Consider the well-known Theis solution for a constant-rate pumping test in a nonleaky confined aquifer of infinite extent:

where

*s*is drawdown [L],

*Q*is pumping rate [L³/T],

*T*is transmissivity [L²/T],

*t*is time [T],

*r*is radial distance [L] and

*S*is storage coefficient [-].

The Cooper and Jacob method approximates the Theis well function,

*w(u)*, by truncating the infinite series in (4) after the first two terms:

The approximation in (5) assumes that

*u*is small, i.e.,

*t*is large and

*r*is small. It is precisely when (5) is valid that infinite-acting radial flow is identified from drawdown data.

How small should

*u*be for (5) to be valid? The threshold for IARF (i.e., when Cooper and Jacob becomes valid) is often given as u ≤ 0.05 (Driscoll 1986) or u ≤ 0.01 (Kruseman and de Ridder 1994); however, we can see for ourselves when (5) is valid by plotting w(u) versus 1/u (dimensionless drawdown versus dimensionless time) on semilog axes and looking for a constant logarithmic derivative plateau (Figure 1).

Figure 1. Theis (1935) solution (blue curve) plotted as w(u) versus 1/u on semilog axes. Logarithmic derivative shown by red curve. |

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**infinite-acting radial flow**.

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