Cooper-Jacob Validity Parameter u

This is the smallest equation in the shard and the most important one on it. $u = r^{2}S/(4Tt)$ is the argument of the Theis well function, and the whole Cooper-Jacob approximation rests on a single condition: $u$ must be below 0.01. That is not a guideline, a rule of thumb, or a conservative recommendation. It is the condition under which the equation is true. Above it, the terms Cooper and Jacob threw away are no longer small, and the straight line on the semi-log plot is not yet straight.

Read the equation as a story about what the aquifer is doing. Early in a test, $t$ is small and $u$ is large: the cone of depression is still a local disturbance near the well, and the pressure transient has not reached the observation point in any meaningful way. As pumping continues, $u$ falls in direct proportion to time. Distance works against you with the square, so an observation well twice as far out needs four times as long before its data can be read this way. Storativity works against you directly, which matters because $S$ is usually the number you are least sure of. And transmissivity works for you: a good aquifer transmits the signal quickly and reaches validity sooner.

The most useful direction on this page is not $u$ at all. Set $u = 0.01$ and solve for $t$, and you get the critical time: the moment after which the straight-line method is allowed at that observation well. Do this before the test, not after. For a well 50 m out in an aquifer with $T = 500$ m²/d and $S = 4\times10^{-4}$, the answer is 1.2 hours, so a two-hour step is fine. Move that same well out to 200 m and the critical time jumps to nineteen hours, because the distance is squared. That single calculation is the difference between a pumping test that can be interpreted and one that cannot.

The practical failure looks like this. Someone runs a four-hour constant-rate test, plots drawdown against the log of time, sees points that look convincingly linear, fits a line, and reports a transmissivity. The points looked linear because early Theis data has gentle curvature that the eye forgives, and the reported $T$ is wrong, usually too high. Computing $u$ takes ten seconds and settles the question. If $u$ is still above 0.01 at the end of the test, the test was too short, and no amount of careful plotting fixes that.