Rapid simulation of the time-dependent diffusion coefficient in complex materials

Abstract

A finite-difference approach is presented for the analysis of the time-dependent diffusion coefficient for general heterogeneous materials that are either cavity-enclosed or periodic. In the bulk material, diffusivity and volume relaxivity are accounted for. The interaction of the diffusive medium with non-diffusive inclusions is modeled via a surface relaxivity. The time dependence is modeled using matrix exponentials that are shown to be efficiently evaluated using a Krylov-subspace approach. For a 3D model grid composed of M voxels of diffusive material (voxels containing non-diffusive material are not stored in the kernel matrix), the memory requirement is 15M and the computational time complexity for two large-scale example models is shown to be of order M1.39 and M1.10. Error estimate formulas are presented that can be used to guide the choice of domain grid resolution. Richardson extrapolation is shown to be effective in lowering simulation error. We apply this approach to modeling the nuclear magnetic resonance response of several subsurface rock pore geometries. They demonstrate the method to be simple and robust in both 2D and 3D complex geometries.