A finite-difference approach was presented for the analysis of the time-dependent diffusion coefficient for general heterogeneous materials that were either cavity-enclosed or periodic. In the bulk material, diffusivity and volume relaxation were accounted for. The interaction of the diffusive medium with non-diffusive inclusions was modelled via a surface relaxation. The time dependence was modelled using matrix exponentials that were shown to be efficiently evaluated using a Krylov-subspace approach. For a three-dimensional model grid composed of M voxels of diffusive material (voxels containing non-diffusive material were not stored in the kernel matrix), the memory requirement was 15M and the computational time complexity for two large-scale example models was shown to be of order M1.39 and M1.10. Error estimate formulas were presented that could be used to guide the choice of domain grid resolution. Richardson extrapolation was shown to be effective in lowering simulation error. This approach was applied to modelling the nuclear magnetic resonance response of several sub-surface rock pore geometries. This demonstrated the method to be simple and robust in both two-dimensional and three-dimensional complex geometries.

Rapid Simulation of the Time-Dependent Diffusion Coefficient in Complex Materials. M.D.Prange, V.Druskin, D.L.Johnson, L.M.Schwartz: Journal of Physics A, 2011, 44[39], 395203