The Cooper-Jacob (1946) method is a simplification of the Theis (1935) method that is valid for greater time values and smaller distances from the pumping well (i.e. smaller values of u). This method involves truncation of the infinite Taylor series that is used to estimate the well function . Due to this truncation, not all early time measured data is considered to be valid for this analysis method. The resulting equation is:

where:

is the drawdown at the observation well [L]

is the discharge from the pumping well [L3/T]

is the transmissivity of the aquifer [L2/T]

is the distance from the well to the observation point [L]

is the elapsed time since the start of pumping [T]

is the storativity of the aquifer [-]

This solution is appropriate for the conditions shown in the following figure.

The Cooper-Jacob Solution assumes the following:

•The aquifer is confined and has an “apparent” infinite extent

•The aquifer is homogeneous, isotropic, and of uniform thickness over the area influenced by pumping

•The piezometric surface was horizontal prior to pumping

•The well is pumped at a constant rate

•The well is fully penetrating

•Water removed from storage is discharged instantaneously with decline in head

•The well diameter is small, so well storage is negligible

•The values of are small (rule of thumb ), where is the dimensionless argument to the well function, , that is is small and/or is relatively large

Kruseman and deRidder (1991) indicate that larger values of (i.e. ) may be acceptable as the difference between the Cooper-Jacob approximation and the full well function are generally less than 5%.

In AquiferTest, it is possible to define different values of for the validity line. For more details, see "Constants tab".

The above equation plots as a straight line on semi-logarithmic paper if the limiting condition is met. Thus, straight-line plots of drawdown versus time can occur after sufficient time has elapsed. In pumping tests with multiple observation wells, the closer wells will meet the conditions before the more distant ones. Time is plotted along the logarithmic X axis and drawdown is plotted along the linear Y axis.

Transmissivity and storativity are calculated as follows:

and

where:

is the amount of drawdown over one log cycle of time (i.e. ) [L]

is the X-axis intercept (i.e. where the extrapolated line of best fit intersects the time axis) [T]

An example of a Cooper-Jacob Time-Drawdown analysis graph has been included below:

An example of a CooperJacob I analysis is available in the project:

"C:\Users\Public\Documents\AquiferTest Pro\Examples\CooperJacob1.HYT"

The data requirements for the Cooper-Jacob Time-Drawdown Solution method are:

•Drawdown vs. time data at an observation well

•Finite distance from the pumping well to the observation well

•Pumping rate (constant)

If simultaneous observations of drawdown in three or more observation wells are available, a modification of the Cooper-Jacob method may be used. The observation well distance is plotted along the logarithmic X-axis, and drawdown is plotted along the linear Y-axis.

Transmissivity and storativity are calculated as follows:

and

where:

is the X-axis intercept (i.e. where the extrapolated line of best fit intersects the distance axis) [L]

An example of a Cooper-Jacob Distance-Drawdown analysis graph has been included below:

An example of a CooperJacob II analysis is available in the project:

"C:\Users\Public\Documents\AquiferTest Pro\Examples\CooperJacob2.HYT"

The data requirements for the Cooper-Jacob Distance-Drawdown Solution method are:

•Drawdown vs. time data at three or more observation wells

•Distance from the pumping well to the observation wells

•Pumping rate (constant)

Both distance and drawdown values at a specific time are plotted, so you must specify this time value in the Results section of the Analysis Navigator Panel.

As with the Distance-Drawdown Method, if simultaneous observations are made of drawdown in three or more observation wells, a modification of the Cooper-Jacob method may be used. Drawdown is plotted along the linear Y-axis and is plotted along the logarithmic X-axis.

Transmissivity and storativity are calculated as follows:

and

where:

is the X-axis intercept (i.e. where the extrapolated line of best fit intersects the axis) [T/L2]

An example of a Cooper-Jacob Type II (Time-Distance-Drawdown) analysis graph has been included in the following figure:

An example of a CooperJacob III analysis is available in the project:

"C:\Users\Public\Documents\AquiferTest Pro\Examples\CooperJacob3.HYT"

The data requirements for the Cooper-Jacob Time-Distance-Drawdown Solution method are:

•Drawdown vs. time data at three or more observation wells

•Distance from the pumping well to the observation wells

•Pumping rate (constant)

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