Multiscale Computation of Compressible Pore-Scale Darcy-Stokes Flow
In the past decade, “digital rock physics”―using high-resolution pore-structure images of rock samples as input for flow simulations—has been used to understand the fluid dynamics in rocks. Flow simulations on digital rock images is challenging due to the cut-off length issue—interstitial voids below the resolution of the imaging instrument (i.e., microporosity) cannot be resolved. A micro-continuum framework can be used to address this problem, which applies to the entire domain the Darcy-Brinkman-Stokes equation that recovers the Stokes equation in the resolved macropores and the Darcy equation in the microporous regions. 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 local correction problems. 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 3D high-resolution digital images of porous materials.