A multiscale method coupling network and continuum models in porous media II – single- and two-phase flows

Modeling and computing transport in the subsurface is a difficult problem that requires good understanding of the relations among different processes at various different length and time scales, and their effective properties. Already at the pore scale, ranging from a few micrometers to millimeters, direct flow simulation in a detailed medium geometry assuming Stokes flow is extremely costly. Network modeling [32, 6, 36] provides an alternative for approximating flows at the pore scale with reduced complexity; in a network model the complicated geometry related to the medium is mapped onto a representative network of idealized pores, throats, and cracks. The fluid displacements within a physical region are then modeled as discrete events that take place in the corresponding pore-throat network. At larger length scales, one typically constructs continuum PDE models involving Darcy’s law. The coefficients in these models are typically derived from grid blocks that contain sufficiently many pores so that the average fluid quantities within evolve smoothly with time. However, the specific microscopic structure of the pore space frequently plays a critical role in determining macroscopic flow properties, and often cannot be ignored. Continuum models accounting for two scales – the so-called