Cooper-Jacob Transient Well Drawdown Formula (Theis Approximation)

An approximation to Theis, not a general validity certificate

For a constant pumping rate Q, the Theis solution has s=Q*W(u)/(4*pi*T), with u=r^2*S/(4*T*t). Cooper–Jacob replaces the well function with its leading logarithmic approximation. The discarded terms become small as u decreases; they do not vanish exactly at a threshold.

This worksheet uses u<=0.01 as a conservative screen. Some references allow 0.05 with a looser error tolerance. Passing either screen only addresses the approximation error, not whether the aquifer satisfies the model assumptions.

The setup assumes a homogeneous, isotropic, nonleaky confined aquifer, constant pumping, a fully penetrating well, radial flow and negligible wellbore-storage and boundary effects during the analysed interval. Leakage, delayed drainage, partial penetration or changing pumping rates require additional analysis.

Enter a consistent set of units

B2 is Q in m^3/day, B3 is transmissivity T in m^2/day, B4 is dimensionless storativity S, B5 is observation distance r in metres, and B6 is elapsed pumping time t in days. Do not mix hours with day-based flow and transmissivity.

The example uses Q=1200, T=300, S=0.0004, r=30 and t=0.5. C2 audits the inputs. Q=0 is permitted and gives zero drawdown; negative Q is excluded because this example models pumping rather than injection. All other inputs must be positive.

The line-source formula is evaluated away from r=0. A valid numeric S is not by itself proof of a plausible confined-aquifer storativity; choose parameters from the site and applicable model.

=IF(COUNT(B2:B6)<>5,"Enter numeric Q, T, S, r and time",IF(OR(B2<0,MIN(B3:B6)<=0),"Require Q >= 0 and T, S, r, time > 0","OK"))

Compute u and the earliest time passing the screen

C3 contains u. Initially it equals 0.0006. C6 contains =IF(C2<>"OK",C2,B5^2*B4/(4*B3*0.01)), giving 0.03 days or 43.2 minutes for this example. This is the time when the small-u screen is first met, not a guarantee that field data are ready for straight-line fitting.

Increasing time decreases u; increasing distance increases it quadratically. At r=500 m and t=0.05 days, u is about 1.6667, so the approximation is rejected.

=IF(C2<>"OK",C2,B5^2*B4/(4*B3*B6))

Calculate unrounded drawdown

C4 uses the following natural-log form, equivalent to the usual LN(10)*Q/(4*pi*T)*LOG10(2.25*T*t/(r^2*S)). Using LN(10) avoids treating the rounded coefficient 2.302585 as exact.

The constant 2.25 is itself a conventional rounded approximation. C4 is about 2.1783 m at 0.5 days and 2.9112 m at 5 days; display three decimals to obtain 2.178 and 2.911 m. Keep the raw values for differences and further calculations.

For u above 0.01, C4 returns a message rather than an unsupported drawdown estimate. Use the full Theis well function where appropriate, and reconsider the physical model if its assumptions fail.

The comparison includes a relative tolerance of 1E-12 solely to keep binary floating-point rounding at u=0.01 from rejecting the boundary; it is not a physical error allowance.

=IF(C2<>"OK",C2,IF(C3>0.01*(1+1E-12),"u > 0.01: use full Theis or another model",B2/(4*PI()*B3)*LN(2.25*B3*B6/(B5^2*B4))))

Check the semi-log slope

C5 gives the drawdown increase per tenfold time increase within this approximation: about 0.732936 m for the initial Q and T. Compare raw drawdowns at t and 10*t; both should pass the small-u screen.

The constant slope is a property of the Cooper–Jacob expression, not an exact statement about the full transient solution at all times. C5 reports the formula’s slope even if the current time fails the screen; it does not independently approve the current drawdown.

If fitting measured data to estimate transmissivity, rearrange the slope relation as T=LN(10)*Q/(4*pi*Delta_s), using a positive fitted drawdown change per log10 cycle and consistent units. This tutorial does not fit field data or infer storativity automatically.

=IF(C2<>"OK",C2,LN(10)*B2/(4*PI()*B3))

Further reading

• USGS Nevada Water Science Center — Cooper–Jacob analysis and small-u criterion
• British Columbia Groundwater Drawdown Tool — methods and assumptions
• USGS — Completion summary for Borehole TAN-2336, aquifer-test assumptions