# Theis' problem

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## Problem description

Theis' problem examines the transient lowering of the water table induced by a pumping well. Theis' fundamental insight was to recognize that Darcy’s law is analogous to the law of heat flow by conduction, i.e., hydraulic pressure being analogous to temperature, pressure-gradient to thermal gradient.

The assumptions required by the Theis solution are:

• the aquifer is homogeneous, isotropic, confined, infinite in radial extent,
• the aquifer has uniform thickness, horizontal piezometric surface
• the well is fully penetrating the entire aquifer thickness,
• the well storage effects can be neglected,
• the well has a constant pumping rate,
• no other wells or long term changes in regional water levels.

## Analytical solution

The analytical solution of the drawdown as a function of time and distance is expressed by $$\begin{eqnarray} h_0 - h(t,x,y) = \frac{Q}{4\pi T}W(u) \label{theis} \end{eqnarray}$$

$$\begin{eqnarray} u = \frac{(x^{2}+y^{2})S}{4Tt} \label{theis_u} \end{eqnarray}$$

where $h_0$ is the constant initial hydraulic head $[L]$, $Q$ is the constant discharge rate [$L^{3}T^{-1}$], $T$ is the aquifer transmissivity [$L^{2}T^{-1}$], $t$ is time $[T]$, $x,y$ is the coordinate at any point $[L]$ and $S$ is the aquifer storage $[-]$. $W(u)$ is the well function defined by an infinite series for a confined aquifer as

$$\begin{eqnarray} W(u) = -\gamma -lnu + \sum^{\infty}_{k=1}{\frac{(-1)^{k+1}u^k}{k\cdot k!}} \label{theis_wu} \end{eqnarray}$$

where $\gamma\approx$ 0.5772 is the Euler-Mascheroni constant. For practical purposes, the simplest approximation of $W(u)$ was proposed as $W(u)=-0.5772-lnu$ for $u <$ 0.05. Other more exact approximations of the well function were summarized by R. Srivastava and A. Guzman-Guzman

## Results and evaluation

The following figure compares the analytical solution, the result by ogs5, and the result by ogs6 (labeled as pressure) within the range that satisfies $u <$ 0.05.

The figure shows that there is a good match between the analytical solution and the numerical solution obtained by using ogs5 or ogs6.

Some of the data of the above curves are given in the following table.

 Distance Analytic OGS5 OGS6 Error Error Solution (ha) (h5) (h6) ($|\frac{h_5-h_a}{h_a}|$) ($|\frac{h_6-h_a}{h_a}|$) 0 12.8141 12.474 12.474 0.0265 0.0272 1.21799 8.9441 8.79341 8.79341 0.0168 0.0171 2.43597 7.48878 7.34717 7.34717 0.0189 0.0192 3.65396 6.59978 6.46176 6.46176 0.0209 0.0213 4.87195 5.94548 5.81072 5.81072 0.0226 0.0231 6.08994 5.43381 5.30267 5.30267 0.0241 0.0247 7.30792 5.00981 4.88283 4.88283 0.0253 0.0260 8.52591 4.65012 4.52793 4.52793 0.0262 0.0269 8.83041 4.56714 4.44623 4.44623 0.0264 0.0271 9.4394 4.4116 4.29344 4.29344 0.0267 0.0275 10.6574 4.12707 4.01501 4.01501 0.0271 0.0279 15.2248 3.28072 3.19698 3.19698 0.0255 0.0261 20.0968 2.61899 2.57517 2.57517 0.0167 0.0170 22.8373 2.31338 2.29626 2.29626 0.0074 0.0074 24.0553 2.18892 2.1846 2.1846 0.0019 0.0019 25.2732 2.07055 2.07958 2.07958 0.0043 0.0043 25.5777 2.04164 2.05407 2.05407 0.0060 0.0060 29.8407 1.67134 1.73518 1.73518 0.0381 0.0367

The analytical solutions are for an ideal problem with point wise pumping term. While for the FEM analysis, the point wise source value is distributed to the surface of a small hole around the source point in order to avoid the singularity. One can see from the table that the precisions of the FEM solutions are still acceptable with such transform of the point pumping term.

This article was written by Wenqing Wang. If you are missing something or you find an error please let us know. Generated with Hugo 0.79.0. Last revision: September 22, 2021
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