Porous Weight Function#

Porous Weight Function = {GALERKIN | SUPG} <float>

Description / Usage#

This required card is used to specify the weight function form on the capacitance term of the Darcy flow equations for partially saturated flow (viz. for Media Type specifications of POROUS_PART_SAT and POROUS_UNSAT, and POROUS_TWO_PHASE.) The standard approach is to use a Galerkin formulation, but often times the SUPG option allows for a more stable time integration algorithm using the classic Streamwise Upwinding Petrov Galerkin weight function (see references below). The model options for this card are as follows:


Name of the weight function formulation. This option requests a standard Galerkin finite element weighted residual treatment. A parameter is required, viz. <float>, but it is not used by Goma; it should be set to zero.

  • <float> - 0.0


Name of the weight function formulation. This option requests a streamwise upwinding Petrov-Galerkin formulation. A floating point parameter is required as a SUPG weighting parameter and it should be set between 0.0 (for no upwinding) and 1.0 (for full upwinding).

  • <float> - a SUPG weighting parameter

The default model if this card is missing is GALERKIN.


An example card

Porous Weight Function = SUPG 1.0

Technical Discussion#

As mentioned above, this card is used to invoke a streamwise upwinding scheme for purposes of stabilizing the solution around steep saturation fronts. Galerkin finite element treatment is often an extremely inaccurate discretization for propagating a discontinuity, such as is the case around these fronts, and often has to be supplemented with streamwise diffusion and/or mass lumping so that the saturation variable remains monotonic and well behaved, viz. to keep it from going below zero. Another expedient to aid in keeping the front smooth and monotonic is to use mass lumping (cf. Mass Lumping card).


GTM-029.0: SUPG Formulation for the Porous Flow Equations in Goma, H. K. Moffat, August 2001 (DRAFT).

Bradford, S. F. and N. D. Katopodes, “The anti-dissipative, non-monotone behavior of Petrov-Galerkin Upwinding,” International J. for Numerical Methods in Fluids, v. 33, 583-608 (2000).

Brooks, A. N. and T. J. R. Hughes, “Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations,” Comp. Math. In Appl. Mechanics and Eng., 32, 199 - 259 (1992).

Gundersen, E. and H. P. Langtangen, “Finite Element Methods for Two-Phase Flow in Heterogeneous Porous Media,” in Numerical Methods and Software Tools in Industrial Mathematics, Morten Daehlen, Aslak Tveito, Eds., Birkhauser, Boston, 1997.

Helmig, R. and R. Huber, “Comparison of Galerkin-type discretization techniques for two-phase flow in heterogeneous porous media,” Advances in Water Resources, 21, 697-711 (1998).

Unger, A. J. A., P. A. Forsyth and E. A. Sudicky, “Variable spatial and temporal weighting schemes for use in multi-phase compositional problems,” Advances in Water Resources, 19, 1 - 27 (1996).