# Theis Step-Drawdown Test Solution for Nonleaky Confined Aquifers

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

Related Solution Methods

Additional Topics

Theis (1935) developed a mathematical solution for determining the hydraulic properties (transmissivity and storativity) of **nonleaky confined** aquifers from pumping tests. Hantush (1961a, b; 1964) later introduced equations for partially penetrating wells.

For step-drawdown tests, the Theis/Hantush solution can be easily modified to include linear and nonlinear well losses occurring at the pumping well (Jacob 1947; Rorabaugh 1953; Ramey 1982). Analysis involves matching a curve to drawdown data collected during a step-drawdown test.

## Assumptions

- aquifer has infinite areal extent
- aquifer is homogeneous and of uniform thickness
- control well is fully or partially penetrating
- aquifer is nonleaky confined
- flow is unsteady
- water is released instantaneously from storage with decline of hydraulic head
- diameter of pumping well is very small so that storage in the well can be neglected

## Equations

The Theis/Hantush model for a partially penetrating pumping well in an infinite anisotropic nonleaky confined aquifer and adapted for step-drawdown tests to include linear and nonlinear well loss is given by the following equation:

$${s}_{\mathrm{w}}=\frac{Q}{4\pi T}[\mathrm{w}\left(u\right)+\frac{2{b}^{2}}{{\pi}^{2}{\left(l-d\right)}^{2}}\sum _{n=1}^{\infty}\frac{1}{{n}^{2}}{\left(\mathrm{sin}\left(\frac{n\pi l}{b}\right)-\mathrm{sin}\left(\frac{n\pi d}{b}\right)\right)}^{2}\xb7\mathrm{w}\left(u\mathrm{,}\beta \right)+\frac{2b}{\left(l-d\right)}{S}_{w}]+C{Q}^{P}\phantom{\rule{1em}{0ex}}\text{(1)}$$ $$u=\frac{{r}_{\mathrm{w}}^{2}S}{4Tt}\phantom{\rule{1em}{0ex}}\text{(2)}$$ $$\beta =\sqrt{{K}_{z}/{K}_{r}}\frac{n\pi {r}_{\mathrm{w}}}{b}\phantom{\rule{1em}{0ex}}\text{(3)}$$where

- $b$ is aquifer thickness [L]
- $C$ is nonlinear well loss coefficient [T
^{P}/L^{3P-1}] - $d$ is the depth to the top of pumping 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]
- $Q$ is pumping rate [L³/T]
- ${r}_{\mathrm{w}}$ is well radius [L]
- ${s}_{\mathrm{w}}$ is drawdown in the pumped well [L]
- $S$ is storativity [dimensionless]
- ${S}_{\mathrm{w}}$ is wellbore skin factor [dimensionless]
- $t$ is elapsed time since start of pumping [T]
- $T$ is transmissivity [L²/T]
- $\text{w}\left(u\right)$ is the Theis well function for nonleaky confined aquifers [dimensionless]
- $\text{w}(u,\beta )$ is the Hantush and Jacob well function for leaky confined aquifers [dimensionless]

The exponent, $P$, in the nonlinear well loss term, $C{Q}^{P}$, is generally taken as **2** as originally proposed by Jacob (1947); however, Rorabaugh (1953) postulated that $P$ may range between **1.5** and **3.5**.

## 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

## Estimated Parameters

AQTESOLV provides visual and automatic methods for matching the Theis nonequilibrium step-drawdown test method to data from step-drawdown tests. The estimated aquifer properties and well-loss parameters are as follows:

- $T$ (transmissivity)
- $S$ (storativity)
- ${S}_{w}$ (wellbore skin factor)
- $C$ (nonlinear well loss coefficient)
- $P$ (nonlinear well loss exponent)

## 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.
- 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.
- Due to correlation in the equations between $S$ (storativity) and ${S}_{w}$ (wellbore skin factor), estimate either $S$ or ${S}_{w}$ for a single-well test but not both as the same time.

## Example

## References

Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage, Am. Geophys. Union Trans., vol. 16, pp. 519-524.

Jacob, C.E., 1947. Drawdown test to determine effective radius of artesian well, Trans. Amer. Soc. of Civil Engrs., vol. 112, paper 2321, pp. 1047-1064.

Ramey, H.J., 1982. Well-loss function and the skin effect: A review. In: Narasimhan, T.N. (ed.) Recent trends in hydrogeology, Geol. Soc. Am., special paper 189, pp. 265-271.

Rorabaugh, M.J., 1953. Graphical and theoretical analysis of step-drawdown test of artesian well, Proc. Amer. Soc. Civil Engrs., vol. 79, separate no. 362, 23 pp.