Numerical advective flux in highly variable velocity fields exemplified by saltwater intrusion

Numerical simulation of advective solute transport requires approximation of two vectors: the concentration gradient and the velocity. Estimation of the concentration gradient by Eulerian solution techniques gives rise to classical ‘numerical dispersionʹ since the truncation error is of the form 2C. Truncation of the velocity vector has escaped close scrutiny because the velocity field generally varies slowly in space in most groundwater environments. However, in many problems, including saltwater intrusion, the velocity field is highly variable and error associated with velocity approximation strongly affects the solution. Particle-tracking (Lagrangian) algorithms create a non-uniform error vector within each numerical block. The non-uniform error gives rise to differential advective flux that mimics the effects of classical numerical dispersion. Since the velocity truncation error vector depends on the size of a numerical block, Lagrangian methods may require extremely fine discretization of the underlying pressure grid. A finite-difference Lagrangian variable-density flow and transport code shows slow convergence to a ‘correctʹ solution of Henryʹs and related problems of saltwater intrusion as the grid density is changed. On the other hand, an Eulerian finite element code is shown to have a uniform velocity error vector within each element and the solution converges quickly as the grid density is changed. This suggests that the usual computational advantage that Lagrangian transport algorithms gain by using coarse grids is lost when modeling transport within highly variable velocity fields.