Study on open‐hole extended‐reach limit in horizontal drilling with wavy wellbore trajectory

Drilling a series of sinuous wellbore trajectories has emerged as a promising operation, offering significant advantages in enhancing reservoir contact. Based on various development cases (Zili and colleagues), it has been observed that horizontal wells with wavy wellbore trajectories tend to exhibit higher production rates. However, extending the length of these wavy trajectories requires careful consideration in terms of safe drilling practices, as the pressure management associated with these intricate trajectories has been seldom explored. In this study, we establish a modified model for annular pressure to predict and assess the open‐hole extend‐reach limit of horizontal wells with wavy wellbore trajectories. The introduced model incorporates dry friction force and annular geometry influenced by cuttings bed, making it more accurate than traditional models based on experimental observations. The wavy wellbore trajectory comprises three types: the up‐dip wellbore, the down‐dip wellbore, and the complex wellbore. Overall, the horizontal interval limits for the down‐dip wellbore and up‐dip wellbore are smaller than those of the corresponding smooth wellbores under varying factors. Notably, the complex wellbore trajectory offers advantages in navigating formations with narrow pressure windows compared to conventionally horizontal wellbores (smooth wellbores), thanks to its longer extended‐reach limit.

practices.Therefore, in this paper, we adopt a quantified definition for wavy wellbores, wherein the wellbore tortuosity undergoes periodic variations along the trajectory, 4 as shown in Figure 1.The wellbore tortuosity references to the summation of the total curvature of the wellbore, which can be defined by inclination angle changes, azimuth angle changes, or overall angle change rate. 5The wavy wellbore trajectory consists of three types: the up-dip wellbore trajectory, the down-dip wellbore trajectory, and the complex wellbore trajectory. 4his classification standard is adopted in this paper.
Owing to the advanced geosteering, drilling a series of wavy wellbore trajectories with real-time data is a promising operation in the productive zone, which can improve the pay zone exposure and avoid dangerous interlayers.However, the unconventional wellbore trajectory is still controversial for the production stimulation.On-site drilling examples show that the well has 1340% of the initial production of the conventionally horizontal wellbore, which uses the wavy wellbore for one-time adjustment and recovery of the trajectory. 1 Based on the transient multiphase analysis, the high prodectivity can avoid severe slug for the toe-down or toe-up wellbore trajectories during the gas lift. 6Upon further investigation, Tran et al. 7 propse that the up-dip wellbore trajectory and complex wellbore trajectory have higer production relative to toe-up and abosolutely horizonal trajectories respectively, if the undulations of trajectories are large enough.The toe-down wellbore trajectory is more conducive to reduce slug and increase production based on a analysis of some production data. 8hetib et al. 9 deduce that the complex wellbore trajectory may lead to severe slug during the gas lift through a towphase flow experiment.Khetib 10 propose that the amplitude and frequency of undulation of the wavy trajectory also have complex effects on production during oil recovery.
Guided by geological directional tools, some wells in China's Sichuan and Xinjiang regions have adopted twodimensional wavy wellbore trajectories. 4However, in drilling or completion designs, there is often a tendency toward conservatism, including the use of additional casing layers and conservative drilling parameters to avoid complications in downhole conditions.The reason behind this lies in the lack of comprehensive research on the extened-reach limits of wavy wellbore trajectories.Gao et al. 11 propose a systematic theory of the extendreach limit including the hydraulic extend-reach limit, mechanical extend-reach limit, and open-hole extendreach limit.Among various extened-reach limits, the open-hole extension limit is often the most crucial onsite, as it is closely related to common downhole complexities such as wellbore leakage and instability of the wellbore wall.In this theory, the open-hole horizontal is the maximum measured depth of an horizontal well when the drilled formation is fractured, and it primarily depends on the annular pressure and the fracture pressure.For coventional horizontal wells, there has been some research on the open-hole extended-reach limit.The conditions of offshore drilling and shale drilling are introduced to the theory of open-hole extend-reach limit. 12Li et al. 13 proposed an open-hole extend-reach limit model considering offshore reelwell drilling conditions.The open hole extended-reach limit model considering the formation and decomposition of hydrate is established by Chen et al. 14 The widely adopted model for the open-hole extendedreach limit in horizontal drilling, accounting for the cuttings bed influence, was initially proposed by Li et al. 15 This model incorporates the cuttings bed coefficient 15,16 and the drilling string eccentricity coefficient to adjust the pressure within the concentric annulus.7][18][19] The impact of cuttings bed on annulus geometry is considered based on the method of Li. 20,21 However, the discussion regarding the accuracy of the model calculation has not garnered sufficient attention.Through a series of calculations, we find a significant error rate resulting from the multiplication of the two coefficients with the pressure gradient.Consequently, a more precise annular pressure model should be proposed to quantify the influence of the cuttings bed on the pressure in eccentric annuli, which holds paramount importance for the accurate prediction of the open-hole extended-reach limit.
In this paper, we introduce a modified model for annular pressure to predict the open-hole extended-reach limit in horizontal drilling with a wavy wellbore trajectory.The inclusion of annular geometry, which takes into account the cuttings bed and dry friction force, is essential for accurate annular pressure prediction within this model.The validity of the predicted annular pressure gradients is confirmed through experiments conducted by Tong et al. 22 To demonstrate the accuracy of the present model, the model of Li et al. 15 is introduced as a contrast.We conduct a sensitivity analysis on key drilling parameters, including flow rate, rate of penetration, drill pipe rotation speed, and drilling fluid density, to illustrate the impact of the wavy wellbore trajectory on the extended-reach limit.This study serves as a foundational contribution to drilling design and pressure management in horizontal drilling scenarios characterized by a wavy wellbore trajectory.

| WAVY WELLBORE TRAJECTORY
The wavy wellbore can be summarized as a special type of "horizontal wellbore" with the wellbore tortuosity changed periodically. 4The wellbore tortuosity can be quantitatively defined by inclination angle changes, azimuth changes, or overall angle change rate. 5For two-dimensional wellbore trajectory, the wellbore tortuosity can be described by the inclination angle appropriately.There are three types of wavy wellbore trajectories including the "down-dip wellbore," "up-dip wellbore," and "complex wellbore." 4The inclination angles of the down-dip wellbores are less than 90°, and the updip wellbores are on the contrary.The "complex wellbore" consists of two types of wavy wellbore sections.In this study, the three types of wavy wellbores are discussed, in which the build-up rate of 1°/30 m as the default is practical for on-site drilling operations.

| PREDICTION MODEL OF OPEN-HOLE EXTENDED-REACH LIMIT
The theory of the open-hole extended-reach limit is originally proposed Gao et al. 11 According to this theory, the open-hole horizontal section can extend forward until the bottom hole of the wellbore is fractured.This implies that wellbore cementation becomes imperative to achieve a greater measured depth when the maximum annular pressure in the wellbore equals the fracture pressure of the formation.The pressure supplied by the pump just offsets the annular pressure loss in the drilling fluid circulation, which is a default setting.In the previous study, the extended section of the wellbore is considered to be absolutely horizontal, and the bottom hole pressure is the maximum annular pressure.However, the location of maximum annular pressure is not necessarily at the bottom of the wellbore under the influence of a wavy wellbore trajectory.For the up-dip wellbore trajectory section, the bottom hole pressure decreases as the measured depth increases in the drilling process.Consequently, ensuring that the drilled formation remains stable and does not collapse becomes an additional constraint in defining the extended-reach limit for a wavy wellbore trajectory.In summary, the annular pressure should be greater than the collapse pressure of the formation and less than the fracture pressure of the formation, which can be expressed as: Some scholars have defined a constraint condition for the wellbore cleanliness, considering a dimensionless cuttings bed height of less than 5% or 10%. 20Within this threshold, the wellbore is deemed to be clean, and drilling parameters such as flow rate, drilling pipe revolving speed, and other related factors are constrained to specific values to ensure hole cleaning.This study specifically explores the impact of a wavy wellbore trajectory on the extended-reach limit.The key characteristic of a wavy wellbore trajectory lies in the differentiation of cuttings bed distribution.The additional pressure loss induced by the cuttings bed becomes a significant factor with substantial implications for predicting the extended-reach limit.Hence, the cuttings bed height does not directly serve as one of the constraint conditions in this study; rather, it is influenced by the drilling parameters.
In this study, the steady-state cuttings bed height during drilling process is calculated as in Equation (2), 21,23 which is an empirical formula for regression based on experimental data. 23Compared with the correlation proposed by Wang et al. 24 the model in this study quantifies the effect of inclination angle on cuttings bed height.(2

( ) ( ) ( )
where the effective viscosity μ e is calculated by use of the method of generalized liquidity index 25 ; v a , the annular velocity of the drilling fluid without cuttings bed, m/s; R p , the rate of penetration, m/s; N, the rotational speed of the drilling pipe; α, the inclination angle, degree.

| MODIFIED MODEL FOR ANNULAR PRESSURE
The bottom hole pressure under the influence of the cuttings bed should be considered, which is crucial for the accurate calculation of the extended-reach limit.For the section without cuttings bed, the model of annular pressure is very common and widely studied.However, the annular pressure model considering cuttings bed is complex and seldom applied in engineering.Typically, the conventional annular pressure model is adjusted using correction coefficients related to the cuttings bed or eccentricity, determined through experimental regression.However, the calculation accuracy of these existing models is often less optimistic, particularly when applied to sections with a cuttings bed.Consequently, this study introduces a novel model for evaluating annular pressure in the presence of a cuttings bed.

| Section with cuttings bed
The influence of cuttings bed on annular pressure drop includes gravity and fluid friction stress.The geometry of the annulus changed by the cuttings bed, such as the hydraulic radius, wetted perimeter, and cross-section area are different from those of the clean annulus.Based on some experimental data, it is found that the annular pressure drops caused by the cuttings bed are greatly affected by the inclination angle of the wellbore.The pressure loss per unit length can be obtained by the difference between the measured total pressure drop and gravitational pressure drop, as shown in Equation (3).A set of experimental results from University of Tulsa et al. 22 demonstrate that the pressure loss decreases with the increase of cuttings bed height when the inclination angle changes from 90°to 75°or 60°, as shown in Figure 2. When the inclination angle is 60°, the cuttings bed height increases greatly, but the annulus pressure loss decreases.
In general, the pressure loss caused by friction stress is very little affected by the inclination angle.And, the higher cuttings bed means the greater pressure loss.The existing models cannot explain this experimental phenomenon.There must be an additional stress to decrease the pressure drop caused by the gravity of the cuttings bed.Therefore, a dry friction force from the theory of solid-liquid two-phase flow is introduced to calculate the pressure drop when the cuttings bed exists in annuli, in which the dry friction force can cancel out some gravity of the cuttings bed when the inclination angle is less than 90°.
When the inclination angle is less than 90°, the pressure gradient can be expressed as Equation (4).
When the inclination angle is greater than or equal to 90°, the pressure gradient can be expressed as: where θ is inclination angle; g is gravitational accelera- tion; A ann is annular cross-sectional area.
The dry friction force F dry between the bed and wall of the drill pipe and the hole depends on positive pressure and dry friction factor, which can be expressed as 26 : The method to calculate the pressure loss employs a new hydraulic radius and wetted perimeter because of the geometry of the annulus changed by the cuttings bed, as shown in Figure 3.
The hydraulic diameter depends on the area of the flow area and the wetted perimeters considering the presence of the cuttings bed, which can be expressed as: Shear stress for the interface between the liquid phase in the flow area and wall including borehole wall and drill pipe wall can be expressed as: where f f is the friction factor proposed by Doron et al.Shear stress at the interface between the liquid phase and cuttings bed is: The f sbsd is the interfacial friction factor, 28 which can be expressed in the following equation: The bed roughness is determined by the particle diameter and reposed angle, which can be expressed as:

| Section without cuttings bed
When there is no cuttings bed in the annulus, the interfacial shear stress τ sbsd and dry friction force no longer exist.The hydraulic radius D hy is equal to The pressure loss gradient is calculated similarly to the conventional method, which can be expressed as: The other calculation steps are the same as in Section 4.1.

| Verification of the model
The data obtained from the large-scale flow-loop system at the University of Tulsa is utilized to validate the modified model. 22The test section consists of a 10 m annulus, where the bed height of cuttings and the annular pressure gradient are measured using a data acquisition system.Four pressure transmitters are strategically placed at intervals of 1.83 m along the test section to capture pressure gradients.The entrance length, allowing for flow development, is set at 3 m.The measured pressure gradient values represent the averages over multiple experimental annular intervals, demonstrating convincing accuracy.However, the slight variation in cuttings bed height across different locations within the experimental annulus introduces an inherent uncertainty to the experiment.
The inclination angle of the test section's end can be adjusted using a hoisting facility.To simulate cuttings transport in the wellbore under drilling conditions, a cuttings injection facility, a mud circulation system, and a cuttings collection facility are employed.The cuttings bed is initially deposited evenly, with a predetermined height of 50% before the experiment.During the experiment, conducted under a steady flow rate of mud and solid particles, the cuttings bed in the test section undergoes flushing to replicate the process of bed erosion in drilling operations.For additional details, refer to Tong et al. 22 The annular pressure model applied to predict the open-hole extended-reach limit is proposed by Li et al. 15 first, in which the influence of the drilling string eccentricity and cuttings bed are considered.
Later scholars in this field have mostly adopted similar methods.Therefore, we compare the Li model with the new model in this paper with experimental data, respectively.The results indicate that the new model has higher accuracy.As shown in Figure 4, the results calculated by the present model are very close to the experimental observations.Compared with the present model, the pressure gradients predicted by the Li model are too small, of which the error rate is more than 50%.Since the drilling fluid displacement and cuttings flow rate is constant in the experiment, the cuttings deposition increases when the inclination angle decreases to 75°, and the pressure gradients predicted by the present model are more accurate as shown in Figure 5.The Li model is more accurate than the present model when the inclination angle decreases to 60°.The error rate of the present model is less than 3% as shown in Figure 6, which is acceptable.The Li model has advantages in pressure prediction for small inclination angles and high cuttings bed height.In general, the build-up wellbore interval with smaller inclination angles is shorter than the horizontal wellbore interval.Therefore, the present model is more suitable for predicting the open-hole extended-reach limit of the wavy wellbore trajectory.
The Reynolds number range of experimental data applied to verify the model is 3547.4~4795.7.It is clear that the predictions of the model are sufficiently reliable in this range.Although no other experimental data has been found to verify the model, the friction coefficients as key parameters is calculated by the widely used and verified formulas in the field of drilling engineering.In this study, there are three types of wavy wellbore trajectories including the down-dip wellbore, the up-dip wellbore, and the complex wellbore, which are adopted and analyzed in this chapter.The casing program of the example well is shown in Table 1.The rheological properties are shown in Table 2.The open hole of the well is thought to employ the wavy wellbore trajectory in this study.To demonstrate the effect of the wavy wellbore trajectory on the open-hole extended-reach limit, the smooth wellbore trajectory is introduced as a contrast.The smooth trajectory has the same horizontal and vertical displacement as the wavy wellbore trajectory, but there is no fluctuation in the inclination angle.For the complex wellbore trajectory, the smooth wellbore is the same as the absolutely horizontal wellbore.For the up-dip wellbore terajectory, the smooth wellbore terajectory can be also called toe-up wellbore terajectory.For the down-dip wellbore terajectory, the smooth wellbore terajectory can be also called toe-down wellbore terajectory.All wellbore trajectories are shown in Figure 7.In this chapter, the build-up rate is 1°/30 m for the wavy wellbore interval, and 5°/30 m is employed for the build-up interval.The wavy wellbore interval and smooth wellbore interval can be thought to be special absolutely horizontal wellbores in horizontal drilling.Therefore, the "horizontal-interval limit" is employed to represent the length of the wavy wellbore interval or smooth wellbore interval in this study.

| Down-dip wellbore
For the down-dip wellbore interval in this case, the inclination angle ranges from 80°~90°.The inclination angle of the smooth wellbore is 85°.Overall, the horizontal-interval limit of the down-dip wellbore is smaller than that of the smooth wellbore under different factors.The cuttings bed is unevenly distributed in the down-dip wellbore because the trajectory fluctuates.Therefore, the pressure loss of the down-dip wellbore is slightly larger than that of the smooth wellbore and the down-dip wellbore has a smaller horizontal-interval limit.
As shown in Figure 8, the horizontal-interval limit first decreases, then increases, and then decreases with the increase of the flow rate, which can be divided into three stages.In the first stage, the increase of the flow rate enhances the pressure loss in the flow area and the reduction of the cuttings bed is not obvious because of the smaller flow rate.The critical flow rates between the first stage and the second stage are 22.7 L/s and 23.2 L/s for the smooth wellbore and down-dip wellbore respectively.For the second stage, the reduction of the cuttings bed is enough to dominate the decrease of the pressure loss, and the effect of the increase of flow rate is offset by the larger flow area.Therefore, the horizontal-interval limit of the down-dip wellbore decreases with the increasing flow rate, of which the critical flow rate is 32.4 L/s.
As shown in Figure 9, the horizontal-interval limit first increases, and then decreases with the increase of the drill pipe rotation speed.Too high or too low a rotation speed brings the horizontal interval limit of the wavy and smooth wellbores closer to each other.The larger drill pipe rotation speed would reduce cuttings bed height which means lower pressure loss caused by dry friction force and the larger rotation speed increases the flow rate in the flow area which increases the pressure loss by fluid flow.Therefore, the horizontal-interval limit rises first.When the drill pipe rotation speed exceeds 50 r/min, the reduction of As shown in Figure 10, the horizontal-interval limit decreases with the increase of the drilling fluid density.The horizontal-interval limit of the down-dip wellbore and smooth wellbore are very close.The regular fluctuation of the curve corresponds to the fluctuation of the wellbore trajectory.The rate of penetration affects the accumulation of the cuttings bed in annuli, which has an important influence on annular pressure.As shown in Figure 11, the horizontal-interval limit decreases with the increase in the rate of penetration.

| Up-dip wellbore
For the up-dip wellbore interval in this case, the inclination angle ranges from 90°~100°.The inclination angle of the smooth wellbore is 95°.The horizontal interval limit of the up-dip wellbore depends on the collapse pressure because gravity reduces the annular pressure in the interval of the up-dip wellbore.As shown in Figure 12, there are four stages for the horizontal-interval limit with the increase of the flow rate.In the first stage, the increase of the flow rate (<18.8L/s) enhances the pressure loss, which means a larger pump pressure and a larger annular pressure.The increasing annular pressure is useful for keeping the wall of the wellbore stable, under which the annular pressure is larger than the collapse pressure.For the second stage, the horizontal-interval limit exceeds the range of the axis, in which the flow rate is larger than 18.8 L/s and smaller than 28.3 L/s.Predictably, enough pump pressure can counteract the pressure caused by gravity, and the annular pressure can be hardly lower than the collapse pressure.When the cuttings bed height is reduced to small enough, the pressure loss decreases and the horizontal-interval limit decreases with the increasing flow rate (28.3 ~32.7 L/s) in the third stage.The increasing flow rate only enhances the annular pressure when the cuttings bed does not exist.Therefore, the horizontal-interval limit rises with the flow rate larger than 32.7 L/s.The results of the up-dip wellbore are opposite in trend to those of the down-dip wellbore, which is obvious in Figure 13.
As shown in Figure 14, the horizontal-interval limit increases with the increase of the drilling fluid density on the whole, because of larger annular pressure.However, the horizontal-interval limit increases more and more slowly with the increasing drilling fluid density.For corresponding smooth wellbore, the limit first increases and then decreases when the density reaches around 1550 kg/m 3  F I G U R E 13 The horizontal-interval limit of up-dip wellbore for different drill pipe rotation speeds.the cuttings bed is too small to affect the annular pressure, in which the gravity dominates.Therefore, the horizontal-interval limit of the smooth wellbore begins to decrease because of wall collapse.As shown in Figure 15, the horizontal-interval limit increases with the increase of the rate of penetration, because of larger annular pressure caused by larger cuttings bed height.

| Complex wellbore
For the down-dip wellbore interval in this case, the inclination angle ranges from 80°~100°.The inclination angle of the smooth wellbore is 90°.Overall, the trend of the horizontal-interval limit of the complex wellbore is almost the same as the up-dip wellbore in trend with the changes of the different factors.Because the limit of two kinds of wellbores depends on the formation fracture pressure.As shown in Figure 16, the horizontal-interval limits of the complex wellbore are larger than those of the smooth wellbore (horizontal wellbore) with low flow rates (<27.1 L/s), which means that the complex wellbore has advantages for pressure management.For high flow rates (>27.1 L/s), the smooth wellbore has larger horizontal-interval limits.As shown in Figure 17, the horizontal-interval limits of the complex wellbore are larger than those of the smooth wellbore with different drill pipe rotation speeds.The situation is most obvious at 40 ~80 r/min.
As shown in Figure 18, the lower drilling fluid density allows the complex wellbore to extend more than the smooth wellbore.As shown in Figure 19, the F I G U R E 15 The horizontal-interval limit of up-dip wellbore for different rates of penetration.
F I G U R E 16 The horizontal-interval limit of complex wellbore for different flow rates.
F I G U R E 17 The horizontal-interval limit of complex wellbore for different drill pipe rotation speeds.
| 1133 horizontal-interval limit increases with the increase of the rate of penetration, because of larger annular pressure caused by larger cuttings bed height.Overall, the horizontal-interval limit of the complex wellbore is larger than that of the smooth wellbore unless there are very few cuttings bed and the pressure loss caused by drilling fluid flow friction dominates the given pump pressure.

| CONCLUSIONS
1.The modified model of annular pressure is proposed, which has more accurate predictions than the previous model and can be employed to predict the extended-reach limit of the wavy wellbore more closely to the drilling site.The annular geometry induced by the cuttings bed and the dry friction force play a significant role in predicting both annular pressure and the extend-reach limit.2. In general, the horizontal interval limits for both down-dip and up-dip wellbores are smaller than those of their respective smooth wellbore counterparts across various influencing factors.However, when the larger drilling fluid density (>1450 kg/m 3 ) is applied, the relationship between the horizontalinterval limits of the up-dip wellbore and the smooth wellbore becomes intricate.The introduction of a complex wellbore trajectory has the potential to increase the extended-reach limit compared to a horizontal wellbore, except when the flow rate surpasses a certain threshold (>27.1 L/s).Drilling a complex wellbore trajectory has advantages for passing through formations with narrow pressure windows compared with the conventionally horizontal wellbore trajectory.3.If the formation fracture is under constrained conditions, such as in down-dip wellbores and complex wellbores, the horizontal interval limit undergoes three distinct stages of change with increasing flow rates.Conversely, the up-dip wellbore, constrained by collapse pressure, exhibits a four-stage limit evolution with rising flow rates, and this trend opposes that observed in down-dip wellbores and complex wellbores.The analyses of drill pipe rotation speed, drilling fluid density, and rate of penetration (ROP) also reveal discrepancies in results based on varying constraint conditions.In essence, the determination of the horizontal interval limit across different influencing factors hinges on the interplay between pressure loss in the cuttings bed and drilling fluid flow friction.4.This study holds significance for the well structure design of wavy wellbore trajectories.The precise prediction of the extended-reach limit serves as a fundamental criterion for minimizing drilling costs and ensuring safe drilling practices.

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I G U R E 2 Measured pressure loss under different inclination angles and cuttings bed.22 ZHANG ET AL.
α a is the contact angle of the bed layer with the hole wall, and β a is the contact angle of the bed layer with the pipe; the dry friction factor η = 0.1.The pressure loss gradient

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I G U R E 3 Geometry of the annulus.

F I G U R E 4
The pressure gradient with different cuttings beds under 90°inclination angle.F I G U R E 5The pressure gradient with different cuttings beds under 75°inclination angle.

F I G U R E 6
The pressure gradient with different cuttings beds under 60°inclination angle.T A B L E 1 Casing program during drilling.Pa cuttings bed is not obvious and the larger flow rate dominates the increase of the pressure loss.So, the horizontal-interval limit decreases with the increasing rotation speed.The reason for the fluctuation of the curve is that the decrease of the cuttings bed height and the increase of the annulus flow rate have different degrees of influence.

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I G U R E 7 The wavy wellbore trajectories and smooth wellbore trajectories.F I G U R E 8 The horizontal-interval limit of down-dip wellbore for different flow rates.F I G U R E 9 The horizontal-interval limit of down-dip wellbore for different drill pipe rotation speeds.
. The larger fluid density means larger gravity, smaller cuttings bed height, and larger fluid friction stress.When the fluid density increases to large enough, F I G U R E 10 The horizontal-interval limit of down-dip wellbore for different drilling fluid densities.F I G U R E 11 The horizontal-interval limit of down-dip wellbore for different rates of penetration.F I G U R E 12 The horizontal-interval limit of up-dip wellbore for different flow rates.

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I G U R E 14 The horizontal-interval limit of up-dip wellbore for different drilling fluid densities.