The finite element mesh is very important in a numerical simulation. A finer finite element mesh commonly gives better calculation results. It has always been a question: what is the resolution of the finite element mesh that provides reasonably accurate results? This article does not answer the question specifically but will provide a sense of the relation between: i) finite element mesh; ii) hydraulic conductivity of the material; and iii) simulation results. A simple model was created to study these relationships. This example helps all modelers to be aware of this issue.
When running this model, the “auto mesh refinement” function in SVFLUX™ was turned off.
The model is made up of a rectangular region containing a Sandy Loam which has a width of 200 m and a height of 100m. The sandy loam has a saturated volumetric water content of 0.3 and a specific gravity of 2.65. The boundary conditions of the model are shown in the figure below.
Figure 1 - Geometry of the model
The soil-water characteristic curve of the Sandy Loam is plotted in the figure below.
Figure 2 - Soil-Water Characteristic Curve for the model
Climate data is described for 0 to 3 days in the table below. The initial condition for this model is the water table is equal to zero (m).
Table 1 - Climate properties used in the numerical model
In this sensitivity study, 5 different cases were implemented (see the table below). For the first three cases the saturated hydraulic conductivity of the material was varied while the finite element mesh is kept the same. The last case uses the lowest saturated permeability and has the finest finite element mesh (i.e., increased number of nodes along the material surface). The Modified Campbell (Fredlund, 1997) is used for the prediction of the unsaturated hydraulic conductivity function. The “p-parameter” is slightly changed between the first three cases to make the unsaturated hydraulic conductivity functions for the three cases parallel to each other (see the figure below).
Table 2 - Hydraulic Conducitivities and Nodes used in the finite element mesh
Figure 3 - Unsaturated Hydraulic conductivity curves
The simulation results for the five cases are presented below. A brief discussion on the results and the effects of the saturated hydraulic conductivity and the finite element mesh to the simulation results are presented. The finite element mesh for the first three cases is shown below. Plots of the simulation results for first three cases are also shown.
Figure 4 - Finite Element Mesh for Cases 1-3
Figure 5 - Simulation Results for Case 1
Figure 6 - Simulation Results for Case 2
Figure 7 - Simulation Results for Case 3
The finite element meshes for cases 4 and 5 (i.e., with 2152 nodes and 4292 nodes) are shown below. The simulation results for cases 4 and 5 are shown. The table that follows shows the summary of the simulation results for the 5 cases.
Figure 8 - Finite Element Mesh for Case 4
Figure 9 - Simulation Results for Case 4
Figure 10 - Finite Element Mesh for Case 5
Figure 11 - Simulation Results for Case 5
As can be seen from the table below, with the same finite element mesh, the higher saturated hydraulic conductivity of the material resulted in lower model error. This table also shows that the finer the finite element mesh, the better the simulation results. Case 1 appears to have a low accuracy due to distances between nodes being too high. Cases 2 and 4 have reasonably good results. It shows that there is a relationship between the accuracy of the model and the ratio: (hydraulic conductivity/distance between nodes near (or at) the material surface). It shows that a ratio of 10 should provide sufficiently good simulation results.