# Theis Solution for Nonleaky Confined Aquifers

The Theis (1935) solution (or *Theis nonequilibrium method*) is useful for determining the hydraulic properties (transmissivity and storativity) of **nonleaky confined aquifers**. Analysis involves matching the Theis type curve for nonleaky confined aquifers to nonequilibrium (i.e., transient) drawdown data collected during a pumping test (aquifer test or aquifer performance test).

**Charles Vernon Theis** (1900-1987) was the first hydrologist to develop a rigorous mathematical model of transient flow of water to a pumping well by recognizing the physical analogy between *heat flow in solids* and *groundwater flow in porous media*. The **Theis type curve method**, used for determining the hydraulic properties of aquifers, is based on the following mathematical equation (i.e., the Theis equation) for flow to a fully penetrating line sink discharging at a constant rate in a homogeneous, isotropic and nonleaky confined aquifer of infinite extent:

where

- $Q$ 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 Theis solution (Equation 1) as the *Theis well function*, abbreviated as ** w(u)**. Therefore, we may write the Theis equation in compact notation as follows:

A partially penetrating pumping well produces vertical components of flow in the aquifer. Hantush (1961a, b; 1964) derived equations extending the Theis method to include partially penetration effects in a nonleaky confined aquifer. In the case of a piezometer, the following equation applies:

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

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

$$\beta =\sqrt{{K}_{z}/{K}_{r}}\frac{n\pi r}{b}\text{(7)}$$The traditional Theis curve-fitting procedure involves matching the Theis type curve to data plotted on a graph with log-log axes. The **modified nonequilibrium method** of Cooper and Jacob (1946), a simplification of the Theis procedure, only involves matching a straight line to drawdown data plotted on semilog graph.

For more complex tests in nonleaky confined aquifers, Dougherty and Babu (1984) introduced a solution which accounts for partial penetration, wellbore storage and wellbore skin.

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

AQTESOLV also includes Jacob's correction for partial dewatering of water-table (phreatic) aquifers, thereby allowing use of the Theis solution for **unconfined aquifers**.

## Assumptions

The following assumptions apply to the use of the Theis type curve 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 nonleaky 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

## Solution

Options

AQTESOLV provides the following options for the Theis nonequilibrium method:

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

## Benchmark

## 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.

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.