Page 185 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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4. CHARACTERIZATION OF RESERVOIR FLUIDS AND CRUDE OILS 165
based on normalized mole fraction becomes
Carbon Number, N
C
∞
1
0 10 20 30 40
0.01 (4.17) F(P)dP = dx c = 1
W. Texas Gas Condensate P ◦ 0
Kuwaiti Crude
If the upper limit of the integral in Eq. (4.17) is at prop-
0.008 Waxy Oil
erty P, then the upper limit of the right-hand side should be
cumulative x c as shown in the following relation:
0.006 (4.18) P F(P)dP
PDF, F x c = P ◦
0.004 Integration of Eq. (4.16) between limits of P 1 and P 2 gives
the mole fraction of all components in the mixture whose
property P is in the range of P 1 ≤ P ≤ P 2 :
0.002
P 2

(4.19) F(P)dP = x c2 − x c1 = x p 1 →p 2
0 P 1
0 200 400 600
where x c1 and x c2 are the values of x c at P 1 and P 2 , respectively.
is sum of the mole fractions for all components having
x p 1 →p 2
Molecular Weight, M
P 1 ≤ P ≤ P 2 . Equation (4.19) can also be obtained by apply-
ing Eq. (4.18) at x c2 and x c1 and subtracting from each other.
FIG. 4.5—Probability density functions for the gas conden-
sate and crude oil samples of Fig. 4.4. Obviously if the PDF is defined in terms of cumulative weight
or volume fractions x represents weight or volume fraction,
respectively. The average value of parameter P for the whole
The continuous distribution for a property P can be ex- continuous mixture, P av ,is
pressed in terms of a function such that
∞

1
(4.16) F(P)dP = dx c (4.20) P av = P(x c )dx c = PF(P)dP
0 P ◦
where P is a property such as M, T b , N C , SG, or I (defined where P(x) is the distribution function for property P in terms
by Eq. 2.36) and F is the probability density function. If the of cumulative mole, weight, or volume fraction, x c . For all
original distribution of P is in terms of cumulative mole frac- the components whose parameters varies from P 1 to P 2 the
tion (x cm ), then x c in Eq. (4.16) is the cumulative mole fraction. average value of property P, P av(P 1 →P 2 ) , is determined as
As mentioned before, parameter P for a continuous mixture
varies from the initial value of P o to infinity. Therefore, for the P 2 PF(P)dP
whole continuous mixture (i.e., C 7+ ), integration of Eq. (4.16) (4.21) P av(P 1 →P 2 ) = P 1 P 2 F(P)dP
P 1
This is shown in Fig. 4.7 where the total area under the curve
600 45 from P o to ∞ is equal to unity (Eq. 4.17) and the area under
curve from P 1 to P 2 represents the fraction of components
W. Texas Gas Condensate
whose property P is greater than P 1 but less than P 2 . Fur-
Kuwaiti Crude
ther properties of distribution functions are discussed when
Waxy Oil
Molecular Weight, M 200 15 Carbon Number, N C 4.5.2 Exponential Model
different models are introduced in the following sections.
400
30
The exponential model is the simplest form of expressing dis-
tribution of SCN groups in a reservoir fluid. Several forms of
exponential models proposed by Lohrenz (1964), Katz (1983),
and Pedersen (1984) have been reviewed and evaluated by
Ahmed [26]. The Katz model [33] suggested for condensate
systems gives an easy method of breaking a C 7+ fraction into
0 0 various SCN groups as [19, 26, 33]:
0 0.2 0.4 0.6 0.8 1
(4.22) x n = 1.38205 exp(−0.25903C N )
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
Cumulative Mole Fraction, x cm
where x n is the normalized mole fraction of SCN in a C 7+
FIG. 4.6—Variation of molecular weight with cumulative fraction and C N is the corresponding carbon number of the
mole fraction for the gas condensate and crude oil samples SCN group. For normalized mole fractions of C 7+ fraction, the
of Fig. 4.4. mole fraction of C 7+ (x 7+ ) is set equal to unity. In splitting a

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