# Exercise 3 Solution: Analytical Solution for Streamflow Depletion

**The problem:**

Estimate streamflow depletion using Glover’s analytical solution (rather than a numerical model). How do these results compare with those from the numerical model used in the Base Case and in Exercise 1? Explain any differences.

**Approach:**

Solve Equation 5 for each year of the 200-year simulation. There are a number of ways to do this, but one reasonable approach is to use formulas in an Excel spreadsheet. This exercise can be completed using information available in this book and in results for Exercise 1.

**Analytical solution:**

We solved Glover’s analytical solution (Equation 5 of this book) using formulas in an Excel spreadsheet. We copied the spreadsheet from Exercise 1 (“RateBudgets.xlsx”) and pasted a copy into the Exercise 3 subfolder “DataSpreadsheets”. Then we deleted the worksheets for the closer well and further well, and added a new worksheet “Glover Soln.”. We entered the known values for parameters in column B, and then solved for values of *z* and *Qs(t)* for every year from 0 to 200 years in columns G and H using the *ERFC* function in Excel for complementary error function. We then plotted these results and the capture values from the Base Case analysis (Figure ExSol 3-1). The zip file “Exercise3.zip”, including the spreadsheet, is available in the online Supplementary Information for this book.

**Comparison of analytical and numerical solutions:**

The two solutions are in close agreement for the first 50 years or so. After that they begin to diverge, with the analytical solution providing smaller values of streamflow depletion at later times. The largest difference, at 200 years, is about 200 m^{3/}d, with the analytical solution value being about 10 percent less than the numerically calculated value.

The analytical solution requires the application of several simplifying assumptions. The Base Case is designed to include many simplifying assumptions also, so the match should be good. The analytical solution assumes a semi-infinite aquifer (i.e., an assumption that the aquifer extends without end in a particular direction). However, the hypothetical aquifer is not semi-infinite in extent as it has impermeable boundaries on the north, west, and south sides (i.e., at finite distances from the well and river); that is, the numerical model assumes the aquifer has a limited areal extent, as defined by the outer impermeable or no-flow boundaries). Figure ExSol 1-7 shows that measurable drawdown occurs at the north, west, and south boundaries. The effect of such boundaries would be to eventually cause increased drawdown in the modeled aquifer compared to having no such boundaries as in a semi-infinite idealization. Increased drawdown would cause increased capture. This is consistent with the results shown in Figure ExSol 3-1, so it is likely that this one difference between the assumptions for the analytical solution and the boundary conditions of the numerical model can explain most or all of the underestimate by the analytical solution. However, the overall excellent agreement, especially for the first 50 years in this case, combined with the efficiency and ease of use of the analytical solution, indicates that this would be a valuable method to apply early in any study of groundwater development in a stream-aquifer system. It will also give you an expectation and basis of comparison for the results of a numerical model that encompasses more complex boundary conditions and heterogeneous properties.