# Well formula according to Dupuit-Thiem

The well formula according to Dupuit-Thiem is an equation in hydrogeology , the sub-area of geology that deals with groundwater and related topics. The well formula relates the water withdrawal by a well to the permeability of the aquifer and the groundwater level. The well formula is formulated in two different cases: once for confined and once for unconstrained groundwater.

The fountain formula is named after Günther Thiem, who published it in 1906, building on works by Jules Dupuit from 1863.

## Framework conditions and model assumptions

A well and two groundwater measuring points are assumed, which for the sake of simplicity are referred to as groundwater measuring point one and groundwater measuring point two. The fountain fulfills the following conditions:

• It is perfect , so it is limited at the bottom by a groundwater non-conductor and the flow reaches it radially.
• There are quasi-stationary flow conditions . If the extraction remains the same, the groundwater level around the well does not change over time.

Furthermore, the aquifer is isotropic, horizontal and of great extent.

## With constrained groundwater

If the above assumptions apply and the aquifer is strained , then the well formula according to Dupuit-Thiem is

${\ displaystyle Q = 2 \ pi \ cdot k_ {f} \ cdot M \ cdot {\ frac {h_ {2} -h_ {1}} {\ ln \ left ({\ tfrac {r_ {2}} {r_ {1}}} \ right)}}}$

Here is

• ${\ displaystyle Q}$ the volume of water taken from the well in cubic meters per second
• ${\ displaystyle \ pi}$the circle number
• ${\ displaystyle k_ {f}}$the permeability coefficient in meters per second
• ${\ displaystyle M}$ the thickness of the aquifer
• ${\ displaystyle r_ {1}}$the distance (as the crow flies) between the groundwater measuring point one and the well in meters
• ${\ displaystyle r_ {2}}$ the distance (as the crow flies) of the groundwater measuring point two from the well in meters
• ${\ displaystyle h_ {1}}$and the height of the standpipe at the groundwater measuring point one or two in meters.${\ displaystyle h_ {2}}$

One denotes the transmissivity of the aquifer, i.e. ${\ displaystyle T}$

${\ displaystyle T = k_ {f} \ cdot M}$,

so this is calculated accordingly

${\ displaystyle T = {\ frac {Q} {2 \ pi \ cdot (h_ {2} -h_ {1})}} \ ln \ left ({\ tfrac {r_ {2}} {r_ {1}} } \ right)}$

## With unconstrained groundwater

If the aquifer is unconstrained , the well formula is according to Dupuit-Thiem

${\ displaystyle Q = \ pi \ cdot k_ {f} \ cdot {\ frac {h_ {2} ^ {2} -h_ {1} ^ {2}} {\ ln \ left ({\ tfrac {r_ {2 }} {r_ {1}}} \ right)}}}$

Here is

• ${\ displaystyle Q}$ the volume of water taken from the well in cubic meters per second
• ${\ displaystyle \ pi}$the circle number
• ${\ displaystyle k_ {f}}$the permeability coefficient in meters per second
• ${\ displaystyle r_ {1}}$the distance between the groundwater measuring point one and the well in meters
• ${\ displaystyle r_ {2}}$ the distance between the groundwater measuring point two and the well in meters
• ${\ displaystyle h_ {1}}$and the height of the standpipe at the groundwater measuring point one or two in meters${\ displaystyle h_ {2}}$

## Individual evidence

1. ^ 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. 289 , doi : 10.1007 / 978-3-8274-2354-2 .
2. Helmut Prinz, Roland Strauss: Engineering Geology . 5th, revised and expanded edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2472-3 , p. 89 .
3. ^ 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. 286-287 , doi : 10.1007 / 978-3-8274-2354-2 .
4. Dieter D. Genske: Engineering Geology . Basics and application. 2nd, revised and updated edition. Springer-Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-55386-8 , p. 280 , doi : 10.1007 / 978-3-642-55387-5 .