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 $r$ < $R$,

$$h_1=b-\frac{Q}{4\pi T}\left(\text{w}\left(u_1\right)-\text{w}\left(\nu \right)\right)\text{(1)}$$

When $r$ > $R$,

$$h_2=H-\frac{\left(H-b\right)\text{w}\left(u_2\right)}{\text{w}\left(\nu S/S_y\right)}\text{(2)}$$

When $r$ = $R$, $h$ = $b$.

$$\text{w}\left(x\right)={\int }_x^{\infty }\frac{e^{-z}}{z}\text{d}z\text{(3)}$$ $$u_1=\frac{r^{2}S}{4Tt}\text{(4)}$$ $$u_2=\frac{r^2{S}_y}{4Tt}\text{(5)}$$ $$\nu =\frac{R^{2}S}{4Tt}\text{(6)}$$ $$\frac{Q}{4\pi T\left(H-b\right)}{e}^{-\nu }-\frac{e^{-\nu S/S_y}}{\text{w}\left(\nu S/S_y\right)}=0\text{(7)}$$ $$h_D=\frac{4\pi T}{Q}h\text{(8)}$$ $$t_D=\frac{Tt}{r^{2}S}\text{(9)}$$

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:

$$s_1=H-h_1\left(r<R\right)\text{(10)}$$ $$s_2=H-h_2\left(r>R\right)\text{(11)}$$ $$s_{\mathrm{R}}=H-b\left(r=R\right)\text{(12)}$$

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.