Cooper and Jacob straight line method

The straight line method developed by Cooper and Jacob , also known as the Cooper-Jacob method for short , is a hydrogeological method for evaluating a pumping test to determine the storage coefficient and transmissivity . The method is based on the Theis method and is therefore suitable for the evaluation of unsteady flow conditions.

Framework

In the Cooper-Jacob process, water is withdrawn from a withdrawal point at a constant withdrawal rate . Depending on the evaluation method ${\displaystyle Q}$

• the lowering measured at one measuring point and at several times (distance lowering method)
• The reduction measured at several measuring points at the same time (time reduction method)
• the lowering measured at several measuring points at several times (distance-time lowering method)

It should always be used for all measurements that are carried out at a groundwater measuring point

${\displaystyle {\frac {r^{2}}{t}}<0{,}08\cdot {\frac {T_{\text{Schätz}}}{S_{\text{Schätz}}}}}$

be valid. Here is

• ${\displaystyle r}$ the distance in meters between the groundwater measuring point at which the measurement is carried out and the extraction point
• ${\displaystyle t}$ the time in seconds from the start of the pumping attempt to the time of the measurement
• ${\displaystyle T_{\text{Schätz}}}$a rough estimate of the transmissivity in square meters per second
• ${\displaystyle S_{\text{Schätz}}}$a roughly estimated value for the dimensionless storage coefficient end

For the origin of this condition see #Herleitungsskizze

Time reduction procedure

With the time lowering method, the lowering is measured at all groundwater measuring points at the same time. The results are then plotted on semi-log paper. The subsidence is plotted on the linear axis and the corresponding distances between the groundwater measuring points are logarithmic. Then a regression line is drawn through the measuring points. This line of best fit is then read off ${\displaystyle t_{0}}$

• the decrease over a logarithmic decade, i.e. the decrease that occurs when the distance increases tenfold ${\displaystyle \Delta h}$
• The distance at which the regression line intersects the distance axis, i.e. the distance from the extraction point at which no further subsidence occurs. ${\displaystyle r_{0}}$

Then applies

${\displaystyle T={\frac {2{,}3}{2\pi }}\cdot {\frac {Q}{\Delta h}}}$

and

${\displaystyle S=2{,}25\cdot {\frac {T\cdot t_{0}}{r_{0}}}}$

It is

• ${\displaystyle T}$the transmissivity in square meters per second
• ${\displaystyle Q}$ the rate of extraction from the well in cubic meters per second
• ${\displaystyle \Delta h}$ the lowering determined on the regression line when the distance from the extraction point is increased tenfold
• ${\displaystyle S}$the dimensionless storage coefficient
• ${\displaystyle t_{0}}$ the time after the start of the pumping test at which the subsidence was measured, in seconds
• ${\displaystyle r_{0}}$ the distance determined on the regression line from the extraction point at which no further subsidence occurs, in meters.

Distance lowering method

At a fixed groundwater measuring point, the time and the corresponding lowering are recorded when the pumping test begins. The measuring points are then plotted on semi-logarithmic paper. The decrease is plotted on the linear axis and the time on the logarithmic axis. A straight line of best fit is drawn through the points created in this way. Read off from this line of best fit

• The decrease over a logarithmic decade, i.e. the decrease that occurs when the time increases tenfold. ${\displaystyle \Delta h}$
• The time after which a first decrease becomes noticeable, i.e. the intersection of the best-fit straight line with the time axis. ${\displaystyle t_{0}}$

Then applies

${\displaystyle T={\frac {2{,}3}{4\pi }}\cdot {\frac {Q}{\Delta h}}}$

and

${\displaystyle S=2{,}25\cdot {\frac {T\cdot t_{0}}{r^{2}}}}$

It is

• ${\displaystyle T}$the transmissivity in square meters per second
• ${\displaystyle Q}$ the rate of extraction from the well in cubic meters per second
• ${\displaystyle \Delta h}$ the decrease determined on the regression line over a logarithmic decade
• ${\displaystyle S}$the dimensionless storage coefficient
• ${\displaystyle t}$ the time read from the regression line until the first lowering in the groundwater measuring point in seconds
• ${\displaystyle r}$ the distance between the groundwater measuring point and the extraction point in meters.

Distance-time lowering method

In the distance-time subsidence method, subsidence in several groundwater points is recorded over time. These measured values ​​are then plotted again semi-logarithmically. The decrease is plotted on the linear axis, the logarithmic axis is plotted, whereby the time is after which the corresponding decrease was measured at the measuring point at a distance from . A regression line is drawn through the measuring points. This is used to read ${\displaystyle t/r^{2}}$ ${\displaystyle t}$ ${\displaystyle r}$

• The decrease over a logarithmic decade of ${\displaystyle \Delta h}$ ${\displaystyle t/r^{2}}$
• The intersection of the regression line with the axis ${\displaystyle k_{0}}$ ${\displaystyle t/r^{2}}$

Then applies

${\displaystyle T={\frac {2{,}3}{4\pi }}\cdot {\frac {Q}{\Delta h}}}$

and

${\displaystyle S=2{,}25\cdot T\cdot k_{0}}$

It is

• ${\displaystyle T}$the transmissivity in square meters per second
• ${\displaystyle Q}$ the rate of extraction from the well in cubic meters per second
• ${\displaystyle \Delta h}$ the lowering determined on the regression line over a logarithmic decade
• ${\displaystyle S}$the dimensionless storage coefficient
• ${\displaystyle k_{0}}$the intersection of the best fit line with the axis determined from the best fit line. ${\displaystyle t/r^{2}}$

Derivation sketch

In the Theis method , the auxiliary variable

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

and the Theis well function introduced. Now it can be shown that under the conditions mentioned at the beginning ${\displaystyle W(u)}$

${\displaystyle {\frac {r^{2}}{t}}<0{,}08\cdot {\frac {T}{S}}}$

which are equivalent to

${\displaystyle u<0{,}02}$

are that approximation

${\displaystyle W(u)=-\gamma -\ln(u)+\underbrace {u-{\frac {u^{2}}{2\cdot 2!}}+{\frac {u^{3}}{3\cdot 3!}}-{\frac {u^{4}}{4\cdot 4!}}+\cdots } _{{\text{ klein, da }}u<0{,}02}\approx -\gamma -\ln(u)}$

applies. Put this into the equation

${\displaystyle \Delta h(r,t)={\frac {Q}{4\pi \cdot T}}\cdot W(u)}$

of the Theis method, the corresponding equations for the Cooper-Jacob method are obtained by rearranging and converting the base of the logarithm into base ten.

swell

• 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. 293-297 , doi : 10.1007 / 978-3-8274-2354-2 .
• Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 240-248 .
• Kevin M. Hiscock, Viktor F. Bense: Hydrogeology . Principles and Practice. 2nd Edition. Wiley-Blackwell, Oxford 2014, ISBN 978-0-470-65663-1 , pp. 277-282 .

Individual evidence

1. ^ Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 240 .
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. 293 , doi : 10.1007 / 978-3-8274-2354-2 .
3. ^ ^{A b} Horst-Robert Langguth, Rudolf Voigt: Hydrogeological methods . 2nd Edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-21126-8 , pp. 241 .