A P-VERSION FINITE-ELEMENT FORMULATION FOR MODELING MAGNETIC-RESONANCE RELAXATION IN POROUS-MEDIA

A new finite element (FE) technique for simulating nuclear magnetic resonance (NMR) relaxation of fluids in porous media has been developed to investigate the effects of pore geometry on NMR response of fluids in reservoir rocks that are not studied readily by other methods. The method uses a polynomial, or ''p-version'', FE formulation derived using the classical variational principle with the hierarchical approximation concept and the Crank-Nicolson time-marching scheme, to solve the governing differential equations for magnetization decay in pores of arbitrary shape. Our p-FE simulation program is applicable to two-dimensional and axisymmetric three-dimensional pore geometries. However, extension of the formulation to a full three-dimensional simulator is straightforward. To validate the model, we compare numerical results to analytical solutions for magnetization decay in planar, cylindrical, and spherical pore geometries under various diffusion conditions in both the time and frequency domains. These tests also demonstrate stability and rapid convergence of the numerical solutions. Because it is based on a continuum representation of the physics of NMR relaxation, the FE program can be used to model local, that is pore-level, variations in the magnetization field, as well as relaxation of the total magnetization for the entire pore network. The capability of yielding a detailed map of the magnetization field at the pore-level could be a powerful tool for investigating some poorly understood controls on pore fluid relaxation, such as rough pore surfaces, interpore diffusion, and spatial variability in NMR surface relaxation rates.