Papadopulos and Cooper Solution for Nonleaky Confined Aquifers

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:

$$s=\frac{2\alpha Q}{{\pi }^{2}T}{\int }_0^{\infty }\frac{\left(1-e^{-{\beta }^2{r}_D^2/4u}\right)\left(J_0\left(\beta r_D\right)A\left(\beta \right)-Y_0\left(\beta r_D\right)B\left(\beta \right)\right)}{\left({\left[A\left(\beta \right)\right]}^2+{\left[B\left(\beta \right)\right]}^2\right){\beta }^2}d\beta \text{(1)}$$ $$A\left(\beta \right)=\beta Y_0\left(\beta \right)-2\alpha Y_1\left(\beta \right)\text{(2)}$$ $$B\left(\beta \right)=\beta J_0\left(\beta \right)-2\alpha J_1\left(\beta \right)\text{(3)}$$ $$u=\frac{r^{2}S}{4Tt}\text{(4)}$$ $$\alpha =\frac{r_w^{2}S}{r_c^2}\text{(5)}$$ $$r_D=\frac{r}{r_w}\text{(6)}$$

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.