Analysis and improvement of Brinkman lattice Boltzmann schemes: Bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media

Abstract

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

Stokes‐Brinkman Flow Simulation Based on 3‐D μ‐CT Images of Porous Rock Using Grayscale Pore Voxel Permeability

“…Two positive eigenvalue functions Λ ± are given as where v e is the effective kinematic viscosity (= μ e / ρ ) expressed by ν / f ( ϕ v ), B is f ( ϕ v )/ k v , Λ is a magic parameter in the standard TRT model, and f ( ϕ v ) is a function of voxel porosity and is set to unity semiheuristically (Brinkman, ; Ginzburg et al, ). As the voxel porosity ϕ v approaches 1 (i.e., as it becomes an apparent pore voxel) and the voxel permeability becomes infinite (∞), Λ + and the viscous friction force become 3 ν e and 0 , respectively.…”

“…Λ is a magic parameter in the standard TRT model, and f (ϕ v ) is a function of voxel porosity and is set to unity semiheuristically (Brinkman, 1949;Ginzburg et al, 2015). As the voxel porosity ϕ v approaches 1 (i.e., as it becomes an apparent pore voxel) and the voxel permeability becomes infinite (∞), Λ + and the viscous friction force become 3ν e and 0, respectively.…”

Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion

“…15 is that err U becomes positive. Our further investigations have shown that by extending interface flow analysis [29] to boundary layers, it becomes possible to analytically predict H -dependent solution for boundary value b (versus bulk value v ) which completely eliminates the bounce-back velocity error in Poiseuille profile. However, a similar solution in plug flow makes b negative.…”

Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

“…Although this value may not be necessarily the optimal choice , it still does not take generality from the analysis. For further information, we refer to works (Ginzburg 2008;Ginzburg et al 2015), while details on the selection of Λ in this class of flow problems are discussed in a parallel work .…”