# The Theis Equation: Evaluation, Sensitivity to Storage and Transmissivity, and Automated Fit of Pumptest Data

### by Carl D. McElwee

#### Prepared by the Kansas Geological Survey in cooperation with the U.S. Geological Survey

Originally published in 1980 as Kansas Geological Survey Ground Water Series 3. This is, in general, the original text as published. The information has not been updated.

## Executive Summary

Water is a valuable resource of the State of Kansas. Much money and effort are expended each year to collect various kinds of data. The data base is massive and still is not complete. It is clear that computer storage and manipulation are necessary to make full use of the data. Many pumping tests have been made through the years to help determine the aquifer transmissivity and storage coefficient (transmissivity is a measure of how easily water flows through the earth; and the storage coefficient is a measure of how much water is stored per unit volume). Undoubtedly, many more pumping tests will be made in the future.

Some means of automating the analysis of these pumping tests would be desirable. That is the goal of this work. The method is simple, quick, and inexpensive. However, access to a computer capable of using the FORTRAN language is necessary. The automated analysis has the advantage that it is always objective and cannot consider any personal bias. The method also gives an indication of how well it could analyze the data. The method is not universally applicable to all pumping tests, but has been designed to analyze data conforming to assumptions implicit in the Theis equation.

## Abstract

Through the years, the Theis equation has played an important role in groundwater hydrology. Comparison of experimental pumptest data with this theoretical curve by graphical means has been a standard method of determining aquifer transmissivity and storage. The purpose of this paper is to present techniques and computer programs to evaluate the Theis equation, to evaluate the sensitivity with respect to transmissivity and storage, and to automatically fit experimental pumptest data to the Theis equation obtaining the "best" transmissivity and storage in the least squares sense. The automated fit for pumptest data developed in this work should be a useful tool for the groundwater hydrologist. It is simple to use, quick and inexpensive. The automated fit has the advantage that it is always objective. As a measure of the error in fitting, the rms error in drawdown is calculated for the "best" transmissivity and storage.

## Introduction

Through the years, the Theis equation has played an important role in groundwater hydrology (Theis, 1935). Comparison of experimental pumptest data with this theoretical curve by graphical means has been a standard method of determining aquifer transmissivity and storage (Jacob, 1940). The purpose of this paper is to present techniques and computer programs to evaluate the Theis equation; to evaluate the sensitivity with respect to transmissivity and storage;

and to automatically fit experimental pumptest data to the Theis equation, obtaining the "best" transmissivity and storage in the least squares sense. For a more detailed discussion of sensitivity coefficients and their uses, see McElwee and Yukler (1973).

The Theis equation involves an integral whose upper limit is infinity. Evaluation of this integral is considered in the section on numerical approximation. After the Theis equation has been evaluated, the sensitivity coefficients can be obtained with little additional work. These sensitivity coefficients are used in the section on least squares fitting to develop an algorithm for fitting the Theis equation to experimental pumptest data. The automated method is simply, quick, and inexpensive. The automated method has the advantage of always being objective and always indicating its error by calculating the rms error in drawdown.

## The Theis Equation

The Theis equation (Theis, 1935) describes radial confined groundwater flow in a uniformly thick horizontal, homogeneous, isotropic aquifer of infinite areal extent.

S = (Q / 4πT) ∫_{(r 2 s/4Tt)} ∞ [(e -u /u) du] (1)

The radius of the pumped well is assumed negligible (line source or sink approximation). The derivation and solution is documented many places (Jacob, 1940) and will not be discussed further here. In the above equation s is drawdown (L), Q is the discharge (L 3 /T), T is the transmissivity (L 2 /T), t is the time (T), S is the dimensionless storage coefficient, and r is the radial observation distance from the pumped well (L). L and T in the preceding parentheses are arbitrary units of length and time.

Usually, the Theis equation is fitted graphically to experimental pumptest data to obtain approximations for the storage coefficient (S) and the transmissivity (T). In this paper an algorithm will be presented for a computer-automated least squares fit to the experimental data, yielding approximations for S and T and giving the rms error for drawdown.

## Numerical Approximation

Many times the integral in equation (1) is symbolically represented by W(u). The drawdown can then be written as

S = (Q / 4πT) W(u) = (Q / 4πT) W(r 2 s/4Tt) (2)

W(u) is the exponential integral and is tabulated in many places (Abramowitz and Stegun, 1968). For specific values of u, table interpolation may be used to obtain the drawdown.

In order to easily evaluate equation (2) in an algorithm, one needs an explicit expression for W(u) involving only simple arithmetic operations. For 0 ≤ u ≤ 1 (Abramowitz and Stegun, 1968)

W(u) = -*ln* u + a_{0} + a_{1} u + a_{2} u 2 + a_{3} u 3 + a_{4} u 4 + a_{5} u 5 + E(u) (3)

|E(u)| < 2 x 10 -7

where a_{0} = -.57721566

a_{1} = .99999193

a_{2} = -.24991055

a_{3} = .05519968

Category: Forex

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