# Hantush (1960) Solution for Leaky Confined Aquifers

- Assumptions
- Equations
- Data requirements
- Solution options
- Estimated parameters
- Curve matching tips
- Benchmark
- References

Related Solution Methods

- Hantush and Jacob (1955)
- Hantush and Jacob (1955) for step tests
- Neuman and Witherspoon (1969)
- Moench (1985)

Additional Topics

Hantush (1960) published a mathematical solution that is useful for determining the hydraulic properties (transmissivity and storativity of pumped aquifer; vertical hydraulic conductivity and storage coefficient of aquitard) of **leaky confined** (semi-confined) aquifers.

Evaluation of aquifer properties involves matching the Hantush well function to water-level drawdown data collected during a pumping test (aquifer test). The solution assumes storage in leaky compressible aquitard(s) and also accounts for partially penetrating wells (Hantush 1961a,b).

You are not restricted to constant-rate tests with the Hantush solution. AQTESOLV incorporates the principle of superposition in time to simulate variable-rate and recovery tests with this method.

## Assumptions

- 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, storage coefficient and uniform thickness
- aquitards are overlain or underlain by an infinite constant-head plane source
- flow in the aquitards is vertical

## Equations

Hantush (1960) derived an analytical solution for unsteady flow to a fully penetrating well in a homogeneous, isotropic leaky confined aquifer assuming aquitard storage. The Laplace transform solution is as follows:

$${\overline{s}}_{D}=\frac{2{K}_{0}\left(x\right)}{p}\phantom{\rule{1em}{0ex}}\text{(1)}$$ $$x=\sqrt{p+{\overline{q}}_{D}}\phantom{\rule{1em}{0ex}}\text{(2)}$$ $${\overline{q}}_{D}=4\sqrt{p}{\beta}^{\prime}\mathrm{coth}\left(\frac{4\sqrt{p}{\beta}^{\prime}}{{\left(r/{B}^{\prime}\right)}^{2}}\right)+4\sqrt{p}{\beta}^{\u2033}\mathrm{coth}\left(\frac{4\sqrt{p}{\beta}^{\u2033}}{{\left(r/{B}^{\u2033}\right)}^{2}}\right)\phantom{\rule{1em}{0ex}}\text{(3)}$$ $${B}^{\prime}=\sqrt{\frac{T{b}^{\prime}}{{K}^{\prime}}}\phantom{\rule{1em}{0ex}}\text{(4)}$$ $${B}^{\u2033}=\sqrt{\frac{T{b}^{\u2033}}{{K}^{\u2033}}}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $${\beta}^{\prime}=\frac{r}{4}\sqrt{\frac{{K}^{\prime}{S}^{\prime}}{{b}^{\prime}TS}}\phantom{\rule{1em}{0ex}}\text{(6)}$$ $${\beta}^{\u2033}=\frac{r}{4}\sqrt{\frac{{K}^{\u2033}{S}^{\u2033}}{{b}^{\u2033}TS}}\phantom{\rule{1em}{0ex}}\text{(7)}$$ $${t}_{D}=\frac{Tt}{S{r}^{2}}\phantom{\rule{1em}{0ex}}\text{(8)}$$ $${s}_{D}=\frac{4\pi T}{Q}s\phantom{\rule{1em}{0ex}}\text{(9)}$$where

- ${b}^{\prime}$ is thickness of first aquitard [L]
- ${b}^{\u2033}$ is thickness of second aquitard [L]
- ${K}^{\prime}$ is vertical hydraulic conductivity of first aquitard [L/T]
- ${K}^{\u2033}$ is vertical hydraulic conductivity of second aquitard [L/T]
- ${K}_{0}$ is modified Bessel function of second kind, order zero
- $p$ is Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $s$ is drawdown [L]
- $S$ is storativity [dimensionless]
- ${S}^{\prime}$ is storativity of first aquitard [dimensionless]
- ${S}^{\u2033}$ is storativity of second aquitard [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]

Hantush (1960) also derived an asymptotic solution ($H(u,\beta )$ well function) for drawdown in the pumped aquifer at *short values of time*:

or

$$\beta =\frac{r}{4}\left[\sqrt{\frac{{K}^{\prime}{S}^{\prime}}{{b}^{\prime}TS}}+\sqrt{\frac{{K}^{\u2033}{S}^{\u2033}}{{b}^{\u2033}TS}}\right]\phantom{\rule{1em}{0ex}}\text{(13)}$$The solution assumes small values of time, i.e., when $t$ < ${b}^{\prime}{S}^{\prime}/10{K}^{\prime}$ and $t$ < ${b}^{\u2033}{S}^{\u2033}/10{K}^{\u2033}$.

The early-time approximate solution can be modified for partial penetrating wells and anisotropy using the approach of Hantush (1961a, b).

At large distances, the effect of partial penetration becomes negligible when

$$r>1.5b\sqrt{{K}_{r}/{K}_{z}}\phantom{\rule{1em}{0ex}}\text{(14)}$$where

- $b$ is aquifer thickness [L]
- ${K}_{r}$ is radial hydraulic conductivity of aquifer [L/T]
- ${K}_{z}$ is vertical hydraulic conductivity of aquifer [L/T]

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- partial penetration depths (optional)
- saturated thickness (for partially penetrating wells)
- hydraulic conductivity anisotropy ratio (for partially penetrating wells)

## Solution Options

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

## Estimated Parameters

- $T$ (transmissivity)
- $S$ (storativity)
- $r/{B}^{\prime}$ (leakage parameter, first aquitard)
- ${\beta}^{\prime}$ (leakage parameter, first aquitard)
- $r/{B}^{\u2033}$ (leakage parameter, second aquitard)
- ${\beta}^{\u2033}$ (leakage parameter, second aquitard)

The Report view shows aquitard properties (${K}^{\prime}/{b}^{\prime}$ and ${K}^{\prime}$; ${K}^{\u2033}/{b}^{\u2033}$ and ${K}^{\u2033}$) computed from the leakage parameters ($r/{B}^{\prime}$ and $r/{B}^{\u2033}$).

Early-Time Solution

- $T$ (transmissivity)
- $S$ (storativity)
- $\beta $ (leakage parameter)
- ${K}_{z}/{K}_{r}$ (hydraulic conductivity anisotropy ratio)
- $b$ (aquifer thickness)

## Curve Matching Tips

- Use the Cooper and Jacob (1946) solution to obtain preliminary estimates of aquifer properties.
- 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 $r/B$, $\beta $ and ${K}_{z}/{K}_{r}$ 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.
- For $b$ < 0.5, all of the type curves in the early-time solution have a similar shape (Kruseman and de Ridder 1994); hence, it is often difficult to obtain a unique match with the early-time solution when 0 ≤ $b$ ≤ 0.5.

## Benchmark

## References

Hantush, M.S., 1960. Modification of the theory of leaky aquifers, Jour. of Geophys. Res., vol. 65, no. 11, pp. 3713-3725.

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.