Convergence analysis of Lattice Boltzmann method for Stokes flow in digital rock characterization

Abstract

The absolute permeability is a key parameter for characterization of reservoir rocks. It can be estimated numerically by combining digital rocks and Stokes flow simulation using Lattice Boltzmann Method (LBM). In previous research, the LBM is usually implemented as an iterative process, which continues until the changes in estimates of the parameters between two consecutive iterations reach below a certain threshold level. We remark that this termination criterion is not appropriate and may lead to low accuracy in the simulation results. Here we analyze the convergence of LBM in various tests including the Poiseuille flow between two parallel plates, and several types of digital rocks (dune sand, sandstone and carbonates). The logarithm of relative error of each estimate compared with the estimate at infinite time (as stable state) is found to have a linear relation with regard to the number of iterations, indicating an exponential rate of convergence. If the difference of errors between two consecutive iterations is used as the termination condition, the result could be far from a stable state. Instead, we recommend using the decay trend of the error difference in the LBM simulation to provide an accurate termination criterion, which serves as a practical and more adequate way for characterization of reservoir rocks. We also provided the theoretical explanation of convergence rate, which is related to the spectral radius of iterative matrix in linear algebra.