# Papadopulos and Cooper Solution for Nonleaky Confined Aquifers

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

Related Solution Methods

Additional Topics

**Stavros S. Papadopulos** (b. 1936) and **Hilton H. Cooper** (1913-1990), groundwater hydrologists with the U.S. Geological Survey, derived a mathematical solution for determining the hydraulic properties (transmissivity and storativity) of **nonleaky confined aquifers** from pumping test data.

Analysis with the Papadopulos and Cooper method involves matching the solution to transient drawdown data collected during a pumping test. The Papadopulos and Cooper solution includes the effect of wellbore storage in a finite-diameter control well.

You are not restricted to analyzing constant-rate tests with the Papadopulos and Cooper solution. AQTESOLV incorporates the principle of superposition in time for the simulation of variable-rate and recovery tests with this method.

**Wellbore storage** has a distinct signature in the early-time response of a pumped well. Use radial flow and derivative plots to detect the wellbore storage effect.

## 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 nonleaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head

## Equations

Prior to the introduction of the Papadopulos and Cooper (1967) method, mathematical models for pumping test analysis such as the Theis 1935 and Hantush 1960 solutions assumed discharge from a line sink of infinitesimal diameter which neglected casing storage in the control well.

The solution by Papadopolus and Cooper extended the well-known Theis (1935) nonequilibrium method by adding a less restrictive boundary condition at the control well. The resulting mathematical model for transient groundwater flow to a fully penetrating, finite-diameter well with wellbore storage which discharges at a constant rate from an infinite homogeneous, isotropic and nonleaky confined aquifer is expressed in the following equations:

$$s=\frac{2\alpha Q}{{\pi}^{2}T}{\int}_{0}^{\infty}\frac{\left(1-{e}^{-{\beta}^{2}{r}_{D}^{2}/4u}\right)\left({J}_{0}\left(\beta {r}_{D}\right)A\left(\beta \right)-{Y}_{0}\left(\beta {r}_{D}\right)B\left(\beta \right)\right)}{\left({\left[A\left(\beta \right)\right]}^{2}+{\left[B\left(\beta \right)\right]}^{2}\right){\beta}^{2}}d\beta \phantom{\rule{1em}{0ex}}\text{(1)}$$ $$A\left(\beta \right)=\beta {Y}_{0}\left(\beta \right)-2\alpha {Y}_{1}\left(\beta \right)\phantom{\rule{1em}{0ex}}\text{(2)}$$ $$B\left(\beta \right)=\beta {J}_{0}\left(\beta \right)-2\alpha {J}_{1}\left(\beta \right)\phantom{\rule{1em}{0ex}}\text{(3)}$$ $$u=\frac{{r}^{2}S}{4Tt}\phantom{\rule{1em}{0ex}}\text{(4)}$$ $$\alpha =\frac{{r}_{w}^{2}S}{{r}_{c}^{2}}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $${r}_{D}=\frac{r}{{r}_{w}}\phantom{\rule{1em}{0ex}}\text{(6)}$$where

- ${J}_{i}$ is Bessel function of first kind, order $i$
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- ${r}_{c}$ is casing radius [L]
- ${r}_{w}$ is well radius [L]
- $s$ is drawdown [L]
- $S$ is storativity [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- ${Y}_{i}$ is Bessel function of second kind, order $i$
- $\beta $ is a variable of integration

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- casing radius and wellbore radius for pumping well(s)
- downhole equipment radius (optional)

## Solution Options

- large-diameter pumping wells
- variable pumping rates
- multiple pumping wells
- multiple observation wells
- boundaries

## Estimated Parameters

AQTESOLV provides visual and automatic methods for matching the Papadopulos and Cooper method to data from pumping tests and recovery tests. The estimated aquifer properties are as follows:

- $T$ (transmissivity)
- $S$ (storativity)
- $\mathrm{r(w)}$ (well radius)
- $\mathrm{r(c)}$ (nominal casing radius)

## Curve Matching Tips

- Use radial flow plots to help diagnose wellbore storage.
- Match the Cooper and Jacob (1946) solution to late-time data to obtain preliminary estimates of aquifer properties.
- Match early-time data affected by wellbore storage by adjusting r(c) with parameter tweaking.
- If you estimate r(c) for the test well, the estimated value replaces the nominal casing radius and AQTESOLV still performs the correction for downhole equipment.
- 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 S, r(w) and r(c) 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.

## Benchmark

## References

Papadopulos, I.S. and H.H. Cooper, 1967. Drawdown in a well of large diameter, Water Resources Research, vol. 3, no. 1, pp. 241-244.