# 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.

Groundwater hydrologists commonly refer to the integral in the
Theis solution (Eq. 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}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