SIG4040 Applied Computing in Petroleum
Newton-Rapson’s Method
1
Finding roots of equations using the Newton-Raphson method
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 eguation
ax 2 + bx + c = 0
(1)
where the roots of the equation are given by
!b ± b ! 4ac
.
2a
2
x=
(2)
These two roots to the quadratic equation are simply the values of x for which the equation is
satisfied, ie. the left side of Eq. (1) is zero.
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
F(x) = 0 .
(3)
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
2. 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.
Here, we will concern ourselves with the Newton-Raphson method. For the derivation of the
fomula used for solving a one-dimensional problem, we simply make a first-order Taylor series
expansion of the function F(x)
F(x + h) = F(x) + hF !(x) .
(4)
Let us use the following notation for the x-values:
xk = x
x k +1 = x + h
(5)
Then, Eq. (4) may be rewritten as
Norges teknisk-naturvitenskapelige universitet
Institutt for petroleumsteknologi og anvendt geofysikk
Professor Jon Kleppe
SIG4040 Applied Computing in Petroleum
Newton-Rapson’s Method
2
F(x k +1 ) = F(x k ) + ( x k +1 ! x k )F " (x k ) .
Setting Eq. (6) to zero and solving for x k +1 yields the following expression:
x k +1 = x k !
(6)
F(x k )
.
F "(x k )
(7)
This is the one-dimensional Newton-Raphson iterative equation, where x k +1 represents the
refined approximation at iteration level k+1, and x k is the approximation at the previous
iteration level (k). Graphically, the method is illustarted 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 ( x 2 ). The tangent at that point intersects at
around 2,4 ( x 3 ), and the fourth value ( x 4 ) is getting very close to the solution at around x=2,7
F(x)
x4
x1
x2
x
x3
Petroleum exercise
Equations of State (EOS) are used for description of PVT-behavior (Pressure-VolumeTemperature) of hydrocarbon gases. One such equation is the Beattie-Bridgeman equation:
P=
RT
!
"
#
+ 2+ 3+ 4
V V
V
V
(8)
where P is pressure (atm), V is molar volume (liter/g mole), T is temperature ( °K ), R is the
universal gas constant (0,08205 liter-atm/ °K -g mole), and the temperature dependent
parameters for the gas, ! , ! and ! , are expressed by the following formulas:
! = RTB0 " A0 "
! = " RTB0 b + A0 a "
!=
Norges teknisk-naturvitenskapelige universitet
Institutt for petroleumsteknologi og anvendt geofysikk
RB0 bc
T2
Rc
T2
RcB0
T2
(9)
(10)
(11)
Professor Jon Kleppe
SIG4040 Applied Computing in Petroleum
Newton-Rapson’s Method
3
where A0 , B0 , a, b, and c are constants (different for each gas).
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):
Z=
PV
RT
(12)
The procedure for the exercise is described in the following. First, we rewrite the BeattieBridgeman equation as:
F(V ) =
RT !
"
#
+ 2 + 3 + 4 $P =0.
V
V
V
V
(13)
Then, we take the derivative of the function F(V) at constant P and T:
F !(V ) = "
RT 2# 3$ 4%
"
"
"
V 2 V 3 V4 V5
(14)
Using the Newton-Raphson formula of Eq. (7), we may find the root of Eq. (13) iteratively:
F (Vk )
F "(Vk )
(15)
RTVk4 + !Vk3 + "Vk2 + #Vk $ PVk5
RTVk3 + 2 !Vk2 + 3"Vk + 4#
(16)
Vk +1 = Vk !
or
Vk +1 = Vk +
where k is the iteration counter. 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):
P=
RT
V
(17)
V1 =
RT
P
(18)
or
The iterative procedure is terminated when the relative change in V is less than a prescribed
convergence criterium, ε, ie.:
Vk +1 ! Vk
"#
Vk +1
Norges teknisk-naturvitenskapelige universitet
Institutt for petroleumsteknologi og anvendt geofysikk
(19)
Professor Jon Kleppe
SIG4040 Applied Computing in Petroleum
Newton-Rapson’s Method
4
Tasks to be completed
1. Make a FORTRAN program that uses the Newton-Raphson method to solve the BeattieBridgeman equation for molar volume (V) for any gas (ie. 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 ouput file.
2. Run the computer program for the data set below.
3. Make a plot of compressibility factor vs. pressure for the two different temperatures (on
the same figure).
Use the following set of data for the gas (methane):
Parameter
Ao
Bo
a
b
c
Value
2,2789
0,05587
0,01855
-0,01587
128000
and make computations for the following pressures and temperatures: (remember that
°K = 273, 13 + °C ):
P (atm)
1
2
5
10
20
40
60
80
100
120
140
160
180
200
1
2
5
10
20
40
60
80
100
120
140
160
180
200
T ( °C )
0
0
0
0
0
0
0
0
0
0
0
0
0
0
200
200
200
200
200
200
200
200
200
200
200
200
200
200
Use a convergence criterium of 0,000001, and set the maximum allowed number of iterations
to 20.
Norges teknisk-naturvitenskapelige universitet
Institutt for petroleumsteknologi og anvendt geofysikk
Professor Jon Kleppe