Dougherty and Babu Slug Test Solution for Confined Aquifers

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:

$${\overline{H}}_{wD}=\frac{W_D\left(A_0+A+S_w\right)}{1+p{W}_D\left(A_0+A+S_w\right)}\text{(1)}$$

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

$${\overline{H}}_D=\frac{W_D\left(E_0+E\right)}{1+p{W}_D\left(A_0+A+S_w\right)}\text{(2)}$$

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

$${\overline{H}}_D^{´}=\frac{W_D\left(E_0^{´}+E^{\prime }\right)}{1+p{W}_D\left(A_0+A+S_w\right)}\text{(3)}$$ $$A_0=\frac{\left(l_D-d_D\right)K_0\left(q_0\right)}{q_0{K}_1\left(q_0\right)}\text{(4)}$$ $$A=\frac{2}{\left(l_D-d_D\right)}\sum _{n=1}^{\infty }\frac{K_0\left(q_n\right)[\sin \left(n\pi l_D\right)-\sin \left(n\pi d_D\right){]}^2}{n^2{\pi }^2{q}_n{K}_1\left(q_n\right)}\text{(5)}$$ $$E_0=E_0^{´}=\frac{(l_D-d_D)K_0\left(q_0{r}_D\right)}{q_0{K}_1\left(q_0\right)}\text{(6)}$$ $$E=2\sum _{n=1}^{\infty }\frac{K_0\left(q_n{r}_D\right)\cos \left(n\pi z_D\right)\left[\sin \right(n\pi l_D)-\sin (n\pi d_D\left)\right]}{n\pi q_n{K}_1\left(q_n\right)}\text{(7)}$$ $$E^{\prime }=2\sum _{n=1}^{\infty }\frac{K_0\left(q_n{r}_D\right)}{(l_D^{´}-d_D^{´})n^2{\pi }^2{q}_n{K}_1\left(q_n\right)}·\left[\sin \text{(}n\pi l_D^{´}\text{)}-\sin \text{(}n\pi d_D^{´}\text{)}\right]\left[\sin \text{(}n\pi l_D\text{)}-\sin \text{(}n\pi d_D\text{)}\right]\text{(8)}$$ $$W_D=\frac{r_c^2}{2r_w^{2}S(l_D-d_D)}\text{(9)}$$ $$q_n=\sqrt{\frac{r_w^2{n}^2{\pi }^2}{b^2}\frac{K_z}{K_r}+p}\text{(10)}$$ $$r_D=\frac{r}{r_w}\text{(11)}$$ $$d_D=\frac{d}{b}\text{(12)}$$ $$l_D=\frac{l}{b}\text{(13)}$$ $$d_D^{´}=\frac{d^{\prime }}{b}\text{(14)}$$ $$l_D^{´}=\frac{l^{\prime }}{b}\text{(15)}$$ $$z_D=\frac{z}{b}\text{(16)}$$ $$t_D=\frac{Tt}{S{r}_w^2}\text{(17)}$$ $$H_D=\frac{H}{H_0}\text{(18)}$$

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.