Step-Drawdown Test Analysis: Well Loss and Efficiency
What is a step-drawdown test?
A step-drawdown test pumps one well at a series of increasing constant rates and records the drawdown in that well, so that the drawdown can be split into a part caused by the aquifer and a part caused by the well. Jacob (1947) ran the first one to predict the drawdown at rates other than the test rate. For transmissivity and storativity from observation wells, see Theis vs Cooper-Jacob.
What are aquifer loss and well loss?
Aquifer loss is the head lost as water flows through the aquifer toward the well, and well loss is the extra head lost at the well itself. Kruseman and de Ridder (1994, Chapter 14) treat aquifer losses as laminar and proportional to the rate, and split well losses in two. Linear well losses come from drilling damage: formation compacted or plugged with mud near the bore, and losses in the gravel pack and screen. Non-linear well losses come from turbulent flow in the screen, the suction pipe and the zone just outside the well.
Jacob wrote the drawdown in the pumped well as sw = BQ + CQ2, where Q is the rate, B = B1 + B2 combines the linear aquifer-loss coefficient B1 and the linear well-loss coefficient B2, and C is the non-linear well-loss coefficient (Jacob 1947; Kruseman and de Ridder 1994, Equation 14.1). B depends on time, because aquifer drawdown keeps growing while pumping continues. BQ is usually called the aquifer loss, although it also holds the linear well loss.
What did Rorabaugh change in Jacob's equation?
Rorabaugh (1953) replaced the fixed square with a fitted exponent, sw = BQ + CQP, because researchers had found considerable variation in the flow in and around wells. Kruseman and de Ridder (1994, Equation 14.2) report that P can take values from 1.5 to 3.5 depending on Q, citing Lennox (1966), and that Jacob's P = 2 is still widely accepted. Their worked example on Sheahan's (1971) data gives P = 2.8.
Rorabaugh's method finds B by trial: assume a value, plot sw/Q − B against Q on log-log axes, and repeat until the points form a straight line. Its slope is P − 1 and its value at Q = 1 is C (Section 14.1.3). The units of C depend on P, so C values fitted with different exponents cannot be compared.
How do you plot s/Q against Q?
Divide Jacob's equation by Q and it becomes a straight line, sw/Q = B + CQ: specific drawdown against rate on arithmetic axes has slope C and meets the Q = 0 axis at B. This is the Hantush-Bierschenk method (Hantush 1964; Bierschenk 1963), which Kruseman and de Ridder (1994, Procedure 14.1) apply to confined, leaky and unconfined aquifers:
- Plot drawdown in the well against the logarithm of time, and extend the curve of each step to the end of the next.
- For each step, read the drawdown increment at a fixed time Δt after the step began, measured from the extended curve of the step before.
- Add the increments to get sw for each step, divide by Q, plot against Q, and read C from the slope and B from the intercept.
If every step reached steady state, the observed end-of-step drawdowns can be used directly. If the points curve, use a method based on CQP instead.
Their example is Clark's (1977) Well 1, in a confined sandstone, pumped in six 180-minute steps from 1306 to 5019 m3/d. With Δt = 100 minutes the line gives B = 3.26 × 10−3 d/m2 and C = 1.45 × 10−7 d2/m5 (Example 14.1).
A line through s/Q against Q gives the low-rate steps extra weight, because dividing by Q stretches their scatter. Fitting s = BQ + CQ2 to the drawdowns themselves avoids this. On the six Well 1 drawdowns in Kruseman and de Ridder's Table 14.2, a least-squares s/Q line misses the drawdowns by 0.119 m (root mean square) and the direct fit by 0.096 m.
How do you calculate well efficiency?
Well efficiency is the share of the drawdown that the aquifer causes: E = BQ / (BQ + CQ2) × 100%. Kruseman and de Ridder (1994, Equation 14.3) define it strictly as B1Q / ((B1 + B2)Q + CQ2), but a step test cannot separate B1 from B2, so only the non-linear losses are measured. The step-test figure therefore overstates the efficiency of a well with a damaged or clogged face.
Efficiency falls as the rate rises, because well loss grows with Q2 and aquifer loss only with Q. For Well 1:
| Rate (m3/d) | BQ (m) | CQ2 (m) | Efficiency |
|---|---|---|---|
| 1306 | 4.26 | 0.25 | 95% |
| 3261 | 10.63 | 1.54 | 87% |
| 5019 | 16.36 | 3.65 | 82% |
B grows with time, so the same well gives a different efficiency at a different Δt. Quote the time with the result.
How many steps does a step test need, and how long should each be?
Kruseman and de Ridder (1994, Section 14.1) ask for at least three steps of equal duration, typically 30 minutes to 2 hours each. Equal steps matter because every increment is read at the same Δt, and B changes with time. Three points barely test whether the line is straight; more steps, as in Clark's six, show curvature and whether P = 2 holds.
What do C values say about the condition of a well?
Walton (1962) proposed, from field experience, a guide linking Jacob's C to well condition. The bands below are as later field reports quote them, with s in metres and Q in m3/min; we could not check them against Walton's bulletin itself, so treat them as a rule of thumb. They apply only with P = 2:
| C (min2/m5) | C (d2/m5) | Well condition |
|---|---|---|
| Below 0.5 | Below 2.4 × 10−7 | Properly designed and developed |
| 0.5 to 1.0 | 2.4 to 4.8 × 10−7 | Mild deterioration or clogging |
| 1.0 to 4.0 | 4.8 × 10−7 to 1.9 × 10−6 | Severe deterioration or clogging |
| Above 4.0 | Above 1.9 × 10−6 | Difficult to restore to original capacity |
The d2/m5 column is our conversion (1 day = 1440 minutes). Well 1's C is about 0.30 min2/m5, in the first band. Treat the bands as a rule of thumb. A negative C means the readings do not follow Jacob's equation; check the rates and step lengths before reading anything into it.
What does a step-drawdown test not tell you?
A step-drawdown test does not tell you how much of B is aquifer and how much is damage at the well face: Kruseman and de Ridder (1994, Section 14.1) note that it gives no separate B1 and B2, and that a recovery test can estimate the skin factor (Section 14.2). Nor does it give:
- The effective well radius Jacob defined, which needs the storativity, and so readings in nearby piezometers.
- Reliable transmissivity and storativity. The Eden-Hazel method (Section 14.1.2) estimates transmissivity from a step test in a confined aquifer, but a constant-rate test with observation wells is the better source.
- The full extent of a well's faults, since not every construction fault shows up as non-linear flow resistance.
- Long-term drawdown. B holds for the Δt at which it was read.
Sources: Jacob (1947), Transactions ASCE 112, 1047-1064; Rorabaugh (1953), Proceedings ASCE 79, separate 362; Bierschenk (1963), IASH Publication 64, 493-507; Hantush (1964), Advances in Hydroscience 1, 281-432; Walton (1962), Illinois State Water Survey Bulletin 49; Clark (1977), Quarterly Journal of Engineering Geology 10, 125-143; Kruseman and de Ridder (1994), Analysis and Evaluation of Pumping Test Data, ILRI Publication 47.
Can you analyse a step test in a browser?
Yes: PumpPilot is a free browser tool whose Step-drawdown tool fits Jacob's s = BQ + CQ2 by least squares to the drawdown at the end of each step of equal length, and reports B, C (always in metres and days) and the efficiency at each rate. It fills the table from the schedule and the pumping well's readings, or takes typed values. It does not extrapolate increments, fit Rorabaugh's exponent or grade C against Walton's bands. Straight-line methods in the PumpPilot docs describes the tool.
Jagadeesh Gaddam
CEO, Smart Bhujal
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Run the analysis described in this article with one prompt. No setup required.
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