Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 535830, 13 pages

http://dx.doi.org/10.1155/2015/535830

## New Analysis Theory and Method for Drag and Torque Based on Full-Hole System Dynamics in Highly Deviated Well

^{1}College of Mechatronic Engineering, Southwest Petroleum University, Chengdu 610500, China^{2}PetroChina Jidong Oilfield Company, Tangshan 063004, China

Received 11 September 2014; Accepted 15 December 2014

Academic Editor: Chenfeng Li

Copyright © 2015 Xiao-hua Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The research on calculation of torque and drag in highly deviated wells has demonstrated a significant gap against oil exploration and development; with the increasingly rigorous situation, the drill string dynamics and the contact or friction of drill pipe with borehole wall under the drill string action of dynamic need more attention and urgent research work. Based on full-hole system dynamics, three-dimensional nonlinear dynamic model and dynamic torque and drag model were established in highly deviated well by using the finite element method. An application of analyzing typical torque and drag problems presented here provides a means to more accurate description of the contact relation between drill string and wellbore. The results show that those models established in this paper have complete adaptability for a complex three-dimensional borehole trajectory. For the actual well application, it will help to evaluate security performance of drill string in complex working conditions.

#### 1. Introduction

With scarcity of resources, great difficulties have been confronted with oil companies during the process of exploration drilling of oil or gas and the complexity of drilling geological environment is now bottlenecking the development of drilling technology. Due to the geographical conditions, most oil and gas wellbore trajectories have to be catenary curve which is used to substitute for straight line; well profiles have to adopt highly deviated well [1]. In recent years, the reach of highly deviated wells has increased; it provides a larger producing low porosity and low permeability reservoir. With highly deviated well drilling, the construction is quite difficult which is often caused by designed path and has a great deviation angle. Frequent contact-rubbing between bending drill string and borehole wall has developed great friction; it put forward the higher requirements of drilling equipment. At the same time, it influences the transmission of drilling string axial load; its major expression is as follows: this substantially increases the drag on the borehole walls; control of the well trajectory becomes more difficult, reducing weight and torque which are applied to the bit; then drilling rate always is decreased. Cuttings bed is formed easily in long open hole interval, which is not obvious under normal inspection and can be difficult to clear up; sometimes there are failure accidents of the drilling equipment in this situation. It is difficult to guarantee the cementing quantity because it may not easily make the casing centralized in wellbore and runs casing in.

In brief, the above-mentioned problems commonly exist in highly deviated wells; on the basis of torque and drag modeling originally started by the works of Johancsik et al. [2], researchers at home and abroad have been improving theoretical basis provided for analyzing and forecasting the frictional resistance and excellent effects are obtained. In 1984, Johancsik et al. [2] assumed that both torque and drag are caused entirely by sliding friction forces that result from contact of the drill string with the wellbore. They then define the sliding friction force to be a function of the normal contact force and the coefficient of friction between the contact surfaces based on Coulomb’s friction model. They wrote the force balance for an element of the pipe considering that the normal component of the tensile force was acting on the element contributing to the normal force. The relationship between the helical buckling of a drill string and torque is discussed by He and Kyllingstad [3]; Maidla and Wojtanowicz [4] presented a method to evaluate an overall friction coefficient between the wellbore and the casing string, assuming that a friction coefficient matches field data and modeling; the equation for predicting surface hook loads is derived from the respective governing differential equations. Schamp et al. [5] suggested some industrial methods to reduce torque in the wellbore during drilling. Mason and Chen [6] pointed out that the change of borehole curvature would affect torque and drag values on a large scale. In 1993, two-dimensional model and three-dimensional model for calculating the friction 8 analyses were established by professor Han [7]. Gao [8, 9] put forward a new calculation model by applying weighted residual method. Shuai et al. [10] analyzed drag and torque by finite element method, and then, on the drilling string system dynamics, professor Zhu [11–13] established a new method of calculation of torque and drag.

Overall, the research on calculation of torque and drag should be deeper and wider in highly deviated wells, resulting in a lot of errors in the torque and drag parameters computed, which has demonstrated a significant gap against oil exploration and development with the increasingly rigorous situation, so the drill string dynamics and the contact or friction of drill pipe with borehole wall under the drill string action of dynamic need more attention and urgent research work, its purposes are to study the drill string vibration rules and the reasons of drill string failure in highly deviated wells, especially to understand the law of torque and drag and promote the development of new drilling technologies in special technology. Therefore, this paper established the models for vertical, lateral, and torsion coupled vibration of full-hole drilling strings in highly deviated well with mud drilling using the finite element method to dynamically analyze torque and drag of string drilling in operation.

#### 2. Model Description

The new model has been developed based on full-hole system dynamics, in which the finite element method is used to model the drill string used in rotational drilling operations. The ABAQUS FEM Explicit solver package was used to develop the dynamic FEM model [14]. In this section, more equations of full-hole system dynamics would be present, but the formulation of equations of motion proceeds along the same lines of the model in [15], So final forms of equations are included here for completeness, without many details.

##### 2.1. Hypotheses

Before establishing the model, the following main assumptions would be needed with synthesizing structure characteristics of full-hole drilling strings, boundary constrains, and load conditions, based on the theory of nonlinear transient:(1)the drill string is simplified to homogeneous beam; cross section shape is circle, ignoring threaded connections between the drill string and partially perforated structures; the geometric size of the drill string and material properties of the drill string can be segmented or grouped depending on drilling tools assembly without considering the impact of temperature;(2)keep the rotation speed constant at the top of the drill string;(3)rock fragmentation process in real time is used for the boundary condition of the end of the drill string, because strength and hardness of bit is greater than that of bottom rock, so set bit to a rigid body in the analysis; rock is assumed to be an isotropic material and rock nonlinearity is simulated by DP elastic-plastic model; simply using finite element method removes the failure elements from rock elements and ignores its influence on subsequent drilling;(4)borehole wall has large rigidity; its cross section is usually in a circular form; its axis is a smooth curve in three-dimensional space with continuous second order derivatives which was made as interpolation to the measured well inclination data.

##### 2.2. Dynamic Finite Element Model of the Drilling

Accordingly, one needs to write the kinetic and strain energy expressions. In this context, the kinetic energy can be expressed in the form as [15] where is the augmented mass matrix with the constituent matrices given by which is the translational mass matrix, is the rotary inertia mass matrix, is the torsional mass matrix, and is the torsional-transverse inertia coupling mass matrix, which is time dependent. The matrix is the damping matrix, and is the constant angular velocity of the rotary table. In this formulation, is the vector of nodal coordinates of the two-node finite string element.

The total strain energy can be written in compact matrix form as where is the augmented stiffness matrix.

Based on the Hamilton theory [16], by utilizing the above energy expressions into the vibrational form of Lagrange’s equation, full well system dynamics equation is obtained as where is the mass matrix, is the viscous term, considering damping, sticky plastic and viscoelastic effect, is the damping matrix, is the stiffness matrix, is the function of external excitation, and , , are the node displacement, velocity, and acceleration vector.

The following boundary conditions are considered. As a result of the limitation of wheel to drill string in well hole, wellhead node lateral displacement and lateral rotation angle are zero, correcting wellhead node generalized displacement in real time, keeping lateral force the same. Twisting vibration is thought to exist when dealing with the drill string twist boundary, but only mainly in lower parts of drill string. For the top node of the drill string, because of the constant power drilling rig, we assume that rotating speed of the top node is constant, but its torque is fluctuating. Axial displacement of the top node is not fixed, and a lift force is joined on it to simulate the hook load:

Based on the interaction of bit and formation, the force from the rock and bit is used for the boundary condition of the end of the drill string. Considering the coupled relations between the drill string and the bit, both of them are linked into a whole unit to make rock failure and drilling hole without constraints of any degrees of freedom. The formation of the hole is relative to the lower longitudinal vibration of drill string and interaction between bit and formation. The drill bit is a direct tool for rock breakage; the longitudinal impact allows it into the bottom hole rock layer and small pieces of the rock that deform and break away due to the lateral scraping action of the bit teeth. Then cuttings are washed from the bottom by drilling fluid.

##### 2.3. Contact Algorithm

Rotating drill string movement along the cross section of the wellbore is hedged about with wellbore wall, when the outer wall of the drill string is closing to borehole wall, and the drill string has a trend of move outwards; then the drill string will collide with the wall with rubbing contact. The contact and friction of a drill pipe string with borehole wall have become the basic research of the drill string dynamics and drag and torque calculation, whose effect on the dynamic characteristics of drill string cannot be ignored, and it is difficult to analyze.

The contact between drill string and borehole wall in the process of highly deviated well drilling is of large area and geometrically nonlinear with unpredictability; this can be seen from the following: inability to forecast the points of contact between borehole wall and drill string, influenced by torsional vibration and axial vibration of drill string, and the value of the impact force and the contact time along the contact position. Even with using the same drilling assembly, the contact position and the value of the contact force are also not the same under different real well trajectory or drilling conditions. So this paper introduces stochastic boundary method for prediction of contact stress.

The annular tolerance between drill string and borehole wall can be used to judge whether the drill string and borehole wall contact in a calculation. Drill string is in a state of freedom of movement without contacting the borehole wall; there is no constraint reaction force. When the drill string connects with borehole wall as shown in Figure 1, the motion of drill string is handicapped in its operation providing a certain amount of normal force , the axial frictional force , the tangent friction force , and the friction torque at the contact interface. After the drill string exerts pressure into the wall, we suppose the contact makes a circular profile of radius on a side; the approaching distance is . The formulae of contact force and other parameters are deduced by Hertzian contact theory and logical-spring-damper theory [15]: