PROBLEM DESCRIPTION
Groundwater flow and especially transport simulations show in some cases numerical artefacts, oscillations or other types of numerical dispersion. These phenomena occur due to poor mesh discretization, inaccurate parameterization, conflicting boundary conditions, inappropriate solver settings or a combination of those. As a result, concentration or temperature distributions show unrealistic values often with positive and negative extreme values adjacent to each other (oscillations).
If FEFLOW users are confronted with such effects limiting reliable results or model convergence, the following actions can be undertaken to better control the numerical performance without re-setting up the entire model.
SOLUTION
Step #1: Plot flow and/or transport error norms
Numerical artefacts or oscillations typically show effects at locations close to their origin. Sometimes, however, non-convergence may have its cause in locations not being obvious to observe from the process variables. In such cases, the error-norm is a useful parameter to identify critical parameterizations or mesh inconsistencies. The error-norm distribution displays the normalized error, which is defined as:
with e being the normalized error, emean the averaged absolute error and ψ represents the maximum occurring value of the variable (head, concentration, temperature).
This parameter can be added from the Data Panel. Open the context menu of /User Data and select Add Predefined Distribution. Here you find all available error-norms for flow, age, mass and/or heat, depending on the pre-defined problem class.
Please note that error-norms must be added before the start of the simulation and show values only as long as the simulation is running or in the DAC file.
Step #2: Local mesh refinement
A common reason for numerical oscillations is a coarse mesh discretization. In Fig. 1 you see a simple 2D mass transport model with a constant contaminant source of 100 mg/l in the left part of the model and two production wells (extraction 500 m³/d) further East. The flow direction is from West to East.
The mesh consists of quite regular triangles with an average element size of 80 m. On the first picture (top left in Fig. 1) you can see no contamination plume as it is cross-faded by extreme values in the areas indicated with red circles.
In the second picture (top right in Fig. 1), the mesh was locally refined in the area where we would expect the contaminant plume to evolve. The resulting element length of approximately 40 m is still too large to support stable numerical calculations. Please note the extreme values of mass concentration which are beyond a reasonable range on both ends.
Fig. 1 - Simulation results of a 2D mass transport model featuring the numerical effects of different mesh discretizations. From top left to bottom right the spatial resolution of the mesh increases while the numerical defects are reduced.
In the third picture (bottom left in Fig. 1), further refinement yields an average element length of 15 m. Now, the contaminant plume is visible with the range of concentration values being more realistic. Close to the source area negative concentrations still exist and an artefact close to the southern boundary is present.
With further refinement of the mesh the results improve. The last picture (bottom right in Fig. 1) features a solid delineation of the plume with no visible numerical artefacts or oscillations.
Step #3: Increase the dispersivities
Besides the spatial resolution of the mesh, the dispersivity of the porous medium is an important factor for numerical stability. Longitudinal and transverse dispersivities can be defined as material property for each mass species. The larger the dispersivities, the faster particles can travel ahead or behind the contamination front. In other words, large dispersivities lead to enhanced dispersion effects reducing locally parameter gradients. In the above example this means that the contamination front becomes blurred or smears along its plume boundaries. This has a positive effect on numerical stability.
Step #4: Activate Upwinding
By manually editing the longitudinal or transverse dispersivities, the user has full control on the applied parameter values. However, the results might not reproduce the exact range of expected values nor eliminate even the smallest numerical artefacts. Applying the upwinding technology, FEFLOW defines arbitrary dispersivity values for each element aiming at the lowest possible numerical errors. This procedure optimizes the modeling results, on the cost of a substantial amount of ‘artificial’ dispersion and an unknown dispersivity distribution. See Fig. 2.
In Fig. 2 you can see the upwinding options in the Problem Settings editor. Please note that there are several options available, of which ‘full’ and ‘streamline’ upwinding are the most common ones. While ‘streamline’ adds dispersion only in the direction of flow, ‘full’ adds it in all directions.
FURTHER INFORMATION AND USEFUL LINKS