Aquifer test

Cross-sectional plot of transient Theis solution for radial distance vs drawdown over time

The Theis equation was created by Charles Vernon Theis (working for the US Geological Survey) in 1935,[1] from heat transfer literature (with the mathematical help of C.I. Lubin), for two-dimensional radial flow to a point sink in an infinite, homogeneous aquifer. It is simply

${\displaystyle {\begin{aligned}s&={\frac {Q}{4\pi T}}W(u)\\[0.5em]u&={\frac {r^{2}S}{4Tt}}\end{aligned}}}$

where s is the drawdown (change in hydraulic head at a point since the beginning of the test in units of distance), u is a dimensionless parameter, Q is the discharge (pumping) rate of the well (volume per unit time), T and S are the transmissivity and storativity of the aquifer around the well (distance squared per unit time and dimensionless, respectively), r is the distance from the pumping well to the point where the drawdown was observed, t is the time since pumping began, and W(u) is the "Well function" (called the incomplete gamma function, ${\displaystyle \Gamma (0,u)}$, in non-hydrogeology literature). The well function is given by the infinite series

${\displaystyle {\begin{aligned}W(u)=\Gamma (0,u)=-\gamma -\ln(u)+u-{\frac {u^{2}}{2\times 2!}}+{\frac {u^{3}}{3\times 3!}}-{\frac {u^{4}}{4\times 4!}}+\cdots \end{aligned}}}$

where γ is the Euler constant (=0.577216...). Typically this equation is used to find the average T and S values near a pumping well, from drawdown data collected during an aquifer test. This is a simple form of inverse modeling, since the result (s) is measured in the well, r, t, and Q are observed, and values of T and S which best reproduce the measured data are put into the equation until a best fit between the observed data and the analytic solution is found.

The Theis solution is based on the following assumptions:

• The flow in the aquifer is adequately described by Darcy's law (i.e. Re<10).
• homogeneous, isotropic, confined aquifer,
• well is fully penetrating (open to the entire thickness (b) of aquifer),
• the well has zero radius (it is approximated as a vertical line) — therefore no water can be stored in the well,
• the well has a constant pumping rate Q,
• the head loss over the well screen is negligible,
• aquifer is infinite in radial extent,
• horizontal (not sloping), flat, impermeable (non-leaky) top and bottom boundaries of aquifer,
• groundwater flow is horizontal
• no other wells or long term changes in regional water levels (all changes in potentiometric surface are the result of the pumping well alone)

Even though these assumptions are rarely all met, depending on the degree to which they are violated (e.g., if the boundaries of the aquifer are well beyond the part of the aquifer which will be tested by the pumping test) the solution may still be useful.

Steady-state radial flow to a pumping well is commonly called the Thiem solution,[2] it comes about from application of Darcy's law to cylindrical shell control volumes (i.e., a cylinder with a larger radius which has a smaller radius cylinder cut out of it) about the pumping well; it is commonly written as:

${\displaystyle h_{0}-h={\frac {Q}{2\pi T}}\ln \left({\frac {R}{r}}\right)}$

In this expression h₀ is the background hydraulic head, h₀-h is the drawdown at the radial distance r from the pumping well, Q is the discharge rate of the pumping well (at the origin), T is the transmissivity, and R is the radius of influence, or the distance at which the head is still h₀. These conditions (steady-state flow to a pumping well with no nearby boundaries) never truly occur in nature, but it can often be used as an approximation to actual conditions; the solution is derived by assuming there is a circular constant head boundary (e.g., a lake or river in full contact with the aquifer) surrounding the pumping well at a distance R.