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Analysis of instability in the lower part of the sucker rod string and design methods for weighted rods – Chen Hongming, China National Petroleum Corporation. During the pumping process using a rod pump, the sucker rod string encounters resistance as it moves downward. At its lower part, the self-weight cannot counteract the resistance, resulting in a compressed state. When the pressure increases to a certain level, the pump rod bends and deforms; under the constraint of the inner diameter of the tubing, this bending takes on a spiral shape. The unstable bending of the rod string has at least three disadvantages: ① It increases stroke loss and reduces pump efficiency ； ②Increase the likelihood of superstress failure ； ③Increased wear between the rod and tubing can easily lead to the breakage of the pumping rod and leakage in the tubing. I. Analysis of the compressed section of the sucker rod string: During its downward movement, the resistance encountered by the lower part of the sucker rod string is counteracted by the weight of a certain length of the rod string; thus, a neutral point is formed in the rod string, and the portion below this neutral point is under compression. The compressive stress on the pump rod increases gradually as it moves downward from the neutral point, reaching its maximum at its lower end. Affected by changes in compressive stress, the compressed rod section undergoes three types of transitions. In the area near the neutral point, due to the stiffness of the rod and the lower compressive stress, the rod remains straight and does not bend. Downward, as the compressive stress increases, the sucker rod undergoes elastic bending deformation. Further down, once the compressive stress exceeds the elastic limit, the rod will undergo irreversible plastic bending deformation. Of course, if the resistance during descent is not sufficient, or if the material and structural properties of the sucker rod are good, the rod string may have no plastic deformation zone or elastic deformation zone. Since the compressive stress is greatest at the very bottom of the rod string, the area near the bottom is the weakest part of the rod string and most susceptible to failure due to unstable bending deformation. This section has the greatest spiral bending deformation and the shortest pitch. Previously, it was believed that when the rod column moved downward, the resistances acting on its lower end mainly included the resistance of the fluid flow through the check valve and the friction between the plunger and the pump cylinder. However, through analysis, the resistances are not limited to these two; the lower end face of the rod also experiences an upward buoyant force. 1. Flow resistance: Flow resistance arises from the head loss that occurs as the fluid passes through the floating valve, and this loss generates a force acting on the annular area between the plunger and the valve seat hole. Pv = nk·Δpv·(F - fo), where Δpv = hv·ρl·g, ω = 2πn/60, and μ = f(Re). By analyzing these relationships, it can be seen that: ⑴. As the value of νl increases, Re decreases, μ decreases, hv decreases, and Pv increases; in other words, the fluid flow resistance Pv is proportional to the dynamic viscosity νl of the liquid. ⑵As F increases, on the one hand, (F-fo) increases (for a standard pilot valve, the value of F/fo remains approximately constant, at (D/do)2≈22=4), and Pv increases as well ； On the other hand, as do increases and Re increases, Pv decreases. Numerical calculations show that as the pump diameter D increases, the fluid flow resistance Pv increases. ⑶As the S*n value increases, Vo increases and hv increases as well; however, Re increases and μ increases, which in turn reduces hv. Numerical calculations show that as the pumping speed (determined by the stroke S and the number of strokes n) increases, the local hydraulic loss hv static through the sliding valve increases, thereby increasing the flow resistance Pv. 2. Plunger friction: The semi-dry friction between the plunger and the bushing is calculated using the formula recommended in the literature. 3. Buoyancy: It can be shown that the buoyant force acting on a circular rod with one end submerged in a liquid is a force directed upward and acting at the center of its lower surface; its magnitude is equal to the product of the liquid pressure at that central point and the cross-sectional area of the rod. If the lower end face of the sucker rod string, including the plunger, is simplified to the cross-section of the sucker rod for analysis, the buoyant force acting on this cross-section is equal to the product of the pressure of the liquid at that section and the cross-sectional area of the sucker rod. Pb = Fr·H·ρl·g 4. Upward resultant force: The resultant force acting upward through the end face on the lower part of the pump rod string during the downward stroke is the sum of these three forces. Pu = Pv + Pf + Pb. Obviously, the resultant force is primarily influenced by the depth of the pump, secondly by the diameter of the lower sucker rod, and is also related to factors such as the pump diameter, liquid viscosity, stroke, and pumping frequency. II. Using weighting rods to prevent the instability of the sucker rod string. There are two basic methods for preventing instability and bending deformation of the sucker rod string: using centralizers or weighting rods. Stabilizers can correct the bending deformation of the rod string and enhance stability, but they are generally not used due to limitations imposed by downhole tools. The best way to prevent this bending deformation is to use weighting rods. During the downward travel of the sucker rod string, the compressed section bends in a spiral shape, with the pitch decreasing gradually as it moves downward. The pitch can be calculated using the following formula. Assuming that when the pitch is not less than a certain length (such as the length of a sucker rod, around 8 meters), the instability and bending of the rod string cause only minimal damage (in this case, the sucker rod couplings can play a role in keeping the rod straight), it can be deduced that the pressure that the sucker rod can withstand is as follows. When the pressure at a certain point on the sucker rod exceeds Pcr, it is possible to replace the section of the rod string below that point with thicker, heavier rods in order to overcome the downward forces and increase the bending strength. The diameter of the cross-section of such a rod string can be calculated using the following formula, and the standard rod diameter dz’ can then be determined as a value not less than dz. Finally, the length of the weighting rod is determined using the formula Pu – Pcr = Fz·Lz·(ρs–ρl)·g. III. Calculation example: In a certain oil pumping well, the depth of the pump below the surface is 720 m, the diameter of the pump is 70 mm; the plunger has two check valves. The sucker rod string consists of φ25.4 mm × 400 m + φ22 mm × 320 m, the stroke length is 2.7 m, and the pumping frequency is 11 times per minute. The dynamic viscosity of the liquid being pumped is 5×10-6 m2/s. Determine the resistance experienced at the lower end of the sucker rod string during the downstroke, and carry out the design of the weighting rod. Through calculation, the values obtained are: Pv=88N, Pf=1102N, Pb=2607N, Pu=3797N, dz=23.5mm; by choosing dz’=25.4mm, Lz=25.7m. There is a pumping well in which the depth of the downhole pump is 1800 m, the diameter of the pump is 38 mm, and the plunger is equipped with two check valves. The sucker rod string consists of φ25.4 mm × 720 m + φ22 mm × 540 m + φ19 mm × 540 m, the stroke length is 5 m, the pumping frequency is 6 times per minute, and the dynamic viscosity of the fluid being pumped is 5×10-6 m2/s. Determine the resistance exerted on the lower end of the sucker rod string during the downstroke, and carry out a design for a weighted rod. Through calculation, the values obtained are: Pv=325N, Pf=534N, Pb=4861N, Pu=5721N, dz=26.0mm; by choosing dz’=28mm, Lz=99.1m. As can be seen from the above two examples, during the downward stroke, the resistance experienced by the lower end of the sucker rod string mainly comes from buoyancy, accounting for 68.7% and 85.0% respectively. IV. Conclusions 1. During its descent, the sucker rod string encounters resistance; its lower portion is under compressive stress, resulting in spiral bending deformation. 2. When the sucker rod string moves downward, the resistances acting on its lower end include the resistance to fluid flow through the sliding valve, the friction between the plunger and the pump cylinder, and the upward buoyant force acting on the lower end surface; among these, the buoyant force is the main factor. 3. Using a weighting rod is an effective method to prevent bending and deformation in the lower part of the sucker rod string. The design of the reinforcing rod should be based on the stability principles of slender compression bars. This paper introduces a pitch-limited design method. Symbol explanation: Pv – Fluid flow resistance generated by the pump plunger, N ； nk──Number of plunger sliding valves in the pumping pump ； Δpv──Pressure loss as the produced fluid passes through the floating valve, Pa ； F──Total area of the pump plunger, m2 ； fo──Cross-sectional area of the floating valve seat hole, m2 ； hv──Hydraulic loss head of the produced fluid flowing through the sliding valve, m ； ρl──Density of the produced liquid, kg/m3 ； g──acceleration due to gravity, m/s2 ； μ──flow coefficient ； Vo──the flow rate of the liquid through the flow valve, in m/s ； Vp──the maximum instantaneous speed of the sucker rod, m/s ； S──Stroke, m ； N──Chōji, min-1 ； ω──Rotational angular velocity of the pump crank, rad/s ； Re──Reynolds number ； do──Diameter of the valve seat hole in the pilot valve, in meters ； νl──Kinematic viscosity of the produced fluid, m2/s ； Pf──Friction force between the plunger and the bushing, N ； D──Diameter of the pumping pump, m ； δ──clearance between the plunger and the bushing, m ； Pb──Buoyant force acting on the lower end of the pump rod string, N ； Fr──Cross-sectional area of the sucker rod, m2 ； H──pump depth, m ； Pu──the upward resultant force acting on the lower end of the sucker rod string, N ； L──pitch, m ； E──elastic modulus, Pa ； I──inertia moment of the member’s cross-section, m4 ； P──Pressure, N ； d──rod diameter, m ； Pcr──critical pressure, N ； Lr──Length of a single sucker rod, in meters ； dr──pump rod diameter, m ； dz──Calculated diameter of the load bar, m ； dz'──diameter of the selected weighting rod, m ； Fz──Cross-sectional area of the load arm, m2 ； ρs──density of the weighting rod, kg/m3 ； Lz —— length of the load-shifting rod, m ；