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Geology 724
Problem Set #2
The Island Recharge Problem-Poisson Equation
Problem set #2 consists of variations of the following problems from Wang and Anderson: 3.1, 3.2, 3.4, and 3.5. Thechanges are described below. We will consider four conceptual models: 1D aquifer with constant thickness (i.e., 1D confined aquifer), 2D aquifer with constant thickness (i.e., 2D confined aquifer), 1D unconfined, and 2D unconfined.
3.1. 1-D Confined Aquifer: Refer to Fig. 3.2 (p. 43) and follow the directions for problem 3.1 on p. 65 of Wang and Anderson. Then, use the analytical solution you derived to solve the inverse problem by calculating a value of R for a system with: T = 10,000 ft2/day; l = 12,000 ft; h(0) = 20 ft (e.g., at the center of the island). Also calculate a value for h(0) if R = 0.00305 ft/day.
3.2 & 3.5. 2-D Confined Aquifer: Construct a spreadsheet model to solve for the distribution of heads using Gauss/Seidel iteration for the upper right hand quadrant of the island pictured in Figure 3.3. Assume a mesh-centered grid. Print out the array of heads to the fourth decimal place. Include a water balance in your program as described in Problem 3.5 in Wang and Anderson. Use the following values for the aquifer parameters: T = 10,000 ft2/day, l = 12,000 ft and R =0.00305 ft/day. Construct two worksheets in the spreadsheet as follows:
4 x 7 grid cells (x = y = 4000 ft) with error tolerance of 1E-4
13 x 25 grid (x = y = 1000 ft) with error tolerance of 1E-4
Compare your solution with the analytical solution on p. 4 of this handout, which was calculated from the equation given in Problem 3.2 of Wang and Anderson.
3.4.Unconfined Aquifer: Follow the directions in Wang and Anderson. Solve the 1D problem analytically and the two-dimensional problem using Gauss/Seidel iteration for the upper right hand quadrant of the island with R = 0.00305 ft/day. Be sure to include a water balance in your spreadsheet. Produce a solution using a 13 x 25 node grid (nodal spacing = 1000 ft). Use an error tolerance of 1E-4 and set K = 100 ft/day.
Add a pumping well to your spreadsheet model of the 2D unconfined aquifer (x = y = 1000 ft). Assume that the well is located in the node at the center of the island (i.e., at the intersection of the groundwater divides). Assume that the well pumps 10% of the water that enters the model via recharge.
Modify the finite difference equation in the cell in your spreadsheet in which the pumping well is located. Also modify the outflow term in your water balance to include the volume of water pumped out of the model by the well.
In your results, report the value of head in the cell that contains the pumping well, i.e. the cell in the center of the island. Also use equation 5.7 on p. 149 of Anderson and Woessner to estimate the head in the well, if we assume the radius of the well is 1 foot.
WRITE-UP. Please type your write-up following the format given below. It is okay to write in equations and figures by hand. Please be brief and concise. Also turn in a disk containing your spreadsheet(s).
Problem Description. Present the modeling objective: to compare the results of all the solutions by focusing on the head at the center of the island. Describe the conceptual model. Present the mathematical model (governing equation and boundary conditions) for each of the following versions of the problem: the 1-D confined aquifer, the 2-D confined aquifer, the 1-D unconfined aquifer and the 2-D unconfined aquifer. Show the grids used for the numerical solutions and identify the numerical solution method.
Results. You have 3 analytical solutions for this problem. Use the template below to tabulate the results from each solution by listing the head in the center of the island when R = 0.00305 ft/day. Also give the results from problem 3.1 for the 1-D confined aquifer for which you assumed that the head was 20 ft and then calculated the recharge rate needed to maintain that head.
You also have 4 numerical solutions. Fill in the tables (below) showing the head in the center of the island (reported to the 4th decimal place), and the water balance fluxes and the water balance error for each numerical solution.
Discussion. Discuss and explain the differences in the solutions as reported in the tables presented in your results section.
Conclusions. Make some conclusions about your modeling work relevant to the modeling objective.
Appendix. Include the derivations of the analytical solutions for the 1-D confined and 1-D unconfined cases. Also include a printout of each numerical model run. Label each printout. Be sure your printout includes the distribution of head in the upper right hand quadrant of the island, convergence criterion (error tolerance), water balance inflow and outflow, difference in the water balance and percent error in the water balance.
Analytical Solutions
Solution Type / Head at center of island (ft)1d confined
1d unconfined
2d confined
Numerical Solutions
Conceptual Model / x, y(ft) / Error Tolerance (ft) / Head at center
of island (ft) / Water Balance
Error (%) / Inflow, Outflow (ft3/day)
2D confined / 4000 / 1E-4
2D confined / 1000 / 1E-4
2D unconfined / 1000 / 1E-4
Pumping Solution
Conceptual Model / x, y(ft) / Error Tolerance (ft) / Head in node at center
of island / Head in well (ft) / Water Balance
Error (%) / Inflow ft3/day / Q
(well)
ft3/day / Outflow to the ocean
(ft3/day)
2D unconfined / 1000 / 1E-4
Analytical solution of the Island Recharge Problem using the equation given in Problem 3.2 (p. 65) in Wang and Anderson. Note that the solution is for the lower right hand quadrant of the problem domain.
20.00 19.86 19.46 18.78 17.82 16.59 15.08 13.29 11.22 8.86 6.20 3.25 .00
19.98 19.85 19.44 18.76 17.81 16.58 15.07 13.28 11.21 8.85 6.20 3.25 .00
19.93 19.80 19.39 18.71 17.76 16.54 15.04 13.25 11.19 8.83 6.18 3.24 .00
19.85 19.71 19.31 18.64 17.69 16.47 14.98 13.20 11.14 8.80 6.16 3.23 .00
19.73 19.59 19.19 18.52 17.58 16.37 14.89 13.13 11.08 8.75 6.13 3.22 .00
19.57 19.43 19.04 18.38 17.45 16.25 14.78 13.03 11.00 8.69 6.09 3.19 .00
19.37 19.24 18.84 18.19 17.27 16.09 14.63 12.91 10.90 8.61 6.04 3.17 .00
19.12 18.99 18.61 17.96 17.06 15.89 14.46 12.76 10.78 8.52 5.97 3.14 .00
18.83 18.70 18.32 17.69 16.81 15.66 14.25 12.58 10.63 8.41 5.90 3.10 .00
18.48 18.36 17.99 17.37 16.51 15.38 14.01 12.37 10.46 8.27 5.81 3.05 .00
18.08 17.96 17.60 17.00 16.15 15.06 13.72 12.12 10.25 8.12 5.70 3.00 .00
17.60 17.49 17.14 16.56 15.74 14.69 13.38 11.83 10.01 7.93 5.58 2.94 .00
17.06 16.95 16.61 16.05 15.27 14.25 12.99 11.49 9.74 7.72 5.43 2.86 .00
16.43 16.32 16.00 15.47 14.72 13.75 12.54 11.10 9.42 7.48 5.27 2.78 .00
15.71 15.61 15.31 14.80 14.09 13.17 12.02 10.65 9.05 7.19 5.07 2.68 .00
14.88 14.79 14.50 14.03 13.37 12.50 11.43 10.14 8.62 6.87 4.85 2.57 .00
13.93 13.85 13.59 13.15 12.54 11.74 10.74 9.55 8.13 6.49 4.60 2.44 .00
12.85 12.78 12.54 12.15 11.59 10.86 9.96 8.86 7.57 6.05 4.30 2.29 .00
11.63 11.56 11.35 11.00 10.51 9.86 9.06 8.08 6.91 5.55 3.96 2.12 .00
10.23 10.17 9.99 9.70 9.27 8.72 8.02 7.17 6.16 4.96 3.55 1.91 .00
8.65 8.60 8.45 8.21 7.86 7.40 6.83 6.13 5.28 4.28 3.08 1.67 .00
6.86 6.82 6.71 6.52 6.25 5.90 5.46 4.92 4.26 3.47 2.53 1.39 .00
4.83 4.81 4.73 4.61 4.43 4.19 3.89 3.52 3.07 2.52 1.86 1.04 .00
2.56 2.54 2.51 2.44 2.35 2.23 2.08 1.89 1.66 1.38 1.04 .60 .00
.00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00