MS-Fundamental Equations for Dynamical Analysis of Rod and Pipe String in Oil and Gas Wells

SPE 56044 Fundamental Equations for Dynamical Analysis of Rod and Gas Wells
Zifend Li, China University of Geoscience, PRC
Copyright 1999, Society of Petroleum Engineers This manuscript was provided to the Society of Petroleum Engineers for distribution and possible publication in an SPE Journal. The contents of this paper (1) are subject to correction by the author(s) and (2) have not undergone SPE peer review for technical accuracy. Thus, SPE makes no claim about the contents of the work. Permission to copy or use is restricted to an abstract of not more than 300 words. Write SPE Librarian, P.O. Box 833836, Richardson, TX 75083-3836 USA– Facsimile: 972-952-9435 Email: papers@
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Abstract A large number of string materials, such as drillstring, pumping rod, tubing, coiled tubing, casing, etc., are used in oil and gas drilling and production. Proper mechanical analyses of these strings are important. This paper analyses motions of these strings, establishes fundamental equations for dynamical analysis of these strings in oil and gas wells. These fundamental equations have successfully unified all the differential equations used in dynamical analysis of rod and pipe string in oil and gas wells. All the differential equations present used in calculation of tension and torque in directional wells, analysis of drillstring vibration, analysis of bottom hole assembly (BHA), analysis of buckling behavior of pipe and casing strings and analysis of sucker rod pumping system can be obtained just by proper simplifying these fundamental equations. Thus, these equations have broad uses in drilling and production. Applications of these fundamental equations in drilling and production have been presented. Introduction Since the pioneering work by Lubinski et al1, the drilling and production industry has gradually come to accept and appreciate the importance of analysis of BHA, drillstring, casing, tubing, pumping rod, etc. A large number of differential equations have been established for special analysis1-17 . But no fundamental equations that unify these differential equations have been established. As an attempt, this paper establishes the fundamental equations for dynamical analysis of drillstring, BHA, casing, tubing, pumping rod, etc. in oil and gas wells. Motions of Strings in Oil and Gas Wells Elastic strings in oil and gas wells include drillstring, tubing string, coiled tubing, casing string, pumping rod string, etc. Each string has its own motion state (Table 1). The drillstring has the most complex motional state, which includes rotation,
whirling motion, longitudinal motion, longitudinal vibration, torsional vibration, lateral vibration. Fundamental Equations Hypotheses. (1)The rod and pipe strings behaves as linear elastic bodies; (2)The rod and pipe strings have annular sections and arbitrary properties that remain constant in a segment; (3)The effect of shear stress on the deformation may be ignored. Coordinate Systems. Two separate right-handed coordinate systems may be employed (Fig.1). (1) Fixed global coordinate system ONED with fixed base i, j, k. The ONED system is fixed with respect to compass directions, i.e., original point O at wellhead, N directs north, E directs east, D directs vertical down. (2) Natural coordinate systems (et , en, eb). This is defined by the rod or pipe string centerline trajectory, where vector r represents a point in the centerline, et is tangent base vector, en is normal base vector, eb is binomial base vector. Geometry Equations. If the centerline of the rod or pipe is r =r (l,t), where l is the arc length measured from a starting point before deformation, t is the time, and s= s (l, t) is the arc length measured from the starting point after deformation, the Frenet-Serren formulas is: ∂r et = ∂s ∂e t = kb en ∂s ......................................(1) ∂e n = kn eb − kbet ∂s ∂e b = −k n e n ∂s where, kb is curvature, kn is torsion: