Assessing the error of geometry-based discretizations in groundwater modelling

The dominant numerical methods for solving partial differential equations, pertaining to groundwater problems, are the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Volume Method (FVM). All these methods rely on a discretization of the flow domain that is guided by the boundary conditions and the locations of interest (mea surements, pumps, etc). The disadvantages of these methods are that the discretization of the FDM is not very adaptable whereas the other two have quite complicated mathematics. Rozos and Koutsoyiannis (2010) suggested the use of a multi-cell modelling approach that discretizes the flow domain based on its geometry (i.e. the flow lines and equipotential lines). This concept is more or less equivalent to the flow-nets, which have been introduced since the beginning of 20th century by Philipp Forchheimer to calculate the leakages under dams (Ettema, 2006). The advantages of this approach are that the discretization can be ac complished using a small number of irregularly shaped cells and that this approach results in simple algebraic equations. This approach is called Finite Volume Method with Simplified In tegration (FVMSI) because it is a simplification of the FVM. In a FVMSI mesh, the cells' boundaries should be either equipotential or flow lines (1 st FVMSI condition). Consequently, all cells between two successive equipotential lines (a row of cells) should have similar simulated hydraulic heads and hence only minimal flux should take place between them (lateral flux). However, because of modelling errors, generally this will not be the case. If there are significant lateral fluxes, then the solution per se manifests an inconsis tency of the mesh. In other words, since the solution indicates significant flux between some cells of the same row, then these cells should have been arranged into different rows (i.e., the mesh design is flawed).