Theis Equation Drawdown Calculator
Theis published his equation in 1935. It gives the drawdown around a well pumped at a constant rate Q in a confined aquifer. The drawdown s at distance r from the well and time t after pumping started depends on the transmissivity T and the storage coefficient S through the dimensionless argument u = r²S/(4Tt), with s = Q·W(u)/(4πT).
W(u) is the well function, the exponential integral of u. It falls as u rises, so drawdown grows with time and shrinks with distance. This calculator evaluates W(u) directly — a power series for u below 1 and a continued fraction for u of 1 or more — after converting every input to m³/day, m²/day, metres and days, so any mix of units gives the same answer.
Cooper and Jacob showed in 1946 that for u up to about 0.01 a straight-line form is enough: sCJ = Q·(−0.5772156649 − ln u)/(4πT). The calculator shows it beside the Theis result with the difference between them, flags it as not valid above u = 0.01, gives the time tCJ after which it applies, and solves for the time needed to reach a target drawdown.
Results
| u (dimensionless) | — |
|---|---|
| W(u) (dimensionless) | — |
| Theis drawdown s | — |
| Cooper-Jacob drawdown sCJ | — |
| Difference sCJ − s | — |
| Cooper-Jacob validity | — |
| tCJ, time after which u ≤ 0.01 | — |
| Time to reach target drawdown | — |
Formulas used
- Inputs are converted to m³/day, m²/day, m and day before computing.
- u = r²S / (4Tt)
- u < 1: W = −0.5772156649 − ln u + u + Σ aₙ for n ≥ 2, with a₁ = u and aₙ = −aₙ₋₁·u·(n−1)/n², until |aₙ| < 10⁻¹⁷
- u ≥ 1: f = u + 201, then for i = 100 down to 1, f = u + 2i − 1 − i²/f; W = exp(−u)/f
- s = Q·W(u) / (4πT)
- sCJ = Q·(−0.5772156649 − ln u) / (4πT), valid for u ≤ 0.01
- tCJ = r²S / (0.04T)
- Wt = 4πT·st / Q; solve W(u) = Wt by bisection on ln u in [ln 10⁻³⁰, ln 50]; t = r²S / (4Tu)
How to use the calculator
- Enter the constant pumping rate Q and choose its unit: gpm, ft³/day, m³/day or L/s.
- Enter the transmissivity T and choose its unit: ft²/day, gpd/ft or m²/day.
- Enter the storage coefficient S, a dimensionless value of at least 10⁻⁹ and below 1.
- Choose the length unit, ft or m, then enter the distance r from the pumped well to the point of interest.
- Enter the time t since pumping started and choose min, h or day.
- Optionally enter a target drawdown to find the time needed to reach it at distance r.
- Read u, W(u), the Theis drawdown s, the Cooper-Jacob drawdown with its difference and validity, tCJ and the time to the target. Results update as you type; Calculate runs the same computation.
Frequently asked questions
What does this Theis drawdown calculator compute?
For a well pumped at a constant rate Q in a confined aquifer with transmissivity T and storage coefficient S, it computes u = r²S/(4Tt), the well function W(u) and the drawdown s = Q·W(u)/(4πT) at distance r and time t since pumping started. Beside it, it shows the Cooper-Jacob drawdown, the time after which that approximation applies, and the time to reach an optional target drawdown.
What is the well function W(u)?
W(u) is the well function of the Theis equation, the exponential integral of u. It falls as u rises. The calculator evaluates it with a power series when u is below 1 and with a continued fraction when u is 1 or more.
When is the Cooper-Jacob approximation valid?
Cooper and Jacob showed in 1946 that for u up to about 0.01 a straight-line form is enough: sCJ = Q·(−0.5772156649 − ln u)/(4πT). The calculator marks sCJ as not valid when u is above 0.01, shows n/a when the formula gives zero or a negative drawdown, and reports tCJ = r²S/(0.04T), the time after which u is 0.01 or less.
How is the time to reach a target drawdown found?
The target drawdown st is turned into a target well function value Wt = 4πT·st/Q. Because W falls as u rises, the calculator finds u by bisection on ln u between 10⁻³⁰ and 50, then converts it to time with t = r²S/(4Tu). If the target needs a well function value above W(10⁻³⁰), it reports an error instead of a time.
Disclaimer: This calculator gives an estimate only and is not a substitute for a qualified hydrogeologist. It assumes a confined, homogeneous, isotropic aquifer of infinite extent, a fully penetrating well of very small diameter, constant discharge, water released instantly from storage, and no recharge, leakage or barriers. It does not include well losses, partial penetration or unconfined delayed yield.