Cooper-Jacob Drawdown

Theis solved the transient well problem in 1935 by borrowing the mathematics of heat conduction, and the answer he got is exact: the drawdown is $s = \frac{Q}{4\pi T} W(u)$, where $W(u)$ is an infinite series. Eleven years later Cooper and Jacob noticed something practical about that series. It runs $W(u) = -0.5772 - \ln u + u - u^2/4 + \dots$, and once $u$ is small every term after the first two is negligible. Drop them, fold the 0.5772 into the constant, and the whole thing collapses into a logarithm you can evaluate by hand: $s = \frac{2.303\,Q}{4\pi T}\log_{10}\!\left(\frac{2.25\,T\,t}{r^{2}S}\right)$. The 2.303 is not a fitted number, it is $\ln 10$, which is why the log-base-10 form and the natural-log form are one equation rather than two.

That collapse is what makes this page possible at all. The Theis equation itself is not on this site, and its absence is deliberate rather than an oversight. $W(u)$ has no closed form, so inverting it for transmissivity or storativity means iterating, and this site only ships solvers that return an answer in one step. Cooper-Jacob is the honest substitute: the same physics, the same aquifer, the same assumptions, with the series truncated once the well has been pumping long enough for the truncation to be legitimate. It is not an approximation in the sense of being sloppy. Within its window it agrees with Theis to well under a per cent.

Notice what is missing from the list of directions this page will solve. There is no brain for $T$, and that is not laziness. Transmissivity sits inside the logarithm and multiplies the coefficient outside it, so solving for it means solving $T$ out of an equation of the form $a = T\ln(bT)$, which is transcendental and has no algebraic answer. Rather than ship a rearrangement that looks like algebra but cannot round-trip to the number it started from, the page sends you to the distance-drawdown form, which is how a hydrogeologist actually extracts $T$ in the field anyway: two observation wells, subtract, and the storage term cancels itself out.

Three mistakes recur. The first is entering the water level instead of the drawdown. Drawdown is the change in head from the pre-pumping level, so a well that stood at 12 m below ground and now stands at 15 m has a drawdown of 3 m, not 15. The second is entering the radius of the pumping well where the equation wants the distance to the observation well. The equation is happy to use the well's own radius, and people do it deliberately to estimate the drawdown in the pumping well, but the answer that comes back is the aquifer's contribution only. It knows nothing about well loss, so it will always be optimistic. The third is checking $u$ after the fact, or not at all. Check it first.