8.2 Using the Theis Equation to Predict Drawdowns in Totally Confined Aquifers
When the properties of a fully confined aquifer are known, T and S can be inserted into the Theis equation (Equation (26)) and the drawdown for any time since pumping began can be computed at any radial distance from the pumping well.
Example
Compute the drawdown at an observation well located 300 m from a well that is pumped at a constant rate of 4000 L/min (4 m3/min = 5,760 m3/d) for 8 hours (0.33 d) when the confined formation T is estimated at 800 m2/d and S=0.00008.
First u is calculated, then the corresponding Theis well function value is derived.
[latex]u = \frac{r^{2}S}{4Tt} = \frac{{(300\ m)}^{2}0.00008}{4\ \left( 800\ \frac{m^{2}}{d} \right)\ 0.33\ d} = u = \ 0.0068\ or\ {6.8x10}^{- 3}[/latex]
Again, it is important to use consistent units such that u has no units because it is a dimensionless value. Using the well function table in Figure 26, W(u) = 4.4204. So, the drawdown 300 m from the pumping well would be calculated as follows.
[latex]s = \frac{5760\ \frac{m^{3}}{d}\ 4.4204}{4\ (3.14)\ 800\ \frac{m^{2}}{d}} = \ 2.53\ m[/latex]
As a check on the table value of W(u), substituting values of u into the infinite series shown in Equation (27) and only extending the calculation to the first two terms that contain u results in W(u) = 4.4204. This correlates with the table value.
Again, this analytical solution is for a fully confined, isotropic and homogeneous system using the general simplifying assumptions and the additional Theis assumptions. The values used in Equation (26) must have compatible units so that drawdown is reported in length units (e.g., ft, m).