SPE 124313

Catenary Well Profiles for Extended and Ultra-Extended Reach Wells Xiushan Liu, Sinopec, and Robello Samuel, Halliburton

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 4–7 October 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract Extended-reach drilling (ERD) technology has rapidly developed during the past two decades and the drilling of ultra-extended-reach (u-ERD) wells to extend their reach to greater depths requires improved models. Wellbore friction is an important issue for ultra-long wells, and optimizing the well path design is an effective means of reducing torque and drag.

This paper presents the dimensionless mathematical model and provides a new method for planning a catenary well profile. The new mathematical model of a catenary well path does not involve hyperbolic functions and uses an exact mathematical solution. The method of explicit solution avoids a trial-and-error procedure and provides excellent maneuverability of planning requirements.

Conversely, bit-walk is a natural tendency of the drill bit to drift sideways while drilling. To reduce azimuth correction frequencies and wellbore tortuosity, effective well path planning and design should account for bit-walk effects. This paper presents a newly developed a 3D mathematical model of the catenary well path and the method for planning bit-walk catenary paths to fit the bit-walk rate by a given rock layer. This paper analyzes the compositive relation of inclination units and azimuth units, provides a method to divide the catenary well profile into many shorter intervals for calculation, discusses the characteristics of planning bit-walk paths, and presents the constraint equations and solutions for ERD wells.

The results show that the essential elements of planning a 2D or 3D catenary profile include determining the position of the catenary section, parameters, such as starting and ending inclinations, and length of the succeeding hold-up section. The model and methods provided use an exact mathematical solution and results. The planned catenary well path completely fits the predetermined bit-walk rates and is absolutely smooth from the wellhead to the given target. Introduction ERD wells not only provide solutions for restricted reservoir production, but also help to eliminate additional platforms. ERD techniques and technologies have rapidly evolved, and step-out wells have incrementally increased since 1993 when the ERD program began accessing offshore reserves under Poole Harbor from land-based wellsites (Robertson et al. 2005). Wytch Farm was at the forefront of extended reach drilling. In 1997, Well M11 broke the 10 km departure milestone to set a new world record. In 2000, Well M16 established another world record at 10,727 m and a measured depth of 11,278 m (Robertson et al. 2005; Meader et al. 2000). Mason and Judzis (1998) indicated that it is possible to drill and complete u-ERD wells in the future.

The comprehensive use of various techniques and technologies is critical to the success of ERD programs. This paper focuses on the models and methods of effectively planning a catenary profile. The catenary profile was first introduced to the oil and gas industry by McClendon and Anders (1985). Early attempts have not had the expected effect because of the constraint of techniques and technologies. Du and Zhang (1987) illustrated two field cases using catenary trajectories drilled in China. Han (1987, 1997) and Liu (2007) discussed the methods for planning a catenary profile. However, most of the ERD well design research has focused on the selection of profile types to reduce well friction and emphasized the analysis of torque and drag (Payne et al. 1994; Aadnoy and Andersen 1998; Aadnoy et al. 2006). Moreover, all of the presented methods focus on planning 2D catenary profiles and do not present a method for planning a 3D catenary profile.

Drilling deviation is the result of rock removal under the complex action of the bit. The “rock-bit interaction” model, which is the kernel of the theoretical analysis of the fundamental problems, is used to predict and control the deviation tendencies of a drill bit. The use of a rock-bit interaction model requires a reliable 3D bottomhole assembly (BHA) analysis program to generate the bit force and bit axis directions. The industry has performed a great deal of research that focuses on the selection and/or design of BHAs and bits (Lubinski and Woods 1953; Walker and Freedman 1977; Millheim et al. 1978; Millheim et al. 1978; Bai 1982; Ma and Azar 1986; Ho 1987; Chen et al. 2008), but provides few published works about how to use the bit-walk tendency to successfully drill a well.

2 SPE 124313

Planeix and Fox (1979) published an algorithm for determining the final angle and direction of a well that turns and builds angle to reach a known target from a known surface location. Unfortunately, their method contains only one turn rate and must determine the turn-end point through calculations. With this method, only part of the well profile involves a turn rate and the turn segment addresses only part of the build section or the whole build section and part of the hold-up section. Based on the radius of curvature equations, McMillian (1981) provided a technique for determining the lead angle to compensate for bit-walk effects. He used a trial-and-error approach to determine the optimum lead angle, and the example showed that the calculated coordinates are inconsistent with the pre-assigned ones at the target. Maidla and Sampaio (1989) determined the bit-walk rate and the estimate of the lead angle based on a rock-bit interaction model. Validated with the data from 15 directional wells, the results show that the bit-walk predictions were good for most of the well trajectories, but additional field examples are needed to further validate the method. Maidla et al. (1991) also presented a method of planning a directional well through two targets by determining the surface location considering bit-walk behaviors, but the example showed that the best surface location still yielded the calculated error of 0.7 m at the first target compared with the given coordinates. Han and Huang (2005) attempted to modify the 2D planned results with the bit-walk rates to yield a 3D well path, but the calculated results of the example well are also inconsistent with the given data.

The above discussion shows that rational and useful ways of planning 3D bit-walk paths will not be possible at such a fundamental level. Liu et al. (1997) developed the natural-curve model that directly accounts for bit-walk effects in trajectory calculations. Liu and Shi (2002) and Liu et al. (2004) developed the methods for planning various 3D bit-walk paths combined with build-up, hold-up, and drop-off sections.

A set of characteristic parameter values uniquely determine the shape of a well profile, and uncountable sets of solutions exist for each profile. The critical issue is that the best solution must be included in the list for comparison when using well friction as the deliberated index. In the drilling industry, the length of the drillstring is much greater than its diameter. The bending rigidity of the entire string is so negligible that it can be regarded as a flexible rope. In this case, the string itself assumes a catenary shape. Theoretically, if the operator drills a well with the same catenary shape, a drillstring inside the well will have no contact with the borehole wall (Aadnoy et al. 2006). Instead, the drillstring will tend to stand off the borehole wall, and the drag and torque applied on the drillstring or casing string can be minimized.

This paper presents a dimensionless 2D model of catenary trajectory and a method for planning a catenary profile. Because the model and method provided avoid the calculation of hyperbolic functions, they are convenient to use. This paper also researches the 3D model of catenary trajectory accounting for bit-walk effects, analyzes the compositive relation of inclination units and azimuth units, and yields a method to divide them into shorter intervals for calculation. The models and methods presented here provide real, optimal techniques for planning 2D and 3D catenary profiles. They use an exact mathematical solution and results, provide excellent maneuverability of planning requirements, and ensure that the planned well paths are smooth from the wellhead to the given target. Model of 2D Catenary Well Path A rope, cable, chain, or any other line of uniform weight that is suspended between two points assumes a shape called a catenary. Common examples are the curves formed by an electrical wire hanging between two telegraph poles and a chain attached to a vessel and an anchor to keep the vessel in place.

In the drilling industry, the length of drillstring is much greater than its diameter. The bending rigidity of the entire drillstring is so negligible that the drillstring can be regarded as a flexible rope. Moreover, it is reasonable to assume that the dead weight per unit length of drillstring remains constant and that the drillstring does not change its length while loaded with the deadweight. In this case, the string obtains a catenary shape and the classical catenary equation is as follows:

axay cosh= . .................................................................................................................................................. (1)

For the deflection curve defined by Equation 1, there is a point (point f in Fig. 1) at which the slope is zero. Fig. 1 gives a standard coordinate system for a catenary curve in which y-axis across point f and x-axis below point f (the offset distance is a).

To simplify the well path design, the classic catenary equations can be converted to dimensionless catenary equations, as shown in Fig. 2. The non-dimensional parameters include the following:

ayY

axX == , . ............................................................................................................................................. (2)

Equation 1 becomes

XY cosh= . .................................................................................................................................................... (3)

The incremental increase of coordinates of the catenary segment can be given in the form of dimensionless abscissa and ordinates as:

( YYaH b −=Δ )). .............................................................................................................................................. (4)

( XXaS b −=Δ .............................................................................................................................................. (5)

SPE 124313 3

where,

⎟⎠⎞

⎜⎝⎛−=

2tanln αX ,

αsin1

=Y

A catenary profile is usually incorporated in a well path profile to achieve a smooth well path. To enable the calculation of the inclination at any point of the catenary section, an algebraic manipulation of Equation 3 yields the following:

aL

b

Δ−

=

α

α

tan1

1tan . ..................................................................................................................................... (6)

This calculation avoids using inconvenient hyperbolic function to estimate the course coordinates position parameters while using the exact mathematical solution. This process also avoids the trial and error of iterative calculations. Design of 2D Catenary Profile A typical catenary profile is shown in Fig.1 It consists of four well sections: an inclined straight section to the kick-off point, an initial buildup section to the start of catenary section, the catenary section, and a straight sail section to the target.

The first section is assumed to be an inclined straight section because a slant rig may be used for land drilling. Generally, it is a vertical section, viz. α1=0. The initial buildup section is required because the catenary section cannot begin vertically. If a slant rig is used, this section can be eliminated and the rig mast should be positioned at the exact inclination to enter the catenary smoothly. Also, the sail section is usually necessary for the design of an extended reach or ultra-extended reach wells.

When planning a well profile, the total vertical depth and horizontal displacement at the target are known. The other geometric parameters of the profile include the following (Fig. 1):

• Inclination of the slanted section, α1 • Length of the slanted section, ΔL1 • Curvature radius of the initial buildup section, R2 • Catenary characteristic parameter, a • Starting inclination of the catenary section, αb • End inclination of the catenary section, αc • Length of the straight sail section, ΔL4 The first three parameters, α1, ΔL1, and R2, are usually given when planning a catenary profile in which R2 is calculated

from the pre-determined buildup rate based on the performance of the BHA design. The end inclination of the catenary section, αc, equals the sail inclination. For sliding pipe or motor drilling, a critical inclination for gravity-driven drilling is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

μα 1tan 1

cr ............................................................................................................................................... (7)

The coefficient of friction between the string and the hole is an important factor for high-angle wells and can be altered by changing the drilling fluid or lubrication. The sail inclination is usually less than the critical inclination, viz. αc<αcr, to ensure that the drillstring or casing slides downward.

One of the essential elements of planning a catenary profile is the determination of the shape and position of the catenary section. Only three parameters, a, αb, and ΔL4, must be determined. Because two constraint equations are required to determine the horizontal displacement and total vertical depth of the target, the catenary profile can be defined if one of them is given. It is convenient and generally used to provide the value of αb, and the solution to a and ΔL4 are as follows:

cc

cc

cbSHa

αααα

cossincossin 00

−−

= ................................................................................................................................... (8)

cc cbcHbSL

αα cossin00

4 −−

=Δ . ................................................................................................................................ (9)

where,

( )12110 sinsincos ααα −−Δ−= bt RLHH

( )bt RLSS ααα coscossin 12110 −−Δ−=

cb

bαα sin

1sin

1−=

4 SPE 124313

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

2tan

2tan

lnb

c

cα

α

Finally, the length of catenary section can be calculated by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=Δ

cb

aLαα tan

1tan

13 . ........................................................................................................................... (10)

Model of 3D Catenary Well Path During directional drilling operations, bit-walk objectively exists along an actual drilled well trajectory. Drilling experience has shown that taking the physical bit-walk into account provides an effective measure to control a well trajectory if the rate of bit-walk can be accurately estimated. Because the planned results of the bit-walk path can evidently reduce the frequency of azimuth correction and drilling-string trips, as well as lessen the difficulty and workload of trajectory control, it will undoubtedly improve the rate of penetration and the quality of well trajectory, and reduce the cost of drilling.

Bit-walk is the natural tendency of the drill bit to drift in a lateral direction during drilling and it occurs in every operation. Operators who plan and design wellbore trajectories must consider planning 3D paths, especially in areas where the amount of bit-walk is considerable. The tendency and degree of bit-walk indicates the bit-walk rate; the industry describes bit-walk behavior as the rate of azimuth change. Assuming that the rates of azimuth change remain constant, the azimuth function vs. measured depth can be expressed by:

LCb Δ+= φκφφ .............................................................................................................................................. (11)

According to the differential model of wellbore trajectory, the relation between coordinate increments and curved-section length, inclination angle, and azimuth angle for a small interval length are given by:

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=Δ

=Δ

=Δ

=Δ

∫

∫

∫

∫

L

L

L

L

L

L

L

L

b

b

b

b

dLS

dLH

dLE

dLN

α

α

φα

φα

sin

cos

sinsin

cossin

................................................................................................................................. (12)

Equation 6 and Equation 11 can be used along with Equation 12 to calculate the incremental course coordinates in the catenary section; for more explicit points, numerical integration must be used. In addition, the borehole curvature and torsion for the catenary section can be estimated using the following equations:

ακκκ φα222 sin+= ...................................................................................................................................(13)

ακκκτ α

φ cos1 2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ................................................................................................................................... (14)

where

απ

κα2sin180

aC

=

Consequently, the relationship between each parameter of the 3D catenary profile and depth can be obtained. Moreover, the catenary profile also includes hold sections (and vertical sections) and build/drop sections, whose 3D drifting trajectory can be expressed by a natural curvature model (Liu and Shi 2002; Liu et al. 2008). Design of 3D Catenary Profile In the vertical profile, the inclination change rate of the hold section is zero, and the rate of the build/drop section is constant. The inclination of the catenary profile has been given by Equation 6. Thus, the relationship of the inclination with the depth changes by sections; every well inclination unit should have its own inclination equation, according to its different inclination characteristics.

The azimuth change of the well trajectory is related to many factors with certain rules, including the formation layer direction, formation anisotropy, bit type, bit anisotropy, BHA, BHA mechanics, weight on bit, bit rotational speed, well

SPE 124313 5

profile, and well geometry parameters. When the formation layer and the bit characteristics are the major effects, the azimuth change can be divided according to the different formation layers, or the well path, from wellhead to the target, can be divided into several sections by true vertical depth (TVD). This process will enable each unit to have the same azimuth drifting rate so that the azimuth units can be obtained.

One well section is a well inclination unit, and one formation layer is usually an azimuth unit. The azimuth units can be divided by the azimuth drifting rates, and so an azimuth unit is not exactly one formation layer. Fig. 4 shows the combined relationship between the inclination unit and the azimuth unit. An azimuth unit may cover an entire well inclination unit, and it is not necessary to divide this inclination unit. An azimuth unit can also cover several inclination units, which can also be divided at the top and bottom edges of the azimuth unit. Therefore, the objectives of combining the inclination units and azimuth units are to divide the well profile into more detailed sections and to create smaller units with a constant inclination change rate and constant azimuth change rate. Obviously, the number of smaller units after combining is often more than the number of well sections.

Because the parameters of a bit-walk profile are interrelated and interdependent, it is too difficult to plan the 3D bit-walk path with the aid of 2D well path planning methods; however, operators must design the well path to reach the predetermined target. Different rock layers show varying natural deflecting behaviors and have varying bit-walk rates. Combining bit-walk units with inclination units divides the well profile into shorter intervals. Thus, the sum of the coordinate increments over every interval must equal the given coordinate difference between the wellhead and target, which can be expressed as follows:

2t

2

1

2

1AEN

m

ii

m

ii =⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ ∑∑

==

. ...................................................................................................................... (15)

∑∑==

Δ=Δm

ii

m

ii NE

1t

1tanϕ .................................................................................................................................. (16)

t1

HHm

ii =Δ∑

=

. ................................................................................................................................................ (17)

Fig. 4(b) shows the schematic drawing of a bit-walk path in the horizontal plot. To compensate a well trajectory for bit-walk effects, a lead angle is necessary. The value of the lead angle may be positive or negative because, although bit-walk generally occurs as right-hand drift, it occasionally occurs as left-hand drift. Well designers must plan a 3D well path and provide the lead angle as one of the calculated parameters accounting for bit-walk effects.

Using these design constraint equations, three unknowns can be solved in the design of the 3D drift path of an extended reach directional well. The solution must be found by an iterative method because the system of equations is nonlinear. In principle, a three-level iterative method should be used with three unknowns. However, it is possible to have fewer iterations and to improve the convergence by selecting the proper iterative method. Examples The first example addresses the design of a 2D catenary profile for an ERD well, given the following input data:

• Target vertical depth, Ht = 2,800 m • Horizontal departure, At = 6,000 m • Azimuth angle of target departure, ϕt = 150° • TVD at kick-off-point, ΔL1 = 300 m • Build-up rate of arc section, κα,2 = 8°/30 m • Initial inclination, α1 = 0° • Starting inclination of the catenary section, αb = 42° • Sail inclination or end inclination of the catenary section, αc = 78° After solving Equations 8 and 9, the results show that a = 3,485.61 m, ΔL4 = 3,417.56 m, and ΔL3 = 3,130.27 m. Table 1

shows the planned results of the catenary profile at nodes. The second example addresses the design of a 3D catenary profile involving bit-walk and uses the same input data as the

first example. Moreover, the bit-walk rates based on the rock layers are: −1.0°/30 m from 300 m to 380 m, 1.2°/30 m from 380 m to 500 m, 0.6°/30 m from 800 m to 1,200 m, 0.5°/30 m from 1,360 m to 1,800 m, −0.4°/30 m from 1,800 m to 2,100 m, 0.3°/30 m from 2,300 m to 2,500 m, and 0.2°/30 m from 2,600 m to 2,800 m, in TVD successively. There are no bit-walks on other rock layers.

After solving Equations 15 through 17, the results show that a = 3,449.22 m, ΔL4 = 3,500.18 m, ΔL3 = 3,097.60 m, and initial azimuth φ0 = 129.48°, with a lead azimuth angle of 20.52°. Table 2 shows the calculated results of the 3D bit-walk path planned at nodes.

6 SPE 124313

Conclusions When a drillstring is suspended between two points, it obtains a catenary shape. If the operator drills a well with the same catenary shape, the drillstring inside the well will have no contact with the borehole wall. Instead, it will tend to stand off the borehole wall, which will minimize the drag and torque applied on the drillstring or casing string. This process provides an attractive option to planning well profiles using catenary model. However, the planned results ensure that no friction occurs for only the catenary section and one specified operation.

Critical components of the successful planning of a catenary profile are the determination of the shape and position of the catenary section. The characteristic parameter of catenary curve, a, describes its shape, and parameters, such as the starting and ending inclination, locate the position of the catenary section in the well profile.

The presented method in the form of dimensionless parameters is easy to use and does not require iterative calculations for the determination of the well path course coordinates.

The purpose of planning a bit-walk path is to enable the use of bit-walk tendency when drilling a directional well, which promotes both safety and more rapid drilling. The process of planning a 3D bit-walk path based on the deflecting behaviors of a formation requires assigning inclination units and bit-walk units, combining bit-walk units with inclination units and dividing them into shorter intervals for calculations, giving constraint equations and determining the characteristic parameters of the well profile to satisfy these constraint equations, and calculating various geometrical parameters of the well path.

Bit-walk occurs everywhere. Well designers, therefore, must consider planning 3D well paths, especially in areas where the amount of bit-walk is considerable. By using the 3D bit-walk regulation of well trajectory to reach the target, operators can apply more weight-on-bit and drill continuously, which is propitious to increasing the penetration rate. Nomenclature

α = Inclination angle, degrees φ = Azimuth angle, degrees κα = Rate of inclination change (dropping off is a negative value), degrees/30m κφ = Rate of azimuth change (decreasing azimuth is a negative value), degrees/30m κ = Curvature of wellbore trajectory, degrees/30m τ = Torsion of wellbore trajectory, degrees/30m μ = Coefficient of friction, dimensionless αcr = Critical inclination, degrees ϕ = Azimuth angle of horizontal departure, degrees Γ = Nondimensional length, dimensionless a = Characteristic parameter of catenary, m A = Horizontal departure, m b, c = Intermediate variable, dimensionless C = Constant related to rate’s unit E = East coordinate (west is negative), m H = Vertical depth, m H0, S0 = Intermediate variable, m l = Length, m L = Measured depth, m m = Interval number of well path N = North coordinate (south is negative), m R = Curvature radius, m S = Horizontal curvilinear departure, m x = x-coordinate, m X = Nondimensional x-coordinate, dimensionless y = y-coordinate, m Y = Nondimensional y-coordinate, dimensionless

Subscripts b = Starting point of catenary section c = End point of catenary section t = Target point i = Variable

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Conference, Dallas, Texas, USA, 3-6 March. Aadnoy, B.S., Fabiri, V.T., and Djurhuus, J. 2006. Construction of Ultra-long Wells Using a Catenary Well Profile. Paper SPE 98890

presented at the IADC/SPE Drilling Conference, Miami, Florida, USA, 21-23 February.

SPE 124313 7

Bai, J.Z. 1982. Bottom Hole Assembly Problems Solved by Beam-Column Theory. Paper SPE 10561 presented at the International Petroleum Exhibition and Technical Symposium, Beijing, China, 17-24 March.

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115024 presented at the IADC/SPE Asia Pacific Drilling Conference and Exhibition, Jakarta, Indonesia, 25-27 August. Lubinski, A. and Woods, H.B. 1953. Factors Affecting the Angle of Inclination and Dog-Legging in Rotary Bore Holes. API Drill and Prod.

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McClendon, R.T. and Anders, E.O. 1985. Directional Drilling Using the Catenary Method. Paper SPE 13478 presented at the SPE/IADC Drilling Conference, New Orleans, Louisiana, USA, 5-8 March.

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of Pressure Vessel Technology 56(4): 367-373. Tables

Table 1–PLANNED RESULTS OF THE 2D CATENARY PROFILE

L, m α, deg. φ, deg. H, m N, m E, m A, m ϕ, deg.

κ, deg./30m

0.00 0.00 ⎯ 0.00 0.00 0.00 0.00 ⎯ 0.00 300.00 0.00 (150.00) 300.00 0.00 0.00 0.00 ⎯ 0.00 457.50 42.00 150.00 443.77 -47.79 27.59 55.19 150.00 8.00

3587.77 78.00 150.00 2089.45 -2301.13 1328.56 2657.12 150.00 0.47 7005.33 78.00 150.00 2800.00 -5196.15 3000.00 6000.00 150.00 0.00

8 SPE 124313

Table 2–PLANNED RESULTS OF THE 3D BIT-WALK CATENARY PROFILE

L, m κ, deg./30m

τ, deg./30mH, m A, m S, mα, deg. φ, deg. ϕ, deg.

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 300.00 0.00 129.48 300.00 0.00 0.00 0.00 0.00 0.00 381.97 21.86 126.75 380.00 15.45 127.66 15.45 8.01 -1.85 457.50 42.00 129.77 443.77 55.18 128.20 55.19 8.04 1.77 533.51 42.57 132.81 500.00 106.28 129.68 106.33 0.84 0.82 952.24 45.96 132.81 800.00 398.20 131.98 398.36 0.26 0.00

1558.76 51.64 144.94 1200.00 851.36 135.70 853.96 0.56 0.26 1824.88 54.47 144.94 1360.00 1061.77 137.54 1066.58 0.33 0.00 2697.07 65.24 159.48 1800.00 1796.71 143.66 1818.15 0.61 0.12 3555.10 78.00 148.04 2072.27 2598.63 146.77 2629.96 0.62 -0.03 3688.46 78.00 146.26 2100.00 2729.07 146.79 2760.41 0.39 -0.08 4650.41 78.00 146.26 2300.00 3669.97 146.65 3701.34 0.00 0.00 5612.36 78.00 155.88 2500.00 4607.57 147.55 4642.27 0.29 0.06 6093.33 78.00 155.88 2600.00 5073.53 148.32 5112.73 0.00 0.00 7055.28 78.00 162.29 2800.00 6000.00 150.00 6053.66 0.20 0.04

Figures

S O

(a) Vertical expansion plot

αc

α1

αb

Catena

St

Ht

R2

ΔL

t

c

b

a

H

ΔL4

a

At

St

c b N

(b) Hor

ϕt

t

φ0 O

E

Fig. 4−3D catenary profile.