Cooper, Bredehoeft and Papadopulos Slug Test Solution for Confined Aquifers

A mathematical solution by Cooper, Bredehoeft and Papadopulos (1967) is useful for determining the hydraulic properties (transmissivity and storage coefficient) of nonleaky confined aquifers. Analysis involves matching a type curve to water-level displacement data from an overdamped slug test.

Hilton H. Cooper, John D. Bredehoeft and Stavros S. Papadopulos, groundwater hydrologists at the U.S. Geological Survey, derived the following Laplace transform solution for an overdamped slug test in a nonleaky confined aquifer assuming a fully penetrating control well:

$$\overline{H}=\frac{H_0{r}_{\text{w}}S{K}_0\left(rq\right)}{Tq\left(r_{\text{w}}q{K}_0\left(r_{\text{w}}q\right)+2\alpha K_1\left(r_{\text{w}}q\right)\right)}\text{(1)}$$ $$q=\sqrt{pS/T}\text{(2)}$$ $$\alpha =\frac{r_{\text{w}}^{2}S}{r_{\text{c}}^2}\text{(3)}$$

where

• $H$ is displacement in the well at time $t$ [L]
• $H_0$ is initial displacement at $t=0$ [L]
• $K_i$ is modified Bessel function of second kind, order $i$
• $p$ is the Laplace transform variable
• $r$ is radial distance [L]
• $r_{\text{c}}$ is casing radius [L]
• $r_{\text{w}}$ is well radius [L]
• $S$ is storativity [dimensionless]
• $t$ is elapsed time since initiation of test [T]
• $T$ is transmissivity [L²/T]

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.

Butler (1998) suggests using the Cooper et al. solution as a screening tool for most overdamped slug tests in confined and unconfined aquifers with the exception of wells screened across the water table.