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appendix iv
Simulation of Ground-Water Discharge to Biscayne Bay, Southeastern FloridaAppendix IV
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The SEAWAT code is relatively new and has not been widely used and tested. For this reason and because the code was modified for this study, SEAWAT was verified by running four different problems and comparing the results to those from other variable-density codes. Voss and Souza (1987) suggest that new variable-density codes should be tested and verified by running four or five benchmark problems that vary in complexity. These benchmark problems are listed below with the reference that describes the problem:
For each of the four problems evaluated, the results from SEAWAT compare reasonably well with the results from other numerical codes. From these comparisons, SEAWAT was determined to be an acceptable code for simulating the variable-density component of ground-water flow. As discussed in the report, the SEAWAT code was also modified to include variable-density sources and sinks. These additional features seem to work properly in the code, but they have not yet been verified against the results from other codes because benchmark problems for these types of features do not currently exist.
The purpose of the box problems is to verify that fluid velocities are properly calculated by the simulation code. Although inconsistent approximations for velocity are more likely to occur with finite-element models, the box problems also provide a good test for variable-density codes based on the finite-difference approximation. There are two different cases of the box problem (Voss and Souza, 1987). In the first case, the variable-density code is tested by simulating flow within a two-dimensional, vertical cross-sectional model with no-flow boundaries on all sides. The size of the model domain and values for hydraulic conductivity and porosity are not important. Dispersivity values should be set to a length similar in size to the length of a model cell, and the diffusion coefficient should be set to zero. The initial conditions within the box consist of a layer of freshwater overlying a layer of saltwater-a stable configuration for fluid density. When this model is run with steady-state conditions, the interface between freshwater and saltwater should remain in the same layer of the model.
For the second case of the box problem, the initial condition and dispersion parameters are the same as the first case, but horizontal flow is induced across the box by setting different hydrostatic pressure boundaries on the left and right sides of the model. If the code is programmed correctly, the interface between freshwater and saltwater should remain in the same layer of the model.
SEAWAT was tested with both cases of the box problem. In each case, the interface remained in the correct model layer. This indicates that the velocity approximation used by SEAWAT is probably valid.
Henry (1964) presented an analytical solution for a problem that is thought to represent fresh ground water flowing toward a seawater boundary. Because an analytical solution was available for the Henry problem, many numerical codes were evaluated and tested with the Henry solution. Segol (1993) showed, however, that the Henry solution was not exact because Henry (1964) eliminated, for computational reasons, mathematical terms from the solution that were thought to be insignificant. When Segol (1993) recalculated Henrys solution with the additional terms, the improved answer was slightly different from the original solution. With the new solution, Segol (1993) showed that numerical codes such as SUTRA (Voss, 1984) could reproduce the correct answer for the Henry problem.
The basic design of the Henry problem is shown in figure A1. The cross-sectional box is 2 m long by 1 m high by 1 m wide. A constant flux of fresh ground water is applied to the left boundary at a rate ( The Henry problem caused further confusion among the modeling community because researchers attempting to verify numerical codes calculated an erroneous value for molecular diffusion that did not correlate with the original value used by Henry (Voss and Souza, 1987). For this reason, some researchers consider there to be two cases of the Henry problem: one in which the value for molecular diffusion ( The finite-difference model grid used to discretize the problem domain consists of 21 columns and 10 layers. Each cell, with the exception of the cells in column 21, is 0.1 by 0.1 m in size. Cells in column 21 are 0.01-m horizontal by 0.1-m vertical. The narrow column of cells at the right side of the model was used to more precisely locate the seawater hydrostatic boundary at a distance of 2 meters.
The comparison between SEAWAT and SUTRA results (Segol, 1993) for the Henry problem are shown in figure A2. Contours of relative salinity are in good agreement for both cases, especially away from the right boundary. Discrepancies in contours of relative salinity between the two models at the right boundary are probably due to differences in the way finite-element and finite-difference models locate and represent constant-head (or pressure) boundaries.
The Elder problem was originally designed for heat flow (Elder, 1967), but Voss and Souza (1987) recast the problem as a variable-density ground-water problem in which fluid density is a function of salt concentration. The Elder problem is commonly used to verify variable-density ground-water codes.
The Elder problem was run with the SEAWAT code for a period of 20 years. Several different methods were used to solve the transport equation. The MOC and the implicit solver with a central-in-space weighting scheme provided solutions closest to those of SUTRA (Voss and Souza, 1987) and those of Elder (1967). Relative concentrations from SEAWAT are compared with results from SUTRA (Voss and Souza, 1987) and the original nondimensional results from Elder (1967) for six different times (fig. A4). For each time, there seems to be a good match between the results from SEAWAT and SUTRA (Voss and Souza, 1987) and those of Elder (1967).
This appendix presents the development and results for four of the benchmark problems listed above. The salt-lake problem has not yet been simulated with SEAWAT.
Box Problems
Henry Problem
Figure A1. (above) Boundary conditions and model parameters for the Henry Problem. [larger image]
Figure A2. (above) Comparison between SEAWAT and SUTRA for the Henry Problem. [click on images above for larger versions]
Elder Problem
The geometry and boundary conditions for the problem are shown in figure A3. A constant-concentration boundary is
specified for part of the upper boundary. During the simulation, salt from the constant-concentration boundary diffuses into the model domain and initiates complex vortices that redistribute salt mass throughout the model. A constant-concentration boundary with a value of zero is specified for the lowest layer in the model. Two outlet cells with constant-head values of zero are specified for the upper left and right boundaries. These constant-head cells allow salt to diffuse into the model by providing an outlet for fluid and salt mass.
Figure A3. Boundary conditions and model parameters for the Elder Problem. [larger image]
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| Figure A4. Comparison between SEAWAT, SUTRA and Elder's solution for the Elder Problem. [click on images above for larger versions] | |
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| Figure A5. (above) Boundary conditions and model parameters for the HYDROCOIN Problem (from Konikow and others, 1997). [larger image] |
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| Figure A6. (above) Comparison between SEAWAT and MOCDENSE for the HYDROCOIN Problem. [larger image] |
The purpose of the Hydrologic Code Intercomparison project (HYDROCOIN) was to evaluate the accuracy of selected ground-water modeling codes. One of the problems used for testing is referred to as the HYDROCOIN problem. The HYDROCOIN problem presented here is based on case 5 that was reevaluated by Konikow and others (1997). The general geometry and boundary conditions for the problem are shown in figure A5. A sloping pressure boundary is imposed across the top of the box that is surrounded on the sides and bottom by no-flow conditions. Along the base of the middle portion of the model, a constant concentration condition is applied to represent the top of a salt dome. As ground water flows along the bottom boundary, salt disperses into the system and collects in the lower right corner of the model domain.
The SEAWAT code was used to simulate the HYDROCOIN problem and uses the general design employed by Konikow and others (1997). Prior to simulating the HYDROCOIN problem, the SEAWAT code was slightly modified to use the equation of state for fluid density as used by Konikow and others (1997). The comparison between SEAWAT and MOCDENSE is shown in figure A6. In general, the relative concentrations simulated by the two codes are consistent with one another. There is a discrepancy toward the upper right part of the model domain. While SEAWAT tends to produce slightly higher concentrations in this region, the comparison between the two codes is considered acceptable.
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Appendix 4