Modelling of highly-heterogeneous media using a flux-vector-based Green element method

One of the modern techniques for solving nonlinear problems encountered in flow in porous media is the Green element method (GEM). It combines the high accuracy of the boundary element method with the efficiency and versatility of the finite element method. The high accuracy of the GEM comes from the direct representation of the normal fluxes as unknowns. However, in the classical GEM procedure the difficulties imposed by a large number of normal fluxes at each internal node are typically overcome by approximating them in terms of the primary variable, and this can lead to a diminution of the overall accuracy, particularly in applications to heterogeneous media. To maintain the high accuracy, another approach was proposed, namely the ‘flux-vector-based GEM’, or ‘q-based’ GEM. According to this approach, only two and three unknown flux components are required for a node in two- and three-dimensional domains, respectively. An important advantage of this approach is that the flux vector at each node is determined directly. We present a comparison between the results obtained using the classical GEM and those obtained using the ‘q-based’ GEM for problems in heterogeneous media with permeability changes of several orders of magnitude. In some simple examples of such situations, the classical GEM fails to produce physically sensible results, whilst our novel development of the ‘q-based’ GEM is in general agreement with both the boundary element method and the control volume method. However, in order to acquire results to the same degree of accuracy, these latter two approaches are less efficient than the ‘q-based’ GEM. This new approach is thus suitable for overcoming the difficulties present when modelling flow in heterogeneous media with rapid and high-order changes in material parameters. Within geological problems, we can therefore apply it to flow through layered sequences, across partially-sealing faults and around wells. The ‘q-based’ GEM approach presented here is developed and applied for rectangular grids, and we provide details of the extension to triangular finite-element-type meshes in two dimensions.