# Moench and Prickett Solution for Nonleaky Confined Aquifers Undergoing Unconfined Conversion

A mathematical solution by Moench and Prickett (1972) is useful for determining the hydraulic properties (transmissivity, confined storativity and specific yield) of **confined** aquifers undergoing conversion to **unconfined** conditions. In the confined region beyond a radial distance $R$ from the control well, flow is described by transmissivity and storativity; upon conversion, flow in the unconfined region is controlled by transmissivity and specific yield. Analysis involves matching the Moench and Prickett solution to drawdown data collected during a pumping test.

## 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
- pumping rate is constant
- aquifer is confined with conversion to unconfined conditions
- 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

Moench and Prickett (1972) derived an analytical solution for unsteady flow to a fully penetrating well discharging at a constant rate in a nonleaky confined aquifer undergoing conversion to water-table conditions.

When $r$ < $R$,

$${h}_{1}=b-\frac{Q}{4\pi T}\left(\text{w}\left({u}_{1}\right)-\text{w}\left(\nu \right)\right)\phantom{\rule{1em}{0ex}}\text{(1)}$$When $r$ > $R$,

$${h}_{2}=H-\frac{\left(H-b\right)\text{w}\left({u}_{2}\right)}{\text{w}\left(\nu S/{S}_{y}\right)}\phantom{\rule{1em}{0ex}}\text{(2)}$$When $r$ = $R$, $h$ = $b$.

$$\text{w}\left(x\right)={\int}_{x}^{\infty}\frac{{e}^{-z}}{z}\text{d}z\phantom{\rule{1em}{0ex}}\text{(3)}$$ $${u}_{1}=\frac{{r}^{2}S}{4Tt}\phantom{\rule{1em}{0ex}}\text{(4)}$$ $${u}_{2}=\frac{{r}^{2}{S}_{y}}{4Tt}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $$\nu =\frac{{R}^{2}S}{4Tt}\phantom{\rule{1em}{0ex}}\text{(6)}$$ $$\frac{Q}{4\pi T\left(H-b\right)}{e}^{-\nu}-\frac{{e}^{-\nu S/{S}_{y}}}{\text{w}\left(\nu S/{S}_{y}\right)}=0\phantom{\rule{1em}{0ex}}\text{(7)}$$ $${h}_{D}=\frac{4\pi T}{Q}h\phantom{\rule{1em}{0ex}}\text{(8)}$$ $${t}_{D}=\frac{Tt}{{r}^{2}S}\phantom{\rule{1em}{0ex}}\text{(9)}$$where

- $b$ is aquifer thickness [L]
- ${h}_{1}$ is the elevation of the water table when $r$ < $R$ [L]
- ${h}_{2}$ is the elevation of the potentiometric surface when $r$ > $R$ [L]
- $H$ is the elevation of the initial potentiometric surface above the base of the aquifer [L]
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $R$ is the radial distance to the point of conversion [L]
- $S$ is storativity [dimensionless]
- ${S}_{y}$ is the storativity in the unconfined zone when $r$ < $R$ [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- $\text{w}\left(x\right)$ is the Theis well function [dimensionless]
- $z$ is a variable of integration

Drawdown, $s$, is computed from the above equations as follows:

$${s}_{1}=H-{h}_{1}\phantom{\rule{0.5em}{0ex}}\left(r<R\right)\phantom{\rule{1em}{0ex}}\text{(10)}$$ $${s}_{2}=H-{h}_{2}\phantom{\rule{0.5em}{0ex}}\left(r>R\right)\phantom{\rule{1em}{0ex}}\text{(11)}$$ $${s}_{\mathrm{R}}=H-b\phantom{\rule{0.5em}{0ex}}\left(r=R\right)\phantom{\rule{1em}{0ex}}\text{(12)}$$When $\nu $ is greater than about 3, which implies $H$ - $b$ is small, the curve generated by the Moench and Prickett solution is essentially the same as the Theis solution using $T$ and ${S}_{y}$. On the other hand when $H$ - $b$ is large and $\nu $ is small, the Moench and Prickett solution is virtually identical to the Theis solution using $T$ and $S$.

## Data Requirements

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

## Solution Options

- constant pumping rate
- multiple observation wells

## Estimated Parameters

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

- $T$ (transmissivity)
- $S$ (storativity)
- ${S}_{y}$ (specific yield)
- $H$ - $b$ (initial head above top of aquifer)

## 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

Moench, A.F. and T.A. Prickett, 1972. Radial flow in an infinite aquifer undergoing conversion from artesian to water-table conditions, Water Resources Research, vol. 8, no. 2, pp. 494-499.