Buckling of Pump Barrel and Rod String Stability in Pumping Wells

Transcription

1 This is a revised version of manuscript PO "Stability of Pump Barrels and Rod String in Pumping Wells" (2015). This manuscript has been submitted to SPE Production & Operations. Manuscript has been subjected to the peer review process. In view of the reviewers' comments, this manuscript has been declined for publication in SPE Production & Operations. Buckling of Pump Barrel and Rod String Stability in Pumping Wells Sh.Vagapov and S.Vagapov, (Scientific and Production Enterprise Limited Liability Company Burintekh, Ufa, Republic Bashkortostan, Russian Federation) Abstract In this paper, the phenomenon of buckling of a pump barrel is explained. For the tubing and the top anchor rod pumps, both the pump barrel and the tubing string are buckled during pump upstroke. The pump barrel will buckle and much more severely in comparison with the tubing string. Possibly the buckling of the pump barrel hastens pump wear. The buckling of the barrel can accelerate wear of the top of the plunger. For the bottom anchor rod pumps, the pump barrel increases the stability in the straight position during pump upstroke. The possibility that the straightening effect can reduce friction and wear of the plunger during pump upstroke is suggested. In some instances, where the crude oil is highly viscous, both the pump barrel and the bottom rods will buckle during pump downstroke. In these cases, the tubing pumps and top anchor rod pumps have the advantage over the bottom anchor rod pumps. In the design of sucker rod pumps it must consider the stability of pump barrel. Accurate analysis of buckling is important because the buckling of the barrel generates bending stresses. The change in thickness may lead to local buckling of the barrel. In the design of sucker rod pumps the pump barrel should also be checked for local buckling. According to Newton's third law, for an upward fictitious force applied to the tubing there is a downward reaction force applied to the rod string. The reaction force exerts a straightening effect which resists rod string buckling. Introduction It is well known that the lower part of freely suspended tubing buckles in a pumping well during pump upstroke (Lubinski 1957, 1962 et al). Extensive researches on this subject have been conducted, both theoretically and experimentally (Mitchell 2008 and others). In the analyses of tubing buckling, Lubinski proposed so-called "fictitious force" as a device to permit easy calculation of the buckling effect of internal pressure. During pump upstroke the tubing will buckle as if subjected to an upward fictitious force. It is commonly considered that the fictitious force is applied to the bottom end of the tubing. Actually, the fictitious force is applied to the pump barrel, because the pump is mounted on a bottom end of the tubing. One might wonder, will the pump barrel be buckled during pump upstroke? It can be expected, that the pump barrel will buckle since the pump barrel has much lower bending stiffness than the bending stiffness of the tubing string (Tables 1 and 2). However, the published works do not take into account buckling of the pump barrels and the influence of internal pressure on pump barrel instability has not been investigated yet.

2 On the other hand, Newton's Third Law states that forces always act in pairs and always act in opposite directions. Obviously, for a fictitious force there must be a reaction force that is equal in size, but opposite direction. However, the tubing and rod strings interact with each other through the fluid column inside tubing. When consideration is given to the reaction force, the question may be asked: How does the reaction force affect the rod string stability? Is a reaction force fictitious or real? The present work investigates the buckling behavior of pump barrels and a rod string in order to give users and manufacturers another point of view in understanding the causes and remedies for the sucker rod pump and rod string failures. The investigation was based upon the following assumptions. 1. The pump barrel and plunger behave as linear elastic bodies. 2. The positive and negative forces will be considered as compressions and tensions, respectively. 3. Wellbore is assumed to be straight and vertical. 4. The tubing string is freely suspended in the well. 5. The pump barrel has the uniform bending stiffness. 6. The dynamic load may be neglected. 7. The fluid level is near the pump intake. 8. No tail pipe below the pump. Buckling Behavior of The Pump Barrels For the purpose pursued in this paper, the basic types of sucker rod pumps (API specification 11AX) were considered. The traveling barrel rod pumps and non API pumps are not considered. Tubing and Top Anchor Rod Pumps. Let us consider a tubing pump as shown diagrammatically in Fig.1. For the tubing pumps, the data and calculated moment of inertia, fictitious and critical forces (see Appendix) are summarized in Table 1. We see, that the pump barrel is an integral part of tubing string because the pump barrel is attached directly to the tubing string. During pump upstroke, the high hydrostatic pressure is applied inside the barrel above the plunger because the standing valve is open and the traveling valve is closed. For this reason, there exists a differential pressure from the barrel ID (inside diameter) to OD (outside diameter). It can be expected, that the portion of the barrel above the plunger will buckle as if subjected to an upward fictitious force. Obviously, the fictitious force must exceed some critical value in order to buckle the pump barrel (see loading scheme for the barrel, Fig.1). Consider now a top anchor rod pump as shown diagrammatically in Fig. 2. For the top anchor rod pumps, the data and calculated moment of inertia, fictitious and critical forces (see Appendix) are summarized in Table 1. The principle of operation of the top anchor rod pump is identical to that described for the tubing pump. Similarly, there exists a differential pressure from the barrel ID to OD during pump upstroke. It should be noted, that the top anchor rod pump is equipped with a centralizer which is located on the top of the barrel (see loading scheme for the barrel, Fig.2). Let us consider the data presented in Table 1 in more detail. The data shows that the fictitious force F is much greater than the critical force F crb for the barrel (the ratio can reach 1,3 6,1).This means, that the pump barrel buckles during pump upstroke. On the other hand, the fictitious force F is greater than the critical force F crt for the tubing string. Therefore, both the pump barrel and the tubing string buckles during pump upstroke.

3 However, the pump barrel has much lower bending stiffness than the bending stiffness of the tubing string. So one can conclude that the pump barrel will buckle and much more severely in comparison with the tubing string. Possibly, the buckling of the barrel hastens sucker rod pump wear. It is evident, that the pump barrel has much lower bending stiffness than the bending stiffness of the plunger. When the plunger starts its upward movement inside the bent barrel, the plunger tries to straighten up the barrel. It can be expected, that the buckling of the barrel can accelerate wear of the top of the plunger. For the tubing pump, the sucker rod string connects directly to the plunger top cage. For the rod pump, the plunger is attached directly to the valve rod. It is evident, that the sucker rod and valve rod has much lower bending stiffness than the bending stiffness of the plunger. The barrel is subjected to the bending moment M during pump upstroke due to the high hydrostatic pressure inside the barrel. Therefore, when the plunger starts its upward movement inside the bent barrel, the force R is applied to the plunger at an angle (Fig.3). The plunger has two reaction forces R a and R b equal in magnitude, oppositely directed. As a result, the forces R a and R b have a turning effect or moment M r = R a(r b) L p (1) where R a(r b) is a reaction force, L p is plunger length. During pump upstroke, the moment M r tries to overcome the resistance of the bending moment M. For a given value of the bending moment M, an increase in the length L p of the plunger causes a decrease in the magnitude of reaction force R a(r b). Consequently, the long plunger can cause significant reduction in the magnitude of friction forces. For this reason, the long plunger should be recommended in deep wells and wells with low static fluid level. Bottom Anchor Rod Pump. Consider now a bottom anchor rod pump as shown diagrammatically in Fig.4. For the bottom anchor rod pumps, the data and calculated moment of inertia, fictitious and critical forces (see Appendix) are summarized in Table 2. During pump upstroke, the high hydrostatic pressure is applied outside the barrel below the plunger because the standing valve is open and the traveling valve is closed. For this reason, there exists a differential pressure from the barrel OD to ID. Therefore, the portion of the barrel below the plunger is subjected to an excessive external pressure. It is well known that the pressure inside the pipe terminated by pistons, exerts a buckling effect in the pipe. The buckling is the same as if the pipe, instead of being subjected to internal pressure, is subjected to the compressive force (a positive fictitious force). On the contrary, an excessive external pressure (Fig.5) exerts a straightening effect which supports the pipe and resists buckling. This is the same as if the pipe, instead of being subjected to an external pressure, is subjected to the tension (a negative fictitious force) as shown in Fig.6. Consider the data presented in Table 2 in more detail. The data shows that in low working fluid level wells, the fictitious force F is greater than the critical force F crt. This means, that the tubing string buckles during pump upstroke. On the other hand, the fictitious force F is negative for the portion of the barrel below the plunger. This indicates that the pump barrel remains straight during pump upstroke. Moreover, a pump barrel tends to increase the stability in the straight position when it is subjected to an external pressures. The bigger an excessive external pressure the more stable the pump barrel. When a pump barrel is perturbed so as to initially deflect and buckle, the external pressure results in forces which act to move the pump barrel back into an equilibrium state by straightening it.

4 During pump upstroke the differential pressure from the barrel OD to ID decreases with the distance from the bottom of the plunger and becomes nil at the top of the plunger (Fig.4). Because of this, the negative fictitious force (straightening effect) decreases with the distance from the bottom of the plunger and becomes nil at the top of the plunger. This means, that the straightening effect is a maximum at the bottom of the plunger during pump upstroke. Therefore, it can be expected, that the contact force between the plunger and the barrel is reduced during pump upstroke. As result, the straightening effect can reduce friction and wear of the plunger during pump upstroke. It is interesting to note, that the buckling of the tubing is counterbalanced by the straightening effect of the pump barrel during pump upstroke. It is known, that bottom anchor rod pump has the advantage of having the hydrostatic tubing pressure applied to the outside of the barrel. The possibility of failures is low because there are not high axial loads on the pump barrel. Therefore, it is the first pump to consider in deep wells. Based upon the foregoing, one may add another argument supporting the use of bottom anchor rod pump in deep wells. Buckling of Pump Barrel During Pumping of Highly Viscous Crude. During pump downstroke, there is a high differential pressure across the plunger due to resistance to fluid flow through the traveling valve. Therefore, the portion of the pump barrel below the plunger is subjected to an excessive internal pressure. For a bottom anchor rod pump, there is no tensile force acting on the barrel. Thus, if an excessive internal pressure is large enough, the lower portion of the barrel will buckle during pump downsroke (Fig.4). The buckling is the same as if the pump barrel, instead of being subjected to an internal pressure, is subjected to a compressive force (a positive fictitious force). On the other hand, an excessive internal pressure results in a compressive force acting on the bottom of the plunger. This compressive force causes the bottom rod to buckle. Consequently, both the pump barrel and the bottom rods will buckle during pump downstroke. It is interesting to note, that the compressive force is a fictitious force for the barrel and is a real force for the bottom rods. For the tubing and the top anchor rod pumps, the barrel is subjected to a tension during pump downstroke (Figs. 1 and 2). This means that the pump barrel remains straight while the bottom rods will buckle during pump downstroke. So one can conclude, that the tubing pumps and top anchor rod pumps have the advantage in the pumping of highly viscous crude. Rod String Stability in Pumping Wells Consider a pipe with ends closed by frictionless pistons connected by an inelastic rod as shown in Fig.7d. It has been mentioned above, the pressure inside the pipe exerts a buckling effect in the pipe. The buckling is the same as if the pipe, instead of being subjected to internal pressure, were subjected to a compressive fictitious force (positive force), as shown in Fig.7b. According to Newton's third law, there must be a reaction force that is equal in size, but opposite direction. Consequently, the rod must be subjected to the reaction tensile force R (negative force) as shown in Fig.7c. One might wonder, whether this force is real or not? Imagine now, that the diameter of the pistons and the rod are the same (Fig.7a). We see, that the compressive fictitious force does not depend upon the diameter of the rod but only upon the diameter of the piston. Therefore, according to Newton's third law, the reaction force also does not depend upon the diameter of the rod. In this case (Fig.7a), the real tensile force does not exist for the rod the rod is subjected to the external

5 pressure only. The reaction force is a fictitious tensile force applied to the rod. Obviously, the rod tends to increase the stability in the straight position. Consider now the same tubing, but the diameter of the rod is much smaller than the diameter of the piston (Fig.7d). The tubing is buckled by the fictitious force, which is the same as in Fig.7b. The rod is subjected to the real tensile force R r due to pressure acting on the annular area marked AB in Fig.7d. R r = A AB P...(2) where P is the pressure differential across the plunger, A AB is the annular area of the piston. At the same time, the rod is subjected to the fictitious tensile force R f due to the external pressure only R f = A r P..... (3) where A r is the rod cross-section area. Therefore, the general reaction force R is equal to the real tensile force R r plus the fictitious tensile force R f for the rod. R = R r + R f (4) It is evident that there is a similarity between a rod string and a rod of Fig.7c. Just as the rod of Fig.7c increases the stability in the straight position, a rod string increases the stability in the straight position during pump upstroke. For the rod string, the real tensile force R r is the weight of the fluid column supported by the net plunger area. The net plunger area is equal to the cross-sectional area of the plunger minus the area of the rod string. At the same time, the rod string is subjected to the fictitious tensile force R f due to the pressure only. The fictitious tensile force R f is equal to the density of the produced fluid multiplied by the cross-sectional area of the rod string, multiplied by the height of the fluid level. Thus, the reaction force R is equal to the weight of the fluid column supported by the net plunger area plus the fictitious tensile force R f for the rod. So one can conclude, that the pressure inside the tubing exerts a straightening effect which resists rod string buckling during pump upstroke. It is worth noting that a reaction force R does not depend upon the diameter of the rod but only upon the diameter of the piston. Conclusions 1. In the design of sucker rod pumps it must consider the stability of pump barrel. Accurate analysis of buckling is important because the buckling of the barrel generates bending stresses. The change in thickness may lead to local buckling of the barrel. In the design of sucker rod pumps the pump barrel should also be checked for local buckling. 2. For the tubing and the top anchor rod pumps, both pump barrel and tubing string are buckled during pump upstroke. The pump barrel will buckle and much more severely in comparison with the tubing string. Possibly buckling of the pump barrel hastens pump wear. The long plunger can cause significant reduction in the magnitude of friction forces. For this reason, the long plunger should be recommended in deep wells and wells with low static fluid level. 3. For the bottom anchor rod pumps, the pump barrel increases the stability in the straight position during pump upstroke. Probable the straightening effect can reduce friction and wear of the plunger during pump upstroke. The straightening effect adds another argument supporting the use of bottom anchor rod pump in deep wells.

6 4. In some instances where the crude oil is highly viscous, both the bottom anchor rod pump barrel and the bottom rods will buckle during pump downstroke. In these cases, the tubing pumps and top anchor rod pumps have the advantage over the bottom anchor rod pumps. 5. According to Newton's third law, for an upward fictitious force applied to the tubing there is a downward reaction force applied to the rod string. The reaction force exerts a straightening effect which resists rod string buckling. The reaction force is equal to the real tensile force plus the fictitious tensile force. The reaction force does not depend upon the diameter of the rod but only upon the diameter of the plunger. References 1. Lubinski, A. and Blenkarn, K. A Buckling of Tubing in Pumping Wells, Its Effect and Means for Controlling It. Trans. AIME (1957) 210, 73. SPE-672-G. 2. Lubinski, A., Althouse, W.S. and Logan, J.L Helical Buckling of Tubing Sealed in Packers. J. Pet. Tech. 14 (6): SPE-178-PA. 3. Mitchell, R.F Tubing Buckling-The State of the Art. SPE Drill & Compl 23 (4): SPE PA. 4. Timoshenko, S. 1st Ed. 1930, 2nd Ed. 1940, 3rd Ed Strength of Materials, Part I, Elementary Theory and Problems, D. Van Nostrand Company. 5. Timoshenko, S. 1st Ed. 1930, 2nd Ed. 1941, 3rd Ed Strength of Materials, Part II, Advanced Theory and Problems, D. Van Nostrand Company. Appendix Let the following designations be made. A - plunger cross-section area P - pressure differential across the plunger H - pump depth (height of fluid above the pump) d b - pump bore (barrel inside diameter; plunger outside diameter) D b - barrel outside diameter d t - tubing inside diameter D t - tubing outside diameter E - modulus of elasticity I b - moment of inertia of the barrel cross-section I t - moment of inertia of the tubing cross-section L - barrel length L b - actual length of the barrel L p - plunger length k - effective length factor F - fictitious force F crb - critical force for the barrel F crt - critical force for the tubing

7 in pump design. end free, k = 2. The fictitious force is F = A P (5) The pressure differential across the plunger (fluid level is near the pump intake) is P = H g.... (6) Where g = 0.5 psi/f (the numerical example from a paper Lubinski and Blenkarn (1957). The plunger cross-section area is A = (d b ) (7) The critical force for the barrel is (Timoshenko 1955, 1956) is E I b F crb = π (8) (k L b ) 2 Where E = 30 x 10 6 psi. For tubing and sucker rod pumps, the effective length factor depends on the constructional differences Fig.1 shows a loading scheme for the tubing pump. For a column with one end built-in and the other Fig.2 shows a loading scheme for the top anchor rod pump. Fig.4 shows a loading scheme for the bottom anchor rod pump. The top and bottom anchor rod pumps are equipped with a centralizer. For a column with pinned ends, k = 1. The actual length of the barrel is L b = L L p 2 For simplicity, L is taken constant and equal 20 ft. For simplicity, L p is taken constant and equal 4 ft. The moment of inertia of the barrel is I b = π D b 4 (1 d 4 b 64 D b 4)..... (9) The moment of inertia of the tubing is I t = π D t 4 (1 d 4 t 64 D t 4)..... (10) The critical force for the tubing (Lubinski 1957, 1962 et al) is 3 2 F crt = 1,94 E I t q t (11) Where q t = 4,089 lb/ft = 0,341 lb/in, tubing size, 2 3/8, fluid specific gravity 1.154, q t = 5,545 lb/ft = 0,462 lb/in, tubing size, 2 7/8, fluid specific gravity 1.154, q t = 8,097 lb/ft = 0,675 lb/in, tubing size, 3 1/2, fluid specific gravity

8 Moment Moment Ratio Wall of Inertia of Inertia Critical Critical Thickness of a of Force For Force For Fictitious Pump Tubing of Pump Barrel a Tubing a Tubing a Barrel Force F I t Pump Bore Size a Barrel depth I b I t F crt F crb F F crb I b Designations (in) (in) (in) (ft) (in 4 ) (in 4 ) (Lb) (Lb) (Lb) Rod Pumps, Stationary Barrel, Top Anchor RHA 1 1/4 2 3/8 3/ RHA 1 1/2 2 7/8 3/ RHA 1 3/4 2 7/8 1/ RHA 2 1/4 3 1/2 1/ RWA 1 1/4 2 3/8 1/ RWA 1 1/2 2 3/8 1/ RWA 2,0 2 7/8 1/ RWA 2 1/2 3 1/2 1/ Tubing Pumps TH 1 3/4 2 3/8 1/ TH 2 1/4 2 7/8 1/ TH 2 3/4 3 1/2 1/ Table 1 The tubing and top anchor rod pumps. Moment of inertia, fictitious and critical forces data Moment Ratio Wall of Inertia Moment Critical Critical Thickness of a of Inertia of Force For Force For Fictitious Pump Tubing Of Pump Barrel a Tubing a Tubing a Barrel Force F I t Pump Bore Size a Barrel depth I b I t F crt F crb F F crb I b Designations (in) (in) (in) (ft) (in 4 ) (in 4 ) (Lb) (Lb) (Lb) Rod Pumps, Stationary Barrel, Bottom Anchor RHB 1 1/16 2 3/8 3/ RHB 1 1/4 2 3/8 1/ RHB 1 3/4 2 7/8 1/ RHB 2 1/4 3 1/2 1/ RWB 1 1/4 2 3/8 1/ RWB 1 1/2 2 3/8 1/ RWB 2,0 2 7/8 1/ RWB 2 1/2 3 1/2 1/ Table 2 The bottom anchor rod pumps. Moment of inertia and critical force data

9 Fig. 1 Tubing pump. Buckling of the barrel (left) and loading scheme (right) Fig. 2 Top anchor rod pump. Buckling of the barrel (left) and loading scheme (right) Fig. 3 Distribution of forces acting on a plunger

10 Fig. 4 Bottom anchor rod pump. Straightening effect in the barrel (left) and loading scheme (center). Buckling of the barrel and rod string during pumping of highly viscous crude (right).

11 Fig. 5 Tubing with piston closed ends subject to external pressure. Fig. 6 - Straightening effect of external pressure

12 (a) (b) (c) (d) Fig. 7 Tubing with piston closed ends subject to internal pressure. Diameter of the piston ends and the rod is the same (a); buckling effect (b); straightening effect (c); diameter of the rod is smaller than diameter of the piston (d)