# Dougherty and Babu Solution for Step-Drawdown Tests in Confined Aquifers

Dougherty and Babu (1984) developed a mathematical solution for interpreting pumping tests in **nonleaky confined** aquifers. Their solution accounts for wellbore storage and wellbore skin in a finite-diameter, partially penetrating well; hydraulic conductivity anisotropy was added to the solution by Moench (1988).

## Assumptions

- aquifer has infinite areal extent
- aquifer is homogeneous, anisotropic and of uniform thickness
- pumping well is fully or partially penetrating
- flow to control well is horizontal when the control well is fully penetrating
- aquifer is nonleaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head

## Equations

The Dougherty and Babu pumping test solution includes drawdown components for **aquifer loss**, **partial penetration loss** and **linear well loss due to wellbore skin**. It is a straightforward matter to add **nonlinear well loss** to drawdown in the pumped well (Bear 1979):

where

- $b$ is aquifer thickness [L]
- $C$ is nonlinear well loss coefficient [T
^{P}/L^{3P-1}] - $d$ is the depth to the top of pumping well screen [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 pumping well screen [L]
- $p$ is the Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- ${r}_{c}$ is casing radius [L]
- ${r}_{w}$ is well radius [L]
- ${s}_{w}$ is drawdown in pumped well [L]
- $S$ is storativity [dimensionless]
- ${S}_{\mathrm{w}}$ is wellbore skin factor [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- ${\mathcal{L}}^{-1}\left\{\right\}$ is the inverse Laplace transform operator

The exponent, $P$, in the nonlinear well loss term, $C{Q}^{P}$, is generally taken as **2** as originally proposed by Jacob (1947); however, Rorabaugh (1953) postulated that $P$ may range between **1.5** and **3.5**.

AQTESOLV performs inverse Laplace transformation, ${\mathcal{L}}^{-1}\left\{\right\}$, through the application of numerical inversion techniques.

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- casing radius and wellbore radius for pumping well(s)
- downhole equipment radius (optional)
- partial penetration depths
- saturated thickness
- hydraulic conductivity anisotropy ratio

## Solution Options

- large-diameter pumping wells
- variable pumping rates
- multiple pumping wells
- multiple observation wells
- partially penetrating wells
- boundaries

## Estimated Parameters

AQTESOLV provides visual and automatic methods for matching the Dougherty and Babu method to data from step-drawdown tests and recovery tests. The estimated aquifer properties are as follows:

- $T$ (transmissivity)
- $S$ (storativity)
- ${K}_{z}/{K}_{r}$ (hydraulic conductivity anisotropy ratio)
- ${S}_{w}$ (dimensionless wellbore skin factor)
- $\mathrm{r(w)}$ (well radius)
- $\mathrm{r(c)}$ (nominal casing radius)
- $C$ (nonlinear well loss coefficient)
- $P$ (nonlinear well loss exponent)

## Curve Matching Tips

- Use radial flow plots to help diagnose wellbore storage.
- Match the Cooper and Jacob (1946) solution to late-time data to obtain preliminary estimates of aquifer properties.
- Match early-time data affected by wellbore storage by adjusting $\mathrm{r(c)}$ with parameter tweaking.
- If you estimate $\mathrm{r(c)}$ for the test well, the estimated value replaces the nominal casing radius and AQTESOLV still performs the correction for downhole equipment.
- 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 ${K}_{z}/{K}_{r}$ 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

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.

Moench, A.F., 1988. The response of partially penetrating wells to pumpage from double-porosity aquifers, Proceedings of the International Conference on Fluid Flow in Fractured Rocks, Atlanta, GA, May 16-18, 1988.

Jacob, C.E., 1947. Drawdown test to determine effective radius of artesian well, Trans. Amer. Soc. of Civil Engrs., vol. 112, paper 2321, pp. 1047-1064.

Ramey, H.J., 1982. Well-loss function and the skin effect: A review. In: Narasimhan, T.N. (ed.) Recent trends in hydrogeology, Geol. Soc. Am., special paper 189, pp. 265-271.

Rorabaugh, M.J., 1953. Graphical and theoretical analysis of step-drawdown test of artesian well, Proc. Amer. Soc. Civil Engrs., vol. 79, separate no. 362, 23 pp.