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A numerical method is formulated for the solution of the advectiveCahn-Hilliard (CH) equation with constant and degenerate mobility inthree-dimensional porous media with non-vanishing velocity on the exteriorboundary. The CH equation describes phase separation of an immiscible binarymixture at constant temperature in the presence of a mass constraint anddissipation of free energy. Porous media/pore-scale problems specificallyentail high-resolution images of rocks in which the solid matrix and porespaces are fully resolved. The interior penalty discontinuous Galerkin methodis used for the spatial discretization of the CH equation in mixed form, whilea semi-implicit convex-concave splitting is utilized for temporaldiscretization. The spatial approximation order is arbitrary, while it reducesto a finite volume scheme for the choice of elementwise constants. Theresulting nonlinear systems of equations are reduced using the Schur complementand solved via Newton's method. The numerical scheme is first validated usingnumerical convergence tests and then applied to a number of fundamentalproblems for validation and numerical experimentation purposes including thecase of degenerate mobility. First-order physical applicability and robustnessof the numerical method are shown in a breakthrough scenario on a voxel setobtained from a micro-CT scan of a real sandstone rock sample.

A finite volume/discontinuous Galerkin method for the advective  Cahn-Hilliard equation with degenerate mobility on porous domains stemming  from micro-CT imaging