# Dougherty and Babu Slug Test Solution for Confined Aquifers

Related Solution Methods

Additional Topics

Dougherty and Babu (1984) presented a mathematical solution for determining the hydraulic properties of **nonleaky confined** aquifers from an overdamped slug test. The solution accounts for wellbore skin as well as hydraulic conductivity anisotropy (Moench 1988). Analysis involves matching a type curve solution to water-level displacement data from the control well and observation wells.

The following Laplace transform solution evaluates dimensionless drawdown in the **control well**:

The Laplace transform solution for dimensionless drawdown in a **piezometer** is as follows:

The Laplace transform solution for dimensionless drawdown in an **observation well** is given by the following equation:

where

- $b$ is aquifer thickness [L]
- $d$ is the depth to the top of control well screen [L]
- ${d}^{\prime}$ is the depth to the top of observation well screen [L]
- $H$ is displacement at time $t$ [L]
- ${H}_{0}$ is initial displacement at $t=0$ [L]
- ${K}_{r}$ is radial (horizontal) hydraulic conductivity [L/T]
- ${K}_{z}$ is vertical hydraulic conductivity [L/T]
- ${K}_{i}$ is modified Bessel function of second kind, order $i$
- $l$ is the depth to the bottom of control well screen [L]
- ${l}^{\prime}$ is the depth to the bottom of observation well screen [L]
- $p$ is the Laplace transform variable
- $r$ is radial distance from control well to observation well [L]
- ${r}_{c}$ is casing radius [L]
- ${r}_{w}$ is well radius [L]
- $S$ is storativity [dimensionless]
- ${S}_{w}$ is wellbore skin factor [dimensionless]
- $t$ is elapsed time since initiation of test [T]
- $T$ is transmissivity [L²/T]
- $z$ is piezometer depth [L]

Note that ${r}_{\text{w}}$ is typically taken as the borehole radius (i.e., extending to the outer radius of the filter pack) when the filter pack is expected to be more conductive than the aquifer.

In a **confined aquifer with no wellbore skin**, the Dougherty and Babu slug test solution is equivalent to Cooper et al. (1967) for **fully penetrating wells** and the KGS Model for **partially penetrating wells**.

The implementation of **wellbore skin** in the Dougherty and Babu solution differs from the KGS Model. Dougherty and Babu assumes an **infinitesimal skin thickness** while the KGS Model has a **finite-thickness skin**.

## Assumptions

The following assumptions apply to the use of the Dougherty and Babu solution:

- aquifer has infinite areal extent
- aquifer is homogeneous and of uniform thickness
- aquifer potentiometric surface is initially horizontal
- wells are fully or partially penetrating
- a volume of water, V, is injected or discharged instantaneously from the control well
- flow to control well is horizontal
- aquifer is nonleaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head

## Solution

Options

AQTESOLV provides the following options for the Dougherty and Babu solution:

- partially penetrating wells
- hydraulic conductivity anisotropy
- wellbore skin effect
- observation wells

## Benchmark

## References

Dougherty, D.E and D.K. Babu, 1984. Flow to a partially penetrating well in a double-porosity reservoir, Water Resources Research, vol. 20, no. 8, pp. 1116-1122.