An efficient finite element solution of the generalised Bloch-Torrey equation for arbitrary domains

Nuclear magnetic resonance (NMR) is an invaluable tool for investigating porous media. Its use allows to study pore size distributions, fiber tortuosity, and permeability as a function of the relaxation time, diffusivity, and flow. This information was shown to be important in many applications, such as medical diagnosis and materials science. 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 generalised BT equation. This method allows to obtain computational simulations of the NMR experiment 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. This framework is flexible enough to deal parametrically with any order of convergence in the spatial domain. 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 method.