The Catenary Method as an Alternative to the Horizontal Directional Drilling Trajectory Design in 2D Space

: An increase in demand for energy natural resources has stimulated the development of gas pipeline networks in Europe, as well as globally. Also, in Poland in recent years there has been a signiﬁcant increase in natural gas consumption. Therefore, it is necessary to build new pipelines networks using dig and no–dig techniques. Horizontal Directional Drilling is one of the most popular trenchless technologies. The aim of this article is to present a new approach in the design of HDD trajectories in two–dimensional space (2D). A review of the trajectories used so far has been provided, o ﬀ ering calculation algorithms to determine their well path. Then, the original catenary method is proposed, taking into account natural deﬂection of casing pipes. Applicable formulas and computational algorithms have been given, together with a computational example which enables comparison of the classical design methodology with the new one. According to the authors, due to natural stress distribution, the catenary method allows the use of smaller pulling forces during installation and ensures longer pipeline life. Therefore, it should be used in industrial practice as an alternative to current designing methods.


Introduction
Horizontal Directional Drilling (HDD) is one of the most popular trenchless technologies. The HDD technology has revolutionized geoengineering drilling in cities and under natural obstacles where the use of classical dig methods is impossible. It enables the ability to perform underground pipeline installations, while overcoming natural or artificial obstacles such as streets, buildings, railways, rivers and lakes, as well as minimizing a negative impact on the natural environment.
Taking into account both environmental and economic factors, the HDD technology can currently be considered one of the few alternatives to the classical dig methods in the construction of underground installations for the transmission of natural resources; in particular, natural gas. Increasing development and interest in this technology is in line with current trends in Polish energy policy and changes taking place in the natural gas market, the consumption of which in Poland increases year by year. The increase in demand is caused, among other reasons, by newly built power plants such as the Combined Cycle Gas Turbine in theŻerań CHP in Warsaw, which will burn approximately 0.65 billion m 3 of natural gas annually [1]. In addition to the energy sector, other industries also consume more and more gas: steel mills, refineries, ceramic factories, glassworks, and nitrogen plants. In 2018, Poland consumed about 18 billion m 3 of natural gas, including 4 billion m 3 from the national resources. The remaining quantity was imported but the import direction changed in comparison to previous  [14,15].

Category
Type of Equipment Casing installation Drilling a pilot hole is the most important part. The pilot hole is made according to the assumed trajectory design. Depending on the rock hardness, drilling works are performed with the use of an asymmetrical hydromonitor bit, a cutter bit, or a cogged bit. The rock is drilled hydraulically and mechanically. The trajectory of the well axis is constantly or periodically measured; the location of the probe in an antimagnetic connection (disposed right after the bit) is verified. Surface or subsurface radiometric cable or cableless telemetric navigation systems are used. The direction of drilling can be corrected by rotating the drill pipe by a predefined angle. In order to obtain the predicted final diameter, the borehole is widened one or more times [15,18,19].
After making a pilot borehole, the drilling bit is substituted in the borehole with a cutter or a milled reamer on the drill pipes. The reamer is introduced into the pilot borehole to enlarge the diameter of the existing well. In the process of reaming, drill pipes are successively added to proceed with reaming operations or to trip the casing [15,18,19]. The final stage of HDD is casing pullback. To do this, a reamer, rotary sub, and a head (tripped with the pipe) are disposed on the string. The rotary sub (swivel) prevents the system against rotations and torque of the drill pipes on the tripped casing [15,18,19].
The key element affecting the effectiveness and success of HDD drilling is the correct well trajectory design. A decision about the trajectory design is made based on the design data.
The following options are considered [20]: • design of Horizontal Directional Drilling trajectory in two-dimensional space, with the following variations: trajectory being a combination of straight and curvilinear sections chain curve trajectory (catenary) an irregular curve trajectory • design of Horizontal Directional Drilling trajectory in three-dimensional space This article is dedicated to the Horizontal Directional Drilling trajectory design in two-dimensional space. A combination of straight and curvilinear sections and the chain curve trajectory (catenary) design will be examined.

Trajectories Being a Combination of Straight and Curvilinear Sections
One of the most commonly applied concepts of the Horizontal Directional Drilling trajectory design is a combination of straight and curvilinear sections. The simplest variant is a trajectory consisting of one curvilinear section with a constant radius of curvature. Next variants are a combination of different numbers of straight and curvilinear sections [18,20]. Figure 1 shows the most common types of trajectories. The final stage of HDD is casing pullback. To do this, a reamer, rotary sub, and a head (tripped with the pipe) are disposed on the string. The rotary sub (swivel) prevents the system against rotations and torque of the drill pipes on the tripped casing [15,18,19].
The key element affecting the effectiveness and success of HDD drilling is the correct well trajectory design. A decision about the trajectory design is made based on the design data.
The following options are considered [20]: This article is dedicated to the Horizontal Directional Drilling trajectory design in twodimensional space. A combination of straight and curvilinear sections and the chain curve trajectory (catenary) design will be examined.

Trajectories Being a Combination of Straight and Curvilinear Sections
One of the most commonly applied concepts of the Horizontal Directional Drilling trajectory design is a combination of straight and curvilinear sections. The simplest variant is a trajectory consisting of one curvilinear section with a constant radius of curvature. Next variants are a combination of different numbers of straight and curvilinear sections [18,20]. Figure 1 shows the most common types of trajectories.  Assuming a reference system (see Figure 2) in a plane perpendicular to the surface of the terrain, it is possible to determine general dependencies for the types of trajectories presented in Figure 1 [18]. Assuming a reference system (see Figure 2) in a plane perpendicular to the surface of the terrain, it is possible to determine general dependencies for the types of trajectories presented in Figure 1 [18]. General dependencies: • vertical displacement of the end point relative to the start point: • horizontal displacement of the end point relative to the start point: • trajectory length: Graphic interpretation of general dependencies is shown in Figure 3. Curvilinear sections are defined by the following parameters: • radius of curvature: General dependencies: • vertical displacement of the end point relative to the start point: • horizontal displacement of the end point relative to the start point: • trajectory length: Graphic interpretation of general dependencies is shown in Figure 3.
Energies 2020, 13, x FOR PEER REVIEW 4 of 15 Assuming a reference system (see Figure 2) in a plane perpendicular to the surface of the terrain, it is possible to determine general dependencies for the types of trajectories presented in Figure 1 [18]. General dependencies: • vertical displacement of the end point relative to the start point: • horizontal displacement of the end point relative to the start point: • trajectory length: • azimuth. Graphic interpretation of general dependencies is shown in Figure 3. Curvilinear sections are defined by the following parameters: • radius of curvature: Curvilinear sections are defined by the following parameters: • radius of curvature: • angle of curvature: Energies 2020, 13, 1112 5 of 14 • projection of a trajectory section on a vertical plane: • projection of a trajectory section on a horizontal plane: • section length: Straight sections are determined by the following parameters: • projection of a trajectory section on a vertical plane: • projection of a trajectory section on a horizontal plane: • section length: At the AGH University of Science and Technology in Krakow at the Drilling and Geoengineering Department, of Faculty Drilling, Oil and Gas, algorithms that enable calculation of characteristic points P j (A j , H j ) of any type of trajectory with different input data variants were created. The Department of Drilling and Geoengineering also presented an algorithm enabling the determination of intermediate Pi characteristic points P i (A i , H i , ε i , β i ) with a given calculation step ∆L [18,20].
The most commonly used trajectory is a combination of straight and curvilinear sections consisting of 5 sections (see Figure 1e). Table 2 presents steps to determine characteristic values of a trajectory with the first variant of input data. Based on the calculated characteristic values of trajectory sections, it is possible to determine spatial coordinates of intermediate points. To do this, the algorithm (see Figure 4) with the following assumptions should be used:  The most important steps of the algorithm determining the trajectory being a combination of straight and curvilinear sections: 1. Determination of input data (A, H, L1, R2, R4, ε1, ε3, ε5) and a calculation step. 2. Calculation of the characteristic points according to the steps in Table 2. 3. Transfer of calculation results from point 2 and definition of additional variables. 4. Beginning of the iteration block associated with the hole section number. 5. Calculation of the current abscissa value rounded down to an integer. 6. Beginning of the nested iteration block. Check if variable X has not exceeded the range, if so, go to point 9. 7. Calculation of spatial coordinates X, Y and the alpha angle depending on the type of section. 8. Incrementation of loop variables and return to point 6. 9. Add to Hsum and Asum values corresponding to a given section. Increment the section. 10. Check if the section scope has not been exceeded, if not, return to step 4. 11. End of algorithm. The value of the angle of deviation from the horizontal plane ε L tangent to the trajectory at point P L lying between the characteristic points P j-1 and P j at the distance L L , measured from the beginning of the wellbore:

Chain Curve Trajectory (Catenary)
• curvilinear section ε j > ε j-1 Spatial coordinates of point PL lying between characteristic points P j-1 and P j : • rectilinear section ε j = ε j-1 • curvilinear section ε j > ε j-1 The most important steps of the algorithm determining the trajectory being a combination of straight and curvilinear sections:

2.
Calculation of the characteristic points according to the steps in Table 2.

3.
Transfer of calculation results from point 2 and definition of additional variables.

4.
Beginning of the iteration block associated with the hole section number.

5.
Calculation of the current abscissa value rounded down to an integer. 6.
Beginning of the nested iteration block. Check if variable X has not exceeded the range, if so, go to point 9.

7.
Calculation of spatial coordinates X, Y and the alpha angle depending on the type of section. 8.
Incrementation of loop variables and return to point 6. 9.
Add to H sum and A sum values corresponding to a given section. Increment the section. 10. Check if the section scope has not been exceeded, if not, return to step 4. 11. End of algorithm.

Chain Curve Trajectory (Catenary)
The second concept for designing Horizontal Directional Drilling trajectories is related to the natural deflection of the casing described by a chain curve (see Figure 5) [18,20]. A catenary curve is a natural curve that a cable, chain, or any other line of uniform weight assumes when suspended between two points. A similar suspension of a drill string would also form a catenary curve. A common example is a curve formed by a telephone line hanging between two utility poles. The catenary method was first introduced by R.T. McClendon and E.O. Anders [21]. In the oil and gas industry, the catenary method is often used to perform extended-reach wells. The application of this method has been described in detail in the literature [22][23][24]. The authors point out the benefits of casing natural deflection. The second concept for designing Horizontal Directional Drilling trajectories is related to the natural deflection of the casing described by a chain curve (see Figure 5) [18,20]. A catenary curve is a natural curve that a cable, chain, or any other line of uniform weight assumes when suspended between two points. A similar suspension of a drill string would also form a catenary curve. A common example is a curve formed by a telephone line hanging between two utility poles. The catenary method was first introduced by R.T. McClendon and E.O. Anders [21]. In the oil and gas industry, the catenary method is often used to perform extended-reach wells. The application of this method has been described in detail in the literature [22][23][24]. The authors point out the benefits of casing natural deflection.
As casings used in HDD have diameters from several centimeters to a meter, and their length is up to over a thousand meters, it can be stated that the length of a pipe column is disproportionately larger than its diameter, which is why such a construction can be treated as a rope hanging between two poles. In the model adopted in this way, there may be slight deviations associated with the assumption that the transverse bending and torsional stiffness is negligible in relation to longitudinal stiffness. Figure 6 shows a fragment of a free-hanging column of casing. In order to determine the relationship between the unit weight of the casing and its forces, section AB of the ds length, which is in equilibrium, is considered. As casings used in HDD have diameters from several centimeters to a meter, and their length is up to over a thousand meters, it can be stated that the length of a pipe column is disproportionately larger than its diameter, which is why such a construction can be treated as a rope hanging between two poles.
In the model adopted in this way, there may be slight deviations associated with the assumption that the transverse bending and torsional stiffness is negligible in relation to longitudinal stiffness. Figure 6 shows a fragment of a free-hanging column of casing. In order to determine the relationship between the unit weight of the casing and its forces, section AB of the ds length, which is in equilibrium, is considered. Energies 2020, 13, x FOR PEER REVIEW 9 of 15 General dependencies: Assuming that the casing is not susceptible to longitudinal deformations and its unit weight q is constant over the entire length, the chain curve equation in the system from Figure 6 is as follows: The relationship between the value of force Npoz and the constant C1 can be determined from the formula (23), using the condition: f(AK) = Hk (see Figure 6), which gives: Assuming the value of force Npoz equal to the pulling force of the drilling device, it is possible to determine the value of constant C1. Equation (24) is an implicit equation of variable Npoz. This type equation can be solved using numerical methods, e.g. bisection methods. The equation transforms into the form: and then the zero of the function is determined.
Using the physical sense of the derivative of function Y at point 0 and the relationship between angles γ0 and ε0, the formula for constant C1 can be obtained: To determine spatial coordinates and the angle of deviation from the horizontal plane of the trajectory described by a chain curve, the algorithm (see Figure 7) and the following assumptions should be used: General dependencies: Assuming that the casing is not susceptible to longitudinal deformations and its unit weight q is constant over the entire length, the chain curve equation in the system from Figure 6 is as follows: The relationship between the value of force N poz and the constant C 1 can be determined from the formula (23), using the condition: f(AK) = H k (see Figure 6), which gives: Assuming the value of force N poz equal to the pulling force of the drilling device, it is possible to determine the value of constant C 1 . Equation (24) is an implicit equation of variable N poz . This type equation can be solved using numerical methods, e.g. bisection methods. The equation transforms into the form: and then the zero of the function is determined. Using the physical sense of the derivative of function Y at point 0 and the relationship between angles γ 0 and ε 0 , the formula for constant C 1 can be obtained: To determine spatial coordinates and the angle of deviation from the horizontal plane of the trajectory described by a chain curve, the algorithm (see Figure 7) and the following assumptions should be used: • global right-handed Cartesian coordinate system with the origin at the entry point of the wellbore. Axis orientation: OX-east geographical, OY-north geographical, OZ-vertical. • specified azimuth of the plane (β) constant for the whole trajectory (β L = β) • global right-handed Cartesian coordinate system with the origin at the entry point of the wellbore. Axis orientation: OX -east geographical, OY -north geographical, OZ -vertical. • specified azimuth of the plane (β) constant for the whole trajectory (βL = β) The value of the angle of deviation from the horizontal plane εL tangent to the trajectory at point PL lying at a distance LL measured from the beginning of the wellbore: ε = arctg N poz Total length of the trajectory (n = Ak): The spatial coordinates of point PL with a given length step (∆L) can be determined using the transformed equations (27) and (28): and: The value of the angle of deviation from the horizontal plane ε L tangent to the trajectory at point P L lying at a distance L L measured from the beginning of the wellbore: Total length of the trajectory (n = A k ): The spatial coordinates of point PL with a given length step (∆L) can be determined using the transformed equations (27) and (28): and:

Calculation Example
Based on the presented equations and algorithms, an example of trajectory calculations was made.

Assumptions:
The

Results
The calculations were made using the designed computational library which implements the proposed algorithms. The simplified results were presented in tabular (see Tables 3 and 4)

Conclusions
1. HDD is a dynamically developing technology for constructing underground pipelines This is due to the fact that this type of drilling can be performed in urban and hard-to-reach areas. Horizontal directional drilling is an alternative to microtunneling, direct pipe and jacking. At the same time, the HDD technology enables minimization of a negative impact on the environment compared to conventional methods. 2. A very attractive alternative to standard design solutions is the concept of a chain curve trajectory (catenary), which enables easier insertion of the pipeline into a wellbore and ensures its longer life due to the natural stress distribution along the length. 3. The algorithms developed by the authors of the article at the AGH University of Science and Technology at the Faculty of Drilling Oil and Gas should be used in practice, as they reduce costs, failures, and complications during well drilling using the HDD technology. 4. Based on given equations and project methodologies, the authors see the potential for expanding the topic in the area of trajectories optimization that will result in a trajectory that combines advantages of both design conceptions.