A parametric finite element solution of the generalised Bloch-Torrey equation for arbitrary domains

Nuclear magnetic resonance (NMR) has proven of enormous value in the investigation of porous media. Its use allows to study pore size distributions, tortuosity, and permeability as a function of the relaxation time, diffusivity, and flow. This information plays an important role in plenty of applications, ranging from oil industry to medical diagnosis. A complete NMR analysis involves the solution of the Bloch-Torrey (BT) equation. However, solving this equation analytically becomes intractable for all but the simplest geometries. We present an efficient numerical framework for solving the complete BT equation in arbitrarily complex domains. In addition to the standard BT equation, the generalised BT formulation takes into account the flow and relaxation terms, allowing a better representation of the phenomena under scope. The presented framework is flexible enough to deal parametrically with any order of convergence in the spatial domain. The major advantage of such approach is to allow both faster computations and sensitivity analyses over realistic geometries. Moreover, we developed a second-order implicit scheme for the temporal discretisation with similar computational demands as the existing explicit methods. This represents a huge step forward for obtaining reliable results with few iterations. Comparisons with analytical solutions and real data show the flexibility and accuracy of the proposed methodology.