Well Drawdown and Cone of Depression

Theory: Well Drawdown

How pumping draws down the water around a well, in steady and transient flow, and how several wells add up.

Confined and unconfined aquifers

A confined aquifer lies under a layer that barely lets water through; its water is under pressure, and stands in a well up to the piezometric level, above the aquifer's top. Pumping lowers that pressure, but the aquifer stays full. An unconfined aquifer has a free water table; pumping drains the soil near the well, so the saturated thickness shrinks with the drawdown.

T = K\, b

T: transmissivity · K: permeability · b: the aquifer's thickness

Steady flow: Thiem

After long pumping, the flow toward a well settles. Darcy's law through cylinders around it gives the drawdown falling off with the log of distance, to nothing at the radius of influence R.

s = \frac{Q}{2\pi T} \ln\frac{R}{r}

s: drawdown at r from the well · Q: pumping rate · R: radius of influence

In an unconfined aquifer the flow passes through a thickness h that falls toward the well. Dupuit took the flow as horizontal, which gives the same law for the square of the saturated thickness.

H^2 - h^2 = \frac{Q}{\pi K} \ln\frac{R}{r}

H: the saturated thickness before pumping · h: at r, from the base

Transient flow: Theis

Before the flow settles, the water pumped comes from storage: the aquifer's compression in a confined aquifer, the soil draining in an unconfined one. Theis solved the flow as heat spreading from a line source; the cone deepens and widens with time.

s = \frac{Q}{4\pi T}\, W(u)

W(u): the well function, the exponential integral E₁(u) · S: storativity, or S_y when unconfined · t: time pumping

Once u is small, under about 0.05, Cooper and Jacob's approximation turns W(u) into a straight line on a log scale of time; it meets s = 0 at the radius of influence.

s \approx \frac{Q}{4\pi T} \ln\frac{2.25\, T t}{r^2 S}

Valid for u < 0.05

In an unconfined aquifer, Theis's solution holds while the drawdown is small against H. Jacob's correction brings the drawdown s of the draining aquifer to the s′ of an equivalent confined one.

s' = s - \frac{s^2}{2H}

s′: the Theis drawdown, with T = KH and S = S_y

Several wells

The flow equation is linear in the drawdown of a confined aquifer, so the drawdowns of several wells add: each pumps as if alone. In an unconfined aquifer, Dupuit's equation is linear in h², so the H² − h² of each well add instead.

s = \sum s_i(r_i)

r_i: distance from well i

A ring of wells draws the water down inside it more than any one well would; dewatering an excavation relies on it.
The drawdown at a well is taken at its radius r_w; losses through the screen and gravel pack add to it in the field.
Pumping tests run the other way: the drawdown measured through time gives T and S, by fitting the Theis curve or Cooper and Jacob's straight line.

References

Thiem, G. Hydrologische Methoden. Leipzig: Gebhardt, 1906.
Theis, C. V. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Transactions of the American Geophysical Union, v. 16, p. 519–524, 1935.
Cooper, H. H.; Jacob, C. E. A generalized graphical method for evaluating formation constants and summarizing well-field history. Transactions of the American Geophysical Union, v. 27, p. 526–534, 1946.
Kruseman, G. P.; De Ridder, N. A. Analysis and Evaluation of Pumping Test Data. 2nd ed. Wageningen: ILRI, 1990.
Fetter, C. W. Applied Hydrogeology. 4th ed. Upper Saddle River: Prentice Hall, 2001.
Powers, J. P. et al. Construction Dewatering and Groundwater Control. 3rd ed. Hoboken: Wiley, 2007.

Place pumping wells in a confined or unconfined aquifer to see the cone of depression they draw in 3D, and find the drawdown at a point of interest, at each well and through time, by Thiem or Theis.

Aquifer

Type

Water table, depthm

Saturated thickness, HmPermeability, Km/sSpecific yield, S_y

H is measured from the water table down to the impervious base. S_y is often 0.1 to 0.3 in sands and gravels; 2e-4 reads as 2 × 10⁻⁴.

Pumping

Method

Time pumping, tdWell radius, r_wm

The drawdown grows with time as the aquifer gives up water from storage. Drag the slider over the scene to watch the cone spread.

Wells

Positions in plan; x and y may be negative. Q is each well's pumping rate.

Point of interest

xmymDrawdown wantedm

Where the drawdown is checked, such as an excavation's center; the 3D block is cut open there.

One horizontal, homogeneous and isotropic aquifer of infinite extent on an impervious base; the confining layer, when there is one, doesn't leak. The wells are fully penetrating and pump steadily from t = 0.
Several wells superpose: the drawdowns add in a confined aquifer; in an unconfined one, the discharge potential H² − h² adds for Thiem, and Jacob's correction s′ = s − s²/2H is applied to the Theis drawdowns added. Both hold while the drawdown stays small against H.
The radius of influence is Thiem's R as given, or Cooper and Jacob's 1.5 √(Tt/S) for Theis. The 3D block runs half of it beyond the wells, kept within 1.5 to 4 times their spread, where the cone still reads.
x and y in plan, in m; depths and levels down from the ground. The 3D scene is stretched vertically, as marked under it, so a thin aquifer reads.