This command solves the Laplace equation (135) in the pore space with Neumann boundary conditions on the pore walls, and a concentration drop in one of the space directions. The equation is solved three times, each time with a concentration gradient in a different space direction, to determine the full relative diffusivity tensor . Recall (see the DiffuDict theoretical background) that is a dimensionless quantity and a property of the porous media alone that, as such, is independent of the diffusing species and surrounding fluid.

Two finite volume-based solvers are available to solve this equation: EJ (Wiegmann and Zemitis) and LIR (Linden et al). Both solvers use harmonic averaging to compute conductivities at voxel faces.

The EJ solver introduces explicit jump variables across material interfaces that represent discontinuities of concentration derivatives. A Schur-complement formulation for the jump variables is derived and solved by using the FFT and BiCGStab methods. The convergence speed of this method is almost independent of the diffusivity contrast which is a very big advantage compared to other approaches, but it depends on the number of material interfaces.

The LIR solver uses an adaptive grid (instead of a regular grid) to reduce the number of grid cells significantly. The adaptive grid basis is a data structure called LIR-tree that is used for spatial partitioning of 3D images. The materials are represented as differently sized rectangular cuboid. The LIR solver is very fast, but the convergence speed depends linearly on the diffusivity contrasts and on the number of material interfaces.