Page 197 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

QC: —/—
T1: IML
P2: KVU/KXT
P1: KVU/KXT
AT029-Manual
AT029-04
4. CHARACTERIZATION OF RESERVOIR FLUIDS AND CRUDE OILS 177
of x wi and T bi for i = 1to N − 1 from Table 4.11, we get
T bN = 787.9K. AT029-Manual-v7.cls June 22, 2007 21:30
Example 4.9—Show how Eq. (4.78) can be derived from
Eq. (4.77).
Solution—From Eq. (4.16): dx w = F(SG)dSG and from defi-
nition of P in Eq. (4.56) we have SG = SG o S + SG o , which
∗
∗
after differentiation we get dSG = SG o dSG . In addition,
∗
from Eq. (4.69), F(SG ) = SG o F(SG) and from Eq. (4.56),
∗
when x cw = 0, we have SG = 0 and at x cw = 1, we have
∗
SG =∞. By combining these basic relations and substitut-
∗
ing them into Eq. (4.77) we get
∞
1 F(SG )dSG ∗
∗
=
∗ ∗ FIG. 4.18—Incomplete gamma function
av
SG ◦ SG + SG ◦ SG ◦ SG + SG ◦
0 Γ Γ(1 + 1/B, q i ) for different values of B. Taken
which after simplification reduces to Eq. (4.78). with permission from Ref. [40].
4.5.4.4 Calculation of Average Properties
of Subfractions
where for the case of Eq. (4.86), a = 1 + 1/B. Values of
In cases that the whole mixture is divided into several pseu- (1 + 1/B, q i ) can be determined from various numerical
docomponents (i.e., SCN groups), it is necessary to calculate handbooks (e.g., Press et al. [38]) or through mathemati-
average properties of a subfraction i whose property P varies cal computer software such as MATHEMATICA. Values of
from P i−1 to P i . Mole, weight, or volume fraction of the groups (1 + 1/B, q i ) for B = 1, 1.5, 2, 2.5, and 4 versus q i are shown
shown by z i can be calculated through Eq. (4.19), which in in Fig. 4.18 [39]. As q i →∞, (1 + 1/B, q i ) → 0 for any value
terms of P becomes of B.At B = 1, Eq. (4.89) gives the following relation for
∗
P i ∗ (1 + 1/B, q i ):
(4.83) z i = F(P )dP ∗ ∞
∗
−t
(4.90) te dt =−(1 + t)e ∞ = (1 + q)e −q
∗
P i−1 (2, q) = −t q
q
Substituting F(P ) from Eq. (4.66) into the above equation
∗
gives Further properties of incomplete gamma functions are given
B ∗B B ∗B
in Ref. [39]. Substitution of Eq. (4.90) into Eq. (4.86) we
(4.84) z i = exp − P i−1 − exp − P i get the following relation to estimate P ∗ for the case of
A A i,av
B = 1:
Average properties of this subfraction shown by P i,av can be
∗
∗
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
calculated from Eq. (4.21), which can be written as P i,av =
A P ∗ P ∗ P ∗ P ∗

P ∗ i−1 i−1 i i
1 i z i 1 + A exp − A − 1 + A exp − A
(4.85) P ∗ = P F(P )dP ∗
∗
∗
i,av
z i (4.91)
P ∗
i−1
by substituting F(P ) from Eq. (4.66) and carrying the inte- where z i is obtained from Eq. (4.84) which for the case of
∗
gration we get B = 1 becomes:
P ∗ P ∗

1 A 1 1 (4.92) i−1 i
1/B

(4.86) P i,av = 1 + , q i−1 − 1 + , q i z i = exp − A − exp − A
∗
z i B B B
∗
∗
where In these relations, P and P i−1 are the upper and lower bound-
i
aries of the subfraction i. One can see that if we set P = M +∗
∗
B ∗B i n
(4.87) q i = P i and P i−1 = M , then Eq. (4.91) is equivalent to Eq. (4.48)
−∗
∗
n
A
for estimated molecular weight of SCN groups through the
z i should be calculated from Eq. (4.84). P i,av is calculated from exponential model.
P i,av through Eq. (4.81) as
∗
(4.88) P i,av = P ◦ (1 + P ∗ ) Example 4.10—For the C 7+ fraction of Example 4.7, com-
i,av
position and molecular weight of SCN groups are given in
In Eq. (4.86), (1 + 1/B, q i ) is the incomplete gamma function Table 4.11. Coefficients of Eq. (4.56) for the molar distribu-
defined as [38] tion of this system are given in Table 4.13 as M o = 89.86,
∞ A = 0.3105, and B = 1. Calculate average molecular weights

(4.89) (a, q) = t a−1 −t of C 12 –C 13 group and its mole fraction. Compare calculated
e dt
values from those given in Table 4.11.
q
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