8.5 Simplification of the Theis Equation
The well function in the Theis equation is shown in Equation (78).
| [latex]W(u) = - 0.5772 - \ln{u + u - \frac{u^{2}}{2*2!}} + \frac{u^{3}}{3*3!} - \frac{u^{4}}{4*4!} + \frac{u^{5}}{5*5!} - and\ so\ on\ldots[/latex] | (78) |
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where:
| [latex]u = \frac{r^{2}s}{4Tt}[/latex] | (79) |
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Cooper and Jacob (1946) noted that, when u becomes small, everything to the right of the term [latex]ln(u)[/latex] in the well function becomes negligible and can be omitted. Omitting this term reduced the well function to a simple term, which in turn reduces the Theis equation to that shown in Equation (80).
| [latex]s = \frac{2.3Q}{4\pi}\log\frac{2.25Tt}{r^{2}S}[/latex] | (80) |
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This is called the Cooper–Jacob simplification of the Theis non‑equilibrium equation. This means that, in certain situations, we can use this simplified version rather than the more complex Theis equation. More specifically, the quantity [latex]u[/latex] becomes small when [latex]t[/latex] becomes large—that is, at long times after the start of pumping—or when [latex]r[/latex] becomes small—that is, very close to the pumping well. This concept can be applied to analyzing aquifer test data to estimate [latex]T[/latex] and [latex]S[/latex] using the methods described in the following two sections.
Time‑Drawdown Method
If time‑drawdown data are available from a single observation well, we can use the Cooper‑Jacob simplification to estimate [latex]T[/latex] and [latex]S[/latex]. The method is as listed here.
- Plot drawdown versus time on semi‑log paper with drawdown on the arithmetic scale and time on the log scale.
- Draw a straight line to fit the data, focusing on the later data, and extend the line to the point where it intersects the time axis.
- Choose two points on the line that are one log‑cycle apart in time—for example, 10 and 100, 1 and 10, or 100 and 1,000 so that the slope of the line is divided by 1.
- Determine the change in drawdown over that interval ([latex]\mathrm{\Delta}s[/latex]).
- Plug that into Equation (81) and solve for [latex]T[/latex].
| [latex]T = \frac{2.3Q}{4\pi\mathrm{\Delta}s}[/latex] | (81) |
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- Determine the value of time where the line crosses the x‑axis ([latex]t_{0}[/latex]).
- Plug [latex]t_{0}[/latex] into Equation (82) and solve for [latex]S[/latex].
| [latex]S = \frac{2.25Tt_{0}}{r^{2}}[/latex] | (82) |
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This procedure is illustrated by Exercise 12⮧.
Distance‑Drawdown Method
If two or more observation wells located at different distances from the pumping well have drawdown data, the distance‑drawdown method can be used to estimate [latex]T[/latex] and [latex]S[/latex]. The method is as follows.
- Plot the drawdown versus distance of each well at a given time after pumping on semi‑log paper (with distance on the log scale).
- Draw a straight line to fit the data and extend the line through the drawdown = zero axis.
- Choose two points on the line that are one log‑cycle apart in distance—for example, 10 and 100, 1 and 10, or 100 and 1,000, so that the slope of the line is divided by 1.
- Determine the change in drawdown over that interval (∆[latex]s[/latex]).
- Substitute ∆[latex]s[/latex]) into Equation (83) and solve for[latex]\ T[/latex] .
| [latex]T = \frac{2.3Q}{4\pi\mathrm{\Delta}s}[/latex] | (83) |
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- Determine the value of distance where the line crosses the x‑axis ([latex]r_{0}[/latex]).
- Substitute [latex]r_{0}[/latex] into Equation (84) and solve for [latex]S[/latex].
| [latex]S = \frac{2.25Tt}{r_{0}^{2}}[/latex] | (84) |
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This procedure is illustrated by Exercise 13⮧.