Hantush (1960) Solution for Leaky Confined Aquifers

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:

$${\overline{s}}_D=\frac{2K_0\left(x\right)}{p}\text{(1)}$$ $$x=\sqrt{p+{\overline{q}}_D}\text{(2)}$$ $${\overline{q}}_D=4\sqrt{p}{\beta }^{\prime }\coth \left(\frac{4\sqrt{p}{\beta }^{\prime }}{{\left(r/B^{\prime }\right)}^2}\right)+4\sqrt{p}{\beta }^{″}\coth \left(\frac{4\sqrt{p}{\beta }^{″}}{{\left(r/B^{″}\right)}^2}\right)\text{(3)}$$ $$B^{\prime }=\sqrt{\frac{T{b}^{\prime }}{K^{\prime }}}\text{(4)}$$ $$B^{″}=\sqrt{\frac{T{b}^{″}}{K^{″}}}\text{(5)}$$ $${\beta }^{\prime }=\frac{r}{4}\sqrt{\frac{K^{\prime }{S}^{\prime }}{b^{\prime }TS}}\text{(6)}$$ $${\beta }^{″}=\frac{r}{4}\sqrt{\frac{K^{″}{S}^{″}}{b^{″}TS}}\text{(7)}$$ $$t_D=\frac{Tt}{S{r}^2}\text{(8)}$$ $$s_D=\frac{4\pi T}{Q}s\text{(9)}$$

where

• $b^{\prime }$ is thickness of first aquitard [L]
• $b^{″}$ is thickness of second aquitard [L]
• $K^{\prime }$ is vertical hydraulic conductivity of first aquitard [L/T]
• $K^{″}$ 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]
• $S^{″}$ 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:

$$s=\frac{Q}{4\pi T}{\int }_u^{\infty }\frac{e^{-y}}{y}\mathrm{erfc}\frac{\beta \sqrt{u}}{\sqrt{y\left(y-u\right)}}\mathrm{dy}\text{(10)}$$ $$u=\frac{r^{2}S}{4Tt}\text{(11)}$$ $$\beta =\frac{r}{4}\sqrt{\frac{K^{\prime }{S}^{\prime }}{b^{\prime }TS}}\text{(12)}$$

or

$$\beta =\frac{r}{4}\left[\sqrt{\frac{K^{\prime }{S}^{\prime }}{b^{\prime }TS}}+\sqrt{\frac{K^{″}{S}^{″}}{b^{″}TS}}\right]\text{(13)}$$

The solution assumes small values of time, i.e., when $t$ < $b^{\prime }{S}^{\prime }/10K^{\prime }$ and $t$ < $b^{″}{S}^{″}/10K^{″}$.

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

$$r>1.5b\sqrt{K_r/K_z}\text{(14)}$$

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)
• $r/B^{″}$ (leakage parameter, second aquitard)
• ${\beta }^{″}$ (leakage parameter, second aquitard)

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

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.