Predicting the Behavior of Sucker-Rod Pumping Systems

S. G. GIBBS

ABSTRACT

A new method for predicting the behavior of sucker-rod pumping systems is presented. The pumping system is described by a flexible mathematical model which is solved by means of partial difference equations with the aid of computers. Polished rod and intermediate-depth dynamom­eter cards can be calculated for various bottom-hole pump conditions. The technique permits simulation of a wide variety of operating conditions, both normal and abnormal. The data generated with the new technique are useful in refining the criteria for design and operation of sucker-rod .I'ystms.

INTRODUCTION

Sucker-rod pumping systems are used in approximately 90 per cent of artificially lifted wells. In view of this wide application, it behooves the industry to have a funda­mental understanding of the sucker-rod pumping process. Oddly enough, our understanding has been rather super­ficial. This is evidenced by the semi-empirical formulas which have been used as the basis for design and opera­tion of sucker-rod installations.

Though we have realized the limitations of our methods for many years, it has not been computationally feasible to use more refined techniques. With the advent and wide­spread use of digital computers, it is now possible to handle the mathematical problems associated with sucker­rod pumping. This paper summarizes a computer-oriented method which can provide greater insight into the sucker­rod pumping process. It is hoped that this technique, and techniques which may evolve from it, will prove to be the tool needed by industry to obtain the most efficient use of rod pumping equipment.

THE MATHEMATICAL MODEL

Prediction of sucker-rod system behavior involves the solution of a boundary value problem. Such a problem includes a differential equation and a set of boundary con­ditions. For the sucker-rod problem, the wave equation is used, together with boundary conditions which describe the initial stress and velocity of the sucker rods, the mo­tion of the polished rod and the operation of the down­hole pump. Of these items, the wave equation, the polished rod motion condition and the down-hole pump conditions are of primary importance. Discussion of the mathemat­ical model centers about these factors.

ROD STRING SIMULATION WITH THE WAVE EQUATION The one-dimensional wave equation with viscous damp-

Original manuscript received in Society of Petroleum Engineers office April 4, 1963. Revised manuscript received June 3, 1963. Paper pre­sented at SPE Rocky Mountain Regional Meeting, May 27-28, 1963, in Denver, Colo.

SPE 588 JULY. 1963

SHELL DEVELOPMENT CO. HOUSTON, TEX.

ing, o'u(x, t) ., o'U(X, t)

at' = a' ox' 71'aV ,ou(x, t)

2L at (1)

is used in the sucker-rod boundary value problem to simu­late the behavior of the rod string. This equation describes the longitudinal vibrations in a long slender rod and, hence, is ideal for the sucker-rod application. Its use incor­porates into toe mathematical model the phenomenon of force wave reflection, which is an important characteristic of real systems.

The viscous damping effect postulated in Eq. 1 yields good solutions, even though non viscous effects such as coulcomb friction and hysteresis loss in the rod material are present. Fortunately, the nonviscous effects are rela­tively small, so the viscous damping approximation used in the wave equation is adequate. The coefficient v is a dimensionless damping factor which is found in field measurements to vary over fairly narrow limits.

For mathematical convenience the gravity term is omitted in Eq. 1. The effect of gravity on rod load and stretch can be treated separately, as will be noted later. Since Eq. 1 is linear, the legitimacy of this procedure is easy to demonstrate.

POLISHED ROD MOTION SIMULATION The motion of the polished rod is determined by the

geometry of the surface pumping unit and the torque­speed characteristics of its prime mover. By determining the motion of the polished rod, we formulate an important boundary condition.

From trigonometrical considerations it can be shown that the position of the polished rod vs crank angle 8 is given by (see Fig. 1)

u(O, (}) = L, [sin-1 (L1 ~n ()) + cos-' (h' + 2i:h- L.')] h = V L/ + L,' + 2L,L2 cos () (2)

These equations are obtained from the general solution of the "four-bar" linkage problem and can be used to de­scribe the kinematics of any modern beam pumping unit. ' If prime mover speed variations are disregarded, the angu­lar velocity of the crank is constant, and Eq. 2 can be used to predict the position of the polished rod vs time. However, the constant-speed condition leading to constant crank angular velocity is only approached in practice; hence, it is better to make provisions for prime mover speed variations in the mathematical model.

The speed at which the prime mover runs is determined by its torque-speed characteristics and the torque imposed upon it. The torque that the prime mover "feels" is the net torque arising from the polished rod load and the opposing torque from the counterbalance effect. The

lReferences given at tnd of paper.

769

torque from the rod load is obtained as the product of the polished rod load and the torque factor. The torque factor as obtained from mechanics is given by

TF = L,L.L5 sin(~ - (J - if;) Lah sin l; ,

5' _ -1 (h' + L; - L: ) ." - cos 2Lah '

_ • -1 (LJ sin l; ) if; - Sill L, . (3)

The counterbalance torque which opposes the torque from the well load is described by the relation

CET = L ,W, cos «(J - d) , (4) where d is the phase angle needed to orient the counter­balance effect with respect to gravity. Thus, the net torque imposed upon the prime mover is the algebraic sum of the well torque and the counterbalance torque:

NT = TF (PRL) + CET (5)

This net torque, when considered along with the torque­speed characteristics of the prime mover, determines the instantaneous speed at which the prime mover rotates. Thus, the instantaneous angular velocity of the crank can be determined in view of the gear box ratio and sheave sizes to reflect the speed variations of the prime mover as it responds to the fluctuating net torque. Inertia effects are not considered in this analysis.

DOWN·HOLE PUMP SIMULATION

The most important boundary condition in the sucker­rod problem is the one which describes the operation of the down-hole pump. Undoubtedly, the mathematical de­scription of the down-hole pump has been the greatest difficulty in analytical treatment of the sucker-rod system. Many attempts have been made to formulate explicit expressions which describe its behavior, but unfortunately these expressions have proved inadequate when applied over the range of conditions encountered in practice. As a matter of fact, it is unlikely that a general pump boundary condition can be conceived which will permit solution of the sucker-rod problem by classical methods.

In order to get around this impasse, it is convenient to write the pump condition as

(l u (L, t) au (L, t) + f3 a = P (t) , (6) ,x

wherein the parameters a, f3 and P (t) depend upon the type of pump operation to be simulated. With the pump condition written in this manner, the flexibility needed to simulate widely varying pumping conditions can be achieved. For example, the choice a = 0, f3 = 1, P(t) = 0 gives Eq. 6 the form

o u (L, t) = 0 ox '

which implies that the down-hole pump is free and unloaded. This situation occurs or is approached in a real sucker-rod installation when the pump is descending with the traveling valve open. As a further example, take a = 1, f3 = 0, P(t) = u,. In this case, the pump condition becomes

u (L, t) = Un

which implies that the pump is stationary at some position u,. This situation is approached in a high-pump-efficiency installation while the fluid load is being transferred from the rods to the tubing or from the tubing to the rods. As a final example, take a = 0, [3 = 1, P (t) = Wt/EA. The pump boundary condition then reduces to

770

EA aU (L, t) _ - - W f ,

'ox which means that a steady load Wf is being applied at the pump. This condition exists while fluid is being lifted to the surface.

It is instructive to list the choices of a, ,[3 and P (t) for the case in which free gas is passing through the pump. In this instance the pump dynagraph card' has the shape depicted in Fig. 2. With regard to Fig. 2, let t, be the time at which the traveling valve closes, t, be the time when the standing valve opens, fa be the time when the standing valve closes, and t, be the time at which the traveling valve opens. The appropriate choices of a, [3 and P (t) are

a = 0 ~ [3 = 1 t, :-:;; t < t" P (t) = G, [u (L, t, ) - u (L, t)]

a=O ~ [3 = 1 1, < 1 < 1", p (t) = Wt/EA

~ =: ~ Ca :-:;; 1 < t" P (t) = Wt/EA - G, [u (L, t) - u (L, ta)]~

a = 0 ~ [3 = 1 t, :-:;; 1 < t,', p (t) = 0

where t,' is the time when the traveling valve again closes, and a new pumping cycle is begun. The functions G, and G, determine the shape of the pump card while the fluid load is being transferred to the rods from the tubing or to the tubing from the rods. These functions depend on the

FIG. I-BEAM PUMPING UNIT SHOW" AS A

FOUR-BAR LINKAGE.

o <.> .. - 0 2 -1 .. 0..

;: 2 o :::>

0..

t 2(STANDING VALVE OPENS) t3 (STANDING VALVE CLOSES)

{ I ~"""'''''''' t, (TtL~vSEELi~G VALVE

PUMP DISPLACEMENT

FIG. 2-TYPICAL PUMP DYNAGRAPH SHOWING VALVE OPERATION.

JOURNAL OF PETROLEUM TECHNOLOGY

pressure-volume relationship of the mixture being pumped. As the amount of free gas passing through the pump diminishes, the volumetric efficiency increases, and the pump dynagraph tends to be rectangular in shape. Suitable choices of a, f3 and P (t) can also be made to simulate this situation.

The timing of valve opening and closing plays the key role. In general it is impossible to predict these times, but a special feature of the numerical method permits auto­matic "sensing" of them by computer tests. These tests are discussed in the next section.

NVMERICAL SOLVTION BY VSE OF PARTIAL DIFFERENCE EQVATIONS

The mathematical model described in the previous sec­tions is fairly complicated, and an analytical solution can be obtained only in restricted instances (and then with great difficulty). It is easier and more efficient to obtain a solution by means of partial difference equations. Owing to a fortunate mathematical circumstance, it is possible to obtain extremely accurate solutions by means of the difference-equation approach.

To obtain the numerical solution, the mathematical model is recast into the language of differences. The wave equation, a partial differential equation, is replaced by an analagous partial difference equation'

T = 0, 1,2, ... , X=0,1,2,···X*,

in which the notation

"vn1_ U (x, t + .At),

"V~ - u (x, t), "V~-1- U (x, t - At),

,,+lV' - u (x + AX, t), "-1V~ - u (x - AX, t),

7raV r = --At

2L

(7)

is used. Eq. 7 is developed by means of difference quotient approximations to the derivatives in the wave equation

3' u (x, t) ,...., u (x, t + At) - 2u (x, t) + U (x, t - At)

of Af 3' u (x, t) u (x + AX, t) - 2u (x, t) + u (x - AX, t)

{lx' .AX'

3 u (x, t) u (x, t + .At) - u (x, t)

in which the special choice of length and time increments

Ax . d k' h' h' - = a IS ma e. By ma mg t IS C Olce, we can obtain At

remarkably accurate solutions to the wave equation by means of its difference analog (Eq. 7). As a matter of fact, for the case of zero damping, solutions to Eq. 7 exactly satisfy the wave equation regardless of the size of .Ax. When damping is considered, the solutions are no longer exact but are very close. Investigation shows that, for low-damping valves normally encountered in pumping installations, the error in pump stroke introduced by the numerical method should be less than 0.5 per cent. This is the fortunate feature which permits rapid and eco­nomical solution by means of digital computers, since the rod need be divided into only a few intervals.

JULY. 1963

Likewise, the boundary conditions are recast into partial difference form. In particular, the down-hole pump con­dition becomes

AX P' + 2/1x*-,vr - V2/1x*-,vr X*V' = (8)

3 IXAx +-'2/1

in which nv' denotes the displacement of the pump. Eq. 8 is obtained directly from Eq. 6 by replacing the derivative by a difference quotient.

The mechanism of the pump condition written in difference form (Eq. 8) is exactly the same as that of Eq. 6. A timely choice of IX, /1 and P (t) is needed, and these choices depend on valve operation. The times of valve opening and closing are "sensed" by the computer with the following tests:

Test for t,-While ~ x"vr - 2x*-,vr + V2 x*-'vr = ° (no

load on pump), the computer senses when x*vr - X"V'-' changes from positive to negative. This indicates that the pump has reached its lowest position, at which time the traveling valve closes. This is the computer's signal to make the appropriate choices IX, /1 and P (t) to simulate the desired pump condition.

Test for t2-While~ x"vr - 2x*-,vr + 1/2x*-,vr > ° (ten­

sion at the pump), the computer makes tests to determine

when EA [~x.vr - 2X*-1V' + 1/2 x,:,_,vr] = WI' At this time Ax 2

the fluid load is completely borne by the rods, and the standing valve opens.

Test for ta-While EA[~x*vr - 2X'-,-IV' + V2 x*-'V']= W, AX 2

(fluid load imposed on the pump), the computer senses when "·V' - "·V'-' changes from negative to positive. At this time the pump has reached its highest position, and the standing valve closes.

Test for t,--While ~X*Vr - 2X*-'V' + 1;2 x*-'vr > ° 2

(tension at the pump), the computer determines when

~ x"vr - 2X*-1vr + V2 X*-'vr = ° At this time the fluid 2 .

load is completely borne by the tubing, and the traveling valve opens.

In this manner the computer continually senses the forces and movements which affect valve action and makes the proper choices in the pump boundary condition to simulate the desired down-hole dynagraph card.

Dynamic loads at the polished rod can be calculated from the difference-equation version of Hooke's law

oF'" = EA[_ ~ovr + 2'V' _ 1;22v r]. AX 2

The wave equation is written without the gravity term, so the effects of gravity must be treated separately. Thus, the total polished rod load, both static and dynamic, is given by

PRL = 'F'" - Wb (9) where Wb is the buoyant weight of the rods. In a similar manner the load and displacement at the pump must be adjusted to incorporate the effects of gravity. The dynamic load at the pump is given by the difference equation

x* F'" = ~A [~x*vr _ 2,,*-1Vr + 112 X*-'V'] Ax 2 '

and the true load is obtained by subtracting the force due

771

to buoyancy from the dynamic load, PL = X*FT + W, . (J 0)

The effects of gravity on pump displacement are accounted for by adding the static stretch of the rods-as they hang in fluid with no attached fluid load-to the pump displace­ments obtained in Eq. 8:

X'"ZT = ":'UT + Static Stretch (11)

Since the differential equation is linear, it is permissible to superimpose the effects of gravity in this manner.

COMPARISON WITH FIELD MEASUREMENTS

The mathematical model as solved with partial differ­ence equations permits accurate prediction of system behavior. This is illustrated for a case in which the down­hole operating conditions are known. Fig. 3a shows a dynagraph card taken from an 8,900-ft well being pro­duced at 12 strokes/min with a 74-in. conventional unit. Since this well has an intake pressure above the bubble point, it is known that the pump dynagraph is generally rectangular. Knowledge of the intake pressure and fluid composition also permits a good estimate of fluid load to be made. Fig. 3b shows synthetic polished rod and pump dynagraph cards produced for the same operating condi­tions. As visual comparison indicates, the polished rod cards compare closely. The pump stroke predicted by the computer is in good agreement with the actual stroke calculated from measured production.

FURTHER APPLICATIONS OF THE TECHNIQUE

To demonstrate the method further, several conditions encountered in practice are simulated. Fig. 4 illustrates the simulation of various down-hole pump volumetric efficiencies. Fig. 4a shows a case wherein the volumetric efficiency approaches 100 per cent. This condition occurs in installations in which little or no gas is coming into the pump and/or the pump intake pressure is above the bubble point. Figs. 4b, 4c and 4d show volumetric effi­ciencies** of 75, 50 and 25 per cent, respectively. Gen­erally, the lower the efficiency the more adverse the com­pression nose on the pump cards, which is caused by greater amounts of free gas in the pump chamber. It is interesting to note the effect of lower volumetric efficiency on the loads imposed at the polished rod. The shape of the card changes as a result of the slower transferral of

:;<>:'These efficiencies do not include losses due to pump slippage.

25,000

20,000

e 15,000

o <t

S !O,OOO

5000

4

WMAX = 21,800 Ib

WMIN = 13,300 Ib

PRODUCTION = 200 bpd

3

MEASURED POLISHED ROD DYNAGRAPH

2 I 0 -I -2 -3 -4

DISPLACEMENT, ft 7

fluid load to and from the rod string. This affects the shape of the force waves which travel within the rod string and can cause significant changes in the maximum and minimum loads imposed at the polished rod.

The difference-equation technique can be usefully ap­plied to the problems of pump spacing. Fig. 5 shows a hypothetical analysis simulating the conditions which exist during the pumping-up period. The spacing is assumed to be such that the pump will hit up if a displacement of -1.45 ft is reached. Fig. 5a shows the conditions soon after the unit is started while the fluid load on the pump is negligible. As indicated by the pump card, the pump is overstroking and hitting up. The corresponding polished rod card shows the reflections of the shock loads imposed when the pump hits up. For this situation the pump boundary condition is arranged to simulate a free-ended rod on the upstroke until the pump has reached a dis­placement of - 1.45 ft. When the pump reaches this point as determined by a computer test, the boundary condition is changed to simulate a fixed end. While the end is fixed, a computer test senses the dynamic pump load, and when this load decreases to zero, the boundary condition is changed to simulate a free-ended rod again. At this time the pump begins to descend with no attached load. Figs. 5b through 5d illustrate the conditions existing while the fluid load is increasing to the full value of 2,600 lb. With increased pump load and the resulting rod stretch, the hitting-up condition no longer occurs.

A variety of pumping malfunctions can also be simu­lated, as shown in Fig. 6. Fig. 6a simulates a gas-locked pump. When the pump is gas-locked, the traveling valve remains closed throughout the pumping cycle. This is incorporated into the digital-computer solution by choos­ing the pressure-volume relationship involved in the pump boundary condition so that the fluid load is never com­pletely removed from the pump, thereby requiring the traveling valve to remain closed. In this condition, energy is stored in the gas as pressure energy on the downstroke and, for the most part, is returned to the system by gas expansion on the upstroke. This results in a small amount of pump work, which is indicated by the small area of the corresponding polished rod card.

The results shown in Fig. 6b simulate the malfunction caused by either a broken rod near the pump or a travel­ing valve which is continuously stuck open. The pump boundary condition used to effect this malfunction pro­vides for a free-ended rod throughout the pumping cycle. As expected, the area of the corresponding polished rod

WMAX = 22,100 Ib

W MIN = 12,900 Ib

PRODUCTION = 195 bpd

SYNTHETIC PUMP DYNAGRAPH

SYNTHETIC POLISHED ROD DYNAGRAPH

6 5 4 3 2 I 0 -I -2 -3 -4 DISPLACEMENT, ft

FIG. 3-COMPARISON OF FIELD DYNAGRAI'H WITH SYNTHETIC DYNAGRAPHS.

772 JOURNAL OF PETROLEUM TECHNOLOGY

card indicates a negligible amount of work being done at the pump.

The cards of Fig. 6c simulate the pumped-off condition which occurs when the capacity of the pumping equip­ment exceeds the inflow capacity of the well. Accordingly, the pump chamber is not completely filled on the upstroke, and on the downstroke the fluid load is quickly released when the plunger encounters the top of the fluid column in the pump chamber. The pump boundary condition set by the computer is similar to the normal condition for

I 20.000

15.000

;;j 10,0,),) -

o --'

(a) 100- PERCENT

EFFICIENCY

°It==============t=r-

20,000

15,000

g 10,000 o -'

I I 7 6 5 4 3 2 I 0 -I -2 -3 -4 -5 -6

DISPLACEMENT, fj

500: __ Lr-_-_-_-------_-_-~---------.. -__ =_==_i"': __ (e) ~~~~~:~~:~_

7 6 5 4 3 2 I 0 -I -2 -3 -4 -5 -6 DISPLACEMENT, fj

gas compression, except that the transferral of fluid load does not begin on the downstroke until the pump reaches a displacement which simulates the top of the fluid column. The corresponding polished rod card shows the abrupt load changes caused by the sudden release of pump load which occurs when the plunger pounds fluid.

The dynagraph cards of Figs. 6d and 6e simulate the hitting-up and hitting-down conditions. In each case the polished rod dynagraphs have the jagged appearance of actual dynagraphs taken in wells with pump malfunctions

r--------------------b-- (b) 75-PERCENT EFFICIENCY

7 6 5 4 3 2 I 0 -I -2 -3 -4 -5 -6 DISPLACEMENT, fj

r-______________ ====-~--(d)25-PERCENT EFFICIENCY

7 6 5 4 3 2 I 0 -I -2 -3 -4 -5 -6 DISPLACEMENT, fj

FIG. ~SIMULATION OF VARIOCS PI-,IP VOU;;Vn:TRlC EFFICIENCIES.

12,000 12,000

10,000 10,000

=::. =::. 8000 8000

c5 0 <t 6000 <t 6000 0 0 -' -'

4000 4000

2000 (a) 2000 (b)

0 0

- 20004 I -I -3

- 20004 I -I -3

DISPLACEMENT, It DISPLACEMENT, ft

12,000 12,000

10,000 10,000 =::.

8000 =::.

8000 c5 c5 <t 6000 <t 6000 0 0 -' -' 4000 4000

2000 2000 (c) (d)

0 0

- 20004 I -3

- 20004 I -I -3

DISPLACEMENT, It DISPLACEMENT, It

FIG. 5-SIMULATION OF CONDITIONS EXISTING DURING PUMPING-UP OPERATIONS.

J I j I. ~/ 1963 77a

of this type. The jagged appearance of the polished rod cards is caused by reflections of the shock loads imposed when the pump hits up or down.

An extremely interesting application is the evaluation of the relative performance of different types of surface pumping units. The mathematical solution is capable of simulating any type of pumping unit simply by use of the appropriate linkage dimensions involved in the pol­ished rod motion condition (Eq. 2). The behavior of the sucker-rod system is significantly affected by the type of driving motion produced by the pumping unit. Signifi­cant effects in polished rod loads, gear box torques and pump strokes can be attributed to the various pumping­unit geometries commercially available. To illustrate a typical application of the technique, Fig. 7 shows hypo­thetical cards produced by a conventional unit and by an air-balanced unit. As indicated, the various unit geometries and configurations produce variations in peak and mini-

12,000

10,000 -~

8000 ci'

-<I 6000 0 ..J

4000 -2000 -

0 .-=- to)

I I I I I 3 2 o -I -2 -3

DISPLACEMENT, ft

�2,000,------------.-----------,

10,000 .0

ci' 8000

~ 6000 ..J

4000

2000 (e)

°r~====~------t_------~~ -20004~-~-~--~1--~--L-~~~-3

DISPLACEMENT, ft

12,000

10,000 ~ 8000 0 <I 6000 0 ..J

4000

2000

0

- 20004 I

mum loads and in dynamometer card shapes for the same operating conditions.

Prime mover speed variations also exert a significant influence on the behavior of sucker-rod systems. The mathematical solution can be used to study these effects, as shown in Fig. S. Fig. Sa shows synthetic dynagraph cards in which prime mover speed variations are neglected. Fig. Sb simulates a medium-slip prime mover for the same condition. It is useful to note that the peak load for the constant-speed condition is slightly too high, and the minimum load is slightly too low. The constant-speed condition is somewhat unrealistic, although it may be closely approached in wells with good counterbalance and large rotary inertia. Prime mover speed variations tend to soften the loads imposed at the polished rod and usually operate to decrease the load range. Some effect on pump stroke is also noted.

The design of tapered rod strings has posed an inter-

12,000

10,000 ~ 8000 ci' <I 6000 0 ..J

4000

2000 (b)

0

- 20004 I -3

DISPLACEMENT, It

12,000..-------------,---------,

10,000 .0

ci' 8000 <I 6000 o ..J

4000

2000 (d)

°r-L===========t---------i -20004~-~-~~-~-~--~-~--J_3

DISPLACEM ENT, ft

(e)

-3

DISPLACEMENT, It

FIG. 6-SIMULATION OF PUMPING MALFUNCTIONS.

20,000

.0 15,000

0 10,000 <I 0 -' 5000

AIR - BALANCED UNIT 0

4 3 2 I 0 -I -2 -3 -4 4 3 2 I 0 -I - 2 -3 -4

DISPLACEMENT, ft DISPLACEMENT, It

FIG. 7-SYNTHETIC DYNAGRAPHS PRODUCED WITH POLISHED ROD MOTIONS OF CONVENTIONAL AND AIR·BALANCED UNITS.

774 JOURNAL OF PETROLEUM TECHNOLOGY

25,000

20,000

15,000

.0

;i 10,000 o -'

5000

(0) NO Sli P

O~~-----------------~~----~--------------

3 I -I - 2 -3 -4

DISPLACEMENT. ft

(b) MEDIUM SLIP

-I -2 -3 -4

FIG. 8-EFFECT OF PRIME MOVER SPEED VARIATION.

esting problem for many years. The question arises in any tapered-rod string installation regarding the percentage of each rod size required to produce balanced stresses. The difference-equation technique (suitably generalized) provides means for studying and making optimum the design of tapered strings. Fig. 9 shows a dynagraph card calculated at the junction of 'Ys - and % -in. rods in a 4,500-ft well. With this card it is possible to calculate the stress at the junction and so determine if the percentage of each rod size is adequate. It is important to note that the peak load always occurs at the polished rod and not at some intermediate depth, as has been conjectured in the past.

DESIGN OF SUCKER-ROD INSTALLATIONS

Another application of the difference-equation tech­nique is in the design of rod pumping installations. The mathematical solution generates entire dynagraph cards for both the polished rod and the pump. Design data such as maximum and minimum rod loads, polished rod horse­power and pump stroke can be obtained from these cards. Since the technique requires a digital computer, it is best to precompute design data for various pumping units, prime movers, rod designs, fluid loads, volumetric effi­ciencies and pumping speeds. With precomputed data, the impracticality of using the digital computer on each design occasion is eliminated.

An efficient way of summarizing the precomputed design data is in graphical nondimensional form. In this scheme the various design factors are incorporated into the fol­lowing groups:

WdSK = nondimensional fluid load, N /N, = nondimensional pumping speed,

Wx/W f = nondimensional maximum dynamic load, W,,jW, = nondimensional minimum dynamic load,

S./S = nondimensional pump stroke, and P /W,SN = nondimensional polished rod horsepower.

The groups are chosen in a manner which makes the calculations as simple as possible. For example, S./S = 0.85 means that the gross pump stroke is 85 per cent of the polished rod stroke. Similarly, Wx/W, = 2 means that the maximum dynamic polished rod load is double the fluid load. As a further step toward simplicity, the effects of the fluid load and the inertial effects of the sucker rod are lumped into the so-called dynamic load. The maximum polished rod load is therefore the weight of rods in fluid plus the maximum dynamic load,

JULY, 1963

DYNAG RAPH CARD AT JUNCTION BETWEEN RODS AND 3/4 -INCH RODS

14.000

12.000

10.000

8000 ,e ci 6000 « 0 ...J 4000

2000 -PUMP CARD

0

-2000 4 3 2 o -I -2 -3

DISPLACEMENT. ft

FIG. 9--CALCULATED DYNAGRAPH CARDS IN A TWO-WAY COMBINATION ROD STRING.

-4

(12)

The corresponding formula for minimum polished rod load is

Wmin == WfJ - Wn (13)

where Wn is the minimum dynamic load. To illustrate the method, a typical set of design curves

and a sample design calculation are shown in Fig. 10. The sample curves are valid for conventional units, uniform rods, 75-per cent volumetric efficiency and negligible prime mover slip. More curves can be developed for tapered strings, other types of units, other volumetric efficiencies, etc.

VISUAL DIAGNOSIS OF OPERATING CONDITIONS

A useful feature of the new method is that entire dyna­graph cards can be predicted. When field dynagraph cards are compared with synthetic dynagraphs generated with the computer, it is possible in many cases to diagnose down-hole operating conditions. This process is illustrated in Fig. 11. Fig. 11 a shows an actual dynagraph card taken in an installation with a 44-in. conventional unit operating 4,700 ft of %-in. rods at 14 strokes/min with a 1 % -in. pump. For this depth and speed, N /N, = 0.27. Fig. 11 b shows a set of synthetic dynagraphs for approxi. mately the same nondimensional speed, N /N, = 0.25. By visual comparison it appears that the field dynagraph corresponds to the condition WdSK = 0.4 with 75 per cent pump efficiency. For this installation the fluid load WI is calculated to be about 4,500 lb. Also, by use of the design curves (Fig. 10), the pump stroKe is calculated as approximately 36 in. Thus, the calculated fluid load indi­cates that the well is virtually pumped-off, the efficiency

775

776

9

C 8

7 7 0..1

G 6

5 - 5

;;: ;-, ,

:;;1< ; 4 - 4

3-

2 :r 0..2

0..5

I r- I~ 0.

(0) cL 17 ---------------,0..1 12

16

1.5

1.4 -

13

1.2

Vl , 1.1 Q.

Vl

1.0.

0..9

0..8

0..7

0..6

0..5 0.

DESIGN ASSUMPTICNS

CCNVENTICNAL UNIT

S= 4.5 It N = 16 spm L =450.0. It 3/4 -INCH RCDS

Wb=65CC Ib

I 1;2 -INCH PUMP

WI = 250.0. Ib

NINo 0..3

EFFICIENCY = 75 PERCENT

PRCDUCTICN REQUIRED = 150. bpd

0..4

II

10.

9

"' 0 7· x

Q.I~ 6

WI 4

SK 3

(c)

0..5 0..1 0..2 NINo 0..3

DESIGN CALCULATICNS

0..4

!!. = __ N_L_ 16 (450.0.) = 0..30. No 240.,0.0.0. 240., 0.0.0.

K = EA = 30.,0.0.0., 0.0.0. (0..442) =2945 Ib/lt L 450.0.

WI 250.0. SK = 4.5 (2945) = 0.189

Wx = 205 WI . W,=2.C5 (250.0.) =51251b

FRCM CURVES

Wn WI = 0..82 , Wn= 0..82 (250.0.)= 20.50. Ib

~ = 0.977 S . , Sp =0..977 (4.5) = 4.4 It

0..3

0.4

0..5

(d)

0..5

W;SN' 10.5= 3.61; P= 3.61 (250.0.) (4.5) (16) 10.-5 = 6.5 hp

WMAX = Wb + W, = 650.0. + 5125 = 11,625 Ib

WMIN = Wb- Wn = 650.0. - 20.50. = 4450. Ib

We = WMAX ~ WM1N = 80.37 Ib

NT =6(S) (WMAX-We) = 96,80.0. In.-Ib

PRCDUCTICN = 1.4 (1.5)2 (4.4) (16) (0..75) = 166 bp.d

FIG. IO-INSTALi.ATION DESIGN BY MEANS OF NON DIMENSIONAL PARAMETERS.

JOURNAL OF PETROLEUM TECH'\"OLOG Y

0.1

0.2

0.4

0.5

JULY, 1963

e=IOO%

/~'~ ---------.--~---

L '4700 ft

N' 14 spm

3/4 - I NCH RODS

I 314-INCH PUMP

1i, 14 (4700) , 0.27 NO 240,000

(0) ACTUAL DYNAGRAPH

75 %

(b) SYNTHETIC DYNAGRAPHS

50 %

FIG. ll-VIS[;AL DIAGNOSIS OF DOWN-HoLE OPERATING CONDITIONS.

25 %

of 75 per cent indicates reasonably good down-hole gas separation and the pump stroke when compared with measured production indicates that the pump is in good mechanical condition.

It should be pointed out that, in comparisons of meas­ured cards with calculated cards, the shape rather than the area must be used. This is because the field card can be made thin or fat by merely changing rings in the dynamometer. Also, the principal diagnostic features of a surface dynagraph lie in the middle two-thirds of the card. A key item is the relative position of the maximum and minimum loads. The features of the card near the ends of the stroke are likely to be unreliable as diagnostic indicators. It should also be stressed that tubing move­ment, prime mover slip, rod design and unit geometry all exert influences on dynagraph card shape. Hence, extreme variations from the conditions used to generate the syn­thetic cards may limit the use of visual diagnosis.

CONCLUSIONS

1. In view of the wide use of sucker-rod pumping, a fundamental understanding of the system is economically essential.

2. The technique of differenc • equations can be used to refine the criteria for designing and operating sucker-rod installations. The method can be useful to manufacturers for predicting the effect of various pumping-unit designs on system behavior. It can also be useful to oil producers for selecting equipment and for determining optimum methods of operation.

778

NOMENCLATURE

a = velocity of force propagation, ftl sec A = area of sucker rods, sq in.

CBT = counterbalance torque, in.-Ib E = modulus of elasticity, psi

oF' = dynamic polished rod load, lb x* pr = dynamic pump load, lb

K = spring constant for rod string, lb/ft

L = length of sucker-rod string, ft N = pumping speed, strokes/min

No = natural frequency of rod string, cycles/min NT = net torque, in.-Ib

P = polished rod horsepower, hp PL = pump load, lb

PRL = polished rod load, lb

S = polished rod stroke, ft Sp = pump stroke, ft

t = time, seconds T = difference notation for t, dimensionless

TP = torque factor, in. u(x,t) = displacement of the sucker rod at

arbitrary depth and time, ft "ur = difference notation for u (x,t), ft Wb = weight of sucker rods in fluid, lb We = counterweight, lb Wi = fluid load, lb

W min = minimum polished rod load, lb W mox = maximum polished rod load, lb

Wn = minimum dynamic polished rod load, lb Wx = maximum dynamic polished rod load, lb

x = distance along unstrained sucker rod measured from polished rod, ft

X = difference notation for x, dimensionless x* = difference notation for L, dimensionless

X* zr = pump displacement including static rod stretch, ft

v = damping factor, dimensionless

REFERENCES

1. Gray, H. E.: "Kinematics of Oil·Well Pumping Units", Paper presented at the API Midcontinent Dist. Meeting, Amarillo, Tex. (March 27·29, 1963).

2. Gilbert, W. E.: "An Oil·Well Pump Dynagraph", Drill. and Prod. Pmc., API (1936).

3. Milne, E.: Numerical Solution of Differential Equations, John Wiley & Sons, Inc., N. Y. (1953). ***

JOURNAL OF PETROLEUM TECHNOLOGY