8.2 Well Hydraulics—A Quick Review
Flow to a well is a special case because flow is radial—that is, converging on a specific point in the aquifer (the well). To handle this mathematically, C.V. Theis derived the equation that governs 2‑D radial flow to a well, as shown in Equation (72).
| [latex]\frac{\partial^{2}h}{{\partial r}^{2}} + \frac{1}{r}\frac{\partial h}{\partial r} + \frac{w}{T} = \frac{S}{T}\frac{\partial h}{\partial t}[/latex] | (72) |
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Theis then ‘stole’ an analytical solution for this equation from the heat flow literature to develop his eponymous equation, which is shown in Equation (73) and where u is defined by Equation (74), and [latex]W(u)[/latex]—the well function—is as shown in Equation (75).
| [latex]S = \frac{Q}{4\pi T}W(u)[/latex] | (73) | |
|---|---|---|
| [latex]u = \frac{r^{2}S}{4Tt}[/latex] | (74) | |
| [latex]W(u) = \int_{u}^{\infty}\frac{e^{- u}du}{u}[/latex] | (75) |
where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]s[/latex] | [latex]=[/latex] | drawdown (L) is defined as the initial head minus the head at time [latex]t[/latex] |
|---|---|---|
| [latex]Q[/latex] | [latex]=[/latex] | pumping rate from the well (L3T-1) is assumed to be constant |
| [latex]T[/latex] | [latex]=[/latex] | aquifer transmissivity ([latex]T = Kb[/latex]) (L2T-1) |
| [latex]r[/latex] | [latex]=[/latex] | radius (distance from pumping well) (L) |
| [latex]S[/latex] | [latex]=[/latex] | storativity (where [latex]S\ = \ S_{s}b[/latex]) (:) |
| [latex]t[/latex] | [latex]=[/latex] | time after the start of pumping ([latex]T[/latex]) |
The Theis equation calculates drawdown ([latex]s[/latex]) at some distance ([latex]r[/latex]) from the pumping well at some time ([latex]t[/latex]) after pumping began for a constant pumping rate ([latex]Q[/latex]). This equation describes the cone of depression that forms around a pumping well. If we know [latex]T[/latex] and [latex]S[/latex], we can use this equation to directly calculate the drawdown at any distance from the well at any time after the start of pumping.
But the problem is how do we measure [latex]T[/latex] and [latex]S[/latex] in the first place? It turns out that one of the best ways that to measure [latex]T[/latex] and [latex]S[/latex] in an aquifer is to pump water from a well at a constant rate, measure drawdown in the aquifer at various times after pumping starts, and then use these data and the Theis equation to calculate [latex]T[/latex] and [latex]S[/latex]. Doing this is called an aquifer test, or more specifically a constant rate aquifer test.