# Murdoch Solution for Flow to an Interceptor Trench in Confined Aquifers

Murdoch (1994) presented an analytical solution for unsteady flow to an interceptor trench based on the Gringarten and Witherspoon (1972) solution for flow to a uniform-flux plane vertical fracture in an anisotropic nonleaky confined aquifer. The trench is represented by a fully penetrating vertical plane source oriented parallel to the x axis. In the uniform-flux formulation of this solution, drawdown is variable and flux is uniformly distributed along the length of the trench.

One may use the Murdoch (1994) solution to determine the hydraulic properties (hydraulic conductivity, specific_storage, hydraulic conductivity anisotropy and trench length) of **nonleaky confined aquifers**. Analysis involves matching the solution to drawdown data collected during pumping from an interceptor trench. The solution estimates hydraulic conductivity anisotropy in a horizontal (x-y) plane.

The early-time response of pumping from an interceptor trench has a distinct signature that you can diagnose with a linear flow plot.

You are not restricted to constant-rate tests with the Murdoch 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 has uniform thickness
- aquifer potentiometric surface is initially horizontal
- trench is fully penetrating
- aquifer is nonleaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head

## Equations

Murdoch (1994) modified the Gringarten and Witherspon (1972) solution for a uniform-flux vertical fracture by replacing the vertical fracture with a trench to arrive at an analytical solution for flow to an interceptor trench in an anisotropic nonleaky confined aquifer:

$$s=\frac{Q}{4\pi \sqrt{{T}_{x}{T}_{y}}}\frac{\sqrt{\pi}}{2}{\int}_{0}^{{t}_{D}}\left[\mathrm{erf}\frac{1-{x}_{D}}{2\sqrt{\tau}}+\mathrm{erf}\frac{1+{x}_{D}}{2\sqrt{\tau}}\right]{e}^{-\frac{{y}_{D}^{2}{T}_{x}}{4\tau {T}_{y}}}\text{d}\tau /\sqrt{\tau}\phantom{\rule{1em}{0ex}}\text{(1)}$$ $${t}_{D}=\frac{{T}_{x}t}{S{x}_{t}^{2}}\phantom{\rule{1em}{0ex}}\text{(2)}$$ $${x}_{D}=\frac{x}{{x}_{t}}\phantom{\rule{1em}{0ex}}\text{(3)}$$ $${y}_{D}=\frac{y}{{x}_{t}}\phantom{\rule{1em}{0ex}}\text{(4)}$$where

- $Q$ is pumping rate [L³/T]
- $s$ is drawdown [L]
- $S$ is storativity [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- ${T}_{x}$ is transmissivity in x direction [L²/T]
- ${T}_{y}$ is transmissivity in y direction [L²/T]
- $\tau $ is variable of integration
- $x$ and $y$ are coordinate directions [L]
- ${x}_{t}$ is the half-length of the trench in the x direction [L]

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- aquifer thickness
- length of trench

## Solution Options

- variable pumping rates
- multiple pumping wells
- multiple observation wells

## Estimated Parameters

- ${K}_{x}$ (hydraulic conductivity in x direction)
- ${S}_{s}$ (specific storage)
- ${K}_{y}/{K}_{x}$ (hydraulic conductivity anisotropy ratio)
- ${L}_{t}$ (length of trench)

## Curve Matching Tips

- Use linear flow plots to help diagnose linear flow.
- 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.
- Select values of ${K}_{y}/{K}_{x}$ 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.

## References

Murdoch, L.C., 1994. Transient analyses of an interceptor trench, Water Resources Research, vol. 30, no. 11, pp. 3023-3031.