Step-Drawdown Test: Aquifer Loss and Well Loss

Jacob's 1947 paper did something no aquifer equation can do: it separated the two completely different reasons a pumping well's water level goes down. Pump at a rate $Q$ and the drawdown inside the well is $s = BQ + CQ^{2}$. The first term is the aquifer. It is laminar flow through the formation toward the well, it obeys Darcy's law, and it is linear in the rate. The second term is the well itself: turbulent flow converging into the gravel pack, accelerating through the slots, and rising up the casing to the pump intake. Turbulent losses go as the square of velocity, so this term goes as the square of the rate.

The distinction matters enormously, because one of those problems is fixable and the other is not. $BQ$ is the aquifer doing what aquifers do, and no amount of money spent on the well changes it. $CQ^{2}$ is the well being badly built, badly developed, incrusted, or plugged, and every metre of it is a metre of pumping head thrown away for nothing. A well drawing down eight metres of which four are well loss is not a poor aquifer; it is a poor well sitting on a perfectly good aquifer, and surging, jetting, or acidising can hand you those four metres back.

To find $B$ and $C$ you run a step-drawdown test: pump at three or four increasing rates, each held long enough for the water level to steady, and record the drawdown at each step. Divide through by $Q$ and the equation becomes $s/Q = B + CQ$, which is a straight line. Plot specific drawdown $s/Q$ against $Q$, and the intercept is $B$ while the slope is $C$. The units follow from whatever $Q$ is in; with $Q$ in m³/d and $s$ in metres, $B$ comes out in d/m² and $C$ in d²/m⁵, which is why those two coefficients are carried on this page as bare numbers with their units printed beside them.

Two cautions. First, the steps must each reach quasi-steady state, or you are fitting the transient and the coefficients are meaningless. Second, $B$ itself is not a pure aquifer constant. It contains a slow time dependence from the Cooper-Jacob logarithm, so a step test run over six hours and one run over sixty will not return quite the same $B$. $C$, the number you actually care about for the well's condition, is far more stable. Rising $C$ between one annual test and the next is the clearest early warning of screen incrustation there is, long before the pump starts struggling.