Hantush-Jacob and Neuman Methods for Pumping Tests

What makes a leaky or unconfined aquifer test different?

In both, the well draws water from a source the Theis solution leaves out: leakage through an aquitard in a leaky aquifer, and slow drainage of the water table in an unconfined one. Hantush and Jacob (1955) solved the first case and Neuman (1975) the second. Choosing between these and Theis from the derivative plot is covered in Theis vs Cooper-Jacob; this post works through each method once you have chosen it.

How do you recognise a leaky aquifer from the readings?

The drawdown stops growing and levels off while the pumping rate stays constant. Against the logarithm of time, the curve rises, bends through an inflection point and flattens toward a maximum drawdown s_m. The readings cannot prove leakage on their own, so look for an aquitard next to the screen; Kruseman and de Ridder (1994, Chapter 4) recommend piezometers in the pumped aquifer, in the aquitard and in the aquifer above it.

The solution also assumes the water table above the aquitard stays put; in a long test where it falls, ignoring that can cause considerable errors (Neuman and Witherspoon 1969b, in Kruseman and de Ridder 1994, Chapter 4).

What equation does the Hantush-Jacob method fit?

It fits the leaky well function W(u, r/L), which adds a second parameter to the Theis u:

s = Q / (4πT) × W(u, r/L), with W(u, r/L) = ∫_u^∞ (1/y) exp(−y − r²/(4L²y)) dy and u = r²S / (4Tt)

(Hantush and Jacob 1955; Kruseman and de Ridder 1994, Equations 4.6 and 4.7). The leakage factor is L = √(Tc), where c = D'/K' is the aquitard's thickness over its vertical hydraulic conductivity (Equation 4.2). For large L the solution becomes Theis.

Kruseman and de Ridder call matching with W(u, r/L) Walton's method, after Walton (1962), and keep the name Hantush-Jacob for a steady-state distance-drawdown method (Section 4.1.2); software and this post use it for the unsteady solution. Beyond the Theis conditions, it assumes the aquitard stores no water and the flow through it is vertical.

How do you match the r/L type curves?

Lay the drawdown against time, on log-log paper, over a family of curves of W(u, r/L) against 1/u drawn at the same scale, one curve per value of r/L (Kruseman and de Ridder 1994, Procedure 4.3). Find the curve that fits, pick any match point and note W(u, r/L), 1/u, s and t there. Then:

• T = Q W(u, r/L) / (4πs);
• S = 4Ttu / r²;
• L = r divided by the r/L of the matching curve, and c = L²/T.

T, S and c from each piezometer should roughly agree. A unique match needs enough readings from before leakage takes effect, or a fairly large r/L.

How does the Hantush inflection-point method work?

It reads the parameters from the point where the semi-log curve turns, and needs the steady-state drawdown s_m, observed or extrapolated (Hantush 1956; Kruseman and de Ridder 1994, Section 4.2.2). At the inflection point P:

• s_p = 0.5 s_m = Q / (4πT) × K₀(r/L), with K₀ the modified Bessel function of the second kind and zero order (Equation 4.8);
• u_p = r²S / (4Tt_p) = r / (2L) (Equation 4.9);
• the slope per log cycle of time is Δs_p = 2.30Q / (4πT) × e^−r/L (Equation 4.10);
• so 2.30 s_p / Δs_p = e^r/L K₀(r/L) (Equation 4.12).

Halve s_m to find s_p, read t_p off the curve, take Δs_p from the tangent at P, and solve Equation 4.12 for r/L from a table of e^xK₀(x). T follows from Equation 4.10, S from Equation 4.9 and c from L²/T. For the piezometer at 90 m in the Dalem test in the Netherlands, this gave L = 600 m, T = 1665 m²/d and S = 1.7 × 10⁻³ (Example 4.4).

Everything rests on the extrapolated s_m, so put the results back into Equation 4.6 at several times; if the computed drawdowns miss the readings, adjust s_m or the slope and repeat.

How far apart are the leaky methods on one test?

Far enough to report a range. On the Dalem test, six methods gave T from 1515 to 2126 m²/d and L from 600 to 1100 m (Kruseman and de Ridder 1994, Table 4.4). The authors blame extrapolated steady-state drawdowns and matching by eye.

How do you recognise delayed yield in an unconfined aquifer?

On log-log axes the drawdown traces an S: steep early, flat in the middle, steep again late (Kruseman and de Ridder 1994, Chapter 5). The early segment, often only the first few minutes, follows a Theis curve as water comes from elastic storage. The flat segment reflects the water table draining and may approach the horizontal. Once flow is mostly horizontal, the late segment follows a Theis curve again.

The flat segment affects drawdown much as leakage does, so a test stopped there can pass for a leaky aquifer. Pumping longer separates them: leaky drawdown stays level, unconfined drawdown climbs again.

What do Neuman's equations and beta mean?

Neuman treats the elastic storativity S_A and the specific yield S_y as separate constants and writes s = Q / (4πT) × W(u_A, u_B, β) (Neuman 1975; Kruseman and de Ridder 1994, Equation 5.1). At early time this reduces to W(u_A, β) with u_A = r²S_A / (4Tt), and at late time to W(u_B, β) with u_B = r²S_y / (4Tt) (Equations 5.2 to 5.5). The third parameter is

β = r²K_v / (D²K_h)

where D is the saturated thickness and K_v and K_h are the vertical and horizontal hydraulic conductivities; for an isotropic aquifer β = r²/D² (Equation 5.6). Because β contains r, each observation well matches a different curve, while K_v/K_h should come out the same.

The conditions include S_y/S_A > 10, a negligible effect of the unsaturated zone, small well diameters and a fully penetrating observation well (Section 5.1.1). Rushton and Howard (1982) noted that such a well short-circuits vertical flow; short-screen piezometers work if their drawdowns are averaged over the thickness (Van der Kamp 1985).

How do you fit the early and late segments?

Match the early readings to a type A curve, then slide the plot horizontally onto the type B curve with the same β (Kruseman and de Ridder 1994, Procedure 5.1). Match point A gives T and S_A, match point B gives T again and S_y, and the two values of T should be close. Then K_h = T/D, and K_v follows from β, r, D and K_h.

The B curves are the weak step. At Vennebulten, with r = 90 m and β = 0.01, they gave T = 1616 m²/d but S_y = 4.9 × 10⁻³, which the authors call unreasonably low (Example 5.1). They attribute such values to neglect of flow in the capillary fringe, and note that a Theis analysis of the late segment yields a fairly realistic S_y (Van der Kamp 1985).

Where drawdown is large compared with D, Jacob (1944) corrected it as s' = s − s²/(2D). Neuman (1975) held that this applies strictly only to late readings on the Theis curve (Kruseman and de Ridder 1994, Chapter 5).

Where do these formulas come from?

• Hantush and Jacob (1955), Transactions AGU 36(1), 95-100.
• Neuman (1975), Water Resources Research 11(2), 329-342.
• Kruseman and de Ridder (1994), Analysis and Evaluation of Pumping Test Data, 2nd edition, ILRI Publication 47, Chapters 4 and 5.

PumpPilot is a browser tool that fits the Hantush-Jacob solution (T, S and the leakage factor, which it labels B) and the Neuman solution (T, S, S_y and K_z/K_r, given the saturated thickness) by least squares over all wells at once, and reports c = B²/T and K_z. It has no inflection-point tool, and no solutions for aquitard storage, boundaries, partially penetrating pumping wells or slug tests. Choosing a solution and Fitting explain both.