On boundary conditions and point sources in the finite element integration of the transport equation

In the finite element integration of the convection‐dispersion equation over a groundwater domain, boundary conditions of the third (or Cauchy) type may or may not be incorporated with the variational statement. While the former development (formulation A) requires the application of Green's first identity to both the convective and the dispersive terms, the latter (formulation B) does not. Although both approaches are equivalent at the theoretical level, they behave differently from a computational standpoint. The numerical solution proves more accurate and stable if Green's lemma is not applied to the convective term, since significant errors arising from the representation of the Darcy velocity over the elements are avoided. Point sources or sinks representing injection or pumping wells can be treated as either sources (sinks) distributed over a small volume or external boundaries with prescribed contaminant flux. It is shown that when the well bore is excluded from the flow field, formulation A yields the same matrix addition for an injection (abstraction) well as formulation B does for an abstraction (injection) well (reciprocity of the results). The computational behaviors of these alternative modeling procedures are illustrated with a number of significant numerical examples.