# Tripability Analysis of Casing Strings in Directional Wells Using the Continuous Beam-Column and Buckling Theory

Authors: Gang Wang; Gang Liu; Kai Wang; Yichen Li; Yinan Hu
Journal: Geofluids
 (2022)

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## Abstract

Hindered casing strings are often encountered in unconventional oil and gas exploration during the casing running process. This not only increases the operating costs and time but can also lead to downhole accidents and even abandonment in serious cases. Due to various assumptions, the calculation results of the existing soft models and hard models are different, which causes confusion for field operators when taking friction reduction measures. Moreover, a lowering force is often applied to assist hindered casing string running in a drilling field. However, its application is mainly based on work experience and lacks mechanistic analysis and theoretical guidance. Thus, in this study, a simulation model for the analysis of casing string tripability in a directional well was established and the model was combined with the continuous beam-column theory and buckling theory. The model was used to study how various factors including the friction coefficient, drilling fluid density, and casing diameter could affect the lowering force required when a casing string was hindered by buckling. The results showed that the maximum lowering force and the maximum effective lowering force decreased with the increase in the friction coefficient and the performance of the drilling fluid could be adjusted rapidly, which would be beneficial for ensuring that the casing string could be tripped smoothly by applying a lowering force. The increase in the drilling fluid density caused the maximum lowering force and the maximum effective lowering force to decrease, which was not conducive to hindered casing string running. The larger the casing diameter was, the greater the maximum lowering force and the maximum effective lowering force were. It was more convenient to apply a lowering force for a casing with a large diameter. In addition, the improved model could identify whether the casing string was in contact with the upper or lower borehole walls. Through finite element method verification, the prediction was in line with the actual casing running operation and the improved model has the smallest prediction error, i.e., 6.58%, compared with the existing models. Therefore, the improved model might provide necessary theoretical guidance for casing running operations in directional wells.


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## Body

## 1. Introduction

Unconventional oil and gas exploration and development have entered an active period. Directional well technologies such as horizontal wells, extended reach wells, and cluster wells have been widely applied in unconventional oil and gas exploration. Casing strings cannot be tripped at the target depth due to the influence of the formation lithology, well trajectory, and casing running technology. Once the casing string running is hindered, it increases the operating cost and time and can lead to downhole accidents and even abandonment in serious cases [1]. Therefore, to ensure that the casing string reaches the target depth smoothly, it is necessary to predict the friction and applied lowering force.At present, several models have been established for calculating the drag and torque of a tubular string. The soft-string torque and drag model was initially developed by Johancsik et al. [2] and later put in a standard differential form by Sheppard et al. [3]. This model assumed that the string was in continuous contact with the borehole wall and did not consider the string stiffness [4–7]. It is generally believed that the soft string model sometimes provides poor results for stiff tubular strings, high dogleg severity (small radius of curvature), or narrow radial clearance in the annulus [8].Because the soft-string model does not consider the influence of the string stiffness, Ho. [9] established a stiff-string model for drag and torque based on the theory of large deformation for a drill string. Mitchell et al. [10] established a differential form that was easy to understand. However, many factors, including the variable stiffness of the tubular strings, the shape of the wellbore, and the clearance, can influence the contact state of the tubular strings and the value of the contact force on the tubular strings. The present stiff string models do not consider these factors, making it difficult to predict accurate results [11–13].To improve the agreement between the calculation results and the real situation of a casing string running in a wellbore, 3D finite element models have been established to analyze the friction, stress, and deformation of the string in the build-up section and the horizontal section during the tubular string running process [14–20]. The well depth is generally several thousand meters. With the increase of the size of the 3D model, it is necessary to take a long time to complete calculations. The results are difficult to apply in real time in the field; it is often used to provide a reference for the optimal design of the wellbore trajectory and tubular string assembly and verify the accuracy of the new model for prediction drag and torque [21].The continuous beam-column theory has been increasingly recognized because of its clear physical meaning and relatively simple algorithm [22–24]. When calculating the contact force between the string and the borehole wall in the traditional continuous beam-column theory, the contact point between the string and the borehole wall is first specified but one must generally be very cautious when specifying the contact point. As a result, some real contact points are missed, and in order to find these contact points, it is necessary to judge whether the deflection of the string between the two contact points exceeds the wellbore clearance. However, the traditional model does not consider the effect of a curved borehole on the deflection of the casing string, which cannot be ignored in the curved section.Therefore, to identify the contact point between the pipe string and the well wall and ensure that the casing string runs smoothly in the directional well, the influence of the initial bending of the casing string on the deflection and deformation of the casing string was considered in this study using the improved continuous beam-column model. In addition, based on buckling theory, a prediction model of the maximum effective lowering force required during hindered casing string running was established. Using the existing friction model for tubular strings and field data verification, the model prediction results were proven to match the actual field data well.


## 2. Casing String Tripability Analysis Model

A model was established to predict whether a casing string could be run smoothly during the casing running process in a directional well. The primary reason that casing strings cannot be tripped in is the friction force. When the total friction of casing strings is greater than the floating weight of the casing strings above the stuck point, the casing strings are hindered during the casing running process. The friction force is generated by the contact between the casing strings and the wellbore wall. Therefore, the contact force is required to be analyzed first.


### 2.1. Basic Assumptions

(a)
The string, which was composed of a casing, coupling, and centralizer, was regarded as an elastic beam column, and its deformation was within the linear elastic range(b)
The wellbore had a uniform circular cross-section without enlargement or hole shrinkage(c)
The coupling, centralizers, and other parts with relatively large diameters on the string or specified points of a certain length were taken as supports that were in contact with the borehole wall(d)
The casing string rotation and vibration were neglected


### 2.2. Three-Dimensional Wellbore Trajectory Plane Decomposition

The three-dimensional wellbore could be approximately represented in two dimensions by decomposing it into two planes, planeP and plane Q [25], as shown in Figure 1. The lateral force and deflection of the string were calculated on the two planes separately. Then, the lateral force and deflection of the string in space could be obtained after the corresponding superposition.Figure 1
Three-dimensional plane decomposition of borehole trajectory.


### 2.3. Establishing a Bottom-Hole Coordinate System

Based on the unit vector of the three coordinate axes in the bottom-hole coordinate system [26], the coordinate transformation matrix M of the vector from the wellhead coordinate system to the bottom-hole coordinate system could be obtained as follows:
(1)M=cosabcosϕbcosabsinϕb−sinab−sinϕbcosϕb0−sinabcosϕb−sinabsinϕb−cosab,where ab and ϕb are the well inclination and the azimuth at the casing shoe, respectively. The coordinate transformation was as follows:
(2)xyz=M⋅N−NbE−EbH−Hb,where Hb, Nb, and Eb are the H, N, and E coordinates of the casing shoe, respectively. H, N, and E are the coordinates of any point in the wellhead coordinate system, and x, y, and z are the x, y, and z coordinates of the corresponding points in the bottom-hole coordinate system, respectively.


### 2.4. Continuous Beam-Column Model

#### 2.4.1. Three-Moment Equations in PlaneP

With the casing string runs in planeP (well inclination plane) as an example, the casing string combination of any number (n) of centralizers was taken as the analysis object. As shown in Figure 2, the centralizers divided the whole casing string into N span beams and the transverse load qi, bending moments Mi–1 and Mi, and the axial force Fi acted on the ith span beam. The right deflection angle of the ith span beam was θiR, the shear force was QiR, the transverse load was qi+1, the bending moments were Mi and Mi+1, and the axial force Fi+1 acted on the i+1th span beam. The right rotation angle of the i+1th span beam was θi+1L, and the shear force was Qi+1L.Figure 2
Force and deformation of casing string running in a curved section. (a) Schematic diagram of casing string running in a curved section. (b) Force and deformation of theith span beam. (c) Force and deformation of the i+1th span beam.
(a)(b)(c)The deflection curves of the two adjacent spans had a common tangent line at the support of the string. Therefore, the rotation angles on both sides of the support were equal:(3)θi,PR=−θi+1,PL.The rotation angle at the support could be obtained from the formula of the end rotation angle and the deformation superposition principle of continuous beam-column theory (AppendixA):
(4)θi,PR=qi,PLi324EIiXui,P+Mi,PLi3EIiYui,P+Mi−1,PLi6EIiZui,P,θi+1,PL=qi+1,PLi+1324EIi+1Xui+1,P+Mi+1,PLi+13EIi+1Yui+1,P+Mi+1,PLi+16EIi+1Zui+1,P,θn+1,PR=qn+1,PLn+1324EIn+1Xun+1,P+Mn+1,PLn+13EIn+1Yun+1,P+Mn+1,PLn+16EIn+1Zun+1,P,where Li is the length of the ith span beam (m), qi,p is the component of the transverse load of the ith span beam in the P plane (N), ai is the well inclination angle at the ith support (°), wi is the weight per meter of the ith span beam in mud (kg/m), Fi is the average axial force on the ith span beam (N), E is the elastic modulus of the string (Pa), and Ii is the section moment of inertia of the ith span beam (m4).


#### 2.4.2. Initial Rotation Angle

For the actual string combination, the diameter of the centralizer was smaller than the diameter of the borehole. Therefore, when the string was tripped in the borehole, the centralizer might have been close to a side of the wellbore wall, which would have caused the displacement of the support due to the elimination of clearance. Therefore, several supports were not in a straight line. This resulted in an additional rotation angle at the end of the beam column due to the displacement of the supports, as shown in Figure3.Figure 3
Schematic diagram of beam supports not in a straight line under the combined force.In planeP, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(5)δθi,P0=arctanδxi0−δxi−10zi−zi−1.In planeQ, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(6)δθi,Q0=arctanδyi0−δyi−10zi−zi−1.


#### 2.4.3. Three-Moment Equations

Considering the initial additional rotation angle generated by the displacement of the support, the corresponding three-moment equation of then+1th span continuous beam-column in the plane P could be obtained:
(7)Mi−1,PLiZui,P6EIi+Mi,PLiYui,P3EIi+Li+1Yui+1,P3EIi+1+Mi+1,PLi+1Zui+1,P6EIi+1=−qi,PLi3Xui,P24EIi−qi+1,PLi+13Xui+1,P24EIi+1+δθi+1,P0−δθi,P0,qn+1,PXun+1,PLn+1324EIn+1+Mn+1,PYun+1,PLn+13EIn+1+Mn,PZun+1,P6EIn+1Ln+1=θn+1,PR−δθn,P0.The boundary conditions at the casing shoe and the upper tangent point in planeP were
(8)M0,P=0,Mn,P=E⋅In⋅kaT,where kaT is the rate of well inclination change at the tangent point.There weren+1 equations for the three bending moments, in which the n+1 unknowns were M1~Mn and Ln+1. Thus, there was a definite solution. For the ith beam support in plane Q, the left and right sides were the ith and i+1th span beams, respectively. Based on the continuous condition that the rotation angle of the beams on both sides of the support were equal, the three bending moment equations in plane Q were obtained.
(9)Mi−1,QLiZui,Q6EIi+Mi,QLiYui,Q3EIi+Li+1Yui+1,Q3EIi+1+Mi+1,QLi+1Zui+1,Q6EIi+1=δθi+1,Q0−δθi,Q0,Mn+1,QYun+1,QLn+13EIn+1+Mn,QZun+1,Q6EIn+1Ln+1=θn+1,QR−δθn,Q0,where the meanings of the parameters are the same as those above discussed.The boundary conditions at the casing shoe and the upper tangent point in plane Q were as follows:
(10)M0,Q=0,Mn,Q=E⋅In⋅kϕTsinaT,where kϕT is the rate of azimuth change at the tangent point and aT is the well inclination at the tangent point.


#### 2.4.4. Contact Force

The relationship between the force and deformation of the string is given. An iterative method could be used to solve the deflection function of each span to obtain the deflection, rotation angle, and bending moment of the string after deformation. Combined with Figures2(b) and 2(c), the contact force at each support in plane P could be calculated:
(11)Ni,P=Mi−1,P−Mi,PLi+qi,pLi2+Mi+1,P−Mi,PLi+1+qi+1,pLi+12,i=1ton−1.Then, the contact force at each support in planeQ was
(12)Ni,Q=Mi−1,Q−Mi,QLi+Mi+1,Q−Mi,QLi+1,i=1ton−1.


### 2.5. Effect of Initial Bending on Casing Deflection

When the casing string was tripped in the curved section of a wellbore, the deformation of the entire casing string had to be generally consistent with the wellbore axis. Therefore, the maximum deflection of the casing string caused by the initial bending could be obtained [27] (Appendix B):
(13)xi,max=xi,0+xi,1=2ζi,Pui21cosui−1=Ki,PEIiFi1cosui−1.


### 2.6. Judgment of Contact Points

Considering the influence of the borehole bending on the casing string deformation, the improved maximum deflectionxi,m in plane P and the improved maximum deflection yi,m in plane Q of the ith span string were calculated as follows:
(14)xi,m=qi,pΔLi416EIiui41cosui−1−ui22+Mi−1,p+Mi,p2FicosΔαi/21−1cosui+KPLi24ui21cosui−1+ei−1,P+ei,P2,yi,m=Mi−1,Q+Mi,Q2FicosΔϕi/21−1cosui+KQLi24ui21cosui−1+ei−1,Q+ei,Q2.Therefore, the criterion for generating the new contact points was(15)xi,m2+yi,m2>Dw−D0i2,where KQ is the curvature on the azimuth plane of the well section where the string is located (°/30 m), Dw is the borehole diameter (m), Dci is the centralizer diameter (m), D0i is the casing diameter (m), ei−1=Dw−Dci−1/2, and ei=Dw−Dci/2.Once the maximum deflection of the casing string between the two supports exceeded the wellbore clearance, there was a new contact point and new support needed to be added to perform the recalculation.


### 2.7. Tripability Judgment of the Casing String

The contact force and the friction at each contact point could be obtained as follows:(16)Ni=Ni,p2+Ni,Q2,Ffi=μNi,where μ is the friction coefficient.When the total friction of the casing string was greater than the floating weight of the casing string above the stuck point, the casing string could be considered hindered during the casing running process:
(17)∑Ffi>wi⋅Ls,where Ls is the length of the casing string above the stuck point (m).At that time, it was difficult for the casing string to run with its own floating weight and it was possible to run the casing string with the help of the other processes. For example, a lowering force, a pick-up and release technique, and casing floating could be applied. Since the casing string buckled during the running process, it was also necessary to consider the case in which the casing string was hindered by buckling.


### 2.8. Critical Load for Helical Buckling of the String

Many scholars have obtained the critical load when a string is subjected to spiral buckling in different well sections through theoretical derivations or experiments, as shown in Table1. Additional friction should be considered when a casing string is spirally buckling and in contact with a borehole wall to determine whether the casing string can be successfully tripped in.Table 1
Critical load for the helical buckling of the string in different well sections, whereFhel is the critical load of helical buckling (N), r is the borehole annulus clearance (m), and R is the string radius (m).
Well sectionCritical load for helical bucklingVerticalFhel=5.55EIw23Straight inclinedFhel=2.83EIwsina/rCurvedFhel=12EI/rR1+1+rR2wsina/8EIHorizontalFhel=222−1EIw/r


### 2.9. Determine the Maximum Lowering Force

Due to the restrictions of a wellbore, a string will undergo multiple deformation processes with the increase in the lowering force after string buckling. With the combined action of the string weight and the buoyancy of the drilling fluid, the string buckling becomes a spatial spiral shape with a varying pitch. The bending of the string causes contact with the borehole wall, resulting in increased friction, which offsets some of the lowering force and reduces the effective lowering force on the stuck point [28]. Therefore, how to determine the maximum lowering force when applying a load to push down the casing is very important for casing running.The maximum lowering forceFmax can be obtained as follows (Appendix C):
(18)Fmax=wmEI3δμr.The maximum effective lowering forceFemax can be obtained as follows:
(19)Femax=Fmax−δμrwmEIFmax3.


### 2.10. Solution Method

An iterative method [23, 24] was used to apply the simulation model in actual wellbores. The flowchart of the procedure is shown in Figure 4.Figure 4
Flowchart of the solution procedure.


## 2.1. Basic Assumptions

(a)
The string, which was composed of a casing, coupling, and centralizer, was regarded as an elastic beam column, and its deformation was within the linear elastic range(b)
The wellbore had a uniform circular cross-section without enlargement or hole shrinkage(c)
The coupling, centralizers, and other parts with relatively large diameters on the string or specified points of a certain length were taken as supports that were in contact with the borehole wall(d)
The casing string rotation and vibration were neglected


## 2.2. Three-Dimensional Wellbore Trajectory Plane Decomposition

The three-dimensional wellbore could be approximately represented in two dimensions by decomposing it into two planes, planeP and plane Q [25], as shown in Figure 1. The lateral force and deflection of the string were calculated on the two planes separately. Then, the lateral force and deflection of the string in space could be obtained after the corresponding superposition.Figure 1
Three-dimensional plane decomposition of borehole trajectory.


## 2.3. Establishing a Bottom-Hole Coordinate System

Based on the unit vector of the three coordinate axes in the bottom-hole coordinate system [26], the coordinate transformation matrix M of the vector from the wellhead coordinate system to the bottom-hole coordinate system could be obtained as follows:
(1)M=cosabcosϕbcosabsinϕb−sinab−sinϕbcosϕb0−sinabcosϕb−sinabsinϕb−cosab,where ab and ϕb are the well inclination and the azimuth at the casing shoe, respectively. The coordinate transformation was as follows:
(2)xyz=M⋅N−NbE−EbH−Hb,where Hb, Nb, and Eb are the H, N, and E coordinates of the casing shoe, respectively. H, N, and E are the coordinates of any point in the wellhead coordinate system, and x, y, and z are the x, y, and z coordinates of the corresponding points in the bottom-hole coordinate system, respectively.


## 2.4. Continuous Beam-Column Model

### 2.4.1. Three-Moment Equations in PlaneP

With the casing string runs in planeP (well inclination plane) as an example, the casing string combination of any number (n) of centralizers was taken as the analysis object. As shown in Figure 2, the centralizers divided the whole casing string into N span beams and the transverse load qi, bending moments Mi–1 and Mi, and the axial force Fi acted on the ith span beam. The right deflection angle of the ith span beam was θiR, the shear force was QiR, the transverse load was qi+1, the bending moments were Mi and Mi+1, and the axial force Fi+1 acted on the i+1th span beam. The right rotation angle of the i+1th span beam was θi+1L, and the shear force was Qi+1L.Figure 2
Force and deformation of casing string running in a curved section. (a) Schematic diagram of casing string running in a curved section. (b) Force and deformation of theith span beam. (c) Force and deformation of the i+1th span beam.
(a)(b)(c)The deflection curves of the two adjacent spans had a common tangent line at the support of the string. Therefore, the rotation angles on both sides of the support were equal:(3)θi,PR=−θi+1,PL.The rotation angle at the support could be obtained from the formula of the end rotation angle and the deformation superposition principle of continuous beam-column theory (AppendixA):
(4)θi,PR=qi,PLi324EIiXui,P+Mi,PLi3EIiYui,P+Mi−1,PLi6EIiZui,P,θi+1,PL=qi+1,PLi+1324EIi+1Xui+1,P+Mi+1,PLi+13EIi+1Yui+1,P+Mi+1,PLi+16EIi+1Zui+1,P,θn+1,PR=qn+1,PLn+1324EIn+1Xun+1,P+Mn+1,PLn+13EIn+1Yun+1,P+Mn+1,PLn+16EIn+1Zun+1,P,where Li is the length of the ith span beam (m), qi,p is the component of the transverse load of the ith span beam in the P plane (N), ai is the well inclination angle at the ith support (°), wi is the weight per meter of the ith span beam in mud (kg/m), Fi is the average axial force on the ith span beam (N), E is the elastic modulus of the string (Pa), and Ii is the section moment of inertia of the ith span beam (m4).


### 2.4.2. Initial Rotation Angle

For the actual string combination, the diameter of the centralizer was smaller than the diameter of the borehole. Therefore, when the string was tripped in the borehole, the centralizer might have been close to a side of the wellbore wall, which would have caused the displacement of the support due to the elimination of clearance. Therefore, several supports were not in a straight line. This resulted in an additional rotation angle at the end of the beam column due to the displacement of the supports, as shown in Figure3.Figure 3
Schematic diagram of beam supports not in a straight line under the combined force.In planeP, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(5)δθi,P0=arctanδxi0−δxi−10zi−zi−1.In planeQ, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(6)δθi,Q0=arctanδyi0−δyi−10zi−zi−1.


### 2.4.3. Three-Moment Equations

Considering the initial additional rotation angle generated by the displacement of the support, the corresponding three-moment equation of then+1th span continuous beam-column in the plane P could be obtained:
(7)Mi−1,PLiZui,P6EIi+Mi,PLiYui,P3EIi+Li+1Yui+1,P3EIi+1+Mi+1,PLi+1Zui+1,P6EIi+1=−qi,PLi3Xui,P24EIi−qi+1,PLi+13Xui+1,P24EIi+1+δθi+1,P0−δθi,P0,qn+1,PXun+1,PLn+1324EIn+1+Mn+1,PYun+1,PLn+13EIn+1+Mn,PZun+1,P6EIn+1Ln+1=θn+1,PR−δθn,P0.The boundary conditions at the casing shoe and the upper tangent point in planeP were
(8)M0,P=0,Mn,P=E⋅In⋅kaT,where kaT is the rate of well inclination change at the tangent point.There weren+1 equations for the three bending moments, in which the n+1 unknowns were M1~Mn and Ln+1. Thus, there was a definite solution. For the ith beam support in plane Q, the left and right sides were the ith and i+1th span beams, respectively. Based on the continuous condition that the rotation angle of the beams on both sides of the support were equal, the three bending moment equations in plane Q were obtained.
(9)Mi−1,QLiZui,Q6EIi+Mi,QLiYui,Q3EIi+Li+1Yui+1,Q3EIi+1+Mi+1,QLi+1Zui+1,Q6EIi+1=δθi+1,Q0−δθi,Q0,Mn+1,QYun+1,QLn+13EIn+1+Mn,QZun+1,Q6EIn+1Ln+1=θn+1,QR−δθn,Q0,where the meanings of the parameters are the same as those above discussed.The boundary conditions at the casing shoe and the upper tangent point in plane Q were as follows:
(10)M0,Q=0,Mn,Q=E⋅In⋅kϕTsinaT,where kϕT is the rate of azimuth change at the tangent point and aT is the well inclination at the tangent point.


### 2.4.4. Contact Force

The relationship between the force and deformation of the string is given. An iterative method could be used to solve the deflection function of each span to obtain the deflection, rotation angle, and bending moment of the string after deformation. Combined with Figures2(b) and 2(c), the contact force at each support in plane P could be calculated:
(11)Ni,P=Mi−1,P−Mi,PLi+qi,pLi2+Mi+1,P−Mi,PLi+1+qi+1,pLi+12,i=1ton−1.Then, the contact force at each support in planeQ was
(12)Ni,Q=Mi−1,Q−Mi,QLi+Mi+1,Q−Mi,QLi+1,i=1ton−1.


## 2.4.1. Three-Moment Equations in PlaneP

With the casing string runs in planeP (well inclination plane) as an example, the casing string combination of any number (n) of centralizers was taken as the analysis object. As shown in Figure 2, the centralizers divided the whole casing string into N span beams and the transverse load qi, bending moments Mi–1 and Mi, and the axial force Fi acted on the ith span beam. The right deflection angle of the ith span beam was θiR, the shear force was QiR, the transverse load was qi+1, the bending moments were Mi and Mi+1, and the axial force Fi+1 acted on the i+1th span beam. The right rotation angle of the i+1th span beam was θi+1L, and the shear force was Qi+1L.Figure 2
Force and deformation of casing string running in a curved section. (a) Schematic diagram of casing string running in a curved section. (b) Force and deformation of theith span beam. (c) Force and deformation of the i+1th span beam.
(a)(b)(c)The deflection curves of the two adjacent spans had a common tangent line at the support of the string. Therefore, the rotation angles on both sides of the support were equal:(3)θi,PR=−θi+1,PL.The rotation angle at the support could be obtained from the formula of the end rotation angle and the deformation superposition principle of continuous beam-column theory (AppendixA):
(4)θi,PR=qi,PLi324EIiXui,P+Mi,PLi3EIiYui,P+Mi−1,PLi6EIiZui,P,θi+1,PL=qi+1,PLi+1324EIi+1Xui+1,P+Mi+1,PLi+13EIi+1Yui+1,P+Mi+1,PLi+16EIi+1Zui+1,P,θn+1,PR=qn+1,PLn+1324EIn+1Xun+1,P+Mn+1,PLn+13EIn+1Yun+1,P+Mn+1,PLn+16EIn+1Zun+1,P,where Li is the length of the ith span beam (m), qi,p is the component of the transverse load of the ith span beam in the P plane (N), ai is the well inclination angle at the ith support (°), wi is the weight per meter of the ith span beam in mud (kg/m), Fi is the average axial force on the ith span beam (N), E is the elastic modulus of the string (Pa), and Ii is the section moment of inertia of the ith span beam (m4).


## 2.4.2. Initial Rotation Angle

For the actual string combination, the diameter of the centralizer was smaller than the diameter of the borehole. Therefore, when the string was tripped in the borehole, the centralizer might have been close to a side of the wellbore wall, which would have caused the displacement of the support due to the elimination of clearance. Therefore, several supports were not in a straight line. This resulted in an additional rotation angle at the end of the beam column due to the displacement of the supports, as shown in Figure3.Figure 3
Schematic diagram of beam supports not in a straight line under the combined force.In planeP, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(5)δθi,P0=arctanδxi0−δxi−10zi−zi−1.In planeQ, the initial additional rotation angle generated by the initial relative displacement of the supports at both ends of the ith span beam was
(6)δθi,Q0=arctanδyi0−δyi−10zi−zi−1.


## 2.4.3. Three-Moment Equations

Considering the initial additional rotation angle generated by the displacement of the support, the corresponding three-moment equation of then+1th span continuous beam-column in the plane P could be obtained:
(7)Mi−1,PLiZui,P6EIi+Mi,PLiYui,P3EIi+Li+1Yui+1,P3EIi+1+Mi+1,PLi+1Zui+1,P6EIi+1=−qi,PLi3Xui,P24EIi−qi+1,PLi+13Xui+1,P24EIi+1+δθi+1,P0−δθi,P0,qn+1,PXun+1,PLn+1324EIn+1+Mn+1,PYun+1,PLn+13EIn+1+Mn,PZun+1,P6EIn+1Ln+1=θn+1,PR−δθn,P0.The boundary conditions at the casing shoe and the upper tangent point in planeP were
(8)M0,P=0,Mn,P=E⋅In⋅kaT,where kaT is the rate of well inclination change at the tangent point.There weren+1 equations for the three bending moments, in which the n+1 unknowns were M1~Mn and Ln+1. Thus, there was a definite solution. For the ith beam support in plane Q, the left and right sides were the ith and i+1th span beams, respectively. Based on the continuous condition that the rotation angle of the beams on both sides of the support were equal, the three bending moment equations in plane Q were obtained.
(9)Mi−1,QLiZui,Q6EIi+Mi,QLiYui,Q3EIi+Li+1Yui+1,Q3EIi+1+Mi+1,QLi+1Zui+1,Q6EIi+1=δθi+1,Q0−δθi,Q0,Mn+1,QYun+1,QLn+13EIn+1+Mn,QZun+1,Q6EIn+1Ln+1=θn+1,QR−δθn,Q0,where the meanings of the parameters are the same as those above discussed.The boundary conditions at the casing shoe and the upper tangent point in plane Q were as follows:
(10)M0,Q=0,Mn,Q=E⋅In⋅kϕTsinaT,where kϕT is the rate of azimuth change at the tangent point and aT is the well inclination at the tangent point.


## 2.4.4. Contact Force

The relationship between the force and deformation of the string is given. An iterative method could be used to solve the deflection function of each span to obtain the deflection, rotation angle, and bending moment of the string after deformation. Combined with Figures2(b) and 2(c), the contact force at each support in plane P could be calculated:
(11)Ni,P=Mi−1,P−Mi,PLi+qi,pLi2+Mi+1,P−Mi,PLi+1+qi+1,pLi+12,i=1ton−1.Then, the contact force at each support in planeQ was
(12)Ni,Q=Mi−1,Q−Mi,QLi+Mi+1,Q−Mi,QLi+1,i=1ton−1.


## 2.5. Effect of Initial Bending on Casing Deflection

When the casing string was tripped in the curved section of a wellbore, the deformation of the entire casing string had to be generally consistent with the wellbore axis. Therefore, the maximum deflection of the casing string caused by the initial bending could be obtained [27] (Appendix B):
(13)xi,max=xi,0+xi,1=2ζi,Pui21cosui−1=Ki,PEIiFi1cosui−1.


## 2.6. Judgment of Contact Points

Considering the influence of the borehole bending on the casing string deformation, the improved maximum deflectionxi,m in plane P and the improved maximum deflection yi,m in plane Q of the ith span string were calculated as follows:
(14)xi,m=qi,pΔLi416EIiui41cosui−1−ui22+Mi−1,p+Mi,p2FicosΔαi/21−1cosui+KPLi24ui21cosui−1+ei−1,P+ei,P2,yi,m=Mi−1,Q+Mi,Q2FicosΔϕi/21−1cosui+KQLi24ui21cosui−1+ei−1,Q+ei,Q2.Therefore, the criterion for generating the new contact points was(15)xi,m2+yi,m2>Dw−D0i2,where KQ is the curvature on the azimuth plane of the well section where the string is located (°/30 m), Dw is the borehole diameter (m), Dci is the centralizer diameter (m), D0i is the casing diameter (m), ei−1=Dw−Dci−1/2, and ei=Dw−Dci/2.Once the maximum deflection of the casing string between the two supports exceeded the wellbore clearance, there was a new contact point and new support needed to be added to perform the recalculation.


## 2.7. Tripability Judgment of the Casing String

The contact force and the friction at each contact point could be obtained as follows:(16)Ni=Ni,p2+Ni,Q2,Ffi=μNi,where μ is the friction coefficient.When the total friction of the casing string was greater than the floating weight of the casing string above the stuck point, the casing string could be considered hindered during the casing running process:
(17)∑Ffi>wi⋅Ls,where Ls is the length of the casing string above the stuck point (m).At that time, it was difficult for the casing string to run with its own floating weight and it was possible to run the casing string with the help of the other processes. For example, a lowering force, a pick-up and release technique, and casing floating could be applied. Since the casing string buckled during the running process, it was also necessary to consider the case in which the casing string was hindered by buckling.


## 2.8. Critical Load for Helical Buckling of the String

Many scholars have obtained the critical load when a string is subjected to spiral buckling in different well sections through theoretical derivations or experiments, as shown in Table1. Additional friction should be considered when a casing string is spirally buckling and in contact with a borehole wall to determine whether the casing string can be successfully tripped in.Table 1
Critical load for the helical buckling of the string in different well sections, whereFhel is the critical load of helical buckling (N), r is the borehole annulus clearance (m), and R is the string radius (m).
Well sectionCritical load for helical bucklingVerticalFhel=5.55EIw23Straight inclinedFhel=2.83EIwsina/rCurvedFhel=12EI/rR1+1+rR2wsina/8EIHorizontalFhel=222−1EIw/r


## 2.9. Determine the Maximum Lowering Force

Due to the restrictions of a wellbore, a string will undergo multiple deformation processes with the increase in the lowering force after string buckling. With the combined action of the string weight and the buoyancy of the drilling fluid, the string buckling becomes a spatial spiral shape with a varying pitch. The bending of the string causes contact with the borehole wall, resulting in increased friction, which offsets some of the lowering force and reduces the effective lowering force on the stuck point [28]. Therefore, how to determine the maximum lowering force when applying a load to push down the casing is very important for casing running.The maximum lowering forceFmax can be obtained as follows (Appendix C):
(18)Fmax=wmEI3δμr.The maximum effective lowering forceFemax can be obtained as follows:
(19)Femax=Fmax−δμrwmEIFmax3.


## 2.10. Solution Method

An iterative method [23, 24] was used to apply the simulation model in actual wellbores. The flowchart of the procedure is shown in Figure 4.Figure 4
Flowchart of the solution procedure.


## 3. Model Validation

An example was adapted from an actual FX160 directional well in China. The finite element method was used for verification. Due to the large size of the actual borehole, the establishment of a full-size three-dimensional finite element model of the borehole trajectory and casing string combination required not only a high amount of computational resources but also a long time to perform the calculations. It was believed that the mechanical behavior of a point in the structure would have no effect on the point in space beyond 15 m [29]. Therefore, the well section at 1210–1410 m was selected. This section included the vertical section and the curved section. Considering the influence of the casing string coupling, the deformation and the contact force when the casing string was tripped in were simulated and calculated. The three-dimensional finite element model is shown in Figure 5. The relevant parameters in the model are shown in Table 2.Figure 5
Finite element model of the casing string.Table 2
Basic calculation parameters.
NameValueWell depth4558 mKickoff point1267 mBuild-up rate4°/30 mRadius of curvature,R430 mVertical section,L1100 mBuild-up section,L2100 mDiameter (borehole),Dw215.9 mmDiameter (casing),Dp177.8 mmDiameter (coupling),Dc200.03 mmWall thickness (coupling),Do11.12 mmYoung’s modulus,E2.1×105MPaPoisson’s ratio,ε0.3Friction coefficient0.2The dynamic explicit analysis method was used, and the casing running speed was 0.8 m/s. When the casing string was tripped in the wellbore, it deformed and came into contact with the borehole wall due to the influence of the curved borehole. Because the model size was large, the contact state of the entire casing string and the wellbore wall could not be clearly displayed. Therefore, only the contact cloud map of the casing string and the wellbore wall in the 1257–1287 m well section is illustrated in Figure6. Figure 6(a) shows the schematic diagram of the casing string combination in this well section, and Figure 6(b) shows the corresponding cloud map of the contact between the 30 m casing string and the wellbore wall. As shown in Figure 6(b), the first and fourth couplings were in contact with the downhole wall and the second and third couplings were in contact with the upper wellbore wall. The simulation showed that due to the influence of the curved wellbore, the casing string occasionally came into contact with the upper or lower hole walls of the wellbore during the running process. The calculation results are illustrated in Figure 7. The magnitude and the direction of the contact force were constantly changing when the casing string was tripped in the 1210–1410 m well section.Figure 6
(a) Schematic diagram of casing string combination. (b) Cloud map of contact between the casing string and the wellbore wall.
(a)(b)Figure 7
Contact force when the casing string was running in the wellbore.The contact force when the casing string was tripped in the wellbore using the improved continuous beam-column model is shown in Figure8. The casing strings are in contact with the upper or lower borehole wall in the curved section, resulting in changes in the direction and magnitude of the contact force. For the casing strings at 1257–1287 m, the direction of the contact force also changed, similar to the results in Figure 6(b).Figure 8
Contact force of the casing string using the improved model.The comparison of the results in Figures7 and 8 showed that the magnitude of the contact force using the improved continuous beam-column model was different from that of the finite element simulation. The magnitude of the contact force calculated using the improved continuous beam-column model represented the contact force of each contact point between the casing string and the borehole wall, while the result of the finite element method was the resultant force of all contact points between the entire casing string and the borehole wall at a certain time. Although the contact force values calculated with the two methods were different, the direction of the contact force reflected the contact state between the casing strings and the wellbore wall when casing strings were tripped in the wellbore, which was more consistent with the actual running operation of the casing strings.


## 4. Field Applications

With the wellbore trajectory in Section3 taken as an example, the established tripability analysis model of the casing string was used to calculate the contact force, hook load, and maximum lowering force required during the casing running in the wellbore. The proposed model was compared with the existing friction model of the string and the field drilling data.


### 4.1. Contact Force on the Casing String

Figure9 shows that the contact force calculated with the soft model was significantly greater than that of the stiff model. The reason for this was that the soft model assumed that the casing string was in continuous contact with the borehole wall, while the stiff model assumed point contact with the borehole wall. However, the contact force calculated based on the improved continuous beam-column model was also greater than that of Fan’s model. The reason for this was that the new model considered the influence of the curved wellbore on the deformation of the casing string. When the string made new contact with the borehole wall due to the initial bending deformation, the additional contact force was included in the total contact force.Figure 9
Contact force on the casing string using different models.


### 4.2. Hook Load of Casing Running

When the casing string was tripped in, its axial load was mainly affected by the floating weight of the string and the friction between the string and the borehole wall. The hook loads calculated by the four models are shown in Figure10. The hook load calculated by the soft model was less than that calculated by the stiff model in the bottom hole. The reason for this was that the friction calculated by the soft model was larger than that of the stiff model and the friction direction was opposite to the movement direction of the casing string, which offset part of the floating weight of the casing string.Figure 10
Axial load on the casing string using different models.The measured hook load was highly affected and sensitive to any change in the drilling operation such as the drill string material gradient, outer diameter, inner diameter, bottom hole assembly, drilling fluid, drilling path, drilling trajectory, or dynamics of the hook. These made the measured hook load data points fluctuate greatly, which was inconvenient for the model verification. Therefore, the hook load data points were fitted with a polynomial function before validation and a fitting curve representing the hook load was obtained, as shown in Figure10. The accuracies of the four models were verified by comparison with the fitted curves. As shown in Table 3, 15 sets of large hook load data points were randomly selected from different well sections for comparison. It can be seen that the improved model had the smallest prediction error, i.e., 6.58%, compared with the existing models.Table 3
Comparison of model prediction errors.
WellboreRelative error (%)Johancsik’s modelHuang’s modelFan’s modelThis paper’s modelVertical section33.2231.6415.9815.9425.0823.5015.8714.0913.8110.56−1.914.60Build-up section17.109.01−2.547.804.63−1.92−5.561.339.202.700.037.34Sail section2.81−1.70−2.013.97−1.34−5.16−2.562.80−5.08−8.38−4.700.21Drop-off section−6.31−9.40−4.270.52−10.70−15.04−8.34−2.72−16.18−19.14−10.58−5.25Inclined section−10.43−14.64−3.750.02−10.43−14.54−3.83−0.33−10.21−14.18−3.70−0.61Root mean square error (%)14.3614.527.416.58


### 4.3. Maximum Lowering Force on the Casing String

With the 7-inch (177.8 mm) casing as an example, the parameters are shown in Table4. The maximum lowering force and the maximum effective lowering force required for the buckled casing string to run in the wellbore were calculated.Table 4
Casing string combination parameters.
NameValueCasing diameter0.1778 mWall thickness0.0115 mUnit weight466.676 N/mMoment of inertia2.09×10−5m4Borehole diameter0.2159 mYoung’s modulus,E2.1×1011PaFigure11 shows that the maximum lowering force and the maximum effective lowering force decreased with the increase in the friction coefficient. Adjusting the performance of the drilling fluid at the appropriate time and reducing the friction coefficient increased the maximum lowering force and the maximum effective lowering force, which was conducive to applying a lowering force to the buckled casing during the casing running process.Figure 11
Lowering force on the casing string for different friction coefficients.As shown in Figure12, the maximum lowering force and the maximum effective running force gradually decreased when the drilling fluid density increased from 1.0 to 2.0 g/cm3, that is, increasing the drilling fluid density was not conducive to applying a lowering force to the buckled casing during the casing running process.Figure 12
Lowering force on the casing string for different drilling fluid densities.As shown in Figure13, with the increase in the casing diameter, the maximum lowering force and the maximum effective lowering force increased and the difference between them also increased. The smaller the casing diameter was, the smaller the maximum lowering force and the maximum effective lowering force were. Therefore, the effect of applying the lowering force to the buckled casing with a smaller diameter was not as good as that for the buckled casing with a larger diameter.Figure 13
Lowering force on the casing string for different casing diameters.


## 4.1. Contact Force on the Casing String

Figure9 shows that the contact force calculated with the soft model was significantly greater than that of the stiff model. The reason for this was that the soft model assumed that the casing string was in continuous contact with the borehole wall, while the stiff model assumed point contact with the borehole wall. However, the contact force calculated based on the improved continuous beam-column model was also greater than that of Fan’s model. The reason for this was that the new model considered the influence of the curved wellbore on the deformation of the casing string. When the string made new contact with the borehole wall due to the initial bending deformation, the additional contact force was included in the total contact force.Figure 9
Contact force on the casing string using different models.


## 4.2. Hook Load of Casing Running

When the casing string was tripped in, its axial load was mainly affected by the floating weight of the string and the friction between the string and the borehole wall. The hook loads calculated by the four models are shown in Figure10. The hook load calculated by the soft model was less than that calculated by the stiff model in the bottom hole. The reason for this was that the friction calculated by the soft model was larger than that of the stiff model and the friction direction was opposite to the movement direction of the casing string, which offset part of the floating weight of the casing string.Figure 10
Axial load on the casing string using different models.The measured hook load was highly affected and sensitive to any change in the drilling operation such as the drill string material gradient, outer diameter, inner diameter, bottom hole assembly, drilling fluid, drilling path, drilling trajectory, or dynamics of the hook. These made the measured hook load data points fluctuate greatly, which was inconvenient for the model verification. Therefore, the hook load data points were fitted with a polynomial function before validation and a fitting curve representing the hook load was obtained, as shown in Figure10. The accuracies of the four models were verified by comparison with the fitted curves. As shown in Table 3, 15 sets of large hook load data points were randomly selected from different well sections for comparison. It can be seen that the improved model had the smallest prediction error, i.e., 6.58%, compared with the existing models.Table 3
Comparison of model prediction errors.
WellboreRelative error (%)Johancsik’s modelHuang’s modelFan’s modelThis paper’s modelVertical section33.2231.6415.9815.9425.0823.5015.8714.0913.8110.56−1.914.60Build-up section17.109.01−2.547.804.63−1.92−5.561.339.202.700.037.34Sail section2.81−1.70−2.013.97−1.34−5.16−2.562.80−5.08−8.38−4.700.21Drop-off section−6.31−9.40−4.270.52−10.70−15.04−8.34−2.72−16.18−19.14−10.58−5.25Inclined section−10.43−14.64−3.750.02−10.43−14.54−3.83−0.33−10.21−14.18−3.70−0.61Root mean square error (%)14.3614.527.416.58


## 4.3. Maximum Lowering Force on the Casing String

With the 7-inch (177.8 mm) casing as an example, the parameters are shown in Table4. The maximum lowering force and the maximum effective lowering force required for the buckled casing string to run in the wellbore were calculated.Table 4
Casing string combination parameters.
NameValueCasing diameter0.1778 mWall thickness0.0115 mUnit weight466.676 N/mMoment of inertia2.09×10−5m4Borehole diameter0.2159 mYoung’s modulus,E2.1×1011PaFigure11 shows that the maximum lowering force and the maximum effective lowering force decreased with the increase in the friction coefficient. Adjusting the performance of the drilling fluid at the appropriate time and reducing the friction coefficient increased the maximum lowering force and the maximum effective lowering force, which was conducive to applying a lowering force to the buckled casing during the casing running process.Figure 11
Lowering force on the casing string for different friction coefficients.As shown in Figure12, the maximum lowering force and the maximum effective running force gradually decreased when the drilling fluid density increased from 1.0 to 2.0 g/cm3, that is, increasing the drilling fluid density was not conducive to applying a lowering force to the buckled casing during the casing running process.Figure 12
Lowering force on the casing string for different drilling fluid densities.As shown in Figure13, with the increase in the casing diameter, the maximum lowering force and the maximum effective lowering force increased and the difference between them also increased. The smaller the casing diameter was, the smaller the maximum lowering force and the maximum effective lowering force were. Therefore, the effect of applying the lowering force to the buckled casing with a smaller diameter was not as good as that for the buckled casing with a larger diameter.Figure 13
Lowering force on the casing string for different casing diameters.


## 5. Conclusions

(1)
The casing string was hindered by buckling during the casing string running process. Improving the performance of the drilling fluid, reducing the friction coefficient, and reducing the drilling fluid density were conducive to applying a lowering force to the casing. The effect of applying the lowering force to the casing with a smaller diameter was not as good as that for a casing with a larger diameter(2)
Compared with the existing models, the proposed model was no longer limited to the specified point contact range between the casing strings and the wellbore wall. Influenced by factors such as the well trajectory, string assembly, and drilling fluid performance, based on the specified point contact, newly added contact points were dynamically identified. The contact point might include contact with the upper or lower wellbore wall, which was close to the actual working conditions of casing strings running in the wellbore(3)
To account for the influence of the curved wellbore on the initial deformation of the casing strings, based on continuous beam-column theory and buckling theory, a comprehensive model for predicting the friction during casing string running and the lowering force required for hindered casing string running was established. A corresponding computer program was developed. Based on the field data verification, the improved model has the smallest prediction error, i.e., 6.58%, compared with the existing model. Therefore, the comprehensive model could be used for predicting the tripability of casing strings and the conclusions provided necessary theoretical guidance for casing running operations in directional wells



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*Source: 9728261-2022-07-31.xml*