Yuting Duan, Xiangzhong YuanSouthwest Petroleum Institute

China

In deviated wells, guides or wheels properly placed on the rod string can prevent both wear on the tubing and the rods.

Proper placement involves integrating information on hole trajectory, rod sting properties, and economics.

### DIRECTIONAL WELLS

Sucker rod pumping units are used in directional oil wells, especially for recovery of heavy oil. When the pump depth is deeper than the deviation kick-off depth, guides or wheels installed on rod strings prevent the rods from direct contact with the tubing.

Guides or wheels in the deviated hole section can lessen rod and tubing wear, lower polished rod loads, and greatly improve the forces on the rod string. Therefore, guides on the rods improve the economics of sucker rod pumping in directional wells.

Too many guides or wheels will increase polished rod load, thereby increasing recovery cost. But too few guides or improper spacing of guides can neither prevent rod and tubing wear nor lower polished rod load. Consequently, optimal rod guide spacing is required.

Few published articles have analyzed the design of sucker rod strings in directional wells. Zengjing Wang discussed the spacing for inclined wells.1 However, his conclusions do not fit the case of highly deviated or horizontal wells.

Force and deformation analyses in deviated wells shows that spacing and location of rod guides depend on well hole geometry, normal deformation of the sucker rod string, and stress level.

### FORCE ANALYSIS

Fig. la shows the configuration of a sucker rod pumping system in a directional well. Guides or wheels are installed along the sucker rod string between kick-off point and pump depth.

Figs. 1b and c show the force analysis of two adjacent guides during the up and downstrokes. The forces on the selected section are as follows:

- Weight of rod string, uniformly distributed along rod
- Weight of guides, concentrated at the middle point of the guides
- Normal forces on guides resulting from contacting the tubing
- Friction between guides and tubing
- Viscous damping force between guides and tubing, concentrated at the middle point of the guides
- Viscous damping force between rods and tubing, distributed along the rods *Inertia force of the rods, uniformly distribution along the rods
- Inertia force of guides, centered at the middle point of the guides

Force calculations are discussed in References 4, 5, and 6.

During the upstroke, the entire sucker rod string is in tension because of the weight of liquid above the plunger. Therefore as the rod straightens and rod deflection decreases because of axial tension, the rod will come into contact with the upper side of the tubing. To prevent this contact, the guide or wheeled coupling should he installed on that side.

During the downstroke, on the other hand, the downward section of the rod string is usually in compression. Therefore, the rod bends and the deflection of the rod increases because of axial compression. Consequently, the rod will contact the lower side of the tubing. To prevent this contact, the guide or wheeled coupling should be installed on the lower side.

During the downstroke, although the upper section of the rod string between the kick-off point and pump depth may be in tension, the rod deflection is relatively large because of minor axial tension.

Proper location and spacing depend on both hole geometry and deflection of the rod.

### SPACING

During the upstroke, rod deflection is very small because of axial tension. Therefore, without properly installed guides or wheeled couplings, the rod may contact the upper side of the tubing.

Fig. 1d shows a rod without deflection. In the span between any two guides, B is the middle point of the span and c is half of the difference between the OD of the guides and the OD of the rods.

Equation 1 (see equation box) is derived from the geometry in Fig. 1d. Equation 2 is derived from Equation 1 if one assumes that the square of c is much less than R.

Equation 3 is obtained if one considers hole curvature instead of radius of curvature. Equation 3 shows that the maximum length between two guides depends on hole curvature and diameter difference between the guides and rods. If the guide diameter is given, the maximum allowable guide spacing depends on hole curvature and rod size. Fig. 2a shows the maximum allowable length for some rod sizes for different hole curvature. The OD of the guide is 7.15 cm.

Fig. 2a shows that the maximum allowable length between two guides decreases greatly as hole curvature increase.

### SPACING AND DEFLECTION

As we analyzed previously, during the downstroke, the sucker rod string is usually in compression or tension with minor axial tension. In this case, the rod contacts the down side of the tubing because the deflection is usually large and is affected by several forces. To prevent the rod and tubing wear during the downstroke, the guides should be spaced based on the rod deflection.

Fig. 1e shows a span between any two adjacent guides with the maximum deflection at the middle point of the span. This is expressed as Equation 4.

The first term in Equation 4 is the load intensity q. The second term describes the internal bending moments at the guides during the downstroke. These moments are usually minor and are caused by the internal forces in the sucker rod string in directional wells. Therefore, the second item in Equation 4 can be eliminated to obtain Equation 5.

Equation 6 is valid if the rod does not contact the tubing. Consequently, Equation 7 includes the maximum allowable length between two guides.

Equation 7 is a nonlinear equation. Iteration should be used to solve the equation to obtain maximum allowable length between two adjacent guides. There are many methods to solve Equation 7.

From Equation 7, the maximum allowable length between two guides depends on the stiffness of rods, weight of rods in liquid, inclination angle, and axial force. As mentioned previously, the axial force is usually small.

If the axial force is equal to zero as a spatial condition, Equation 8 is obtained from Equation 7. This is the maximum allowable guide spacing controlled by rod deflection in the downstroke without axial force.

Fig. 2b shows the maximum spacing of some sizes of rods for different hole inclinations. Axial force is assumed to be zero.

### OPTIMAL SPACING

To avoid rod and tubing wear, guide spacing should satisfy the conditions that the rod does not come into contact with both the upper side and downward side of the tubing during the up and downstrokes. Therefore, the spacing is given by Equation 9.

The allowable spacing may be any value within a range. But in practice, it is not convenient or possible to have any rod length. However, the conventional length series is too limited. For example, the conventional rod lengths in China are 8, 6, and 3 m. These rod lengths do not satisfy the needs for directional wells.

A new rod length series is needed for sucker rod pumping of directional wells. We suggest that rods of 1, 1.5, and 8 m can satisfy the requirements.

Internal force analysis shows that the normal stress, produced by the axial force and internal bending moment, on the upstroke varies with the change of hole curvature. Therefore, the normal stress peaks in the section with a large hole curvature. As a result, the stress level in some positions is high while in other positions the stress is low. Analysis shows that peak stress can be reduced by shortening guide spacing.

However, more guides increase polished-rod load because the guides are relatively heavy compared to the rod. A compromise should be made between spacing and peak stress to obtain an optimal design.

Equation 10 shows the API recommended allowable stress on sucker rods. The maximum working stress during the upstroke should satisfy Equation 11.

Fig. 3 and Table 1 illustrate the analysis of a horizontal well that pumps heavy crude.

### REFERENCES

- Zengjing Wang, "The Calculation of Rod Guide Spacing in Inclined Wells," Oil Drilling & Production Technology, February 1991.
- Xiaoping Zhu, "The calculation of stress and deformation of sucker rod string in straight wells," Petroleum Machinery, February 1992.
- Lukasiewicz, S.A., "Dynamic Behavior of Sucker Rod String in Inclined Wells," SPE Paper No. 21665.
- API RP 11BR, 1977.
- Timoshenke, S., Theory of Elastic Stability.
- Hongxun Wang, Qi Zhang, Principle of Petroleum Production, 1990.