Page 172 - Petroleum Production Engineering, A Computer-Assisted Approach

Guo, Boyun / Computer Assited Petroleum Production Engg 0750682701_chap12 Final Proof page 168 4.1.2007 2:43pm Compositor Name: SJoearun

12/168 ARTIFICIAL LIFT METHODS
2pN
v ¼ (rad=sec): (12:3) involves only acceleration of the rods. Also, the friction
60 term and the weight of the plunger are neglected. We ignore
The maximum downward acceleration of point B (which the reflective forces, which will tend to underestimate the
occurs when the crank arm is vertically upward) is maximum PRL. To compensate for this, we set the up-
2 thrust force to zero. Also, we assume the TV is closed at
cN c
2
a max ¼ 1 þ (ft=sec ) (12:4) the instant at which the acceleration term reaches its maxi-
91:2 h mum. With these assumptions, the PRL max becomes
or A p A r ) g s DA r
2
cN g c PRL max ¼ S f (62:4)D( 144 þ 144
2
a max ¼ 1 þ (ft=sec ): (12:5)
2936:3 h g s DA r SN M
2
þ , (12:13)
Likewise the minimum upward (a min ) acceleration of point 144 70,471:2
B (which occurs when the crank arm is vertically down- where
ward) is
2
cN g c S f ¼ specific gravity of fluid in tubing
2
a min ¼ 1 (ft=sec ): (12:6) D ¼ length of sucker rod string (ft)
2936:3 h A p ¼ gross plunger cross-sectional area (in: )
2
2
It follows that in a conventional pumping unit, the max- A r ¼ sucker rod cross-sectional area (in: )
3
imum upward acceleration of the horse’s head occurs at g s ¼ specific weight of steel (490 lb=ft )
the bottom of the stroke (polished rod) and is equal to M ¼ Eq. (12.11).
2
d 1 cN g c Note that for the air-balanced unit, M in Eq. (12.13) is
2
a max ¼ 1 þ (ft=sec ), (12:7)
d 2 2936:3 h replaced by 1-c/h.
Equation (12.13) can be rewritten as
where d 1 and d 2 are shown in Fig. 12.5. However,
DA p DA r g s DA r
2cd 2 PRL max ¼ S f (62:4) S f (62:4) þ
¼ S, 144 144 144
d 1 2
g s DA r SN M
where S is the polished rod stroke length. So if S is mea- þ : (12:14)
sured in inches, then 144 70,471:2
2cd 2 S If the weight of the rod string in air is
¼
d 1 12 g s DA r
W r ¼ , (12:15)
or 144
cd 2 S which can be solved for A r , which is
¼ : (12:8)
d 1 24 144W r
A r ¼ : (12:16)
So substituting Eq. (12.8) into Eq. (12.7) yields g s D
2
SN g c Substituting Eq. (12.16) into Eq. (12.14) yields
2
a max ¼ 1 þ (ft=sec ), (12:9)
70471:2 h DA p W r
PRL max ¼ S f (62:4) S f (62:4) þ W r
or we can write Eq. (12.9) as 144 g s

2
2
SN g SN M
2
a max ¼ M(ft=sec ), (12:10) þ W r : (12:17)
70,471:2 70,471:2
where M is the machinery factor and is defined as The above equation is often further reduced by taking the
c fluid in the second term (the subtractive term) as an 50 8API
M ¼ 1 þ : (12:11) with S f ¼ 0.78. Thus, Eq. (12.17) becomes (where g s ¼ 490)
h
2
Similarly, PRL max ¼ S f (62:4) DA p 0:1W r þ W r þ W r SN M
2
SN g c 144 70,471:2
2
a min ¼ 1 (ft=sec ): (12:12)
70471:2 h or

2
For air-balanced units, because of the arrangements of the SN M
levers, the acceleration defined in Eq. (12.12) occurs at the PRL max ¼ W f þ 0:9W r þ W r 70,471:2 , (12:18)
bottom of the stroke, and the acceleration defined in Eq. DA p
(12.9) occurs at the top. With the lever system of an air- where W f ¼ S f (62:4) 144 and is called the fluid load (not to
balanced unit, the polished rod is at the top of its stroke be confused with the actual fluid weight on the rod string).
when the crank arm is vertically upward (Fig. 12.5b). Thus, Eq. (12.18) can be rewritten as
PRL max ¼ W f þ (0:9 þ F 1 )W r , (12:19)
12.4 Load to the Pumping Unit where for conventional units
c
2
The load exerted to the pumping unit depends on well SN (1 þ )
h
depth, rod size, fluid properties, and system dynamics. F 1 ¼ 70,471:2 (12:20)
The maximum PRL and peak torque are major concerns
for pumping unit. and for air-balanced units
2
c
SN (1 )
F 1 ¼ h : (12:21)
12.4.1 Maximum PRL 70,471:2
The PRL is the sum of weight of fluid being lifted, weight
of plunger, weight of sucker rods string, dynamic load due
to acceleration, friction force, and the up-thrust from 12.4.2 Minimum PRL
below on plunger. In practice, no force attributable to The minimum PRL occurs while the TV is open so that
fluid acceleration is required, so the acceleration term the fluid column weight is carried by the tubing and not