# Hantush and Jacob r/B Solution for Leaky Confined Aquifers

A mathematical solution by Hantush and Jacob (1955) is useful for determining the hydraulic properties (transmissivity and storativity of pumped aquifer; vertical hydraulic conductivity of aquitard) of **leaky confined** (semi-confined) aquifers.

Analysis involves matching the Hantush-Jacob **w(u,r/B)** well function for leaky confined aquifers to drawdown data collected during a pumping test (aquifer test). The solution assumes no storage in **incompressible** leaky aquitard(s) and can account for partially penetrating wells (Hantush 1961a,b).

**Mahdi S. Hantush** (1921-1984) and **Charles Edward Jacob** (1914-1970) developed the first rigorous mathematical model of transient flow of water to a pumping well in a leaky confined aquifer. Their mathematical equation for flow to a fully penetrating line sink discharging at a constant rate in a homogeneous, isotropic and leaky confined aquifer of infinite extent is as follows:

where

- ${b}^{\prime}$ is aquitard thickness [L]
- ${K}^{\prime}$ is vertical hydraulic conductivity of the aquitard [L/T]
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $s$ is drawdown [L]
- $S$ is storativity [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- $y$ is a variable of integration

Groundwater hydrologists commonly refer to the integral in the Hantush and Jacob solution (Equation 1) as the *Hantush and Jacob well function for leaky confined aquifers*, abbreviated as **w(u,r/B)**. Therefore, we may write the Hantush and Jacob equation in compact notation as follows:

Note that Kruseman and de Ridder (1994) refer to the Hantush and Jacob curve-fitting procedure as **Walton's method** (Walton 1962) and use the symbol **w(u,r/L)** to represent the Hantush and Jacob well function for leaky confined aquifers.

A partially penetrating pumping well produces vertical components of flow in the pumped aquifer. Hantush (1961a, b) derived equations which extend the Hantush and Jacob method to include partially penetration effects in a leaky confined aquifer. In the case of a piezometer, the following equation applies:

$$s=\frac{Q}{4\pi T}[\mathrm{w}\left(u\text{,}r/B\right)+\frac{2b}{\pi \left(l-d\right)}\sum _{n=1}^{\infty}\frac{1}{n}(\mathrm{sin}\left(\frac{n\pi l}{b}\right)-\mathrm{sin}\left(\frac{n\pi d}{b}\right))\xb7\mathrm{cos}\left(\frac{n\pi z}{b}\right)\xb7\mathrm{w}\left(u\mathrm{,}\sqrt{{(r/B)}^{2}+{K}_{z}/{K}_{r}{\left(\frac{n\pi r}{b}\right)}^{2}}\right)]\text{(5)}$$The following equation computes drawdown for a partially penetrating observation well:

$$s=\frac{Q}{4\pi T}[\mathrm{w}\left(u\text{,}r/B\right)+\frac{2{b}^{2}}{{\pi}^{2}\left(l-d\right)\left({l}^{\prime}-{d}^{\prime}\right)}\sum _{n=1}^{\infty}\frac{1}{{n}^{2}}\left(\mathrm{sin}\left(\frac{n\pi l}{b}\right)-\mathrm{sin}\left(\frac{n\pi d}{b}\right)\right)\xb7\left(\mathrm{sin}\left(\frac{n\pi {l}^{\prime}}{b}\right)-\mathrm{sin}\left(\frac{n{\pi d}^{\prime}}{b}\right)\right)\xb7\mathrm{w}\left(u\mathrm{,}\sqrt{{(r/B)}^{2}+{K}_{z}/{K}_{r}{\left(\frac{n\pi r}{b}\right)}^{2}}\right)]\text{(6)}$$where

- $b$ is aquifer thickness [L]
- $d$ is the depth to the top of pumping well screen [L]
- ${d}^{\prime}$ is the depth to the top of observation well screen [L]
- ${K}_{r}$ is the radial (horizontal) hydraulic conductivity [L/T]
- ${K}_{z}$ is the vertical hydraulic conductivity [L/T]
- $l$ is the depth to the bottom of pumping well screen [L]
- ${l}^{\prime}$ is the depth to the bottom of observation well screen [L]
- $\text{w}(u,\beta )$ is the Hantush and Jacob well function for leaky confined aquifers [dimensionless]
- $z$ is piezometer depth [L]

and

$$\beta =\sqrt{{(r/B)}^{2}+{K}_{z}/{K}_{r}{\left(\frac{n\pi r}{b}\right)}^{2}}\text{(7)}$$Other solution methods for leaky confined aquifers include the following:

- Hantush (1960) - line source with compressible aquitard
- Neuman and Witherspoon (1969) - line source with compressible aquitard and variable-head in source aquifer
- Moench (1985) - finite-diameter well with wellbore storage and wellbore skin for Hantush Cases 1, 2 and 3

## Assumptions

The following assumptions apply to the use of the Hantush and Jacob r/B pumping test solution:

- aquifer has infinite areal extent
- aquifer is homogeneous, isotropic and of uniform thickness
- control well is fully or partially penetrating
- flow to control well is horizontal when control well is fully penetrating
- aquifer is leaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head
- diameter of control well is very small so that storage in the well can be neglected
- aquitards have infinite areal extent, uniform vertical hydraulic conductivity and uniform thickness
- aquitards are overlain or underlain by an infinite constant-head plane source
- aquitards are incompressible (no storage)
- flow in the aquitards is vertical

## Solution

Options

AQTESOLV provides the following options for the Hantush and Jacob r/B solution for leaky confined aquifers:

- variable pumping rates
- multiple pumping wells
- multiple observation wells
- partially penetrating pumping and observation wells
- boundaries

## Benchmark

## References

Hantush, M.S. and C.E. Jacob, 1955. Non-steady radial flow in an infinite leaky aquifer, Am. Geophys. Union Trans., vol. 36, no. 1, pp. 95-100.

Hantush, M.S., 1961a. Drawdown around a partially penetrating well, Jour. of the Hyd. Div., Proc. of the Am. Soc. of Civil Eng., vol. 87, no. HY4, pp. 83-98.

Hantush, M.S., 1961b. Aquifer tests on partially penetrating wells, Jour. of the Hyd. Div., Proc. of the Am. Soc. of Civil Eng., vol. 87, no. HY5, pp. 171-194.