# Cooper and Jacob Solution for Unconfined Aquifers

Related Solution Methods

Additional Topics

**C. E. Jacob** is credited with a procedure (Jacob 1944) that corrects drawdown data for the reduction in an unconfined aquifer's saturated thickness resulting from groundwater withdrawal by a pumping well and thereby enables pumping tests in water-table aquifers to be interpreted by methods for nonleaky confined aquifers such as the Cooper and Jacob (1946) solution.

Jacob's correction applies to pumping tests with discharging wells and is not appropriate for injection tests.

## Assumptions

- aquifer has infinite areal extent
- aquifer is homogeneous, isotropic and of uniform thickness
- control well is fully penetrating
- flow to control well is horizontal
- aquifer is confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head
- diameter of pumping well is very small so that storage in the control well can be neglected
- values of u are small (i.e., r is small and t is large)
- no delayed gravity response in aquifer
- low velocity is proportional to tangent of the hydraulic gradient instead of the sine (which is actually the case)
- flow is horizontal and uniform in a vertical section through the axis of the well
- drawdown is small relative to saturated thickness of aquifer

## Equations

The Cooper and Jacob (1946) solution may be used for the interpretation of pumping tests in unconfined aquifers through the application of the following simple correction to drawdown data measured during a test (Kruseman and de Ridder 1994):

$${s}^{\prime}=s-{s}^{2}/2b\text{(1)}$$where

- $b$ is saturated thickness [L]
- $s$ is obsesrved drawdown [L]
- ${s}^{\prime}$ is corrected drawdown [L]

For more information regarding this method or to apply the solution without the dewatering correction, refer to the Cooper and Jacob (1946) solution for confined aquifers.

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- saturated thickness

## Solution Options

- variable pumping rates
- multiple observation wells

## Estimated Parameters

AQTESOLV provides visual and automatic methods for matching the Cooper and Jacob modified nonequilibrium method for unconfined aquifers to pumping test data. The estimated aquifer properties are as follows:

- $T$ (transmissivity)
- $S$ (storativity)

## References

Cooper, H.H. and C.E. Jacob, 1946. A generalized graphical method for evaluating formation constants and summarizing well field history, Am. Geophys. Union Trans., vol. 27, pp. 526-534.

Jacob, C.E., 1944. Notes on determining permeability by pumping tests under water-table conditions, U.S. Geological Survey, 25p.

Kruseman, G.P. and N.A. de Ridder, 1994. Analysis and Evaluation of Pumping Test Data (2nd ed.), Publication 47, Intern. Inst. for Land Reclamation and Improvement, Wageningen, The Netherlands, 370p. [pdf]