Large Scale Voxel-Based FEM Formulation for NMR Relaxation in Porous Media

Nuclear magnetic resonance (NMR) techniques are key in the study of porous reservoir rocks. They can provide valuable insight into the pore size distribution of the pore space of a given rock sample due to its dependence on the magnetic fluid/matrix interaction. The pore space is often studied at the μm scale through the use of micro-CT images, which are often composed of hundreds of millions of voxels, posing significant challenges to numerical simulations. In this paper, we present an image-based, fully explicit, and matrix-free finite element implementation for the simulation of NMR relaxation process that is capable of handling such large 3D problems in single GPUs. The chosen explicit time-integration scheme uses a lumped capacitance formulation and stabilization via hyperbolization, and it is capable of handling arbitrary time-step sizes with controllable error levels. The image-based representation of the pore space is used for a memory-efficient, matrix-free formulation of the time integration using massively parallel processes on a single GPU. In addition, we propose the substitution of a global digital roughness correction factor that depends on the porous space’s geometry for a problem-independent local correction factor, based on nodal neighborhoods. We show that the numerical scheme converges with successive refinements as expected and that our local correction coefficient is capable of estimating the correct S/V parameter of several different classical geometries. We tested our formulation against an image-based Random Walk simulation of four digital rock core samples, achieving good agreement between them. We manage to simulate a giga-voxel image-based model on a personal use GPU (less than 10GB of memory use) in 33 min with our FEM implementation.