Permeability from a pumping test: the Dupuit formulas

Updated: 2026-08-13 · ISKRIT

A pumping test measures what a laboratory permeameter cannot: the permeability of the aquifer as it actually lies, over the whole volume the well drains. The work-up is a pair of steady-state formulas that go back to Dupuit, and the whole method stands on the assumptions behind them.

The two formulas

For a fully penetrating well pumped to steady state, with Q the discharge, R the radius of influence and r the well radius, the unconfined case is:

kf = Q · ln(R/r) / (π · (H² − h²))

where H is the initial saturated thickness and h = H − s is the thickness left at the well after a drawdown s. The confined case is:

kf = Q · ln(R/r) / (2π · m · s)

with m the thickness of the confined aquifer. Getting these two mixed up is the common error, because both take a discharge, a drawdown and a thickness, and only the unconfined one squares the thickness.

Why the unconfined formula squares the thickness

In a confined aquifer the flow section is fixed: the thickness m stays the same however far the head is drawn down, so the drawdown enters linearly. In an unconfined aquifer the water table itself is the top of the flow section, so the section shrinks towards the well and the transmissivity varies with radius. Integrating that gives a difference of squares. The two forms are related, since H² − h² = s · (2H − s): when the drawdown is small next to the thickness, the unconfined formula collapses to the confined one with m ≈ H, and the difference between them only becomes large when the aquifer is heavily dewatered.

Worked

Unconfined aquifer, Q = 500 m³/day, well radius r = 0.15 m, radius of influence R = 150 m, saturated thickness H = 12 m, drawdown at the well s = 3 m, so h = 9 m:

ln(R/r) = ln(1000) = 6.908
H² − h² = 144 − 81 = 63 m²
kf = 500 · 6.908 / (π · 63) = 17.4 m/day

The same well in a confined aquifer of thickness m = 8 m with the same drawdown:

kf = 500 · 6.908 / (2π · 8 · 3) = 22.9 m/day

The radius of influence

R is the distance at which drawdown has died out. It is measured, not calculated: observation wells around the pumped well show where the cone of depression flattens. Where there are no observation wells it is estimated by an empirical relation such as Sichardt's, which is awkward because that relation already contains the permeability being sought, so it has to be iterated.

The saving grace is that R enters as a logarithm. In the worked example, doubling R from 150 m to 300 m raises ln(R/r) from 6.908 to 7.601, that is kf from 17.4 to 19.2 m/day, about 10 percent. Halving it costs the same 10 percent the other way. A radius of influence that is out by a factor of two therefore moves the answer less than the scatter between two wells on the same site, and arguing about R is rarely the best use of the time.

The assumptions, and which way each error pushes

The formulas assume steady flow, a fully penetrating well, a homogeneous aquifer of constant thickness, and horizontal flow. Each violation has a direction:

• Reading before steady state. Drawdown is still growing, so the s used is too small and the permeability comes out too high.
• A partially penetrating well. Flow has to converge vertically to reach the screen, which adds drawdown that the formula reads as low permeability, so kf comes out too low.
• Drawdown measured inside the pumped well. It includes entrance and screen losses that are not aquifer drawdown, again biasing kf low. Where observation wells exist, take the drawdown from them.

Field test against laboratory permeameter

A falling head or constant head test measures one specimen a hundred millimetres across, in the direction the specimen sits in the cell, usually vertical, and only where the sample survived transport intact. A pumping test measures a horizontal permeability averaged over the metres of aquifer the well drains, fissures, sand partings and all. In layered ground the field value is routinely higher, often by an order of magnitude, and that is not a discrepancy to be reconciled: they are answers to different questions. Dewatering and inflow calculations want the field number.

Questions

Can a pumping test result be used for a settlement calculation? Not directly, since it gives permeability rather than stiffness, though it does set the drainage path and therefore the rate at which consolidation settlement occurs.

Does the well have to be pumped at a constant rate? Yes, for this work-up, because steady state means a constant discharge with the drawdown no longer changing. A test with a stepped discharge is worked up step by step and each step is checked for stabilisation on its own.

Enter the discharge, the drawdown and the geometry and get kf in metres per day, with the substituted values shown, in the pumping test permeability calculator, part of the field testing tool.