7.3 Well hydraulics & aquifer testing

Pumping wells and drawdown

When a well pumps water from an aquifer, the water level around it declines, forming a cone of depression — a downward-flaring funnel in the water table (unconfined) or potentiometric surface (confined) centered on the well. Drawdown (s) is the vertical distance between the original (static) level and the pumped (dynamic) level at any point. Drawdown is greatest at the well and decreases outward, approaching zero at the radius of influence (r₀), the distance beyond which pumping produces no measurable decline. Higher pumping rates, lower transmissivity, and longer pumping times all deepen and widen the cone. Where cones from adjacent wells overlap, well interference adds their drawdowns and reduces the available head at each well.

Transmissivity and storativity

Two parameters govern how the cone develops.

Transmissivity (T) is the rate at which water moves through the entire saturated thickness of an aquifer under a unit hydraulic gradient:

T = K · b

where K is hydraulic conductivity and b is the saturated thickness. Units are L²/T (e.g., m²/day). A 20-m-thick sand with K = 15 m/day has T = 15 × 20 = 300 m²/day. Transmissivity describes the whole aquifer's ability to deliver water; two aquifers of equal K but different thickness have different T.

Storativity (storage coefficient, S) is the volume of water released from (or taken into) storage per unit surface area of aquifer per unit change in head. It is dimensionless.

• In confined aquifers, water is released by expansion of the water and compression of the aquifer skeleton as pressure drops; S is very small, typically 10⁻⁵ to 10⁻³.
• In unconfined aquifers, water is released mainly by physical drainage of the pores, so S is approximately equal to the specific yield, typically 0.05 to 0.30 — hundreds of times larger than confined storativity.

Worked example — transmissivity and storage

An unconfined aquifer has K = 25 m/day and saturated thickness b = 12 m, giving T = 25 × 12 = 300 m²/day. If its specific yield is 0.20, then a 1-m head drop over a 1,000-m² area releases 0.20 × 1,000 × 1 = 200 m³ of water from storage.

Aquifer (pumping) tests

An aquifer test stresses the aquifer by pumping a well at a constant rate while measuring drawdown over time in the pumping well and in nearby observation wells. Analyzing the time–drawdown data yields T and S — the parameters needed to predict future drawdown, sustainable well yields, and contaminant travel.

The Theis method

C. V. Theis (1935) produced the first transient (time-dependent, or non-equilibrium) solution. It treats the aquifer as confined, homogeneous, isotropic, of infinite extent and uniform thickness, fully penetrated by the well, with water released instantaneously as head drops. Drawdown is:

s = (Q / 4πT) · W(u), with u = r²S / (4Tt)

Here W(u) is the "well function," r is radial distance from the pumped well, and t is time. Field data (s versus t) are matched to a type curve of W(u) versus 1/u; the match point yields T and S. Theis captures drawdown that keeps changing as the cone of depression expands.

The Cooper–Jacob approximation

Cooper and Jacob (1946) simplified Theis for later pumping time (small u, u < about 0.05), when the well function reduces to a logarithm. Plotting drawdown against the logarithm of time then yields a straight line. From the slope and intercept:

T = 2.30 Q / (4π Δs), and S = 2.25 T t₀ / r²

where Δs is the drawdown change per log cycle of time and t₀ is the time-axis intercept of the straight line. Cooper–Jacob is a graphical shortcut to Theis, valid only after enough pumping time that the early-time (large-u) data can be excluded.

Specific capacity and well efficiency

A well's specific capacity is its pumping rate divided by the drawdown it produces (Q/s, e.g., m³/day per m of drawdown); it is a quick field index of well performance and increases with transmissivity. Actual drawdown in the pumping well exceeds the aquifer (formation) loss because of well loss — extra head loss from turbulent flow through the screen and gravel pack. A step-drawdown test, in which the rate is raised in successive steps, separates linear aquifer loss from this non-linear well loss and defines well efficiency. Poorly developed or encrusted wells show low efficiency and large well loss.

Steady-state (equilibrium) analysis

When the cone stabilizes and drawdown stops changing, steady-state methods such as the Thiem equation relate discharge to the drawdown difference between two observation wells. Steady-state analysis yields T but not S, because storage is no longer changing; a transient test is required to obtain storativity.

Aquifer-test parameter summary

For the exam, know that T = Kb, that confined S is tiny while unconfined S ≈ specific yield, that the Theis method is the transient type-curve solution, and that Cooper–Jacob is its late-time straight-line (semilog) simplification.