Enforcing global constraints for the dispersion closure problem: τ2-SIMPLE algorithm

Permeability and effective dispersion tensors are critical parameters to characterize flow and transport in porous media at the continuum scale. Homogenization theory defines a framework in which such effective properties are first computed from solving a closure problem in a repeating unit cell of the periodic microstructure and then used in a macroscopic formulation for efficient computation. The closure problem is formulated as a local boundary value problem subjected to global constraints, which guarantee the uniqueness of the solution and can be difficult to satisfy for complex geometries and at high flow conditions. These constraints also ensure that pore-scale pressure, velocity, and concentration fields can be accurately reconstructed from the closure variable. Building on previous work, here we present a framework that allows to satisfy global constraints associated to both the permeability and the dispersion closure problems by introducing two artificial time scales. The algorithm, called ${\tau }^2$-SIMPLE, computes both permeability and effective dispersion given an arbitrarily complex geometry and flow condition. This algorithm is demonstrated to be accurate for both 2D and 3D geometries across varying flow conditions, and thus it can be used to quickly characterize effective properties from porous media images in many applications.