Hu Yongquan,Yuan Xiangzhong,
Southwest Petroleum Institute
Nanchong, Sichuan, China
Determination of the minimum rod diameter, from statistical relationships, can decrease the time needed for designing a sucker-rod string for a directional well.
A tapered rod string design for a directional well is more complex than for a vertical well.
Based on the theory of a continuous beam column, the rod string design in a directional well is a trial and error method. The key to reduce the time to obtain a solution is to rapidly determine the minimum rod diameter. This can be done with a statistical relationship.
Sucker rods
Sucker-rod pumping units are often used in directional wells for artificial lift, but because the service conditions are different than in vertical wells, the rod string design is more complicated.
Usually, the internal forces and deflections along the string are calculated according to the theory of a continuous beam column. After preliminarily determining the minimum rod diameter, one can design a tapered rod string by strength and stiffness criterion. This trial and error method can be protracted unless the minimum rod diameter is first obtained.
Design method
In directional wells, sucker-rod pumping units have advantages such as being of simple structure, easy to handle, low cost, and having high efficiency. However, the working conditions of the rod string in a directional well are more complex because:
- Trajectory is a complicated curve, making it more complex to calculate internal forces and deflections along the string.
- Guide centralizers must be installed in the string to avoid damage between the tubing and rod. Because the centralizer diameter is greater than the rods, the rod string is a variable-stiffness, continuous-beam column with many supports. This is a statical indeterminable problem.
- The action of the transversal component of gravity causes the downward section of the string to be a continuous beam column rather than a "pure bend."
Research on design and analysis has been done for vertical wells, and relevant technical specifications are available, the most common being API RP 11L.1 But little study has been done on designing and analyzing strings in directional wells.
The usual method determines a preliminary rod diameter and then calculates the internal forces and deflections along the string based on the theory on a continuous beam column.2 For a taper rod string, this is a tedious calculation.
In this article, we consider the effect of parameters on the rod string and present an approximation method which, based on the characteristics of the string, may rapidly estimate the rod diameter in a directional well. With this method, an approximate result can be obtained that will greatly reduce computing time.
Basic analysis
Because the real rod string is very complicated, some assumptions are made for simplification. These are:
- The well azimuth is constant and the string is considered as a two-dimensional continuous beam column.
- The pump is set in the section with a small deviation, and the string is composed of vertical and curve sections (Fig. 1)(52751 bytes). The terms are defined in the nomenclature box.
- It is assumed that that middle and small curvature are suitable, and an analysis is made according to the theory of small deflections.
- The stiffness difference between the rod and centralizer and the rod's stiffness is neglected and regarded as the same as the whole beam.
- The centralizer's gravity and inertia force may be regarded as a concentration of forces because the centralizer's length is comparatively smaller than that of the rod.
- The centralizers are installed at the string couplings below the kickoff point (KOP).
The forces then acting on the rod string between any two adjacent centralizers (cutoff from the middle point of centralizer) are illustrated in Fig. 2 (54562 bytes) and are as follows:
- Gravity and inertia force of the rod.
- Gravity and inertia of the centralizer.
- Normal force on centralizer resulting from the reaction of the tubing.
- Mechanical friction force between the tubing and centralizer.
- Viscous damping force between the tubing and well fluid on the upstroke.
- Viscous damping force between the rod and well fluid on the downstroke.
- Axial forces acting on both ends.
In addition, there are semidry friction forces between the pump plunger and the sleeve, hydrostatic pressure acting on the plunger during the upstroke, and fluid flow resistance through the traveling valve during the downstroke.
These equations are discussed in Reference 3.
Rod design
For a continuous beam analysis, assume that the centralizers are joined with a hinge. The string, except for the vertical section, may then be treated as a continuous beam column. Based on the theory of continuous beam column, the equation of three moments which is used to solve the internal forces and deflections can be obtained by Equation 1 in the equation box.
This is a statically indeterminable problem, in which boundary conditions at the pump and KOP must be used (Equations 2 and 3).
Thus, the axial forces and deflections along the string may be obtained by iteration. Further, the stress along the string may also be determined.
The deflection along the string is determined by Equation 4, and the strength criterion4 and stiffness criterion along the string must agree with Equations 5 and 6.
From these equations the distribution of the centralizers may be obtained and also the span distance between any two centralizers.
Minimum rod diameter(77176 bytes)
Because of the severe force conditions, usually the minimum span distance (Smin) is at the bottom of the rod string.
The Smin is influenced by the well bore structure, tapered rod string, pumping parameters, and fluid viscosity. Conversely, if we know a set of well conditions, pumping parameters, and Smin, the tapered rod string can be determined. That is, we can obtain the relation between Smin and these factors and thus determine the minimum rod diameter.
Because of the string characteristics in a directional well, no rigorous, theoretical analytic solution can be obtained, but a statistical relation can be used to predict the minimum rod diameter.
The parameters in Table 1(16404 bytes) affect Smin, therefore, many possible combinations can be obtained by an orthogonal design method. In these cases, Smin is obtained according to the continuous beam column, then, a statistical relation is determined by regression analysis (Equation 7). The multiple correlation factor is equal to 0.847.
In addition, the length of a rod must meet the specification and engineering requirement. In fact, h, R, dt, are based by well conditions. If S, N, and Smin are determined, minimum rod diameter can be obtained from Equation 8. The minimum rod diameter is the dr in Equation 7.
Data from the three wells in Liaho oil field in China verify the reliability of the statistical model. The general well parameters and calculated minimum rod diameters are listed in Table 2(35231 bytes).
From Table 2(35231 bytes), that minimum rod diameter determined by the statistical model is very close to that obtained from the continuous beam column model. Therefore, the minimum rod diameter predicted by the statistical model has good reliability, and may be used to estimate the minimum diameter of a tapered rod string. The use of this minimum rod diameter as an initial value will greatly decrease the design time of a tapered rod string in a directional well.
References
- Recommended Practice for Design Calculation for Sucker-rod Pumping Units, API RP 11L, February 1977.
- Hu Yongquan, Yuan Xiangzhong, Duan Yuting, "Design of Sucker Rod String for Directional Well," China Petroleum Machinery, Vol. 22, No. 10, October 1994.
- Wang Hongxun, Zhang Qi, The Principle of Oil Well Production (revised version), Petroleum Industry Publishing House, Beijing, July 1989.
- Recommended Practice for Care and Handling of Sucker Rods, 7th Edition, API-RP 11BR, 1986.
The Authors
Hu Yongquan is a lecturer at the Southwest Petroleum Institute, Nanchong, China. He is involved in conducting research on production, stimulation, and completions. Hu Yongquan graduated from Southwest Petroleum Institute (SWPI) in 1985 with a BS. He received an MS in petroleum engineering in 1988.
Yuan Xiangzhong is a professor of mechanics and the dean of the Southwest Petroleum Institute. His research interests include applied mechanics and petroleum engineering. He is a graduate of the civil engineering university of Chongching in Sichuan, China, and he majored in engineering structures.
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