# Theis Solution for Nonleaky Confined Aquifers

## Description

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** was the first 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 *s* is drawdown [L], *Q* is pumping rate [L³/T],
*T* is transmissivity [L²/T], *r* is radial
distance from pumping well to observation well [L], *S* is storativity
[-] and *t* is elapsed time since start of pumping [T].

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:

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:

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

$$s=\frac{Q}{4\pi T}[\mathrm{w}\left(u\right)+\frac{2{b}^{2}}{{\pi}^{2}\left(l-d\right)\left(l\text{'}-d\text{'}\right)}\sum _{n=1}^{\infty}\frac{1}{{n}^{2}}\left(\mathrm{sin}\left(\frac{n\pi l}{b}\right)-\mathrm{sin}\left(\frac{n\pi d}{b}\right)\right)\xb7$$ $$\left(\mathrm{sin}\left(\frac{n\pi {l}^{\prime}}{b}\right)-\mathrm{sin}\left(\frac{n{\pi d}^{\prime}}{b}\right)\right)\xb7\mathrm{w}\left(u\mathrm{,}\sqrt{{K}_{z}/{K}_{r}}\frac{n\pi r}{b}\right)]\text{(6)}$$where *w(u,β)* is the Hantush
well function for leaky confined aquifers [-], *d* and *l* are the depths to the
top and bottom of pumping well screen [L], respectively, *z* is piezometer depth
[L], *d'* and *l'* are the depths to the top and bottom of
observation well screen [L], respectively, *b* is aquifer thickness
[L], *K _{z}* is the vertical hydraulic conductivity [L/T],

*K*is the radial (horizontal) hydraulic conductivity [L/T], and

_{r}The effect of a partially penetrating pumping well is to produce vertical components of flow in the aquifer.

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**.

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.

Assumptions | Solution Options | Benchmark | Example | References

## 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

Comparison of **AQTESOLV** (blue line=Theis type curve) and published
Theis (1935)
w(u) well function values (symbols).

## Example

Contours of drawdown around a partially penetrating well in an isotropic nonleaky confined aquifer (t=100 min, b=100 ft).

## 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.

** See also**:
solution methods,
pumping tests