Macroscale lattice‐Boltzmann methods for low Peclet number solute and heat transport in heterogeneous porous media

Abstract

This paper introduces new methods for simulating subsurface solute and heat transport in heterogeneous media using large‐scale lattice‐Boltzmann models capable of representing both macroscopically averaged porous media and open channel flows. Previous examples of macroscopically averaged lattice‐Boltzmann models for solute and heat transport are only applicable to homogeneous media. Here, we extend these models to properly account for heterogeneous pore‐space distributions. For simplicity, in the majority of this paper we assume low Peclet number flows with an isotropic dispersion tensor. Nevertheless, this approach may also be extended to include anisotropic‐dispersion by using multiple relaxation time lattice‐Boltzmann methods. We describe two methods for introducing heterogeneity into macroscopically averaged lattice‐Boltzmann models. The first model delivers the desired behavior by introducing an additional time‐derivative term to the collision rule; the second model by separately weighting symmetric and anti‐symmetric components of the fluid packet densities. Chapman‐Enskog expansions are conducted on the governing equations of the two models, demonstrating that the correct constitutive behavior is obtained in both cases. In addition, methods for improving model stability at low porosities are also discussed: (1) an implicit formulation of the model; and (2) a local transformation that normalizes the lattice‐Boltzmann model by the local porosity. The model performances are evaluated through comparisons of simulated results with analytical solutions for one‐ and two‐dimensional flows, and by comparing model predictions to finite element simulations of advection isotropic‐dispersion in heterogeneous porous media. We conclude by presenting an example application, demonstrating the ability of the new models to couple with simulations of reactive flow and changing flow geometry: a simulation of groundwater flow through a carbonate system.