1. Finding roots of equations using the Newton-Raphson method

2. Home Quiz Introduction Petroleum Exercise: Preliminaries
Useful Info
Resources
Quiz

3. Learning objectives in this module
Develop problem solution skills using computers and numerical methods
Review of the Newton-Raphson method
Develop programming skills using FORTRAN
FORTRAN elements in this module
input/output
loops
format

4. Introduction
Finding roots of equations is one of the oldest applications of mathematics, and is required for a large variety of applications, also in the petroleum area. A familiar equation is the simple quadratic equation
(1)
where the roots of the equation are given by 
(2)
These two roots to the quadratic equation are simply the values of x for which the equation is satisfied, i.e. the left side of eq. (1) is zero.
How would you do this in Fortran? Think it through and see this
of how it could be done
Example
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5. Fortran Sample PROGRAM QUADRATIC REAL A, B, C, D, SOL1, SOL2
WRITE(*,*) ’Type the constants A,B and C’
READ* ,A,B,C
D = B**2 - 4*A*C
IF (D .LT. 0) THEN
WRITE(*,*) ’The equation has a complex solution'
ELSE
SOL1 = (-B+SQRT(D)) / (2*A)
SOL2 = (-B-SQRT(D)) / (2*A)
WRITE(*,*) 'X1=',SOL1,'X2=',SOL2
ENDIF
END

6. Introduction
In a more general form, we are given a function of x, F(x), and we wish to find a value for x for which:
The function F(x) may be algebraic or transcendental, and we generally assume that it may be differentiated.
In practice, the functions we deal with in petroleum applications have no simple closed formula for their roots, as the quadratic equation above has. Instead, we turn to methods for approximation of the roots, and two steps are involved:
1 Finding an approximate root
Refining the approximation to wanted accuracy
The first step will normally be a qualified guess based on the physics of the system. For the second step, a variety of methods exists. Please see the textbook for a discussion of various methods.
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7. Introduction
Here, we will concern ourselves with the Newton-Raphson method. For the derivation of the formula used for solving a
one-dimensional problem, we simply make a first-order Taylor series expansion of the function F(x)
(4)
Let us use the following notation for the x-values:
(5)
Then, eq. (4) may be rewritten as
(6)
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8. View Graphic Illustration
Introduction
Setting Eq. (6) to zero and solving for xk+1 yields the following expression
(7)
This is the one-dimensional Newton-Raphson iterative equation, where represents the refined approximation at iteration level k+1, and is the approximation at the previous iteration level (k).
View Graphic Illustration

9. See the Newton-Raphson method - animated
Graphic Illustration
Graphically, the method is illustrated in the figure below. The first approximation (qualified guess) of the solution (x1) is around 1,6. The tangent to the function at that x-value intersects the x-axis at around 3,3 (x2). The tangent at that point intersects at around 2,4 (x3), and the fourth value (x4) is getting very close to the solution at around x=2,7
F(x) x
See the Newton-Raphson method - animated
HERE

10. Petroleum Exercise:Preliminaries
Equations of State (EOS) are used for description of PVT-behavior (Pressure-Volume-Temperature) of hydrocarbon gases. One such equation is the Beattie-Bridgeman equation:
(8)
P is pressure in atmospheres (atm)
V is molar volume (liter/g mole)
T is temperature (oK)
R is the universal gas constant (0,08205 liter-atm / oK-g mole)
Next

11. Beta
Equations of State (EOS) are used for description of PVT-behavior (Pressure-Volume-Temperature) of hydrocarbon gases. One such equation is the Beattie-Bridgeman equation:
(8)
P is pressure in atmospheres (atm)
V is molar volume (l/g mole)
T is temperature (oK)
R is the universal gas constant (0,08205 liter-atm / oK-g mole)
A0, B0, a, b, c are gas specific constants
Next

12. Gamma
Equations of State (EOS) are used for description of PVT-behavior (Pressure-Volume-Temperature) of hydrocarbon gases. One such equation is the Beattie-Bridgeman equation:
(8)
P is pressure in atmospheres (atm)
V is molar volume (l/g mole)
T is temperature (oK)
R is the universal gas constant (0,08205 liter-atm / oK-g mole)
A0, B0, a, b, c are gas specific constants
Next

13. Delta
Equations of State (EOS) are used for description of PVT-behavior (Pressure-Volume-Temperature) of hydrocarbon gases. One such equation is the Beattie-Bridgeman equation:
(8)
P is pressure in atmospheres (atm)
V is molar volume (l/g mole)
T is temperature (oK)
R is the universal gas constant (0,08205 liter-atm / oK-g mole)
A0, B0, a, b, c are gas specific constants
Next

14. Petroleum Exercise:Preliminaries
After solving for the root of the eq. (8), ie. for the value for V that satisfies the equation for one particular set of pressure and temperature, we may find the corresponding compressibility factor (Z-factor) for the gas using the formula (the gas law for a real gas):
(12)
PV = ZRT
Which can be rearranged to ...
(obviously)
Z = PV RT
Next

15. Petroleum Exercise:Preliminaries
The procedure for the exercise is described in the following. First, we rewrite the Beattie-Bridgeman equation as:
(13)
Then, we take the derivative of the function F(V) at
constant P and T
Try yourself! What is the derivative of F (V)?
then check the
Answer (click)

16. Quiz
Did you really try??
Click the correct solution and see if you still know your maths A) B)

17. Right Answer
Right Answer!!
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18. Wrong Answer
Sorry, Wrong Answer!!
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19. Petroleum Exercise:Preliminaries
Now, the derivate of the function F(V) at constant P & T being
(14)
using the Newton-Raphson formula of eq. (7), we may find the root of Eq. (13) iteratively
and after reducing the fraction, we get
where k is the iteration counter
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20. Petroleum Exercise:Preliminaries
For the first iteration (k=1) we need a start value V1.
Here, we estimate a value using the ideal gas law (assuming Z=1):
PV=RT or
V1=RT / P
The iterative procedure is terminated when the relative change in V is less than a prescribed convergence criterion, e, i.e.

21. Petroleum Exercise Tasks to be completed
Make a FORTRAN program that uses the Newton-Raphson method to
solve the Beattie-Bridgeman equation for molar volume (V ) for any
gas (i.e.. for any set of the parameters A0, B0, a, b, c) at a given
pressure (P ) and a given temperature (T ). After finding the volume
(V ), the compressibility factor (Z ) should be computed. The computer
program will read all parameters, and pressure and temperature, from
the input file, and should write the computed parameters, pressure
and temperature, and computed compressibility factor, for each set of
pressure and temperature to the output file.
Run the computer program for This data set
Make a plot of compressibility factor vs. pressure for the two different
temperatures (on the same figure). 1 2 3

22. Petroleum Exercise (continued)
P (atm)
T (C) 1 2 5 10 20 40 60 80
100
120
140
160
180
200
Use the following set of data for the gas (methane)
Make computations for the following pressures and temperatures. (Remember that K=273,15+0C)
Parameter
Value Ao
2,2789 Bo
0,05587 a
0,01855 b
-0,01587 c
128000
Use a convergence criterion of , and set max allowed number of iterations to 20
Useful info and tip on the Fortran Program
HERE

23. For the Newton-Raphson program
Useful Info
For the Newton-Raphson program
Include an input and an output file.
For the iterative operation, the DO loop is recommended.
The convergence criterion should be tested with the IF statement
Find a reasonable format in which to present your calculations
All clear? Go to Work!!
Resources

24. Resources Introduction to Fortran Fortran Template here
The whole exercise in a printable format here
Web sites
Numerical Recipes
Fortran Tutorial
Professional Programmer's Guide to Fortran77
Programming in Fortran77

25. Quiz
This section includes a quiz on the topics covered by this module.
The quiz is meant as a control to see if you have learned some of the most important features
Hit object to start quiz

26. General information About the author Title:
Finding roots of equations using the Newton-Raphson method
Teacher(s):
Professor Jon Kleppe
Assistant(s):
Per Jørgen Dahl Svendsen
Abstract:
Provide a good background for solving problems within petroleum related topics using numerical methods
4 keywords:
Newtons Method, Loops, Fortran, Input/Output
Topic discipline:
Level: 2
Prerequisites:
None
Learning goals:
Develop problem solution skills using computers and numerical methods
Size in megabytes:
0.7 MB
Software requirements:
MS Power Point 2002 or later, Flash Player 6.0
Estimated time to complete:
Copyright information:
The author has copyright to the module and use of the content must be in agreement with the responsible author or in agreement with
About the author

27. FAQ
No questions have been posted yet. However, when questions are asked they will be posted here.
Remember, if something is unclear to you, it is a good chance that there are more people that have the same question
For more general questions and definitions try these
Dataleksikon
Webopedia
Schlumberger Oilfield Glossary

28. References
W. H. Preuss, et al., “Numerical Recipes in Fortran”, 2nd edition
Cambridge University Press, 1992
References to the textbook :
Newton-Raphson Method Using Derivative: page 355
The Textbook can also be accessed online:
Numerical Recipes in Fortran

29. Summary Subsequent to this module you should...
be able to translate a problem to Fortran code
write and handle DO loops
have a feel for the output format
know the conditional statements and use the IF structure

30. Newton-Raphson Method Movie
(Just click on image to start movie)
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