# Cooley and Case Solution for Leaky Confined Aquifers

- Assumptions
- Equations
- Data requirements
- Solution options
- Estimated parameters
- Curve matching tips
- References

Related Solution Methods

Additional Topics

A mathematical solution by Cooley and Case (1973) is useful for determining the hydraulic properties (transmissivity, storativity and leakage parameters) of a **leaky confined** aquifer with a water-table aquitard. Analysis involves matching the Cooley and Case solution to drawdown data collected during a constant- or variable-rate pumping test.

You are not restricted to constant-rate tests with the Cooley and Case solution. AQTESOLV incorporates the principle of superposition in time to simulate variable-rate and recovery tests with this method.

## Assumptions

- aquifer has infinite areal extent
- aquifer is homogeneous, isotropic and of uniform thickness
- potentiometric surface is initially horizontal
- aquifer is confined
- flow is unsteady
- wells are fully penetrating
- water is released instantaneously from storage with decline of hydraulic head
- diameter of control well is very small so that storage in the well can be neglected
- confining bed has infinite areal extent, uniform vertical hydraulic conductivity, storage coefficient, specific yield and thickness
- flow is vertical in the aquitard

## Equations

Cooley and Case (1973) derived a solution for unsteady flow to a fully penetrating well in a homogeneous, isotropic leaky confined aquifer overlain by a water-table aquitard. The Laplace transform solution is as follows:

$${\overline{s}}_{D}=\frac{2{K}_{0}\left(x\right)}{p}\text{(1)}$$ $$x=\sqrt{p+{\overline{q}}_{D}}\text{(2)}$$ $${\overline{q}}_{D}=4\sqrt{p}\beta \mathrm{coth}\left(\frac{4\sqrt{p}\beta}{{\left(r/B\right)}^{2}}\right)+\left[p{\mathrm{sech}}^{2}\left(\frac{4\sqrt{p}\beta}{{\left(r/B\right)}^{2}}\right)\right]\times $$ $${\left[p\frac{L/{b}^{\prime}}{{\left(r/B\right)}^{2}}+\frac{{\left(r/B\right)}^{2}}{16{\beta}^{2}}\frac{{S}^{\prime}}{{S}_{y}}+\frac{\sqrt{p}}{4\beta}\mathrm{tanh}\left(\frac{4\sqrt{p}\beta}{{\left(r/B\right)}^{2}}\right)\right]}^{\mathrm{-1}}\text{(3)}$$ $$B=\sqrt{\frac{T\left({b}^{\prime}+L\right)}{{K}^{\prime}}}\text{(3)}$$ $$\beta =\frac{r}{4}\sqrt{\frac{{K}^{\prime}{S}^{\prime}}{{b}^{\prime}TS}}\text{(5)}$$ $${t}_{D}=\frac{Tt}{S{r}^{2}}\text{(6)}$$ $${s}_{D}=\frac{4\pi T}{Q}s\text{(7)}$$where

- ${b}^{\prime}$ is thickness of aquitard [L]
- ${K}^{\prime}$ is vertical hydraulic conductivity of aquitard [L/T]
- ${K}_{0}$ is modified Bessel function of second kind, order zero
- $L$ is height of capillary fringe [L]
- $p$ is Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $s$ is drawdown [L]
- $S$ is storativity [dimensionless]
- ${S}^{\prime}$ is storativity of aquitard [dimensionless]
- ${S}_{y}$ is specific yield of aquitard [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]

## Data Requirements

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

## Solution Options

- constant or variable pumping rate with recovery
- multiple pumping wells
- multiple observation wells
- boundaries

## Estimated Parameters

- $T$ (transmissivity)
- $S$ (storativity)
- $r/B$ (leakage parameter)
- $\beta $ (leakage parameter)
- ${S}^{\prime}/{S}_{y}$ (storage ratio in aquitard)
- $L/{b}^{\prime}$ (dimensionless height of capillary fringe)

The Report view shows aquitard properties, $K\prime /b\prime $ and $K\prime $, computed from the leakage parameter, $r/B$.

## Curve Matching Tips

- Use the Cooper and Jacob (1946) solution to obtain preliminary estimates of aquifer properties.
- Choose Match>Visual to perform visual curve matching using the procedure for type curve solutions.
- Use active type curves for more effective visual matching with variable-rate pumping tests.
- Select values of $r/B$ and $\beta $ from the
**Family**and**Curve**drop-down lists on the toolbar. - Use parameter tweaking to perform visual curve matching and sensitivity analysis.
- Perform visual curve matching prior to automatic estimation to obtain reasonable starting values for the aquifer properties.

## References

Cooley, R.L. and C.M. Case, 1973. Effect of a water table aquitard on drawdown in an underlying pumped aquifer, Water Resources Research, vol. 9, no. 2, pp. 434-447.