this model.

[1]  Li-Shi Luo,et al.  Accuracy of the viscous stress in the lattice Boltzmann equation with simple boundary conditions. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Y. Nagatsu,et al.  Investigation of reacting flow fields in miscible viscous fingering by a novel experimental method , 2009 .

[3]  Albert J. Valocchi,et al.  Pore-scale simulation of liquid CO2 displacement of water using a two-phase lattice Boltzmann model , 2014 .

[4]  He Zhenmin,et al.  EMC effect on p-A high energy collisions , 1991 .

[5]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  Qinjun Kang,et al.  Immiscible displacement in a channel: simulations of fingering in two dimensions , 2004 .

[7]  George M. Homsy,et al.  Viscous fingering in porous media , 1987 .

[8]  J. Azaiez,et al.  Fully implicit finite difference pseudo‐spectral method for simulating high mobility‐ratio miscible displacements , 2005 .

[9]  Eckart Meiburg,et al.  High-Accuracy Implicit Finite-Difference Simulations of Homogeneous and Heterogeneous Miscible-Porous-Medium Flows , 2000 .

[10]  Dominique Salin,et al.  Simulations of viscous flows of complex fluids with a Bhatnagar, Gross, and Krook lattice gas , 1996 .

[11]  Anke Lindner,et al.  Viscous fingering in a shear-thinning fluid , 2000 .

[12]  Jonathan Chin,et al.  Lattice Boltzmann simulation of the flow of binary immiscible fluids with different viscosities using the Shan-Chen microscopic interaction model , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  R. Juanes,et al.  Fluid mixing from viscous fingering. , 2010, Physical review letters.

[14]  Albert J. Valocchi,et al.  Lattice Boltzmann simulation of immiscible fluid displacement in porous media: homogeneous versus heterogeneous pore network , 2015 .

[15]  E. Koval,et al.  A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media , 1963 .

[16]  Yongchen Song,et al.  Lattice Boltzmann simulation of viscous fingering phenomenon of immiscible fluids displacement in a channel , 2010 .

[17]  Albert J. Valocchi,et al.  Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method , 2013, Transport in Porous Media.

[18]  Takaji Inamuro,et al.  A lattice kinetic scheme for incompressible viscous flows with heat transfer , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  George M. Homsy,et al.  Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations , 1997 .

[20]  George M. Homsy,et al.  Simulation of nonlinear viscous fingering in miscible displacement , 1988 .

[21]  M. R. Todd,et al.  Development, testing and application of a numerical simulator for predicting miscible flood performance , 1972 .

[22]  D. Salin,et al.  Miscible displacement between two parallel plates: BGK lattice gas simulations , 1997, Journal of Fluid Mechanics.

[23]  Zhenhua Chai,et al.  A novel lattice Boltzmann model for the Poisson equation , 2008 .

[24]  D. Bonn,et al.  Viscous fingering in non-Newtonian fluids , 2002, Journal of Fluid Mechanics.

[25]  S. Hill,et al.  Channeling in packed columns , 1952 .

[26]  Finite element simulation of viscous fingering in miscible displacements at high mobility-ratios , 2010 .

[27]  R. Juanes,et al.  Quantifying mixing in viscously unstable porous media flows. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Mukul M. Sharma,et al.  Experimental study of the growth of mixing zone in miscible viscous fingering , 2015 .

[29]  B. Shi,et al.  Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method , 2002 .

[30]  Christie,et al.  Detailed Simulation of Unstable Processes in Miscible Flooding , 1987 .

[31]  Zhaoli Guo,et al.  Multiple-relaxation-time lattice Boltzmann model for incompressible miscible flow with large viscosity ratio and high Péclet number. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Dominique Salin,et al.  Asymptotic regimes in unstable miscible displacements in random porous media , 2002 .

[33]  F. J. Fayers,et al.  Detailed validation of an empirical model for viscous fingering with gravity effects , 1988 .

[34]  Edo S. Boek,et al.  Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries , 2010, Comput. Math. Appl..

[35]  Z. Chai,et al.  Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Chen,et al.  Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  Eckart Meiburg,et al.  Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion , 1999 .

[38]  Alvaro L. G. A. Coutinho,et al.  Stabilized finite element methods with reduced integration techniques for miscible displacements in porous media , 2004 .