Page 80 - Fundamentals of Reservoir Engineering

SOME BASIC CONCEPTS IN RESERVOIR ENGINEERING 19

both extremely accurate and very simple to program, even for small desk calculators,
since it requires only five storage registers.

The Hall-Yarborough equations, developed using the Starling-Carnahan equation of
state, are

−
0.06125p t e − 1.2(1 t) 2
Z = pr (1.20)
y

where p pr = the pseudo reduced pressure
t = the reciprocal, pseudo reduced temperature (T pc/T)

and y = the "reduced" density which can be obtained as the solution of the
equation.

2
3
−
3
2
−
− 0.06125 p t e − 1.2(1 t) 2 + y + y + y − y 4 − (14.76t 9.76t + 4.58t )y 2
pr
(1 y) 3
−
3 (2.18 +
2
+ (90.7t − 242.2t + 42.4t )y 2.82t) = 0 (1.21)
This non-linear equation can be conveniently solved for y using the simple Newton
Raphson iterative technique. The steps involved in applying this are:
k
1) make an initial estimate of y , where k is an iteration counter (which in this case is
1
unity, e.g. y = 0.001)
2) substitute this value in equ. (1.21); unless the correct value of y has been initially
k
selected, equ. (1.21) will have some small, non-zero value F .
3) using the first order Taylor series expansion, a better estimate of y can be
determined as

k
y k1 = y − F k dF k (1.22)
+
dy
where the general expression for dF/dk can be obtained as the derivative of
equ. (1.21), i.e.

3
2
dF 1 4y + 4y − 4y + y 4 2 9.16t )y
+
3
−
dy = (1 y) 4 − (29.52t 19.52t +
−
+
3
2
+ (2.18 2.82t) (90.7t − 242.2t + 42.4t )y (1.18 2.82t) (1.23)
+
4) iterate, using equs. (1.21) and (1.22), until satisfactory convergence is obtained
k
(F ≈ 0).
5) substitution of the correct value of y in equ. (1.20) will give the Z−factor.
15
(N.B. there appears to be a typographical error in the original Hall-Yarborough paper ,
3
in that the equations presented for F (equ. 8) and dF/dy (equ. 11), contain 1−y and