Determination of the diffusivity, dispersion, skewness and kurtosis in heterogeneous porous flow. Part II: Lattice Boltzmann schemes with implicit interface

Abstract

Abstract A simple local two-relaxation-time Lattice Boltzmann numerical formulation (TRT-EMM) of the extended method of moments (EMM) is proposed for analysis of the spatial and temporal Taylor dispersion in d-dimensional streamwise-periodic stationary mesoscopic velocity field resolved in a piecewise-continuous porous media. The method provides an effective diffusivity, dispersion, skewness and kurtosis of the mean concentration profile and residence time distribution. The TRT-EMM solves a chain of steady-state heterogeneous advection–diffusion equations with the pre-computed space-variable mass-source and automatically undergoes diffusion-flux jump on the abrupt-porosity streamwise-normal interface. The temporal and spatial systems of moments are computed within the same run; the symmetric dispersion tensor can be restored from independent computations performed for each periodic mean-velocity axis; the numerical algorithm recursively extends for any order moment. We derive an exact form of the bulk equation and implicit closure relations, construct symbolic TRT-EMM solutions and determine specific relation between the equilibrium and the collision degrees of freedom viewing an exact parameterization by the physical non-dimensional numbers in two alternate situations: “parallel” fracture/matrix flow and “perpendicular” Darcy flow through porous blocks in “series”. Two-dimensional simulations in linear Brinkman flow around solid and through porous obstacles validate the method in comparison with the two heterogeneous direct LBM-ADE schemes with different anti-numerical-diffusion treatment which are proposed and examined in parallel. On the coarse grid, accuracy of the three moments is essentially determined by the free-tunable collision rate in all schemes, and especially TRT-EMM. However, operated within a single periodic cell, the TRT-EMM is many orders of magnitude faster than the direct solvers, numerical-diffusion free, more robust and much more capable for accuracy improving, high Peclet range and free-parameter influence reduction with the mesh refinement. The method is an efficient predicting tool for the Taylor dispersion, asymmetry and peakedness; moreover, it allows for an optimal analysis between the mutual effect of the flow regime, Peclet number, porosity, permeability and obstruction geometry.