Fundamental Equations for Dynamic Analysis of Rod and Pipe String in Oil-Gas Wells and Application in Static Buckling Analysis
Authors: Li Z. , Li J. 
Published: March 2010

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 the motions of these strings, and establishes the fundamental equations for the dynamic analysis of these strings in oil and gas wells. The fundamental equations have successfully unified all of the differential equations used in the dynamic analysis of rod and pipe strings in oil and gas wells. All of the differential equations present used in the calculation of tension and torque in directional wells, analysis of drillstring vibration, analysis of bottom hole assembly (BHA), analysis of the buckling behaviour of pipe and casing strings, and analysis of sucker rod pumping systems can be obtained just by proper simplification of the fundamental equations. Thus, these equations have broad uses in drilling and production.
The buckling of coiled tubing string, drillstring and casing etc., plays an important role in oil drilling and production operations. As a very important application of the fundamental equations, this paper also establishes the differential equations for the buckling of the strings in inclined straight holes. The linear and non-linear mathematical models for buckling of these strings in horizontal sections have been established. The critical forces for helical and sinusoidal buckling in inclined straight holes have been found. The linear and non-linear buckling of the weightless strings has been analysed. Other applications of the fundamental equations in drilling and production have been presented.

Introduction
Since the pioneering work by Lubinski et al.,(1) 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 analysis(1–20). However, no fundamental equations that unify these differential equations have been established. As an attempt, this paper establishes the fundamental equations for the dynamic analysis of drillstring, BHA, casing, tubing, pumping rod, etc. in oil and gas wells.
Lubinski et al.(1) defined the helical buckling behaviour of pipes in vertical wells. However, the post-buckling behaviour of pipe in inclined and horizontal wells is different from that in nearby vertical wells. Knowledge of the configuration of buckled tubular is important to prevent costly failures, to predict whether a tool can be forced along a directional or horizontal well, and to optimize the tubular string.
As an important application of the fundamental equations, this paper also establishes the differential equations for the buckling of the string in inclined straight holes. The linear and non-linear mathematical models for buckling of these strings in horizontal sections have been established. The critical forces for helical and sinusoidal buckling in inclined straight holes have been found. The linear and non-linear bucking of the weightless strings has been analysed.

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

Motions of Strings in Oil and Gas Wells