# Butler Solution for Confined Aquifers with Nonuniform Properties

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

Related Solution Methods

Additional Topics

A mathematical solution by Butler (1988) is useful for determining the hydraulic properties (transmissivity and storativity of two zones) of **nonleaky confined aquifers** with **nonuniform properties**.

The solution assumes the pumping well is located at the center of a disk embedded within an infinite matrix. Hydraulic properties are assumed uniform within each zone (disk and matrix), but may differ between the two zones. The solution assumes a line source for the pumped well and therefore neglects wellbore storage.

Analysis involves matching the Butler solution to drawdown data collected during a pumping test.

You are not restricted to constant-rate tests with the Butler solution. Using the principle of superposition in time, AQTESOLV can simulate variable-rate and recovery tests with this method.

## Assumptions

- aquifer has infinite areal extent
- aquifer is heterogeneous with a disk embedded within a matrix
- 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 control well is very small so that storage in the well can be neglected

## Equations

Butler (1988) derived a solution for unsteady flow to a fully penetrating line source in a heterogeneous, isotropic confined aquifer. The solution assumes the pumping well is located at the center of a disk of radius $R$ embedded within an infinite matrix. The Laplace transform solution is as follows:

$${s}_{1}=\left(\frac{Q}{2\pi {T}_{1}}\right)\frac{{K}_{0}\left(Nr\right)}{p}+\left(\frac{Q}{2\pi {T}_{1}}\right)\frac{\left[{K}_{1}\left(NR\right){K}_{0}\left(AR\right)-\frac{{T}_{2}}{{T}_{1}}\frac{A}{N}{K}_{0}\left(NR\right){K}_{1}\left(AR\right)\right]{I}_{0}\left(Nr\right)}{p\left[\frac{{T}_{2}}{{T}_{1}}\frac{A}{N}{I}_{0}\left(NR\right){K}_{1}\left(AR\right)+{I}_{1}\left(NR\right){K}_{0}\left(AR\right)\right]}\text{(1)}$$ $${s}_{2}=\left(\frac{Q}{2\pi {T}_{1}}\right)\frac{\left[{K}_{0}\left(NR\right){I}_{1}\left(NR\right)+{K}_{1}\left(NR\right){I}_{0}\left(NR\right)\right]{K}_{0}\left(Ar\right)}{p\left[\frac{{T}_{2}}{{T}_{1}}\frac{A}{N}{I}_{0}\left(NR\right){K}_{1}\left(AR\right)+{I}_{1}\left(NR\right){K}_{0}\left(AR\right)\right]}\text{(2)}$$ $$N=\sqrt{{S}_{1}p/{T}_{1}}\text{(3)}$$ $$A=\sqrt{{S}_{2}p/{T}_{2}}\text{(4)}$$ $${s}_{D}=\frac{4\pi T}{Q}s\text{(5)}$$ $${t}_{D}=\frac{Tt}{{r}^{2}S}\text{(6)}$$where

- ${I}_{i}$ is modified Bessel function of first kind, order i
- ${K}_{i}$ is modified Bessel function of second kind, order i
- $p$ is Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $R$ is radial distance to disk-matrix interface [L]
- ${s}_{1}$ is drawdown in the disk [L]
- ${s}_{2}$ is drawdown in the matrix [L]
- ${S}_{1}$ is storativity in the disk [dimensionless]
- ${S}_{2}$ is storativity in the matrix [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- ${T}_{1}$ is transmissivity in the disk [L²/T]
- ${T}_{2}$ is transmissivity in the matrix [L²/T]

## Data Requirements

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

## Solution Options

- variable pumping rates
- multiple observation wells

## Estimated Parameters

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

- ${T}_{1}$ (transmissivity in disk)
- ${S}_{1}$ (storativity in disk)
- ${T}_{2}$ (transmissivity in matrix)
- ${S}_{2}$ (storativity in matrix)
- $R$ (radial distance to disk-matrix interface)

## Curve Matching Tips

- Match the Cooper and Jacob (1946) solution to late-time data 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.
- 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

Butler, J.J., Jr., 1988. Pumping tests in nonuniform aquifersâ€”the radially symmetric case, Journal of Hydrology, vol. 101, pp. 15-30.