Image-based multiscale formulation for pore-scale compressible Darcy-Stokes flow

Digital Rock Physics (DRP) has become a routine technology to understand fluid dynamics in rocks. A challenge with DRP is the so-called cut-off length, where features (e.g., pore size) below the resolution of a rock image cannot be resolved. Such subresolution porosity (i.e., microporosity) can be critical for fluid flow and transport because they could provide important flow pathways. This becomes even more challenging for organic-rich shale rocks, where the majority of the pores are below the resolution of microCT or FIB-SEM images. A so-called micro-continuum framework can be used to address the cut-off length issue. It applies to the entire domain a single momentum equation (i.e., Darcy-Brinkman-Stokes equation) that recovers Stokes equation in the macropore and Darcy equation in the microporous region. However, the DBS-based micro-continuum framework is computationally demanding, if not impossible, for high-resolution large images.

Here, we develop a multiscale method to speed up the pore-scale Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the void space and the microporous region into subdomains that either belong to the macropore or the microporous region, where we solve Stokes or Darcy problems locally to build basis functions. We then, by imposing mass flux continuity at the interfaces of the subdomains, superimpose the local bases to obtain an approximate global fine-scale solution. The approximate multiscale solution and the reference fine-scale single-domain solution agree very well (within a few percent). The multiscale solution can be further improved by an iterative correction scheme to converge to the fine-scale single-domain solution. The iterative correction is very efficient, and we show that one correction reduces the error by roughly an order of magnitude. Overall, the multiscale method is much more computationally efficient than the fine-scale single-domain solution while being accurate, which makes it attractive for pore-scale simulations on high-resolution large rock images with microporosity.