Theis Solution Calculator
Transient drawdown around a pumping well in a confined aquifer — and whether the Cooper–Jacob straight-line approximation may be used instead.
✓ Reviewed by a hydrogeologist (Ph.D.) SI and US customary units Cooper–Jacob validity check Sources: Theis 1935 · Cooper & Jacob 1946
What to enter for what you want
Every derived quantity is computed on every keystroke, so Solve for does not decide what you can see — it only decides which variable is treated as the unknown.
| To find | Enter | Read it in |
|---|---|---|
| Drawdown at an observation well | Q, T, S, r, t | Headline |
| Transmissivity from a pumping test | Q, s, S, r, t — set Solve for to T | Headline |
| The rate that produces a target drawdown | T, S, r, t, s — set Solve for to Q | Headline |
| Whether Cooper–Jacob is usable | any of the above | u and the validity note |
| How far the cone reaches | Q, T, S, t | Radius of influence |
Solving for T is the pumping-test case, and it cannot be done algebraically — T appears both outside the well function and inside its argument. The calculator solves it numerically by bisection on a logarithmic scale, which is why the value settles rather than appearing instantly.
What the equation says
By 1935 the steady-state well equations of Thiem had a well-known defect: real aquifers never reach steady state, because the cone of depression keeps expanding for as long as the pump runs. Charles Theis, working with the mathematician Clarence Lubin, noticed that this expanding cone is formally the same problem as heat conducting away from a line source — a problem already solved. Transplanting that solution gave hydrogeology its first transient analysis.
s \;=\; \frac{Q}{4\pi T}\,W(u), \qquad u \;=\; \frac{r^{2}S}{4Tt}
W(u) is the well function, which is the exponential integral E_1(u):
W(u) \;=\; \int_{u}^{\infty}\frac{e^{-y}}{y}\,dy \;=\; -\gamma - \ln u + \sum_{n=1}^{\infty}\frac{(-1)^{n+1}u^{n}}{n\cdot n!}
with \gamma = 0.5772157 (Euler’s constant). This calculator evaluates it with the rational approximations of Abramowitz & Stegun (5.1.53 and 5.1.56), which agree with published well-function tables to within 0.02 % over the range 10^{-5} \le u \le 5.
The variables
| Symbol | Quantity | SI unit | Note |
|---|---|---|---|
| s | Drawdown at the observation well | m | Measured down from the pre-pumping level |
| Q | Pumping rate | m³/s | Held constant — the solution assumes it never varies |
| T | Transmissivity, Kb | m²/s | Conductivity times aquifer thickness |
| S | Storativity (storage coefficient) | – | Dimensionless; typically 10⁻⁵–10⁻³ when confined |
| r | Distance from the pumping well | m | Radial |
| t | Time since pumping began | s | From the instant the pump starts |
| u | Dimensionless time parameter | – | Small u means late time or short distance |
What u actually measures
Everything about the shape of the response is carried by the single group u = r^2S/4Tt. It is small when the test has run a long time or the observation well is close, and large early on or far away. This is why u — not time alone — decides which analysis applies: an observation well at 20 m may already be in the late-time regime while one at 300 m is still in the early-time regime at the same instant.
When Cooper–Jacob can replace it
For small u the series for W(u) is dominated by its first two terms, and the solution collapses into a straight line on a semi-logarithmic plot:
s \;\approx\; \frac{2.303\,Q}{4\pi T}\,\log_{10}\!\left(\frac{2.25\,T t}{r^{2}S}\right) \qquad (u \lesssim 0.01)
That is the Cooper–Jacob (1946) approximation, and it is what makes graphical pumping-test analysis possible. The threshold matters: at u = 0.01 the error is about 1 %, but it grows quickly. The calculator reports the Cooper–Jacob value and its error alongside the exact Theis result, so you can see for yourself whether the shortcut is safe at your r and t.
Above u ≈ 1 you are in the earliest moments of the test at that radius. The Theis solution is still exact there, but wellbore storage — water drawn from the casing rather than the aquifer — distorts real measurements, so early data are usually discarded rather than fitted.
Worked example
Problem. A confined aquifer has T = 1.2×10⁻² m²/s and S = 2.0×10⁻⁴. A well is pumped at 0.020 m³/s. Find the drawdown (a) after one day at an observation well 50 m away, and (b) after 30 minutes at one 200 m away. In each case, check whether Cooper–Jacob could have been used.
(a) t = 86 400 s, r = 50 m
u = r²S/(4Tt) = (2500 × 2.0×10⁻⁴) / (4 × 1.2×10⁻² × 86400) = 1.206×10⁻⁴W(u) = 8.446s = Q/(4πT) · W(u) = 0.020/(4π × 1.2×10⁻²) × 8.446 = 0.1326 × 8.446 = 1.120 m- Cooper–Jacob gives 1.121 m — an error of 0.04 %. Since u ≪ 0.01, the straight-line method is entirely safe here.
(b) t = 1 800 s, r = 200 m
u = (40000 × 2.0×10⁻⁴) / (4 × 1.2×10⁻² × 1800) = 9.26×10⁻²W(u) = 1.893s = 0.1326 × 1.893 = 0.251 m- Cooper–Jacob gives 0.239 m — an error of −4.7 %, an underestimate.
The difference between the two cases is not the aquifer; it is u. At 200 m and 30 minutes the test has not yet reached the late-time regime, and using the straight-line method there would bias the fitted transmissivity. This is the single most common error in pumping-test analysis: applying Cooper–Jacob to early data because the plot happens to look straight.
Frequently asked
What assumptions am I accepting by using this?
That the aquifer is confined, homogeneous, isotropic, of uniform thickness, and infinite in extent; that the well fully penetrates it; that the pumping rate is constant; and that water is released from storage instantaneously. Real aquifers violate several of these routinely. The Theis curve is nevertheless the reference against which departures are diagnosed — leakage, boundaries, and delayed yield are all identified by how measured drawdown deviates from it.
Can I use this for an unconfined aquifer?
Not directly. In an unconfined aquifer the saturated thickness itself decreases as water is removed, and drainage from the pores is not instantaneous. Use the Neuman solution, or apply the Jacob correction s′ = s − s²/2b when the drawdown is a small fraction of the saturated thickness b.
Why does drawdown never stabilise?
Because the solution assumes an infinite aquifer with no source of recharge. The cone must keep expanding to supply the pumped water. If measured drawdown does level off, that is informative: it means a boundary or a source of leakage has been reached, and the Theis assumptions no longer hold.
What is a realistic storativity?
For confined aquifers, 10⁻⁵ to 10⁻³ — the water comes from expansion of water and compression of the matrix, so very little is available per unit head. Unconfined values are three orders of magnitude larger (0.05–0.30) because the water is physically drained from pores. A “confined” test returning S = 0.1 is telling you the aquifer is not confined.
Read further
Pumping Tests & the Theis Equation: Measuring Aquifer Properties in Python works through a full pumping test in the Groundwater Science series — fitting the type curve to real drawdown data rather than evaluating the solution at a single point, which is what this calculator does.
References
- Theis, C.V. (1935) The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Transactions, American Geophysical Union 16, 519–524.
- Cooper, H.H. & Jacob, C.E. (1946) A generalized graphical method for evaluating formation constants and summarizing well-field history. Transactions, American Geophysical Union 27, 526–534.
- Abramowitz, M. & Stegun, I.A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington DC. Equations 5.1.53 and 5.1.56.
- Freeze, R.A. & Cherry, J.A. (1979) Groundwater. Prentice-Hall, Englewood Cliffs, NJ.
- Kruseman, G.P. & de Ridder, N.A. (1990) Analysis and Evaluation of Pumping Test Data, 2nd ed. ILRI, Wageningen.