Theis Recovery Test Analysis: Residual Drawdown Method

What is a recovery test?

A recovery test records how water levels rise after the pump is switched off at the end of a pumping test, and the Theis recovery method turns those readings into a transmissivity. What is measured is the residual drawdown s′: the difference between the water level before pumping started and the level at a time t′ after pumping stopped (Kruseman and de Ridder 1994, Chapter 13). Levels rise fastest in the first hour, and Kruseman and de Ridder advise the same reading schedule as during pumping (Section 2.6.1). For the pumping half of the test, see Theis vs Cooper-Jacob.

What equation describes residual drawdown?

Residual drawdown follows from superposition: after the pump stops, the well is treated as if it kept pumping at Q while an imaginary recharge of the same rate Q is injected at the same point, so the two cancel and the well is idle (Kruseman and de Ridder 1994, Chapter 13). Theis (1935) wrote the result as

s′ = Q / (4πT) × [W(u) − W(u′)], with u = r²S / (4Tt) and u′ = r²S′ / (4Tt′)

where t is the time since pumping started, t′ the time since it stopped, r the distance from the pumped well, S the storativity during pumping and S′ the storativity during recovery (Kruseman and de Ridder 1994, Equation 13.1). When u and u′ are both small, each well function reduces to its logarithm, as in the Cooper-Jacob method, and

s′ = Q / (4πT) × [ln(4Tt / (r²S)) − ln(4Tt′ / (r²S′))] (Equation 13.2).

If S and S′ are equal and constant, the distance and the storativity drop out:

s′ = (2.30Q / (4πT)) × log10(t / t′) (Equation 13.3).

How do you get T from the s′ vs log(t/t′) plot?

Plot s′ against t/t′ on semi-log axes and take T from the slope of the straight line. Procedure 13.1 in Kruseman and de Ridder (1994) runs:

1. For each reading of s′, calculate t/t′.
2. Plot s′ against t/t′ for one well, t/t′ on the log scale.
3. Fit a straight line and read Δs′, the fall in s′ over one log cycle.
4. Calculate T = 2.30Q / (4πΔs′), from Δs′ = 2.30Q / (4πT) (Equation 13.4).

With Q in m³/d and Δs′ in metres, T is in m²/d. As an illustration, a well pumped at 1000 m³/d whose line falls 0.5 m per log cycle gives T = 2300 / (4π × 0.5), about 366 m²/d.

The plot reads backwards in time: t/t′ is large just after shutdown and falls toward 1, so the late readings that meet the small-u′ condition sit at the left.

Why are recovery data often cleaner than pumping data?

Recovery data are often cleaner because the recovery rate is constant, while a constant discharge is often hard to hold in the field (Kruseman and de Ridder 1994, Chapter 13). Kruseman and de Ridder add that if the discharge was not constant during the test, the water table recovers at a constant rate equal to the average pumping rate (Section 2.6.1). Recovery readings also give an independent check on the pumping analysis at little extra cost.

In the pumped well itself, influences on residual drawdown that are or become constant with time, such as well losses and partial penetration, do not change the slope, so they do not affect T (Kruseman and de Ridder 1994, Chapter 15). Influences that change with time, such as storage in the well, still do.

When can S not be estimated from a recovery test?

The Theis recovery method gives transmissivity only; Equation 13.3 contains neither r nor S, so the slope says nothing about storativity. If S and S′ are constant but unequal, the line crosses s′ = 0 at a value (t/t′)₀ = S/S′ (Kruseman and de Ridder 1994, Remark to Section 13.1.1). That is a ratio, not S itself. A line that reaches zero residual drawdown at t/t′ = 1 is consistent with S = S′.

In a leaky aquifer, too, it yields only transmissivity (Section 13.1.2). For storativity, fit the pumping-phase drawdown at an observation well.

What if the pumping rate was not constant?

If the rate varied, one Q no longer describes the test, and Kruseman and de Ridder (1994, Section 13.3.1) point to the Birsoy and Summers (1980) method for step-wise or intermittently changing rates. It plots s′/Q_n, where Q_n is the rate in the last pumping period, against an adjusted time of recovery built from every pumping period and its rate (their Equation 12.2). The slope of that line is 2.30 / (4πT).

After a constant-drawdown test, such as a free-flowing well, use the discharge just before the well was shut in as Q (Section 13.2, citing Rushton and Rathod 1980). After a step-drawdown test, the Eden-Hazel method applies to the recovery phase (Section 15.3.3); see step-drawdown test analysis for the steps themselves.

What conditions must the readings meet?

The readings must meet the Theis assumptions, adjusted for recovery, with unsteady flow and both u and u′ below 0.01 (Kruseman and de Ridder 1994, Section 13.1.1):

• pumping time t_p > 25r²S/T, so u was small by the time the pump stopped;
• t′ > 25r²S/T, so only later recovery readings belong on the line.

Other settings add limits:

• Leaky aquifers: short times, t_p + t′ ≤ L²S/(20T) or cS/20 (Section 13.1.2).
• Unconfined aquifers: only late-time recovery data, once the elastic storage effects after shutdown have dissipated (Section 13.1.3, citing Neuman 1975).
• Partially penetrating wells: both t_p and t′ greater than 10D²S/T, where D is the aquifer thickness (Section 13.1.4, citing Uffink 1982).
• The pumped well: storage in the casing can distort early recovery, so t_p and t′ should exceed 25r_c²/T, where r_c is the radius of the casing in which the level moves. For a line at least one log cycle long, Uffink (1982) recommends both be at least 500r_c²/T (Section 15.3.1).

What checks should you run on a recovery analysis?

Check the line, then compare it with the pumping analysis:

• Fit only the readings that satisfy the u′ condition, near the left of the plot, and see that they form a straight line over at least one log cycle.
• Compare T from recovery with T from the pumping data; a large difference suggests that one record departs from the assumptions.
• Look at where the line crosses s′ = 0. Far from t/t′ = 1, S and S′ differ.
• Check the water level returns to its pre-test level. If it does not, the head may have a natural rise or fall that should be corrected for before analysis (Section 2.8.2).

Sources: Theis (1935), Transactions AGU 16, 519-524; Birsoy and Summers (1980), Ground Water 18, 137-146; Rushton and Rathod (1980), Ground Water 18, 61-69; Kruseman and de Ridder (1994), Analysis and Evaluation of Pumping Test Data, 2nd edition, ILRI Publication 47, Chapters 2, 13 and 15.

Can you run the Theis recovery method in a browser?

Yes: PumpPilot is a browser tool whose Theis recovery tool fits a least-squares line to residual drawdown against log(t/t′) at one well and reports T and the number of readings used. It needs a step at rate 0 in the schedule where pumping stopped, takes Q from the first pumping step and uses every reading after the stop, including early ones that may not meet the u′ condition. It does not estimate S or S/S′, check u′, or run Birsoy-Summers, and it has no slug tests. Its Theis, Hantush-Jacob, Papadopulos-Cooper and Neuman fits handle the schedule by superposition, so recovery readings can be fitted with the pumping ones. Straight-line methods in the PumpPilot docs describes the tool.