Theis Equation for Transient Well Flow
The Theis equation is the exact transient solution for drawdown around a pumping well in a confined aquifer, valid at all times and distances.
The equation
s = (Q / 4πT) W(u), u = r²S / 4Tt
- W(u) is the well function, an exponential integral
- u combines distance, storativity, transmissivity, and time
- All other symbols as in the Cooper Jacob equation
Worked example
With Q of 1200, T of 350, S of 0.0002, r of 50 meters, and t of 0.1 days, u = 0.00357 and W(u) = 5.061, giving s = 1.38 meters. Run the same numbers at one day and the answer converges on the Cooper Jacob value of 2.01 meters.
When to use it
The reference solution for confined aquifer tests. Real aquifers leak, have boundaries, and store water in aquitards, so treat departures of field data from the Theis curve as information rather than error.
Related Calculators and Reference
- Groundwater Flow Simulator: watch this equation animate a cone of depression.
- Demo Valley Worked Project: a full site project using these methods.
- USGS Water Science School: authoritative background on groundwater.