Inverse method to determine hydraulic conductivity from a velocity field using graph theory

A numerical inverse method called FlowPaths is presented to solve for the hydraulic conductivity field of an isotropic heterogeneous porous medium from a known specific discharge field (and constant-head boundary conditions). This method makes possible a new approach to reactive transport experiments, aimed at understanding the dynamic spatial and temporal evolution of hydraulic conductivity, which simultaneously record the evolving reaction and the evolving flow geometry. This inverse method assumes steady, two-dimensional flow through a square matrix of grid blocks. A graph-theoretical approach is used to find a set of flow paths through the porous medium using the known components of the specific discharge, where every vertex is traversed by at least one path from the upstream high-head boundary to the downstream low-head boundary. Darcy’s law is used to create an equation for the unknown head drop across each edge. Summation of these edge equations along each path through the network generates a set of linearly independent head-drop equations that is solved directly for the hydraulic conductivity field. FlowPaths is verified by generating 12,740 hydraulic conductivity fields of varying size and heterogeneity, calculating the corresponding specific discharge field for each, and then using that specific discharge field to estimate the underlying hydraulic conductivity field. When estimates from FlowPaths are compared to the simulated hydraulic conductivity fields, the inverse method is demonstrated to be accurate and numerically stable. Accordingly, within certain limitations, FlowPaths can be used in field or laboratory applications to find hydraulic conductivity from a known velocity field.