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, pressuregradient to thermal gradient.
The assumptions required by the Theis solution are:
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 EulerMascheroni constant. For practical purposes, the simplest approximation of $W(u)$ was proposed as $W(u)=0.5772lnu$ for $u <$ 0.05. Other more exact approximations of the well function were summarized by R. Srivastava and A. GuzmanGuzman
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.
Some of the data of the above curves are given in the following table.
Distance  Analytic  OGS5  OGS6  Error  Error 
Solution (h_{a})  (h_{5})  (h_{6})  ($\frac{h_5h_a}{h_a}$)  ($\frac{h_6h_a}{h_a}$)  
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.
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