Theis vs Cooper-Jacob: Which Pumping Test Method to Use

What is the difference between Theis and Cooper-Jacob?

The Theis method fits the whole drawdown curve of a confined aquifer, while the Cooper-Jacob method fits only the late part of it, where drawdown plotted against the logarithm of time becomes a straight line. Cooper and Jacob (1946) derived their method from the Theis (1935) solution, so both rest on the same assumptions and should give the same transmissivity when those assumptions hold. The choice between them is about which readings you trust and how many you have. If the record follows neither, the aquifer may be leaky or unconfined, or the well may be storing water, and Hantush-Jacob, Neuman or Papadopulos-Cooper is the better place to start.

What does the Theis solution assume?

The Theis solution assumes an ideal confined aquifer pumped at a constant rate by a well too thin to store water. Kruseman and de Ridder (1994, Chapter 3) list the conditions:

• the aquifer is confined and of seemingly infinite areal extent;
• it is homogeneous, isotropic and of uniform thickness over the area the test reaches;
• the piezometric surface is horizontal, or nearly so, before pumping;
• the well is pumped at a constant rate and penetrates the full thickness of the aquifer;
• water is released from storage instantaneously as the head declines;
• the well diameter is small, so storage in the well can be neglected.

The drawdown s at distance r and time t is then s = Q / (4πT) × W(u), with u = r²S / (4Tt), where Q is the pumping rate, T the transmissivity, S the storativity and W(u) the Theis well function (Theis 1935; Kruseman and de Ridder 1994, Section 3.2.1).

How does Cooper-Jacob get T and S from a straight line?

Cooper-Jacob takes transmissivity from the slope of the line and storativity from where the line crosses zero drawdown. The well function expands as W(u) = -0.5772 - ln u + u - u²/(2·2!) + ..., and when u is small every term after ln u can be dropped, giving

s = (2.3Q / (4πT)) × log10(2.25Tt / (r²S))

(Cooper and Jacob 1946; Kruseman and de Ridder 1994, Section 3.2.2). At one well this is a straight line against log t. If Δs is the change in drawdown over one log cycle of time and t0 is the time at which the extended line crosses zero drawdown, then

• T = 2.3Q / (4πΔs)
• S = 2.25Tt0 / r²

With Q in m³/d and t0 in days, T is in m²/d.

When is u small enough for the straight line?

The usual textbook condition is u < 0.01, which is the same as t > 25r²S/T. Kruseman and de Ridder (1994, Section 3.2.2) call it "rather rigid" and give the error of the approximation as less than 1, 2, 5 and 10 per cent for u below 0.03, 0.05, 0.1 and 0.15. Since it is hard to see on a plot where readings leave the line anywhere between u = 0.01 and 0.1, they suggest u < 0.1 as the working condition.

Distance enters as r², so a well ten times further from the pumping well needs a hundred times longer to reach the line. And u depends on T and S, so the check comes after the fit: compute u at the first reading on the line and move the start later if it is too large.

The logarithmic derivative shows the straight-line period directly. On the Cooper-Jacob line, ds/d(ln t) = Q/(4πT), a constant (Renard et al. 2009, Equation 2), so a flat derivative on a log-log plot marks the readings where the line is valid. Renard et al. recommend running a test until the derivative has stayed flat for at least 1 to 1.5 log cycles.

When should you use Theis instead of Cooper-Jacob?

Use Theis when most readings come before u is small, as in a short test or at a distant observation well. Cooper-Jacob discards every reading before the line starts, which can leave few points on the line, or none.

Use Cooper-Jacob when you have a long late-time record and want a value of T that does not lean on early readings, which Kruseman and de Ridder (1994, Section 3.2.1) advise weighting less because the pumping rate takes time to settle. They also prefer a straight late segment to matching by eye (Section 2.9.1). That advice dates from fitting on paper; a least-squares Theis fit uses every reading and needs no match point, so run both. If they disagree, the record departs from the Theis assumptions somewhere, and the derivative usually shows where.

How do you tell a leaky aquifer from an unconfined one?

Plot drawdown and its logarithmic derivative on log-log axes and look at what happens after early time (Renard et al. 2009; Kruseman and de Ridder 1994, Figure 2.12):

• Confined (Theis): the derivative rises, then stays flat.
• Leaky (Hantush-Jacob): early readings follow Theis, then drawdown levels off as water leaks through the aquitard, and the derivative falls toward zero. A constant-head boundary such as a river looks similar, though Renard et al. note that in theory the derivative falls faster for leakage.
• Unconfined with delayed yield (Neuman): drawdown follows Theis briefly, flattens, then steepens again, and the derivative dips before returning to its flat level. Double-porosity fractured aquifers give the same shape.

Renard et al. stress that more than one model can produce the same curve, so the geology decides: an aquitard above the screen points to leakage, a water table in the tested unit to delayed yield, a river nearby to a boundary.

What do Hantush-Jacob and Neuman add?

Hantush-Jacob adds a leakage factor L = √(Tc), where c is the aquitard's thickness divided by its vertical hydraulic conductivity (Hantush and Jacob 1955; Kruseman and de Ridder 1994, Chapter 4). It assumes the aquitard releases no water from storage, which Kruseman and de Ridder (Section 4.2), citing Neuman and Witherspoon (1972), warn can overestimate the aquifer's hydraulic conductivity and underestimate the aquitard's.

Neuman (1972, 1975) treats the early elastic storativity and the specific yield as separate constants, and needs the saturated thickness and the ratio of vertical to horizontal conductivity. Kruseman and de Ridder (Chapter 5) note that a storativity from the early segment cannot predict long-term drawdown, and that Neuman's method often returns unreasonably low specific yields.

What if the readings come from the pumping well?

Pumping-well readings include water stored in the casing and losses at the well face, so use Papadopulos-Cooper or treat the straight line with care. Wellbore storage shows on a log-log plot as an early segment with a slope of one (Kruseman and de Ridder 1994, Section 2.9.2), then a hump in the derivative (Renard et al. 2009). Papadopulos and Cooper (1967) found its effect negligible once t > 25rc²/T, where rc is the radius of the casing in which the water level moves (Kruseman and de Ridder, Chapter 15). After that the Cooper-Jacob slope still gives T, but S depends on the effective well radius and well losses, so take S from an observation well.

Which method should you choose?

Choose by what the derivative shows and where the readings were taken:

What you see	Start with	Check
Confined, observation well, long record, flat late derivative	Cooper-Jacob and Theis together	u at the first point on the line
Confined, short test or distant well	Theis	Early readings disturbed by the pump settling
Drawdown levels off, derivative falls toward zero, aquitard present	Hantush-Jacob	A river or other recharge boundary nearby
Derivative dips and recovers, water-table aquifer	Neuman	Saturated thickness; a plausible specific yield
Pumping-well readings, slope of one at early time	Papadopulos-Cooper	S from an observation well

Where do these formulas come from?

All of them are in Kruseman and de Ridder, which cites the original papers:

• Theis (1935), Transactions AGU 16, 519-524.
• Cooper and Jacob (1946), Transactions AGU 27, 526-534.
• Hantush and Jacob (1955), Transactions AGU 36, 95-100.
• Papadopulos and Cooper (1967), Water Resources Research 3, 241-244.
• Neuman (1972), Water Resources Research 8, 1031-1045; Neuman (1975), Water Resources Research 11, 329-342.
• Kruseman and de Ridder (1994), Analysis and Evaluation of Pumping Test Data, 2nd edition, ILRI Publication 47.
• Renard, Glenz and Mejias (2009), Understanding diagnostic plots for well-test interpretation, Hydrogeology Journal 17, 589-600.

Can you run these methods in a browser?

Yes: PumpPilot is a free browser tool that fits the Theis, Hantush-Jacob, Papadopulos-Cooper and Neuman solutions to readings from any number of wells, plots the logarithmic derivative beside the drawdown, and ranks solutions by AICc. Unless you set the start yourself, its Cooper-Jacob tool begins the line on the later half of the record and moves it later until u is at most 0.05 at every point used. It has no solutions for boundaries, aquitard storage or slug tests yet. Choosing a solution explains the derivative shapes, and Straight-line methods covers Cooper-Jacob, Theis recovery and step-drawdown.