Modeling multi-phase fluid flow in the subsurface is a notoriously difficult challenge. One must account for processes occurring on a broad range of scales; typically different modeling approaches are needed at different length scales so that the underlying physics can be properly described. At the pore scale (with grain size ranging from micron to milimeter, direct simulation of Stokes flow in a medium with rich geometry is extremely costly. Model reduction at the pore scale is normally done by mapping the pore space onto a representative network of idealized pores and throats and then modeling fluid displacements as discrete events on the pore-throat network. At larger scales (ranging from meters and up), Darcy's law based on permeability is the most common model. A challenge lies in adequately compressing the pore scale features into the permeability function used in Darcy's law.
In this talk, I present a new multiscale model and algorithm for computing pressure of flow in porous media. The algorithm has the form of the heterogeneous multiscale method (HMM), and couples network models with continuum scale PDE models defined on the same physical domain. The coupling method uses local simulations on sampled microscale domains to evaluate the continuum equation and
thus solve for the pressure in the full domain. Nonlinearity in the network model is allowed as well as the mass conservation equation. In the case where classical homogenization applies, one can prove convergence of the proposed multiscale solutions to the homogenized equations. I further present preliminary work on two phase flow.