Well Hydraulics and Pumping Tests: Method

Well Functions, Straight-Line and Least-Squares Analysis, Superposition, Slug Tests and Their Limits

Scope

The first dropdown chooses one of nine analyses. Four take field data: an aquifer test read as drawdown against time in one observation well, as drawdown against distance in several wells at one time, or as residual drawdown during recovery, and a slug test. Five predict: the drawdown of one well (Theis, Cooper-Jacob, a leaky aquifer, a boundary, recovery), steady flow between two observation wells (Thiem, Dupuit), a well field with stepped rates by superposition, well loss and specific capacity from a step test, and the capture zone of a well in uniform regional flow. Every analysis opens with a worked case; the cases from U.S. Geological Survey publications are named below.

Not covered, and named so that a result is not stretched over them: partial penetration of the pumped well (its effect on the drawdown is insignificant only beyond a distance of about 1.5 b √(K_h/K_v) from the well, b the aquifer thickness; nearer than that the formulas here do not hold), delayed yield of an unconfined aquifer (Boulton, Neuman), storage in the well bore of a large-diameter well, storage in the confining bed of a leaky aquifer (the Hantush-Jacob function assumes none, and Lohman warns that leakance fitted on that assumption comes out too large), anisotropy in plan, sloping or thinning aquifers, multiple boundaries (one image well only), the Cooper, Bredehoeft and Papadopulos type-curve method for slug tests and oscillating slug tests in very permeable formations.

Conventions and units

Drawdown is positive downward from the static level. Distances, drawdowns, radii and thicknesses typed in single fields go through the site's unit dictionary. Pumping rates, times, transmissivity and hydraulic conductivity are plain numbers in the units chosen in the four unit dropdowns; the unit switch does not convert them. Pasted tables carry lengths in feet with U.S. units and in metres with SI, and rates and times in the units of the dropdowns; slug-test times are always seconds. Internally everything is in metres and seconds. The constant written 2.25 in the Cooper-Jacob formulas is 4 e^−γ = 2.2458 and 2.303 is ln 10; both are used at full precision.

Theis equation and the well function

For a well pumping at the constant rate Q from a confined, homogeneous, isotropic aquifer of wide extent that it fully penetrates, with water released from storage at once as the head falls, the drawdown at the distance r after the time t is

s = (Q/(4 π T)) W(u),   u = r² S/(4 T t),   W(u) = ∫_u^∞ e^−y dy/y

(Lohman, equations 44 to 47). W(u) is the exponential integral. For u ≤ 1 the page sums the series −0.577216 − ln u + u − u²/(2·2!) + u³/(3·3!) − …; above 1 it uses the continued fraction e^−u/(u + 1 − 1/(u + 3 − 4/(u + 5 − …))). Both are shown where their ranges overlap. The functions reproduce the 1,181 cells of Lohman's Table 5 that can be read from the scanned text and agree with a reference library to 13 figures.

Cooper-Jacob approximation

The first two terms of the series give s = (Q/(4 π T)) ln(2.25 T t/(r² S)), a straight line of drawdown against the logarithm of time or of distance. It is within 0.25 % of the Theis value for u ≤ 0.01 and about 2 % at u = 0.05; the page flags a case outside u ≤ 0.01 and gives the time after which the approximation holds at the distance entered. The radius at which this expression gives zero drawdown, r₀ = 1.5 √(T t/S), is reported as the radius of influence; it grows with the square root of time and is a property of the approximation, not a physical edge of the cone.

Leaky aquifer

When the confining bed leaks from a source bed whose head stays constant, and stores no water itself, the well function becomes (Hantush and Jacob; Lohman, equation 85, with r/B = 2v)

W(u, r/B) = ∫_u^∞ exp(−y − (r/B)²/(4y)) dy/y,   B = √(T b′/K′)

with b′ and K′ the thickness and vertical conductivity of the confining bed. The page integrates it numerically after the substitution y = e^x (Simpson's rule, 2,000 intervals). At long times it tends to 2 K₀(r/B), a steady drawdown. All 202 printed cells of Reed's Table 4.1 (after Hantush) are reproduced within two units of the fourth decimal; ten of them differ from the exact value by one or two units, which is the accuracy of the 1961 table.

Boundaries, recovery and superposition

The flow equation is linear, so drawdowns add. A straight barrier is replaced by an image well at the mirror position pumping at the same rate; a recharge boundary (a stream or lake in full connection) by an image well recharging at that rate. Enter the distance from the point of interest to the image well. Near a recharge boundary the drawdown levels off at (Q/(2 π T)) ln(r_i/r).

When pumping stops, the well is treated as still pumping while an equal recharge well starts at the moment of shutdown; the residual drawdown is the difference. The same idea handles a well field and stepped rates: every line of the pasted list is a well with its place, rate, start time and optional stop time, and a change of rate is a further line at the same place carrying the change.

Steady flow: Thiem and Dupuit

Between two observation wells in steady radial flow, T = Q ln(r₂/r₁)/(2 π (s₁ − s₂)) for a confined aquifer (Thiem; Lohman, equation 32) and K = Q ln(r₂/r₁)/(π (h₂² − h₁²)) for an unconfined one with saturated thickness h = b − s (Dupuit; equation 31). The second is the first applied to drawdowns corrected to s − s²/(2b) (Jacob; equation 33). The distance r₀ at which the line through the wells reaches zero drawdown gives S = 2.25 T t/r₀² when the time of the readings is known (equation 59), and for a thin unconfined aquifer the corrected value S′ = S (b − s)/b with s the drawdown at the geometric mean distance (equation 64). The default is the Wichita test of Professional Paper 708 with each pair of wells averaged: Lohman's graphical line gives 20,700 ft²/day and S = 0.35 (0.29 corrected); the page gives 20,820 ft²/day, 0.345 and 0.293.

Well loss and specific capacity

The drawdown in the pumped well is written s_w = B Q + C Q² (Jacob), a formation loss and a well loss. A step test read at equal times gives a straight line of s_w/Q against Q whose intercept is B and slope C, fitted by least squares. The page then gives the specific capacity Q/s_w and the well loss at an operating rate. With the Cooper-Jacob drawdown at the well face, Q/s_w = 4 π T/ln(2.25 T t/(r_w² S)) (Lohman, equation 139), which the page solves for T by iteration, once with the measured drawdown and once with the well loss removed. The estimate needs a storage coefficient, to which it is only weakly sensitive, and assumes a fully penetrating well.

Capture zone

A sink of strength Q in a uniform flow T i per unit width has the stream function ψ = T i y − (Q/2π) θ, with θ the angle about the well from the downgradient direction. The streamline ψ = 0 divides the water that reaches the well from the water that passes: y = (Q/(2 π T i)) θ. It gives the stagnation point Q/(2 π T i) downgradient, the half-width Q/(4 T i) abreast of the well and Q/(2 T i) far upgradient. The result is for steady flow in a confined aquifer, or an unconfined one whose drawdown is small against its thickness.

Aquifer-test analysis

Straight line (time-drawdown). A line s = a + Δs log₁₀ t is fitted by least squares; T = 2.303 Q/(4 π Δs) and S = 2.25 T t₀/r² with t₀ the time of zero drawdown on the line (Lohman, equations 56 and 59). The readings that belong on the line are chosen automatically: starting from the later half of the record, the page fits, computes u for every reading from the fitted T and S, keeps the readings with u at or below the limit (0.01 by default), and repeats until the set stops changing. If fewer than three readings qualify, the limit is relaxed to 0.05 and then 0.1 and the page says so. A number in "Late Points" overrides the choice.

Theis fit. The Theis curve is fitted to all readings by least squares on the drawdown, with ln T and ln S as the unknowns (Levenberg-Marquardt, analytic derivatives, several starting points). This is the numerical equivalent of matching the type curve by eye and uses the early readings that the straight line must discard. The 95 % confidence intervals come from the linearised covariance at the solution with Student's t; they assume independent errors of equal variance and say nothing about whether the Theis model fits the aquifer, so look at the residuals in the table and at the two charts. For the default data (Lohman, Table 6, well N-1) the fit gives 13,370 ft²/day and 2.02 × 10⁻⁴; Lohman's match by eye gives 13,700 and 2 × 10⁻⁴.

Distance-drawdown. A line of drawdown against log₁₀ r through all the wells gives T = 2.303 Q/(2 π Δs) and S from the distance of zero drawdown, with Jacob's correction when a saturated thickness is entered. The default is the six wells of the Wichita test.

Recovery. For small u′ the residual drawdown is s′ = (2.303 Q/(4 π T)) log₁₀(t/t′), so a line through the late readings gives T. The page uses the later half of the recovery record unless a number of late points is entered. The ratio t/t′ at which the line reaches zero should be near 1; the page notes a large departure.

Slug tests

Both methods fit a straight line to ln(y/y₀) against time over the window of y/y₀ entered; its slope is the decay rate λ and 1/λ is Hvorslev's basic time lag T₀. Real records are often slightly curved, so the window matters: for the default test the window 0.2 to 1 gives 5.0 ft/day by Bouwer and Rice and the whole record gives 4.5 ft/day, the value the U.S. Geological Survey spreadsheet reports from a line drawn by hand.

Hvorslev (Figure 18 of the bulletin, variable-head and basic-time-lag columns): K_h = d² ln[mL/D + √(1 + (mL/D)²)]/(8 L T₀) for a well point in uniform soil (case G), with 2mL/D in place of mL/D for an intake against an impervious boundary (case F); d and D are the diameters of the standpipe and of the intake and m = √(K_h/K_v). In radii this is the expression printed on the page, and for mL/D above 4 it reduces to the familiar K = r_c² ln(L/r_w)/(2 L T₀).

Bouwer and Rice: K = r_c² ln(R_e/r_w)/(2 L_e) (1/t) ln(y₀/y_t) with ln(R_e/r_w) = [1.1/ln(H/r_w) + (A + B ln((D − H)/r_w))/(L_e/r_w)]⁻¹ for a partially penetrating screen and [1.1/ln(H/r_w) + C/(L_e/r_w)]⁻¹ for a screen reaching the base of the aquifer; the term ln((D − H)/r_w) is not taken above 6. A, B and C are functions of L_e/r_w given by curves in the paper; the page reads them from the fourteen-point look-up table of the U.S. Geological Survey spreadsheet, interpolated linearly in log₁₀(L_e/r_w) as the spreadsheet does, for L_e/r_w from 3.2 to 1,890. The fully penetrating form is used when the screen bottom is within 2 % of the saturated thickness. When the water table lies within the screen, the casing radius of a rising-head test should be increased for the porosity of the gravel pack; the page does not do this.

Checks

The test script compares the library with Lohman's Tables 5, 18 and 19 and his Wichita, Table 6 and leaky examples, with Reed's Table 4.1, with the cell values of four U.S. Geological Survey spreadsheets, with Hvorslev's formulas, and with an independent numerical reference written separately (exponential integral, quadrature, least squares, root finding). The fits are also run on 20 synthetic tests with 1 % noise, where they return the transmissivity within 4 % and agree with the reference fit to five figures. Two misprints in Lohman's Table 18 (the rows for 4,000 ft and 30,000 ft) are recorded with the sources.