Prediction Model of Mechanical Extending Limits in Horizontal Drilling and Design Methods of Tubular Strings to Improve Limits

Mechanical extending limit in horizontal drilling means the maximum horizontal extending length of a horizontal well under certain ground and down-hole mechanical constraint conditions. Around this concept, the constrained optimization model of mechanical extending limits is built and simplified analytical results for pick-up and slack-offoperations are deduced.Thehorizontal extending limits for kinds of tubular strings under different drilling parameters are calculated and drawn. To improve extending limits, an optimal design model of drill strings is built and applied to a case study. The results indicate that horizontal extending limits are underestimated a lot when the effects of friction force on critical helical buckling loads are neglected.Horizontal extending limits firstly increase and tend to stable values with vertical depths. Horizontal extending limits increase faster but finally become smaller with the increase of horizontal pushing forces for tubular strings of smaller modulus-weight ratio. Sliding slack-off is the main limit operation and high axial friction is the main constraint factor constraining horizontal extending limits. A sophisticated installation of multiple tubular strings can greatly inhibit helical buckling and increase horizontal extending limits. The optimal design model is called only once to obtain design results, which greatly increases the calculation efficiency.


Introduction
Extending limit in horizontal drilling is a constrained optimization problem in which horizontal extension of the wellbore is the objective function, and kinds of ground and down-hole constraint conditions and operation conditions in horizontal drilling are considered.There are two main issues in extending limits including prediction and control.Prediction refers to "what are the extending limits," and control means "how to increase the extending limits."Until now, some researchers have built models by considering several factors or analyzed the extending limits with drilling data, and a systematic theoretical framework of extending limits in rotary drilling is forming.
Wu and Juvkam-Wold [1] studied the effect of helical buckling on the slack-off extension limit of tubing in horizontal wells and pointed out that helical buckling can limit wellbore extension a lot.Mason and Judzis [2] analyzed the extending limits in shallow, mid-deep, and deep wells with drilling data and the results show that friction force is main constraint factor for shallow wells, drill rig capacity and tubular strength are for mid-deep wells, and tubular strength is for deep wells.Wang and Guo [3] built the models of extending limits by considering the effects of formation property, hydraulic equipment capacity, drill fluid property, and so on.Gao et al. [4] proposed the concepts of extending limits of extended-reach wells.They pointed out that extending limits can be further divided into three categories according to the properties of constraint conditions, including mechanical extending limits, open-hole extending limits, and hydraulic extending limits, for the convenience of analysis.Mechanical extending limit means the maximum wellbore extension under the constraints of drilling mode, tubular strength, drill rig capacity, and so on, open-hole

Helical buckling Nonbuckling
Sinusoidal buckling Figure 1: Buckling state of tubular strings in a horizontal well while slack-off.
extending limit means the maximum wellbore extension within which either collapse or fracture of the wellbore does not happen, and hydraulic extending limit means the maximum wellbore extension in which cutting removing is available under certain hydraulic parameters.Yan et al. [5] built theoretical models to predict the extending limits by considering the effects of rock breaking threshold, tubular strength, friction factor, and so on.Li et al. [6,7] studied the open-hole extending limits by considering the effects of well path, drill fluid property, wellbore collapse and fracture, and so on in kinds of drilling scenarios.However, the above studies mainly focus the value of extending limit for a certain well and neglect the distributions of extending limits for kinds of wells.In other words, the law of the envelope of the extending limits for certain drill rig and down-hole constraint conditions is not revealed.Until now, methods to improve mechanical extending limits have been discussed in previous studies.Peng and Zhao [8] proposed that drill collars or heavy weight drill pipes should be installed in the vertical wellbore to provide enough weight on bit, and common and heavy weight drill pipes should be adopted on the horizontal or holding section to reduce the axial friction.Han [9] proposed that the design of tubular string components should include the effects of rock breaking threshold, tubular failure, and tubular buckling.According to Han's studies, the objective functions of tubular string in sliding and rotary drilling modes should be, respectively, set to minimizing axial friction and minimizing tubular string weight.The tubular string components are constantly revised in iteration process.Allen et al. [10] studied the measures from the aspects of well path plan, drag reduction, and hydraulic parameters selection to improve extending limits with the case study of extendedreach wells in Wytch Farm.Xia et al. [11] proposed the design idea for tubular selection in extended-reach wells with trial and error method.The previous mechanical analysis of drill string behavior and design of tubular string components must be conducted several times to obtain an appropriate result in the previous methods, which means the design process is time-consuming and inefficient.Moreover, the previous design methods contain too much personal experience and lack of rigorous mathematical derivations.
In this paper, the mechanical extending limit model in horizontal drilling, in which kinds of ground and down-hole constraint conditions and operation conditions are included, is built.The proposed model is compared with other models in an example of four kinds of tubular strings.The law of distributions of mechanical extending limits is analyzed and the main factors constraining the wellbore extension are identified.A high-efficiency design method of drill strings components is built.At last, the extending limit model and design method are applied to a case study.

Prediction Model of Mechanical
Extending Limits

Integral Mechanical Model.
A typical horizontal well trajectory shown in Figure 1 includes three parts, namely, vertical, build-up, and horizontal wellbores.If a tubular string is in the pick-up process, the tubular strings in wellbores are in axial tensile state and the axis of tubular string is parallel to that of the wellbore.In the horizontal section, the tubular string lies on the bottom of wellbore due to gravity.
In the build-up section, the tubular string touches the top or the bottom of the wellbore.In the vertical section, the tubular string stays on the wellbore axis and does not touch the inner surface of the wellbore.If the tubular string is in the slack-off operation, the mechanical behaviors of tubular strings become more complex.In the horizontal section, the tubular string is in axial compression state and the axial compressive force on the tubular string increases from the drill bit to the starting point of the horizontal section.If the axial force is smaller than the critical buckling load, the tubular string lies on the bottom of the wellbore.However, if the axial force exceeds the critical sinusoidal or helical buckling load, the tubular string enters into buckling state and the deflection curve of the tubular string is depicted by a sinusoidal curve or a helix.Therefore, buckling tends to occur on the starting part of the horizontal section shown in Figure 1.In the build-up section, the tubular string is usually in nonbuckling state for the buckling load is increased due to the effect of the curved wellbore configuration.Then, the tubular string tends to touch the bottom of the buildup wellbore.In the vertical wellbore, the axial compressive force increases with the increase of vertical depth.If the axial force is smaller than the critical helical buckling load, the tubular string keeps in straight line state.However, if the axial force exceeds the critical helical buckling loads, the tubular string becomes into helix state.Therefore, helical buckling tends to occur on the bottom part of the vertical wellbore.
Until now, down-hole mechanical model is the most effective method to calculate the mechanical behavior of down-hole tubular strings.The conventional integral mechanical model of down-hole tubular strings mainly developed from soft rope model (Johancsik et al. [12]), stiff rod model (Ho [13]), integral model based on basic equations of down-hole tubular mechanics (Gao et al. [14]), integral model including the tubular buckling effect (Wu and Juvkam-Wold [1]), and discrete integral model including the effect of connectors (Mitchell et al. [15]) to combined integral and mechanical model including the combined effects of connectors and buckling (Huang and Gao, [16]).In this paper, the buckling effect is included and connector is neglected.Further considering that the azimuth angles and bending curvatures of the wellbore axes are both equal to 0 for vertical, horizontal, and build-up sections of a horizontal wellbore, the axial force transfer on the tubular string is simplified as where "∓" represents pick-up and slack-off operations,  is the axial compressive force on the tubular string,  is the well depth,  is tubular weight per unit length,  is the inclination angle,   is the contact force between tubular string and wellbore per unit length of the tubular string, and  1 is the axial friction factor.The torque transfer on the tubular string is calculated by where   is the torque on the tubular string,  2 is the circumferential friction factor, and   is the outer diameter of the tubular string.
The bending moment on the tubular string is calculated by where   is the bending moment,   is the bending curvature of the wellbore axis, and  is the bending stiffness of the tubular string.
The contact force on the tubular string per unit length is expressed by where   is the radial clearance between the tubular string and wellbore and calculated by   = (  −   )/2 and  sin and  hel are the critical sinusoidal and helical buckling loads.
The mechanical results of tubular strings in vertical, build-up, and horizontal wellbores under pick-up and slackoff operations are given in Appendix A.

Mechanical Extending Limit
Model.Mechanical extending limit is a constrained optimization model with the length of well trajectory as the objective function, pick-up and slackoff in sliding and rotary modes as operation conditions, and drilling rig parameters, rock breaking thresholds, tubular strength, and so on as constrained conditions.Thus, the mechanical extending limit model is expressed by where  is the well depth, ( ) is an implicit function, namely, the integral mechanical model of down-hole tubular strings,  is the constraint parameters such as drill rig capacity, tubular strength,  is the allowable region for constraint parameters, namely, constraint conditions,  is the design parameters such as well trajectory and tubular string components, and  is the operation conditions including pick-up and slack-off in sliding and rotary modes.The integral mechanical model for a tubular string constrained in a horizontal well under kinds of operation conditions is given in Section 2.1 and Appendix A. In this section, the constraint conditions are given as follows.
For a certain drill rig, the rated pick-up force is a limited value.The axial tensile force on the top of the vertical section during pick-up is not larger than the rated pick-up force; namely, where  rig  is the rated pick-up force of the drill rig and  vs is the axial force on the top of the vertical wellbore.
Meanwhile, the axial tensile force on the top of the vertical section during slack-off is not smaller than the minimum slack-off force; namely, where  rig  is the minimum slack-off force of the drill rig and usually taken as 0.
The torque on the top of the vertical section while rotating tubular string is not larger than the rated torque of the drill rig; namely, where  rig  is the rated torque of the drill rig and  vs is the torque on the top of the vertical wellbore.
In the drilling process, the axial force and torque on the drill bit must be not smaller than the threshold values of rock breaking; namely, where  ths  and  ths  are the threshold values of axial force and torque for rock breaking,  is the ratio of torque to axial force on the bit, and  he and  he are the axial force and torque on the end of the horizontal wellbore.
The average axial stress on the cross section of the tubular string due to axial force is calculated by where   is the average axial stress on the tubular string and   is the inner diameter of the tubular string.The maximum shear stress on the cross section of the tubular string due to torque is calculated by where   is the maximum shear stress on the tubular string cross section.The maximum bending stress on the cross section of the tubular string due to bending moment is calculated by where   is the maximum bending stress on the tubular string cross section.
According to the fourth strength theory (Han, [17]), the equivalent stress is calculated by To ensure the safety of tubular string, the equivalent stress must not be larger than the allowable stress; namely, where  is the equivalent stress, [] is the allowable stress,   is the yielding strength of the tubular string, and  4 is the safety factor.

Horizontal Extending Limits.
A horizontal well can be divided into vertical-build and horizontal wellbores by the starting point of the horizontal wellbore.If the axial force and torque at the starting point of the horizontal wellbore are given, the axial force and torque transfer in the vertical-build wellbore can be obtained.Substituting the axial force and torque in the vertical-build wellbore, it can be determined whether the constraint expressions mentioned above are satisfied or not.Here three critical loads at the starting point of the horizontal section are defined, including minimum axial force  hs  , maximum axial force  hs  , and maximum torque  hs  , under which the constraint conditions for tubular strings in vertical-build wellbore are just satisfied.The term  hs  means the minimum axial force under which the tubular strings in the vertical-build wellbore are not yielded, and the axial tensile force on ground is not larger than the rated axial force on hook in the pick-up operation.The term  hs  means the maximum axial force under which the axial tensile force on ground is not smaller than the minimum axial force on hook in the slack-off operation.The term  hs  means the maximum torque under which the tubular strings are not yielded and torque on ground is not larger than the rated torque on rotary table.The calculation expressions of these three terms are given in Appendix B.
These three critical loads can also be seen as the constraint conditions in the mechanical analysis of tubular strings in the horizontal wellbores.According to the constraint conditions including  hs  ,  hs  ,  hs  , rock breaking thresholds ( 9) and the axial force and torque transfer expressions of the extending limits of the horizontal wellbores can be determined.
For the pick-up operation in the sliding mode, the minimum axial force limits the wellbore extension, and the extending limit of the horizontal section is calculated by where  ℎ is the limit length of the horizontal wellbore.
For the pick-up operation in the rotary mode, the maximum torque limits the wellbore extension and the extending limit is calculated by For the slack-off process in the sliding mode, the maximum axial force limits the wellbore extension and the extending limit is calculated by For the slack-off process in the rotary mode, the extending limit is equal to the minimum value of ( 16) and (17).

Comparison with Other Models.
Here, the horizontal extending limits of a SDP constrained in horizontal wellbore for different models are studied.Friction factor and mud density are assumed to be 0.3 and 1677 kg/m 3 .The diameter of the horizontal wellbore is 0.2159 m.The values of other parameters are given in Table 1. Figure 2 shows the horizontal extending limits of SDP for Wu and Juvkam-Wold's [1] and Mehrabi and Miska's [18] models and the proposed model.Wu and Juvkam-Wold [1] have deduced the extending limit of coiled tubing in horizontal well.Similar to (17), the calculation expressions of horizontal extending limits for Wu's model are given by in which  ℎsin is the critical sinusoidal buckling load in horizontal wellbore for no friction case.Note that  ℎhel is the critical helical buckling load in horizontal wellbore considering friction force.The amendement factor   =   () means the ratio of critical helical buckling load for friction case to that for nonfriction case.The calculation expression of   is given in (A.8).The expression   = 1 means the critical helical buckling load for zero friction factor case, namely, friction-free critical helical buckling load.
The results show that the extending limits increase with the axial forces at the starting point of the horizontal section.If helical buckling does not occur on the horizontal section ( hs <  hs_hel ), the horizontal extending limits increase linearly.If helical buckling occurs ( hs ≥  hs_hel ), the horizontal extending limits increase slowly and tend to stable values when axial forces tend to infinity.Note that horizontal extending limits can increase a lot even after helical buckling occurs.Therefore, the occurrence of helical buckling is too conservative to predict the horizontal extending limits.
For the friction-free critical helical buckling load, the extending limits in the new model are larger than that of Wu and Juvkam-Wold's [1] model.In Wu's derivation, the contact force is calculated by   =    2 /4 +  sin  while  ≥  hel and it is larger than the contact force in (4) in helical buckling mode.Therefore, the axial force loss for Wu's model is larger than that in the new model and the horizontal extending limits for Wu's model are smaller than that in the new model when helical buckling occurs.However, when the helical buckling does not occur, the horizontal extending limits for these two models are equal.
For the friction-related critical helical buckling load, the horizontal extending limits are larger than that for frictionfree model.Therefore, if the effect of friction force on critical helical buckling load is neglected, the horizontal extending limits are underestimated a lot.In Mehrabi and Miska's [18] derivation, the effect of friction force on critical buckling load is considered, so the results for these two models are very close to each other.However, Mehrabi and Miska's [18] have not given the explicit calculation expressions of horizontal extending limits while considering the axial force and torque transfer in vertical and build-up wellbores.In this paper, a case study in which kinds of ground and down-hole constraint factors are comprehensively included is conducted and given in Section 2.5.
Here, several tubular strings of different parameters including SDP (steel drill string), TDP (titanium drill string), ADP (aluminum drill string), and CDP (composite drill string) are adopted and the extending limits are, respectively, calculated.The values of relevant parameters for these tubular strings are shown in Table 1.The term /  means the ratio of Young's modulus of tubular string to tubular string weight per unit length (modulus-weight ratio).Note that modulusweight ratio decreases for tubular strings transform from  SDP, TDP, and ADP to CDP.The horizontal extending limits for different tubular strings are shown in Figure 3.
According to (17), horizontal extending limits increase faster for tubular strings of smaller weight per unit length when helical buckling does not occur.The horizontal extending limits when helical buckling just occurs increase with the increase of Young's modulus of tubular string.Meanwhile, the final extending limits when axial forces tend to infinity also increase with Young's modulus.Therefore, the distributions of horizontal extending limits are closely related to the combination of Young's modulus and tubular string weight per unit length, namely, modulus-weight ratio.Figure 3 shows that, for tubular strings of smaller modulus-weight ratio, the horizontal extending limits increase faster but the final extending limits are smaller.Therefore, there is an intersection of extending limit curves for every two tubular strings.At the intersection, the extending limits and axial forces are both equal for these two tubular strings.For example,  hs1 and  ℎ1 are the intersection for CDP and ADP.
When the axial force is smaller than  hs1 or the designed horizontal length is smaller than  ℎ1 , CDP is superior to ADP.Conversely, ADP is superior to CDP.Other intersections are shown in Table 2. Therefore, the intersection provides a criterion for optimal section of tubular strings in horizontal drilling.
Overall, TDP is the most optimal solution to increase horizontal extending limits.However, the cost of TDP is also the highest, which limits its application.CDP is not recommended for application for its rather small finial extending limits.Therefore, SDP and ADP are the two common kinds of tubular strings in engineering application.The results show that SDP is more suitable for high axial forces even when helical buckling occurs and ADP is more suitable for small forces when helical buckling does not occur.Therefore, a more optimal solution is combination of ADP and SDP.The optimal design of multiple kinds of tubular strings is studied in Section 3.

Case Studies.
In the above section, only the horizontal section is considered.In this section, the entire horizontal well including vertical, build-up, and horizontal wellbores is taken as the research object.A virtual case study is conducted to analyze the extending limits of a SDP in horizontal drilling.The axial friction factor in sliding mode is  1 = 0.25, and the axial and circumferential friction factors in rotary mode are  1 = 0.1 and  2 = 0.25.The values of the other parameters are given in Table 3.The extending limits in slack-off and pickup conditions in the sliding and rotary drilling modes are studied, in which the friction-free helical buckling model is adopted.Figures 4 and 5 show the slack-off and pick-up extending limits of horizontal wells in sliding and rotary modes.The red line represents the extending limits in the slack-off operation and the green line represents the extending limits in the pick-up operation.The left region which is surrounded by the slack-off and pick-up limits represents the areas that the drill rig can drill to and the boundaries of this region are the extending limits of the horizontal well.The term  1 means the critical vertical depth within which the tubular string in vertical wellbore is in the nonbuckling state in the slack-off mode.The term  2 means the critical vertical depth exceeding which helical buckling occurs on the horizontal section.When the vertical well depth  V is smaller than  2 and larger than  1 , helical buckling occurs on the vertical section but does not on the horizontal section.When  V is larger than  2 , helical buckling occurs on both vertical and horizontal sections.
As is shown in Figure 4, slack-off is the operation condition which limits the wellbore extension in horizontal drilling except that the vertical depth is very large.For the slack-off process, the extending limits increase with the vertical well depth and approach the yielding values, namely, line  1 .Since contact force increases in square function with axial force on helical buckling section shown in (4), helical buckling increases the contact force and decreases the axial force transfer, which leads to the yield of extending limit curve.Note that  1 corresponds to the case that the axial force on the end of the vertical section  ve reaches its maximum value  Vmax .If  ve tends to infinity, the extending limits are the line  2 .If  hs  is assumed to be infinity, the extending limits are the line  3 .Increasing the axial force transfer in the vertical and built-up wellbores which means  hs  is increased has very limited contribution to increase the extending limits.The most important reason is that helical buckling can lead to high axial friction on tubular strings or even "lockup" in the slack-off process in the horizontal wellbore.The most important measure is to inhibit helical buckling in the horizontal well, which is discussed in Section 3.
Compared with the sliding mode, the extending limits in the rotary mode become larger in the rotary mode shown in Figure 5, since rotating tubular strings can decrease the axial friction forces.However, the extending limits also approach the yield values similar to the sliding mode.When the rotated tubular string reaches its extending limit, helical buckling is a rather serious problem.Serious helical buckling leads to high contact force, which further leads to high torque loss.Considering that serious helical buckling is caused by high axial friction on the horizontal wellbore, the coupled effect of helical buckling and axial friction is the main factor constraining the rotary extending limits.Therefore, the extending limits caused by minimum axial force are very close to that by maximum torque on the starting point of the horizontal section.To further increase the extending limits, inhibiting helical buckling on the horizontal section and increasing the rated torque on the rotary table are indispensable.
Figure 6 shows the extending limits under different friction factors.The results show that the extending limits increase with the decrease of friction factor.Since the friction factor is decreased, the axial force transfer loss is also decreased.The decrease of axial force loss weakens the helical buckling phenomenon on the helical buckling section, which decreases the contact force and further decreases the axial friction and torque loss.Therefore, the increasing rate of extending limits becomes larger with the decrease of friction factor shown in Figure 6.
Figure 7 shows the extending limits of horizontal wells under different drill rigs and the results indicate that drill rig capability affects vertical extending limits a lot.This is because the minimum axial force  hs  which is closely related to the rated axial force of the drill rig is the main factor for wellbore extension.For shallow wells, drill rig capability affects the extending limits very little, for the main factor  hs  is nearly irrelevant to the drill rig capacity.However, the maximum torque  hs  which is close to the rate torque of the drill rig constraints the rotary extending limits when the rated torque of the drill rig is small.Therefore, axial friction    and helical buckling are the primary factors and drill rig capacity is the secondary factor constraining the wellbore extension in shallow horizontal drilling.
Figure 8 shows the extending limits of horizontal wells under different rock breaking threshold values.The results indicate that the horizontal extending limits decrease with the increase of rock breaking threshold values.The rock breaking thresholds correspond to the axial forces on the bit  he , and the change of  he is proportional to that of the horizontal extending limits  ℎ shown in (B.10) when  he >  ℎhel .Therefore, drill bits which match underground rock which means  ths is small should be adopted to increase extending limits.
Figure 9 shows the extending limits of horizontal wells under different tubular strengths.When the vertical depth is large, pick-up is the operation condition corresponding to the extending limits and tubular strength is the constraint condition.However, when the vertical depth is small, slackoff is the operation condition corresponding to the extending limits and high friction force is the constraint condition.Therefore, tubular strength affects vertical extending limits a lot but does not affect horizontal extending limits.
Figure 10 shows the extending limits of horizontal wells under different tubular diameters.Increasing tubular diameter can inhibit tubular buckling and reduce axial friction on the helical buckling section, so the extending limits increase with the increase of tubular diameter.However, the axial friction in the nonbuckling section can also be increased with the increase of tubular diameter, so the increasing rate of the extending limits becomes lower.With the increase of tubular diameter, the effect of inhibiting tubular buckling becomes smaller and meanwhile the effect of increasing axial friction in the nonbuckling section becomes larger.When the tubular diameter is rather large, the extending limits become smaller with the increase of tubular diameter.Therefore, several kinds of tubular string components should be adopted and optimized to increase extending limits.

Optimal Design Method of Down-Hole Tubular Strings
According to the above studies, tubular strings of multiple types and sizes are superior to that of single type and size.
The previous studies on the section of tubular strings are most empirical or too complicated for application.In this section, a more high-efficiency design method of tubular strings is proposed.
Considering that an integral tubular string in a horizontal well can be divided into many local tubular strings, design problem for an integral tubular string can also be correspondingly divided into many subdesign problems of local tubular strings.The local and integral design problems of tubular string components are studied as follows.

Local Design Method of Tubular Strings.
The above analyses show that slack-off is the most common operation condition which corresponds to the extending limits in horizontal drilling.Therefore, the optimal design of downhole tubular components mainly considers the slack-off case.Rotating tubular string can increase the extending limits a lot, for friction force vector is moved from axial direction to circumferential direction in the rotary mode and the axial friction factor decreases a lot.Some other methods to reduce friction factor such as lubricate drilling mud and installing drag reducing tools also can increase the extending limits.Figures 4 and 5 indicate that the occurrence of helical buckling on the horizontal section leads to the extending limits approaching the yield values.Therefore, inhibiting helical buckling on the horizontal section is an important factor to increase the extending limits.According to (B.10), increasing the maximum axial force on the starting point of the horizontal section ( hs  ) also contributes to the increase of extending limits.Therefore, there are four principles which can be adopted to increase the horizontal extending limits: (1) rotating tubular string, (2) reducing friction factor, (3) inhibiting helical buckling, and (4) increasing the pushing force  hs .In this paper, the design method of tubular string components based on principles ( 3) and ( 4) to improve the mechanical extending limits is focused.
For a local tubular string, the design model for maximizing the axial force and torque transfer efficiency is expressed by where (/)(  /) is the axial force (torque) transfer efficiency on the tubular string.The term  tub presents the relevant parameters for specified tubular strings.Considering that only several types (such as   in ( 19)) of drill strings are manufactured,  tub distributes discretely.The term  tub is depicted by the outer diameter, inner diameter, Young's modulus, and so on of the drill string.The optimal results ( tub * ) of ( 19) can be obtained with exhaustion method.

Integral Design
where  vs −  he is the axial force transfer from the ground to drill bit,  vs −  he is the torque transfer from the ground to drill bit, and Δ  is the length of the th local tubular string.
Note that, there are  design variables in ( 20) and there are   possible values for the design variables.If the exhaustion method is used to obtain the optimal solutions, the local mechanical model is called (  )  times, which takes two much calculation time when  is large.
Take the th local tubular string adjacent to the drill bit as the research object.Considering that the axial force and torque on the right end of the th local tubular string, which are also the axial force and torque on the drill bit, are given in advance, the optimal design results can be obtained with the local design model on the th local tubular string.After the optimal results for th local tubular string are obtained, the axial force and torque on the left end of the th local tubular string, which are also the axial force and torque on the right end of the ( − 1)th local tubular string, are also calculated.Then, the optimal results for the (−1)th local tubular string can be calculated similar to the th local tubular string.The above procedure is conducted from the Nth local tubular string near the drill bit to the first local tubular string near the drill.Then, the integral design model of tubular string components is expressed by Equation ( 21) means that the local design problem of th local tubular string is affected by the ( + 1)th to th local tubular strings but irrelevant to first to ( − 1)th local tubular strings.The integral design problem is equivalent to sequential local design problems (19) from th to first local tubular string.That means the integral design problem is a simple accumulation of  local design problems.Then, the local mechanical model is called (  ) ×  times to obtain the optimal solution, which greatly decreases the calculation complexity.

Case Studies.
The above models are applied to a virtual case study.A horizontal well is drilled, of which the kick-off depth is 2500 m.The other parameters have been given in Section 2.5. Figure 11 shows the axial force distributions for different 5  drill string components including SDP, HWSDP (heavy weight steel drill string), and ADP.With the increase of well depth, the axial force firstly increases and then decreases.The left end represents the axial force on hook and the right end the axial force on drill bit.When the horizontal section length reaches 1960 m, the axial force on hook increase to 0, which means the drill string cannot move forward further and reaches its extending limit, namely, 1960 m. Figure 12 shows the buckling state distributions for different drill string components, in which "0" means nonbuckling, "1" sinusoidal buckling, and "2" helical buckling.For the ordinary drill string, helical buckling occurs on the vertical wellbore and the starting part of the horizontal wellbore.If HWSDP is adopted and installation position is optimized, the axial force on hook decreases by about 1.4 * 10 6 N. Figure 13 shows the tubular type distributions along the wellbore, in which 1 means SDP, 2 HWSDP, and 3 ADP.The results show that HWSDP should be installed on vertical wellbore, build-up wellbore, and the starting section of the horizontal wellbore to inhibit helical buckling and increase the maximum axial force on the starting point of the horizontal wellbore.Then, the horizontal wellbore extension can exceed the initial limit 1960 m and become larger.If ADP is further adopted, the axial force on hook can be decreased further.ADP should be installed on the end part of the horizontal wellbore to reduce the drill string weight and axial friction force [19].Figure 14 shows the axial forces on hook under different horizontal wellbore lengths for different drill string components in the sliding drilling mode.The axial forces on hook increase faster and faster with horizontal wellbore length due to the existence of helical buckling.The horizontal extending limit is equal to the horizontal wellbore length at which the axial force on hook is 0. When the horizontal wellbores approach the extending limits, the axial forces on the hook change a lot, but the horizontal extension lengths nearly keep unchanged.This phenomenon is also called "lock-up" in drilling engineering.If the weight of hook is further applied on the drill string, which means the axial forces on the hook are further increased, the horizontal wellbore cannot be further extended.To extend the horizontal extending limit, drill string components should be optimized.The horizontal extending limits for the 5  SDP, SDP + HWSDP, and SDP + HWSDP + ADP are, respectively, about 1900 m, 3100 m, and 3400 m and for 5.5  SDP, SDP + HWSDP, and SDP + HWSDP + ADP are, respectively, about 2400 m, 3700 m, and 4500 m.Figures 15 and 16, respectively, show the axial forces on hook and torques on rotary table under different horizontal wellbore lengths for different drill string components in the rotary drilling mode.The results show that the horizontal extending limits due to torques on rotary table are smaller than that due to axial forces on hook.This is because helical buckling is serious when the horizontal wellbore approaches its extending limit, which further increases the contact force between the drill string and wellbore.The increase of contact force further increases the axial friction and torque loss on the drill string.When the axial force on hook is 0, the torque loss is also very large which usually exceeds the rated torque on the rotary table for a certain drill rig (such as 80 kN⋅m in Figure 16).The horizontal extending limits for the 5  SDP, SDP + HWSDP, and SDP + HWSDP + ADP are, respectively, about 6200 m, 8200 m, and 8900 m and for 5.5  SDP, SDP + HWSDP, and SDP + HWSDP + ADP are, respectively, about 7200 m, 8900 m, and 10600 m.The results indicate that the horizontal extending limits can be greatly increased by optimizing drilling mode and drill string components.By adopting rotary drilling mode and using 5.5  SDP + HWSDP + ADP, the horizontal extending limit can break through 10000 m in this case study.

Discussions on Design Method of Down-Hole Tubular Strings
The above studies show that extending limits in horizontal drilling are affected by kinds of factors.To improve extending limits, rotary drilling mode, drill rig of high rated torque on rotary table, and high-strength drill string, an appropriate selection of drill string components is indispensable.Among the above factors, high friction force is the primary factor constraining the horizontal wellbore extension and the optimal design of tubular string components is the more effective method to improve extending limits.
Compared with SDP, HWSDP can transfer more weight in the vertical wellbore and inhibit buckling in the horizontal wellbore, but it also leads to more tubular string weight and axial friction force on the nonbuckling section in the horizontal wellbore.Therefore, SDP should be used on the nonbuckling section and HWSDP should be installed on the buckling-prone section.Compared with SDP, ADP can decrease the tubular string weight and axial friction force on the nonbuckling section.However, ADP tends to buckle on the high axial force section, since its critical buckling load is smaller than that of SDP.So, ADP should be installed on the nonbuckling section to reduce the tubular string weight.However, the buckling distribution is unknown before the design of tubular string components and must be calculated with the integral mechanical model.So, one possible way is that the initial drill string components are given, integral mechanical model is called, and then the design model is called.After the drill string components are updated with the design model, the integral mechanical results must be recalculated with the integral mechanical model.Next, the design model is called to obtain new design results of drill string components.The process is repeated several times until the difference of the previous and current design results is small.This method is time-consuming and inefficient, which limits the application of this method.In this paper, the integral mechanical analysis and tubular design are combined together and called only once according to the objective function in horizontal drilling, and the integral mechanical results and design results of drill strings can be obtained simultaneously.Therefore, the new method is rather fast and effective, which can be easily applied to drill design in field operation.

Conclusions
In this paper, a theoretical model for predicting mechanical extending limits in horizontal drilling is built and the design method of tubular string components for improving extending limits is presented.The main conclusions are summarized as follows: (1) The horizontal extending limits are underestimated a lot when the effects of friction force on critical helical Mathematical Problems in Engineering buckling load are neglected.Therefore, the effects of friction force on critical buckling load and load transfer should be both included in the prediction model of extending limits.
(2) The distributions of horizontal extending limits are closely related to the combination of Young's modulus of the tubular string and tubular string weight per unit length.The horizontal extending limits increase faster with the axial force on the starting point of the horizontal wellbore but the final extending limits are smaller for the tubular strings of smaller modulus-weight ratio.Therefore, tubular strings of smaller modulus-weight ratio should be selected for short designed horizontal wellbore and that of larger modulus-weight ratio for long designed horizontal wellbore.
(3) Sliding slack-off is the main limit operation condition and high axial friction factor is the main factor constraining the horizontal extending limits.Rotating drill strings and optimizing the arrangement of multiple kinds of tubular string components in horizontal drilling can greatly increase the horizontal extending limits.
(4) Horizontal extending limits for tubular string components of multiple sizes and types are obviously larger than that of single size and type.The design results indicate that tubular strings of larger diameters and Young's modulus should be installed on the vertical wellbore and the starting section of the horizontal wellbore and tubular strings of smaller diameters and Young's modulus on the end section of the horizontal wellbore.
(5) The optimal design model is called only once to obtain the design results, which greatly increases the design efficiency compared with the traditional design method.The next step is to apply the proposed method to drilling practice.
ve =  hs + Δ   vs =  hs + Δ  + Δ V , (B.5) where Δ V and Δ  are the torque losses in the vertical and build-up sections.
According to the constraint conditions of the rated torque of the drill rig and tubular yield strength, the maximum torque at the starting point of the horizontal section is calculated by where the allowable shear stresses at the starting point of the horizontal section, the end point of the vertical section, and the starting point of the vertical section are, respectively, calculated by (B.7) Then, the extending limit of the horizontal section during pick-up is calculated by (A.When the axial force on the top of the vertical section reaches the minimum slack-off force of the drill rig, the down-hole tubular string reaches its extending limit.Then, the expression − hs =  rig  is satisfied and the maximum axial force on the starting point of the horizontal section is calculated by For tubular strings under slack-off process in horizontal wells, it is the maximum axial force  hs  rather than the minimum axial force  hs  which limits the horizontal wellbore extension.This is to say, the minimum axial force  hs  and maximum torque  hs  loss are the available constraint factors in horizontal drilling.Further introducing the axial force transfer in the horizontal section, the extending limit of the horizontal section is calculated by (B.10) Note that the extending limit in the slack-off operation due to the maximum torque is identical with (B.8).

ADP:
Aluminum drill string CDP: Composite drill string HWSDP: Heavy weight steel drill string SDP: Steel drill string   : A r e a o f t u b u l a r s t r i n g c r o s s s e c t i o n , m 2   ,   , and   : Diameters of wellbore, tubular string body, and connector, m  1 ,  2 : The critical vertical depth within which the tubular string in vertical wellbore is in the nonbuckling state in the slack-off mode, m; the critical vertical depth exceeding which helical buckling occurs on the horizontal section, m  tub : Relevant parameters of a specified tubular string : Bending stiffness of the tubular string, N⋅m 2 : Axial force on the tubular string, N  vs ,  ve : Axial forces at the top and bottom points of the vertical wellbore, N  hs ,  he : Axial forces at the starting and end points of the horizontal wellbore, N  hs  : The minimum axial force on the starting point of the horizontal section under which the tubular strings in the vertical-build wellbore are not yielded, and the axial tensile force on ground is not larger than the rated axial force on hook in the pick-up operation, N T u b u l a r s t r i n g w e i g h t p e r u n i t length, N/m   : Radial clearance between the tubular string and wellbore, m : C u r v a t u r e r a d i u s o f t h e build-up wellbore, m : Axial distance along the wellbore axis, m  rig  ,  rig  : Maximum and minimum pick-up forces of the drill rig, N⋅m : I n c l i n a t i o n a n g l e o f t h e wellbore, rad   : The amendment factor of critical helical buckling load due to the existence of friction factor   : Bending curvature of the tubular string, m −1  1 ,  2 : The axial and circumferential components of the friction factor : Maximum equivalent stress on the tubular string, N/m   : Maximum bending stress on the tubular string, N/m 2 []: Allowable stress of tubular string failure, N/m 2   : A v e r a g e a x i a l s t r e s s o n t h e cross section of the tubular string due to axial force, N/m A l l o w a b l es h e a rs t r e s s e sa tt h e starting and end points of the vertical section and starting point of the horizontal section, N/m 2 .

Figure 2 :
Figure 2: Horizontal extending limits for different models.

Figure 3 :
Figure 3: Horizontal extending limits for different tubular strings.

Figure 4 :Figure 5 :
Figure 4: Extending limits of horizontal wells under sliding mode.

Figure 6 :
Figure 6: Extending limits of horizontal wells under different friction factors.

Figure 7 :F
Figure 7: Extending limits of horizontal wells under different drill rigs.

Figure 8 :
Figure 8: Extending limits of horizontal wells under different rock breaking threshold values.

Figure 9 :
Figure 9: Extending limits of horizontal wells under different tubular strengths.

Figure 10 :
Figure 10: Extending limits of horizontal wells under different tubular diameters.

Figure 11 :Figure 12 :
Figure 11: Axial force distributions for different drill string components.

Figure 13 :Figure 14 :
Figure 13: Tubular type distributions for different drill string components.

Figure 15 :Figure 16 :
Figure 15: Axial forces on hook for different drill string components in rotary drilling mode.

Table 1 :
Values of the relevant parameters for different tubular strings.

Table 2 :
Axial forces and extending limits when helical buckling occurs.

Table 3 :
Values of the relevant parameters in a case study.
Method of Tubular Strings.The design objective of an integral drill string is to maximum the axial force or torque from the ground to drill bit.As mentioned above, an integral tubular string is divided into  local tubular strings, so the axial force and torque transfer on an integral tubular string are the sum of those on  local tubular strings.Therefore, the objective function of the integral tubular string is the sum of those of  local tubular strings.Then, the integral design model of tubular string components is expressed by min  vs −  he (,  vs −  he ) ∀,  tub, ∈ { tub 1 ⋅ ⋅ ⋅  tub   } , Torque on the tubular string, N⋅m  vs ,  ve : Torques at the top and bottom points of the vertical wellbore, N⋅m  hs ,  he : Torques at the starting and N⋅m  rig : Rated torque of the drill rig, N⋅m Δ V , Δ  :T o r q u e l o s s e s i n t h e v e r t i c a l and build-up wellbores, N⋅m  max,lat ,  max,inter ,  max,intra : Dimensionless maximum 2 [  ] vs , [  ] ve : Allowable axial stresses at the starting and end points of the vertical section, N/m 2   : Yield strength of the tubular string, N/m 2   : M a x i m u m s h e a r s t r e s s o n t h e tubular string, N/m 2 [  ] vs , [  ] ve , [  ] hs :