A Weakly Compressible SPH Method for the Brinkman Equation and Its Application to Estimating the Bulk Permeability of Digital Cores

The Brinkman equation provides a convenient continuum description of flow through digital cores containing resolved pores, unresolved porous voxels, and solids. Nevertheless, two issues are usually encountered in explicit solvers. The first is the instability in the discrete form of the viscous term when the porosity is violently changing. The second is the extremely small time-step restriction resulting from the stiff Darcy drag. This study addresses the two issues by developing a weakly compressible smoothed particle hydrodynamics (WCSPH) formulation in which the variable-porosity viscous term is expanded and its velocity-linear contribution is combined with the Darcy drag. The instability in the discrete formulation after this operator reorganization is greatly mitigated. For moderately stiff drag, the Strang splitting method with an exact integration of the drag is used to get rid of the drag restriction to the time step. Hence, the time step is considerably enlarged. For severely stiff drag, the particle is integrated exactly with the enlarged time step after neglecting the viscous contribution. A local selector based on the local Darcy number chooses between the two integration schemes. Constant-porosity channel tests show errors below 8% over the investigated parameter range, while variable-porosity channels, periodic porous obstacles, and two digital-core examples demonstrate numerical feasibility.