Mathematical Modeling of Friction Reduction in Drilling Long Horizontal Wells Using Smooth Catenary Well Trajectories

Abstract

Drilling long horizontal wells in naturally cracked/fractured unconventional shale gas/oil formations presents a huge challenge to the energy industry because of wellbore clogging complications that cause pipe sticking problems. This work proposes to use smooth catenary well trajectories to reduce drilling friction to mitigate these problems. A mathematical model was developed in this study for designing well trajectory profiles with a smooth transition from the kick-out point (KOP) to the catenary section. This model consists of closed-form equations for the radius of curvature and inclination angle in the catenary section. Using the radius of curvature at the top point of the catenary section to design the arc section below the KOP eliminates the trial-and-error procedure required for achieving the smooth transition between the two sections. The result of a field case study with Tuscaloosa Marine Shale (TMS) data shows that the drilling drag (hook load) can be reduced by 15% to 30% with the use of smooth catenary well trajectories to replace the conventional arc-type well trajectories. Model-calculated reduction in the hook load drops linearly with the horizontal borehole friction coefficient (clog indicator). The reduction increases non-linearly from 15% to 30% with drill collar weight increasing from 20 lb/ft to 92 lb/ft.

1. Introduction

The primary aim of engineering in well trajectory design is to significantly reduce drilling operation challenges and costs. One of the main issues in drilling operations is the loss of drag and torque caused by friction between the wellbore and the drill string. This friction is influenced by the well trajectory design. Several suggestions for optimizing well trajectory design involve calculating operational parameters such as path length, torque and drag, and friction between the drill string and wellbore wall [1,2,3]. The ideal well path trajectory is expected to reach the pay zone with the highest production rate while minimizing well path length, torque, and drag loss. Proper well trajectory design can minimize torque and drag loss, leading to significant cost savings and operational efficiency [4,5]. Additionally, friction-induced torque and drag loss are a primary concern during drilling operations [6,7].

The typical well trajectories in directional well design include L-shapes, also known as Build and Hold Trajectory, S-shapes (which include a vertical section, a kick-off point, a build-up section, a tangent section, and a hold and drop section), J-shapes (which include a vertical section, a deep kick-off, and a build-up section), and Horizontal Directional Wells (including a vertical section, a build section, and horizontal section). However, in most directional well trajectory designs, the inclination can be increased or decreased, creating a sine wave curve like a series of dog-leg shapes. This can result in the sticking of pipes and increased frictional forces between the wellbore and the drill string. The increased frictional forces will cause significant difficulties in freeing the pipes if they get stuck. The mechanical contact and friction force between the drill string and wellbore can be minimized if the wellbore trajectory takes a catenary shape [8]. When tension is applied to the drill string in case the catenary shape is used, the drill string tends to move away from the wellbore wall and suspend in the borehole. This is a special and exclusive characteristic of the catenary shape. Therefore, the drag (friction force) and torque are substantially decreased, and liberating stuck pipes is much more reasonable and straightforward. Due to not forming the sine wave curve while drilling in the catenary trajectory, the probability of pipes getting stuck is reduced.

Catenary designs have been studied in various industries, including railroad design [9], architectural design, and building construction [10,11], due to their ability to reduce stresses. A few drilling trials were made using the catenary trajectory in China [12]. Although the catenary trajectory design was proposed in the 1980s, the early attempts of this technology were not successful due to the absence of technology for dynamic building inclination angles. Another issue that limited the application of the catenary well trajectory is the lack of a design method that can give a wellbore section smoothly linking the kick-out point (KOP) to the catenary section. Numerical trials requiring computational power have been employed to obtain reasonable results [13,14,15,16]. A summary of trajectory design methods was presented by [17], covering arc curve, pendulum curve, catenary curve, and the modified catenary curve. Because the catenary profile has a starting point of non-zero inclination angle, the problem of designing a complete catenary profile with a smooth transition from the vertical wellbore section to the catenary section remains unsolved [18].

In this study, we developed an analytical method for designing a catenary well trajectory with a smooth transitional section between the KOP and the catenary section. The mathematical solution takes a closed form and does not require numerical trials to solve. A case study is presented to compare the frictional drilling drags in a smooth catenary well trajectory and in a conventional arc-type well trajectory. The result of the field data study with the TMS data shows that the drilling drag (hook load) can be reduced by 15% to 30% with the use of smooth catenary well trajectories to replace the conventional arc-type well trajectories.

2. Smooth Catenary Trajectory Design Procedure

A catenary is a curve that describes the shape of a flexible hanging cable/rope/cable in mathematics. Any freely hanging cable assumes this shape if the body is of uniform mass per unit of length and is acted upon solely by gravity. The long drilling string used in well drilling may be approximated by such a cable. If a wellbore is drilled along a catenary curve, it is expected that the contact between the drill string and wellbore is minimal.

Figure 1 shows a typical catenary well trajectory that involves a vertical section, an arc section, a catenary section, and a slant/horizontal section. The arc section is needed because the upper end of the catenary curve has a non-zero inclination angle. The arc section is used to provide a transition from the kick-off point (KOP), where the inclination angle is zero, to the beginning of the catenary curve. The radius of curvature of the arc section should be equal to that of the catenary section at the connection point so that the connection of the two sections is smooth. Equations for designing the smooth catenary trajectory are derived in Appendix A.

The procedure for designing a well trajectory with a catenary section is outlined as follows:

(1)

Design the horizontal/slant wellbore length based on the well productivity requirement.

(2)

Design the length of the catenary section by specifying $S_{end}$ and $V_{end}$.

(3)

Numerically solve for a-value from the following equation:

$$V_{end}={\displaystyle \frac{a}{2}}\left[e^{\left({\displaystyle \frac{-S_{end}}{a}}\right)}+e^{\left({\displaystyle \frac{S_{end}}{a}}\right)}\right]-a$$

(1)

where a is the intercept of the catenary curve with the y-axis,

S_end is the total horizontal displacement in the catenary section,

V_end is the total vertical displacement in the catenary section.

(4)

Calculate the radius of curvature (R_top) and inclination angle (θ_top) at the top end of the catenary section using the following equations:

$$R_{top}={\displaystyle \frac{{\left(1+\frac{1}{4}{\left[e^{\left(\frac{-S_{end}}{a}\right)}-e^{\left(-\frac{-S_{end}}{a}\right)}\right]}^2\right)}^{\frac{3}{2}}}{-\frac{1}{2a}\left[e^{\left(\frac{-S_{end}}{a}\right)}+e^{\left(-\frac{-S_{end}}{a}\right)}\right]}}$$

(2)

$${\theta }_{top}={\displaystyle \frac{\pi }{2}}-arctan \left\{-{\displaystyle \frac{1}{2}}\left[e^{\left({\displaystyle \frac{-S_{end}}{a}}\right)}-e^{\left(-{\displaystyle \frac{-S_{end}}{a}}\right)}\right]\right\}$$

(3)

(5)

Calculate the vertical displacement of the arc section above the catenary section using the radius of curvature at the top-end of the catenary section, i.e.,

$$V_{arc}=R_{top} sin\left({\theta }_{top}\right)$$

(4)

(6)

Determine the KOP by

$${KOP=V_{tgt}-V_{end}-V}_{arc}$$

(5)

(7)

Calculate the vertical and horizontal coordinates in the arc section using the following inclination angle build rate expressed by

$$B={\displaystyle \frac{5730}{R_{top}}}$$

(6)

where B is inclination angle build rate in degrees per unit length of arc.

(8)

Calculate the vertical and horizontal coordinates in the catenary section using the following inclination angle equation:

$$\theta ={\displaystyle \frac{\pi }{2}}-arctan \left\{-{\displaystyle \frac{1}{2}}\left[e^{\left({\displaystyle \frac{S-S_{end}}{a}}\right)}-e^{\left(-{\displaystyle \frac{S-S_{end}}{a}}\right)}\right]\right\}$$

(7)

(9)

Calculate the vertical and horizontal coordinates in the horizontal/slant section based on the final inclination angle.

3. Prediction of Drilling Drag

The drilling drag is defined as the maximum hook load when a drill string is in the upward motion during trip-out operations. The drag is due to the weight of the drill string and the friction force occurring in the contact area of the string and wellbore. Ref. [19] presented an analytical friction model for 3-dimensional arc-type well trajectory sections. Ref. [20] used a numerical model to compute wellbore frictions. Applications of these models are limited due to their complex mathematical nature. Figure 2 illustrates forces acting on a simplified drill string in a 2-dimentional horizontal well trajectory involving an arc section, where W_v, W_c, and W_h are weights of vertical, curve, and horizontal sections, respectively; N_c and N_h are borehole contact forces in the curve and horizontal sections, respectively; and f_c and f_h are frictional forces in the curve and horizontal sections, respectively.

Statics gives the following relation for predicting the tension force at the beginning of the horizontal section:

$$F_{\pi /2}=\mu W_h$$

(8)

Because the friction force in the curve section is inclination-angle-dependent, the tension force in the curve section is formulated by integration shown in Appendix B:

$$F_{{\theta }_1}=F_{{\theta }_2}+w_{c}R\left[sin\left({\theta }_2\right)-\mu cos\left({\theta }_2\right)-sin\left({\theta }_1\right)+\mu cos\left({\theta }_1\right)\right]$$

(9)

If the curve section is only an arc of a quarter circle where ${\theta }_1$ = 0 and ${\theta }_2$ = π/2, this equation degenerates to

$$F_0=F_{\pi /2}+w_{c}R\left[1+\mu \right]$$

(10)

If the curve section contains an arc section between the KOP and a catenary section, the tension at the top of the catenary section should be calculated based on the free-body diagram shown in Figure 3. The force balance in the horizontal directions gives

$$F_c={\displaystyle \frac{F_{\pi /2}}{sin\left({\theta }_c\right)}}$$

(11)

The tension force at the KOP can be obtained from Equation (9):

$$F_0=F_c+w_{c}R\left[sin\left({\theta }_c\right)-\mu cos\left({\theta }_c\right)+\mu \right]$$

(12)

The tension force at the surface (hook load) is expressed as

$${T=W_v+F}_0$$

(13)

4. Field Data Study

The Tuscaloosa Marine Shale (TMS) across Louisiana and Mississippi in the U.S. is a sedimentary formation consisting of organic-rich, fine-grained sediments deposited during the Upper Cretaceous with later-formed natural cracks. The TMS thickness ranges from 500 ft in southwestern Mississippi to more than 800 ft in southeastern Louisiana at depths between 11,000 ft and 15,500 ft. The Middle TMS is composed of dark gray, fissile, and sandy marine shale with a matrix porosity between 2.3% and 8.0% and a permeability between 0.01 md and 0.06 md [21]. Eighty-seven horizontal wells were drilled with arc-type trajectory and completed with multi-stage hydraulic fracturing technology between 2012 and 2014. Some wells produced oil at the initial rates of more than 1000 stb/d, while the three-year oil production of TMS wells was generally comparable to that of Eagle Ford Shale wells [22]. The drilling activities in TMS stopped in 2014 due to low oil prices and high drilling costs. The drilling complications include loss of circulation, well kick, wellbore collapse, and pipe sticking. It is believed that all these issues are related to the natural cracks in the formation. The well kicks were caused by a loss of circulation. If the low-pressure zones with natural cracks had been drilled with gaseous fluids to reduce loss of circulation and sealed with casing, the well kick would not have occurred in deeper drilling. Although the wellbore stability issue was believed to be due to the swelling of water-sensitive shale, it could also be induced by the mechanical erosion of the drill string that dynamically contacts the naturally cracked shale formation. The dynamic impact of the drill string on the naturally cracked formation should make it easier for shale debris to fall into and clog the wellbore, resulting in a stuck pipe situation, which is a condition where the drilling drag is beyond the rig’s pulling capacity.

For typical horizontal wells in the TMS, the average target base depth is about 12,500 ft. Well trajectories can be designed to include different lengths of the catenary section, depending on the vertical and horizontal displacements $S_{end}$ and $V_{end}$. The greater the vertical/horizontal displacement ratio ($V_{end}$/$S_{end}$), the larger the proportion of the catenary section. A catenary well trajectory design is outlined as follows:

(1)

Based on the past well productivity data, a horizontal wellbore length of 7500 ft is designed.

(2)

A catenary section is designed to have V_end = 2000 ft and S_end = 4000 ft.

(3)

Numerical solution of Equation (1) gives a = 4297 ft.

(4)

Equations (2) and (3) yield a radius of curvature of 9228 ft and an inclination angle of 43° at the top end of the catenary section.

(5)

Equation (4) gives the vertical displacement of the arc section of 6292 ft.

(6)

Equation (5) results in a KOP at 4208 ft.

(7)

Equation (6) gives the inclination angle build rate of 0.62°/100 ft. The vertical and horizontal coordinates in the arc section are calculated based on the B-value and tabulated measured depth.

(8)

The vertical and horizontal coordinates in the catenary section are calculated based on the inclination angle given by Equation (7) as a function of tabulated measured depth.

(9)

The vertical and horizontal coordinates in the horizontal section are calculated based on the maximum inclination angle of 90°.

Figure 4 presents a side view of the designed catenary well trajectory. The total well measured depth is 24,732 ft. The length of the catenary section is 5450 ft.

If the same KOP at 4208 ft is used to design an arc-type well trajectory, the radius of curvature is 8292 ft. The designed arc-type trajectory is shown in Figure 5.

Table 1. Base Data for Drilling Drag Calculations.

Parameters	Vertical Section	Curve Section	Horizontal Section
Length (ft)	4208		7500
Pipe outer diameter (in.)	5.00	6.50	5.00
Pipe inner diameter (in.)	4.21	2.81	4.35
Pipe unit weight (lb/ft)	19.50	19.5~91.69	16.25
Pipe cross-sectional area (in2)	5.73	26.96	4.78
Fluid density (ppg)	9.00	9.00	9.00
Pipe bottom depth (ft)	4208	12,500	12,500
Steel density (lb/ft3)	490	490	490
Fraction coefficient (clog indicator)	0.35	0.35	0.50~2.0
Buoyant factor	0.89	0.89	0.89

presents the base data for comparison of the theoretical drilling drags in the two designed well trajectories. The friction coefficient in the horizontal section is assumed to be higher than normal to simulate tight-hole situations in clogged-hole conditions. The pipe unit weight in the curve section varies depending on drill collar geometry. The buoyancy factor is calculated based on steel and fluid densities.

Figure 6 shows the model-calculated effect of horizontal borehole friction coefficient on hook load in two well trajectories when the drill collar weight is 91.69 lb/ft. It indicates that hook load increases linearly with the horizontal borehole friction coefficient for both the arc trajectory and the catenary trajectory. However, the hook load in the catenary trajectory is much lower than that in the arc trajectory. Figure 7 illustrates the effect of horizontal borehole friction coefficient (clog indicator) on the reduction in hook load by using the catenary trajectory. The reduction is at a level of 30% and linearly drops to 20% in the range of friction coefficients from 0.25 to 2.0, which are considered as severe well clogging conditions before pipe sticking occurs.

Figure 8 presents the model-calculated effect of drill collar weight on hook load in two well trajectories when the horizontal borehole friction coefficient is 0.5. It demonstrates that hook load increases linearly with the drill collar weight for both the arc trajectory and the catenary trajectory. However, the hook load in the catenary trajectory is significantly lower than that in the arc trajectory when heavy drill collars are used. Figure 9 shows the effect of drill collar weight on the reduction in hook load by using a catenary trajectory. The reduction non-linearly increases from 10% to nearly 30% in the range of drill collar weight from 20 lb/ft to 90 lb/ft.

Further analysis was conducted to compare the total hook load between the catenary trajectory design and the arc trajectory design in order to confirm the superiority of the catenary trajectory design. Figure 10 and Figure 11 continue illustrating the comparison between the two models and the reduction in total hook load, respectively, when the drill collar weight is 55 lbf/feet. The reduction was reaffirmed to be up to 28%. When the Horizontal Borehole Friction Coefficient is 1, the comparison and reduction in total hook load between the two models are described in Figure 12 and Figure 13, respectively, indicating a reduction of up to 25%.

5. Discussion

The catenary shape design reduces the total hook load compared with arc design trajectories due to the reduction of friction. The reduced curvature is the main reason for the reduced friction when using catenary trajectories compared with arc trajectories. The curvature along the length of the curve is reduced in the catenary trajectory design, which helps to minimize sharp changes and leads to reduced friction and tension. This also allows for a more even distribution of tension along the curve. On the other hand, in arc trajectory design, where the curvature is constant, the direction of the curve changes rapidly, leading to an increase in normal force and friction.

The reduction in friction can also be explained mathematically. The drag force profile, F(s), can be expressed by [4]

$$F\left(s\right)=K{\left\{{\left[{\sigma }_e\left(s\right){\displaystyle \frac{\partial \theta }{\partial s}}+W_{b}sin\theta \left(s\right)\right]}^2+{\left[{\sigma }_e\left(s\right){\displaystyle \frac{\partial \beta }{\partial s}}sin\theta \left(s\right)\right]}^2\right\}}^{1/2}$$

(14)

where

K is sliding friction coefficient,

s is pipe length from bit,

${\sigma }_e\left(s\right) \text{is} \text{effective} \text{tension} \text{at} \text{s}$,

$\theta \text{in} \text{an} \text{inclination} \text{angle}$,

W_b is the buoyed weight of steel per unit length,

$\beta \text{is} \text{azimuth} \text{angle}$.

For planar wells (wells that have a small degree of turn, such as catenary wells, etc.)

$${\displaystyle \frac{\partial \beta }{\partial s}}=0$$

(15)

Therefore,

$$F\left(s\right)=K\left[{\sigma }_e\left(s\right){\displaystyle \frac{\partial \theta }{\partial s}}+W_{b}sin\theta \left(s\right)\right]$$

(16)

With curvature, C(s), can be defined as follows:

$$C\left(s\right)=-{\displaystyle \frac{\partial \theta }{\partial s}}$$

(17)

Thus,

$$F\left(s\right)=-K\left[{\sigma }_e\left(s\right)C\left(s\right)-W_{b}sin\theta \left(s\right)\right]$$

(18)

For catenary curves, this includes the case where the effective tension in a uniform-density string satisfies

$${\sigma }_e\left(s\right)={\displaystyle \frac{W_{b}sin\theta \left(s\right)}{C\left(s\right)}}$$

(19)

Therefore, the drag force can be reduced.

6. Conclusions

Drilling long horizontal wells in unconventional shale gas/oil formations with natural cracks and fractures presents a huge challenge to the oil and gas industry because of wellbore clogging situations that cause pipe sticking problems. This work proposes to use a catenary well trajectory for reducing drilling drag to minimize the pipe sticking problems. A mathematical model was developed in this study for designing well trajectory profiles with a smooth transition from the kick-out point (KOP) to the catenary section. The following conclusions are drawn.

(1)

The mathematical model of catenary trajectory contains closed-form equations for the radius of curvature and inclination angle in the catenary section. Using the radius of curvature at the top point of the catenary section to design the arc-section below the KOP eliminates the trial-and-error procedure required for achieving smooth transition between the two trajectory sections.

(2)

The result of the field case study with Tuscaloosa Marine Shale (TMS) data shows that the drilling drag (hook load) can be reduced by 15% to 30% with the use of a catenary-type trajectory to replace an arc-type trajectory.

(3)

The reduction in the hook load depends on the degree of wellbore clogging. The reduction drops linearly from 30% to 20%, with the clog indicator increasing from 0.35 to 2.0.

(4)

The reduction in the hook load increases non-linearly from 15% to 30% with drill collar weight increasing from 20 lb/ft to 90 lb/ft.