A characteristic-conservative model for Darcian advection

A numerical method based on the modi®ed method of characteristics is developed for incompressible Darcy ̄ow. Fluid elements modeled as grid cells are mapped back in time to their twisted forms and a strict equality of volumes is imposed between the two. These relations are then cast in terms of potentials using Darcy's law and a nonlinear algebraic problem is solved for potentials. Though a general technique for obtaining Darcy ̄ow, this method is most useful when the solute advection problem also is solved with the modi®ed method of characteristics. The combined technique (referred to as the characteristic-conservative method) using the same characteristics to obtain both velocities and concentrations is then a direct numerical approximation to the Reynolds transport theorem. The method is implemented in three dimensions and a few sample problems featuring nonuniform ̄ow-®elds are solved to demonstrate the exact mass conservation property. In ̄ow and out ̄ow boundaries do not cause any problems in the implementation. In all cases, the characteristic-conservative method obtains velocities that preserve ̄uid volume and, concentrations that achieve exact local and global mass balance; a desirable property that usually eludes characteristics based methods for solute advection in multidimensional, nonuniform ̄ow®elds. Ó 1999 Elsevier Science Limited. All rights reserved