Reading an Aquifer Backward: What the Theis Solution Tells You From a Pump Test

The problem with a well nobody can see into

You can't watch groundwater move. Everything a hydrogeologist knows about an aquifer's transmissivity and storage capacity comes from indirect evidence, and the single most common source of that evidence is the pumping test: turn on a well at a known, constant rate, and watch the water level drop in one or more nearby observation wells over time. The shape of that drawdown curve — how fast it falls, how far it travels, how it flattens out — is a fingerprint of the aquifer's hydraulic properties, and in 1935 Charles Theis published the equation that turned that fingerprint into a number.

The Theis solution treats the aquifer as an idealized confined layer: infinite in extent, uniform thickness, uniform properties, with a fully penetrating well pumping at a constant rate from time zero. Real aquifers rarely match that description perfectly, but the solution is close enough, often enough, that it remains the default starting point for pump test analysis nearly a century later. The five inputs that drive it — Pumping rate Q, Transmissivity T, Storage coefficient S, Observation distance r, and elapsed time — map directly onto the parameters of any Theis calculator, alongside a couple of convenience inputs like a Max pumping time cap and an Elapsed time (scrub) control for stepping through the drawdown curve as it develops.

The equation and the well function

Theis's core result is deceptively compact:

s(r,t) = Q / (4*pi*T) * W(u)

where the dimensionless time parameter u is:

u = r^2 * S / (4 * T * t)

Here s is the drawdown s at obs. well — the amount the water level has fallen at radial distance r from the pumping well, measured at elapsed time t. Q is volumetric pumping rate, T is transmissivity (a measure of how easily water moves horizontally through the full thickness of the aquifer), and S is the dimensionless storage coefficient (how much water the aquifer releases from storage per unit drop in head, per unit area).

W(u), the well function, is where the real work happens. It's an exponential integral:

W(u) = -0.5772 - ln(u) + u - u^2/(2*2!) + u^3/(3*3!) - ...

For the small values of u typical of a pump test observed reasonably far from the well, the series after the first two terms contributes almost nothing, so W(u) ≈ -0.5772 - ln(u) is often accurate enough for a quick hand estimate — though a proper tool evaluates the full series rather than truncating.

Notice what u actually encodes: it's small when t is large (drawdown has had time to spread and stabilize) and when r is small (the observation point is close to the pumping well). Small u means large W(u), which means large drawdown — exactly the intuitive behavior you'd expect. The elegance of Theis's approach is folding distance, time, and both aquifer properties into that single dimensionless number.

Working a test by hand

Suppose a confined aquifer is pumped at Q = 1,500 m³/day. An observation well sits r = 50 m away. After t = 1 day of continuous pumping, the water level in that well has dropped by some amount we want to predict, given a transmissivity T = 250 m²/day and storage coefficient S = 2×10⁻⁴ — values you might have assumed going in, or be trying to back-calculate by matching predicted drawdown to what's actually measured in the field.

First, the dimensionless time parameter:

u = r^2 * S / (4 * T * t)
  = (50^2) * (2e-4) / (4 * 250 * 1)
  = 2500 * 0.0002 / 1000
  = 0.0005

That's a small u, as expected for an observation point reasonably close to the well after a full day of pumping. Evaluating the well function:

W(u) ≈ -0.5772 - ln(0.0005) + 0.0005
     = -0.5772 - (-7.6009) + 0.0005
     ≈ 7.0242

And the drawdown itself:

s = Q / (4*pi*T) * W(u)
  = 1500 / (4 * 3.14159 * 250) * 7.0242
  = 1500 / 3141.59 * 7.0242
  ≈ 0.4775 * 7.0242
  ≈ 3.35 m

So after one day, this observation well 50 m away should show roughly 3.35 m of drawdown. Two more derived numbers round out the picture. The specific capacity Q/s, a quick field metric of well performance, comes out to 1500 / 3.35 ≈ 447 m³/day per meter of drawdown — useful for comparing this well's efficiency against others in the same field survey. And a rough influence radius estimate, using the common Cooper-Jacob approximation R ≈ 1.5·√(T·t/S), gives:

R ≈ 1.5 * sqrt(250 * 1 / 0.0002)
  = 1.5 * sqrt(1,250,000)
  ≈ 1.5 * 1118
  ≈ 1677 m

That's the rough distance beyond which the pumping well's effect on head is negligible after one day — a number that matters enormously for well-spacing decisions and for predicting interference between neighboring wells on the same aquifer.

Where the idealized model breaks

The Theis solution assumes an infinite, homogeneous, confined aquifer, and every one of those assumptions is a place real data can diverge from the curve. A nearby impermeable boundary — a fault, a bedrock high — makes drawdown accelerate faster than Theis predicts, because the cone of depression can't spread in that direction and has to deepen instead. A nearby recharge boundary, like a connected river, does the opposite: drawdown flattens out and can even stabilize completely, something the pure Theis equation, with its infinite aquifer assumption, can never produce on its own.

Unconfined aquifers are another common mismatch. Theis was derived for confined conditions where storage release comes from elastic compression of the aquifer matrix and water, a fast and nearly instantaneous process. In an unconfined (water table) aquifer, storage release also comes from actual gravity drainage of pore spaces, which happens on a much slower timescale — this is the delayed yield effect, and it produces an S-shaped drawdown curve that a naive Theis fit will badly misread, often producing a storage coefficient estimate an order of magnitude too small if you fit only the early-time data.

Finally, remember that T and S here are properties of the aquifer between the pumping well and the observation point, not universal constants for the whole formation. A single pump test with one observation well gives you one averaged estimate along one radial path. Real aquifer characterization studies pump multiple tests, or use multiple observation wells at different distances and directions, precisely because heterogeneity means the Theis fit from one well pair doesn't necessarily generalize to the next.

Fitting T and S from field data, not just predicting drawdown

Everything above runs the Theis equation forward: given T and S, predict s at some r and t. In practice, a field engineer usually needs to run it backward — given a logged drawdown curve from the observation well, back out the T and S that produced it. The classic technique is Cooper and Jacob's straight-line method, a simplification that holds once u is small enough (roughly u < 0.01, comfortably satisfied in the worked example above) that the higher-order terms of the well function series can be dropped entirely, leaving:

s ≈ (2.3 * Q) / (4*pi*T) * log10(2.25 * T * t / (r^2 * S))

Plotted against log(t), this becomes a straight line, and the slope of that line gives T directly, while the time-axis intercept (where s = 0) gives S. It's a far faster hand-calculation than evaluating the exponential integral, which is exactly why it became the field standard for decades before software made the full Theis curve-match trivial. The tradeoff is that the straight-line approximation only holds at the smaller u values reached later in a test — early-time data, where u is still large, has to be excluded from the fit or it visibly bends the line and biases both T and S.

The two methods should agree closely when both are applicable, and checking that agreement is itself a useful quality check on a field dataset: if the Cooper-Jacob straight line and a full Theis type-curve match disagree by more than a small margin, that's usually a sign of a boundary effect, a partially penetrating well, or noisy early-time readings distorting the fit, rather than a real difference in aquifer behavior.

Try it yourself

Stepping the Elapsed time (scrub) control forward and watching drawdown propagate outward in real time makes the u-versus-W(u) relationship far more intuitive than working through the exponential integral by hand every time. You can run your own Q, T, S, and r combinations, including the influence radius and specific capacity outputs, using the Aquifer Pump Test Theis Simulator here. If the site also involves soil classification work upstream of the hydrogeology, the Atterberg plasticity index tool covers a common companion calculation in geotechnical site investigations.