Once the cone of depression is well developed, drawdown plots as a straight line against the logarithm of time. The slope gives the transmissivity and the intercept gives the storativity — two numbers from one ruler, which is why this method outlived the curve-matching it replaced.
Cooper-Jacob is Theis with the tail of a series thrown away, and that is only legitimate once u = r²S/4Tt falls below about 0.01. Early readings, and readings from a distant observation well, do not satisfy it — and fitting them anyway is the commonest way to get a transmissivity that is quietly wrong. The panel above checks each reading and says which ones it used.
The full solution, with no approximation thrown away. Every reading is usable, including the earliest — which matters when the test is short or the observation well is far out. This fits T and S to all of them at once.
W(u) is the exponential integral, and it is the whole of Theis. Everything else is bookkeeping.
When the pump stops, the water comes back — and the rate it comes back at carries the same transmissivity, measured with no pump running, no rate to hold steady, and no observation well needed. Residual drawdown against log(t/t′) is a straight line through the origin, and the slope is all you need.
Not all the drawdown in a pumped well belongs to the aquifer. Some of it is the well fighting its own screen and gravel pack, and that part grows with the square of the rate while the aquifer's part grows only linearly. Separating them tells you whether the well needs developing or the aquifer is simply poor.
Drop a slug in, or bail one out, and watch the head come back. No pump, no discharge to dispose of, and a result in minutes. It only tests the ground within a metre or so of the screen, which is both its great convenience and its great limitation.
With T and S in hand, the same equation runs forwards. How far the drawdown reaches, how deep it gets, and what happens when the neighbour's well is pumping too — the last of which is the question that actually ends up in court.
What is dissolved in the water that comes up — and whether it is safe or aggressive — is Water chemistry. How fast water moves through the ground between wells is Groundwater flow, and the strength of the ground you are dewatering belongs to Foundations and tunnels.