﻿ Pumping Test: Theory and Analysis Methods > Pumping Test Analysis Methods - Fixed Assumptions > Cooper-Jacob Method (confined; small r or large time)

# Cooper-Jacob Method (confined; small r or large time)

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.

## Assumptions

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".

## Cooper-Jacob I: Time-Drawdown Method

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]

### Example

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"

### Data Requirements

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)

## Cooper-Jacob II: Distance-Drawdown Method

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]

### Example

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"

### Data Requirements

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.

## Cooper-Jacob III: Time-Distance-Drawdown Method

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]

### Example

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"

### Data Requirements

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)