Theis procedure

The Theis method is a hydrogeological method for determining the transmissivity and the storage coefficient of an aquifer . It is based on carrying out a pumping test under unsteady flow conditions . Water is taken from a well at a constant rate and the lowering of the groundwater level over time is measured in a suitable groundwater measuring point. The subsidence rates can then be used to draw conclusions about the properties of the aquifer. A method based on the Theis method is the straight line method by Cooper and Jacob .

The method is named after Charles Vernon Theis , who published the underlying equations in 1935.

Framework

The Theis method is based on the assumption that there is a perfect well with a negligible diameter in confined groundwater. The following should apply to the aquifer:

1. it lies horizontally
2. it is of infinite dimensions with constant thickness
3. it is homogeneous and isotropic.
4. before the start of the pumping test, the pressure level is horizontal

For the pumping test, water is taken from the well at a constant delivery rate and the lowering of the water level over time is measured in one or more groundwater measuring points. With this measurement data, conclusions can be drawn about the storage coefficient , the permeability coefficient and the transmissivity .

Be it

• ${\displaystyle Q}$ the flow rate of the well in cubic meters per second
• ${\displaystyle S}$ the storage coefficient of the aquifer
• ${\displaystyle k_{f}}$ the permeability coefficient of the aquifer in meters per second
• ${\displaystyle h_{M}}$ the thickness of the aquifer in meters
• ${\displaystyle T=k_{f}\cdot h_{M}}$ the transmissivity of the aquifer in square meters per second
• ${\displaystyle t}$ the time in seconds since the start of the pumping attempt
• ${\displaystyle r}$the distance (as the crow flies ) from the well in meters
• ${\displaystyle h_{s}(r,t)}$the groundwater lowering at a distance from the well according to the time in meters ${\displaystyle r}$ ${\displaystyle t}$

The Theis fountain function

An auxiliary variable and an auxiliary function are introduced for a more compact representation of the analytical relationships . The auxiliary variable is defined as ${\displaystyle u}$ ${\displaystyle W(u)}$ ${\displaystyle u}$

${\displaystyle {\begin{aligned}u=u(r,t)&={\frac {r^{2}\cdot S}{4\cdot k_{f}\cdot h_{M}\cdot t}}\\&={\frac {r^{2}\cdot S}{4\cdot T\cdot t}}\end{aligned}}}$.

The auxiliary function is called the (Theis) well function and is given as the infinite series ${\displaystyle W(u)}$

${\displaystyle {\begin{aligned}W(u)&=-\gamma -\ln(u)+\sum _{j=1}^{\infty }(-1)^{j+1}{\frac {u^{j}}{j\cdot j!}}\\&=-\gamma -\ln(u)+u-{\frac {u^{2}}{2\cdot 2!}}+{\frac {u^{3}}{3\cdot 3!}}-{\frac {u^{4}}{4\cdot 4!}}+\cdots \end{aligned}}}$.

Here is the Euler-Mascheroni constant . ${\displaystyle \gamma \approx 0{,}5772}$

For the application, the values ​​of the Theis well function for a given table are taken. ${\displaystyle u}$

Analytical statement

Under the above conditions, the groundwater lowering is based on the time at a distance from the well ${\displaystyle h_{\text{s}}}$ ${\displaystyle t}$ ${\displaystyle r}$

${\displaystyle {\begin{aligned}h_{\text{s}}(r,t)&={\frac {Q}{4\pi \cdot k_{f}\cdot h_{M}}}\cdot W(u)\\&={\frac {Q}{4\pi \cdot T}}\cdot W(u)\end{aligned}}}$. (1)

If you switch to the transmissivity, you get

${\displaystyle T={\frac {Q}{4\pi \cdot h_{\text{s}}(r,t)}}\cdot W(u)}$. (2)

The storage coefficient is then obtained

${\displaystyle S={\frac {4T\cdot t\cdot u}{r^{2}}}}$. (3)

evaluation

Action

In the application one is usually determined by a determination of the transmissivity and the storage coefficient. However, this is not directly possible because it occurs both in the well function and outside of it. Therefore a graphic method is used. For this purpose, the values ​​of the well function (y-axis) are first plotted against the values ​​of (x-axis) on transparent double-logarithmic paper . As mentioned above, the values ​​of are taken from a table. The resulting curve is referred to as the Theis type curve or Theis standard curve. ${\displaystyle T}$ ${\displaystyle {\frac {1}{u}}}$ ${\displaystyle W(u)}$

In a second step, the results of the pumping test are also drawn on log-logarithmic paper. The x-axis is plotted against the y-axis. ${\displaystyle {\frac {t}{r^{2}}}}$ ${\displaystyle h_{s}(r,t)}$

In the third step, the Theis type curve is placed over the data curve. By moving the Theis-type curve along the axes, it is brought to coincide with the data curve over the largest possible section. The shift of the Theis-type curve with respect to the data curve along the x-axis then corresponds to the shift along the y-axis . ${\displaystyle \log _{10}\left({\frac {S}{4T}}\right)}$ ${\displaystyle \log _{10}\left({\frac {Q}{4\pi T}}\right)}$

For more precise analysis selects one called a match point in the overlapping region and reads out there , in the Journal of the Theis-type curve and the reduction and onto the leaf of the data curve from. The selected point itself does not have to be on the curve. ${\displaystyle W_{0}(u)}$ ${\displaystyle {\tfrac {1}{u_{0}}}}$ ${\displaystyle h_{s}}$ ${\displaystyle {\tfrac {t}{r^{2}}}}$

The transmissivity is determined from the coordinates determined in this way

${\displaystyle T={\frac {Q}{4\pi }}\cdot {\frac {h_{s}}{W_{0}(u)}}}$

and based on this the storage coefficient as

${\displaystyle S=4T\cdot {\frac {t}{r^{2}}}\cdot u_{0}}$

Derivation

Taking the logarithm of equations (1) and (3), one obtains

${\displaystyle \log _{10}(h_{s}(u))=\underbrace {\log _{10}\left({\frac {Q}{4\pi \cdot T}}\right)} _{\text{konstant}}+\log _{10}(W(u))}$

and

${\displaystyle \log _{10}\left({\frac {t}{r^{2}}}\right)=\underbrace {\log _{10}\left({\frac {S}{4T}}\right)} _{\text{konstant}}+\log _{10}\left({\frac {1}{u}}\right)}$

The two terms in parentheses remain constant during the pumping attempt. If the results of the pumping test are plotted on log-logarithmic paper as described above, the constants have no influence on the slope, but only on the vertical and horizontal shift of the data curve compared to the Theis-type curve.

swell

• Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 229-239 .
• Bernward Hölting, Wilhelm Georg Coldewey: Hydrogeology . Introduction to General and Applied Hydrogeology. 8th edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-8274-2353-5 , pp. 290-293 , doi : 10.1007 / 978-3-8274-2354-2 .

Individual evidence

1. ^ ^{A b} Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 232 .
2. ^ Bernward Hölting, Wilhelm Georg Coldewey: Hydrogeology . Introduction to General and Applied Hydrogeology. 8th edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-8274-2353-5 , pp. 290 , doi : 10.1007 / 978-3-8274-2354-2 .
3. ^ ^{A b} Bernward Hölting, Wilhelm Georg Coldewey: Hydrogeology . Introduction to General and Applied Hydrogeology. 8th edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-8274-2353-5 , pp. 291 , doi : 10.1007 / 978-3-8274-2354-2 .
4. ^ Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 234 .