Convergence Behavior of Lattice Boltzmann Method for Pore-scale Modeling

The characterization of reservoir rocks depends on the absolute permeability as a crucial parameter. To estimate this property numerically, one can employ a combination of digital rocks and Stokes flow simulation through the Lattice Boltzmann Method (LBM). In previous studies, the LBM has typically been implemented as an iterative process, wherein iterations are repeated until the parameter estimates between consecutive iterations reach a certain threshold level. However, we argue that this termination criterion is unsuitable and may compromise the accuracy of simulation results.

In this study, we investigate the convergence of the LBM through various tests, including the Poiseuille flow between parallel plates and different types of digital rocks (such as dune sand, sandstone, and carbonates). We find that the logarithm of the relative error, when compared to the estimate at infinite time (representing a stable state), demonstrates a linear relationship with the number of iterations. This linear relationship suggests an exponential rate of convergence. On the other hand, if we rely on the difference in errors between consecutive iterations as the termination condition, the simulation may not reach a stable state.

Instead, we propose a more accurate termination criterion for the LBM simulation by analyzing the decay trend of the error difference. This criterion provides a practical and appropriate approach for the characterization of reservoir rocks. Additionally, we offer a theoretical explanation for the convergence rate, which is linked to the spectral radius of the iterative matrix in linear algebra.