Multiscale formulation of pore-scale compressible Darcy-Stokes flow

Direct numerical simulation (DNS) of fluid dynamics in digital images of porous materials is challenging due to the cut-off length issue where interstitial voids below the resolution of the imaging instrument cannot be resolved. Such subresolution microporosity can be critical for flow and transport because they could provide important flow pathways. A micro-continuum framework can be used to address this problem, which applies to the entire domain a single momentum equation, i.e., Darcy-Brinkman-Stokes (DBS) equation, that recovers Stokes equation in the resolved void space (i.e., macropores) and Darcy equation in the microporous regions. However, the DBS-based micro-continuum framework is computationally demanding. Here, we develop an efficient multiscale method for the compressible Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the domain into subdomains that either belong to the macropores or the microporous regions, on which Stokes or Darcy problems are solved locally, only once, to build basis functions. The nonlinearity from compressible flow is accounted for in a local correction problem on each subdomain. A global interface problem is solved to couple the local bases and correction functions to obtain an approximate global multiscale solution, which is in excellent agreement with the reference single-scale solution. The multiscale solution can be improved through an iterative strategy that guarantees convergence to the single-scale solution. The method is computationally efficient and well-suited for parallelization to simulate fluid dynamics in large high-resolution digital images of porous materials.