# Tartakovsky and Neuman Solution for Unconfined Aquifers

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

Related Solution Methods

Additional Topics

A mathematical solution by Tartakovsky and Neuman (2007) is useful for determining the hydraulic properties (transmissivity, elastic storage coefficient, specific yield and vertical hydraulic conductivity) of **unconfined** aquifers. Analysis involves matching the solution to drawdown data collected during a pumping test. The Tartakovsky and Neuman solution accounts for delayed gravity response, partial penetration and three-dimensional unsaturated flow.

AQTESOLV provides visual and automatic methods for matching the Neuman solution to pumping test and recovery test data. This easy-to-use and intuitive software promotes rapid and accurate determination of aquifer properties.

## Assumptions

- aquifer has infinite areal extent
- aquifer is homogeneous, anisotropic and of uniform thickness
- aquifer potentiometric surface is initially horizontal
- pumping well is fully or partially penetrating
- aquifer is unconfined
- the unsaturated zone has infinite thickness
- flow is unsteady
- diameter of control well is very small so that storage in the well can be neglected

## Equations

Tartakovsky and Neuman (2007) developed a mathematical model for unsteady flow to a partially penetrating well in an unconfined aquifer with three-dimensional flow in the saturated and unsaturated zones.

The Tartakovsky and Neuman solution computes drawdown in an unconfined aquifer as the sum of three components:

$$s={s}_{T}+{s}_{H}+{s}_{U}\phantom{\rule{1em}{0ex}}\text{(1)}$$where

- $s$ is drawdown in the unconfined aquifer [L]
- ${s}_{T}$ is drawdown due to a fully penetrating well in a nonleaky confined aquifer [L]
- ${s}_{H}$ is a drawdown correction for partial penetration in a nonleaky confined aquifer [L]
- ${s}_{U}$ is a drawdown correction for saturated-unsaturated flow in an unconfined aquifer [L]

Equations for ${s}_{T}$ and ${s}_{H}$ due to Theis (1935) and Hantush (1964), respectively, are presented with the Theis method. The Laplace transform of ${s}_{U}$ is given as follows:

$${\overline{s}}_{U}=\frac{Qt}{2\pi T{p}_{D}}{\int}_{0}^{\infty}\frac{\mathrm{cosh}\left(\mu {z}_{D}\right)\gamma}{\mathrm{cosh}\left(\mu \right)\gamma -\mu \mathrm{sinh}\left(\mu \right)}\xb7\frac{\mathrm{sinh}\left[\mu \left(1-{l}_{D}\right)\right]-\mathrm{sinh}\left[\mu \left(1-{d}_{D}\right)\right]}{{\mu}^{2}\left({l}_{D}-{d}_{D}\right)\mathrm{sinh}\left(\mu \right)}y{J}_{0}\left(y\sqrt{\beta}\right)dy\phantom{\rule{1em}{0ex}}\text{(2)}$$ $$\beta =\frac{{r}^{2}{K}_{z}}{{b}^{2}{K}_{r}}\phantom{\rule{1em}{0ex}}\text{(3)}$$ $$\gamma =\frac{{\kappa}_{D}}{2}-\sqrt{\frac{{\kappa}_{D}^{2}}{4}+{\nu}^{2}}\phantom{\rule{1em}{0ex}}\text{(4)}$$ $${\mu}^{2}={y}^{2}+\frac{{p}_{D}}{{t}_{s}\beta}\phantom{\rule{1em}{0ex}}\text{(5)}$$ $${\nu}^{2}={y}^{2}+\frac{{p}_{D}}{{t}_{s}\phi \beta}\phantom{\rule{1em}{0ex}}\text{(6)}$$ $$\phi =\frac{S}{{\kappa}_{D}{S}_{y}}\phantom{\rule{1em}{0ex}}\text{(7)}$$ $${d}_{D}=d/b\phantom{\rule{1em}{0ex}}\text{(8)}$$ $${l}_{D}=l/b\phantom{\rule{1em}{0ex}}\text{(9)}$$ $${z}_{D}=z/b\phantom{\rule{1em}{0ex}}\text{(10)}$$ $${\kappa}_{D}=\kappa b\phantom{\rule{1em}{0ex}}\text{(11)}$$ $${t}_{s}=\frac{Tt}{S{r}^{2}}\phantom{\rule{1em}{0ex}}\text{(12)}$$ $${p}_{D}=tp\phantom{\rule{1em}{0ex}}\text{(13)}$$ $${s}_{D}=\frac{4\pi T}{Q}s\phantom{\rule{1em}{0ex}}\text{(14)}$$where

- $b$ is aquifer saturated thickness [L]
- $d$ is distance from water table to top of pumping well screen [L]
- ${J}_{0}$ is Bessel function of first kind, zero order
- ${K}_{r}$ is radial hydraulic conductivity of aquifer [L/T]
- ${K}_{z}$ is vertical hydraulic conductivity of aquifer [L/T]
- $\kappa $ is Gardner parameter for the unsaturated zone [L
^{-1}] - $l$ is distance from water table to bottom of pumping well screen [L]
- $p$ is Laplace transform variable
- $Q$ is pumping rate [L³/T]
- $r$ is radial distance from pumping well to observation well [L]
- $S$ is elastic storage coefficient ($={S}_{s}b$) [dimensionless]
- ${S}_{s}$ is specific storage [L
^{-1}] - ${S}_{y}$ is specific yield [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- $y$ is a variable of integration
- $z$ is elevation of piezometer above base of aquifer [L]

Expressions for fully and partially penetrating observations wells are obtained by vertically averaging the drawdown computed with (1) over the length of the observation well screen.

As ${\kappa}_{D}$ approaches ∞, the effect of the unsaturated zone dies out and the solution behaves like the model of Neuman (1974).

## Data Requirements

- pumping and observation well locations
- pumping rate(s)
- observation well measurements (time and displacement)
- partial penetration depths (optional)
- saturated thickness

## Solution Options

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

## Estimated Parameters

- $T$ (transmissivity)
- $S$ (elastic storage coefficient)
- ${S}_{y}$ (specific yield)
- ${K}_{z}/{K}_{r}$ (hydraulic conductivity anisotropy ratio)
- $k\left(D\right)$ (dimensionless Gardner parameter)

## Curve Matching Tips

- 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.
- 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.
- Change convergence levels to optimize the speed and accuracy of computations for the solution. The default setting is suitable for all cases. The fastest level works well for fully penetrating wells or partially penetrating wells with longer screens. Use higher convergence levels to check the performance of lower settings.

## References

Tartakovsky, G.D. and S.P. Neuman, 2007. Three-dimensional saturated-unsaturated flow with axial symmetry to a partially penetrating well in a compressible unconfined aquifer, Water Resources Research, W01410, doi:1029/2006WR005153.