US20150019134A1 - Wellbore Positioning System and Method - Google Patents

Wellbore Positioning System and Method Download PDF

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US20150019134A1
US20150019134A1 US14/370,913 US201314370913A US2015019134A1 US 20150019134 A1 US20150019134 A1 US 20150019134A1 US 201314370913 A US201314370913 A US 201314370913A US 2015019134 A1 US2015019134 A1 US 2015019134A1
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wellbore
ellipse
ellipses
expansion factor
determining
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Steven James Sawaryn
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BP Exploration Operating Co Ltd
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    • E21B41/0092—
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • E—FIXED CONSTRUCTIONS
    • E21—EARTH OR ROCK DRILLING; MINING
    • E21B—EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B47/00—Survey of boreholes or wells
    • E21B47/02—Determining slope or direction
    • E21B47/022—Determining slope or direction of the borehole, e.g. using geomagnetism
    • E—FIXED CONSTRUCTIONS
    • E21—EARTH OR ROCK DRILLING; MINING
    • E21B—EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B7/00—Special methods or apparatus for drilling
    • E21B7/04—Directional drilling

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  • the present invention relates to a computer-implemented method and a system for determining the relative positions of a wellbore and an object. Spatial relationships between two ellipses, each of which represents the positional uncertainty of a wellbore, are utilized to determine the conditions governing osculation between the two ellipses, expressing the determination as an expansion scale factor.
  • the positional uncertainty at any point in a well is dependent on a number of factors, including the positional uncertainty of the surface location, the well's geographical location and trajectory and the various instruments used to survey the well.
  • positional uncertainty is meant positional uncertainty of the well's geographical location, positional uncertainty of its trajectory etc.
  • the expected behaviours of these instrument types are presented as instrument performance models. Application of these models quantifies the uncertainty of the true wellbore position for a stated confidence.
  • the positional uncertainty 1 about a point representing the calculated position of the centre of a wellbore is commonly represented as an ellipsoid with its principal axes aligned with the high-side, right-side and along-hole directions.
  • high-side is the direction normal to the wellbore in the vertical plane
  • right-side is the direction normal to both the wellbore direction and the high-side and so lies in the horizontal plane.
  • the ellipsoid usually also accounts for the dimensions of the casing or open hole of the wellbore. The size of the ellipsoid varies according to the wellbore trajectory shape, survey instruments used in calculating the position, and selected confidence limits.
  • the resulting positional uncertainty about a wellbore along its trajectory is the envelope of the ellipsoids; a curved, continuous cone 2 with a curved end.
  • the interference between two adjacent wells can be visualised as the interference between the two cones.
  • the positional uncertainty can change over time as data is re-processed or more data is acquired. It also changes when the wellbore is resurveyed using a more accurate instrument system, for example when a high accuracy gyroscope is run at a casing point 3 . This then narrows the cone, as shown in FIG. 1 . Therefore, if a new measurement is taken at a subsequent point along the trajectory of the wellbore, the positional uncertainty decreases, then increases as the distance from the measurement point increases.
  • the separation between wellbores in 3D can be represented in 2D using a collision avoidance plot, also known as a travelling cylinder or normal plane diagram.
  • a collision avoidance plot also known as a travelling cylinder or normal plane diagram.
  • the intersection of, for example, an existing and a planned well (or two planned wells) is displayed on a plane, constructed normal to the planned well.
  • the planned well is kept at the centre of the plot and therefore the relative separation between the planned well and the adjacent well is indicated by the locus of points obtained at successive depths.
  • the plane also intersects the curved cone and at low or modest angles of incidence between wells the intersections with the cones appears, to good approximation as two ellipses.
  • the planned well is also referred to as the subject or reference well.
  • the separation ⁇ between the wells can be calculated using Eqs. 1 to 3, below. In the absence of bias this is also the separation between the error ellipses. Further adjustments can be made if required.
  • ⁇ min between the wellbores must be greater than the sum of the open hole and casing radii, ⁇ min >(d h +d c )/2, where d h is the hole diameter and d c is the casing diameter. This criterion automatically satisfies the mathematical constraint ⁇ 0.
  • CVM centre vector method
  • PCM pedal curve method
  • FIG. 2 shows how the currently used “centre vector method” (CVM) is used to calculate a separation factor between two wellbores.
  • CVM centre vector method
  • FIG. 3 shows an alternative method of calculating a separation factor between two wellbores, the “pedal curve method” (PCM).
  • PCM the characteristic lengths s 1 and s 2 are determined from the line that is both tangent to the ellipse and is orthogonal to the line ⁇ joining their centres.
  • the first step is to determine the points of tangency, marked A and B in FIG. 3 . In this case the tangent lines meet and the separation factor k PCM reaches unity before the ellipses touch. Therefore the separation factors calculated using this method may be too conservative, leading to unnecessary shut-in of wells or missed opportunities.
  • a computer-implemented method for determining the relative positions of a wellbore and an object comprising the steps of:
  • Embodiments of the present invention utilize spatial relationships between two ellipses for determining the conditions governing osculation between the two ellipses (where osculation is the case in which the ellipses touch), expressing the determination as an expansion scale factor.
  • Each expansion factor calculation involves using the smallest positive root of the quartic equation.
  • the explicit schemes of the present invention offer improvements in both calculation efficiency and reliability over known methods of calculating a separation factor and over iterative methods of calculating an expansion factor.
  • the wellbore is a first wellbore
  • the object is a second wellbore
  • the object may be a sub-surface hazard that is to be avoided when drilling the wellbore.
  • FIG. 1 shows a three-dimensional representation of a cone which represents the positional uncertainty of a wellbore
  • FIG. 2 shows a “centre vector method” for estimating the separation between two ellipses
  • FIG. 3 shows a “pedal curve method” for estimating the separation between two ellipses
  • FIG. 4 a shows the determination of an “expansion factor” by the simultaneous and equal expansion (k>1) of two ellipses;
  • FIG. 4 b shows the determination of an “expansion factor” by the simultaneous and equal contraction (k ⁇ 1) of two ellipses;
  • FIG. 5 a shows the steps involved in a first method of calculating an expansion factor
  • FIGS. 5 b - 5 e show an expansion of the ellipses carried out in the first method of calculating the expansion factor
  • FIG. 6 a shows the steps involved in a second method of calculating an expansion factor
  • FIG. 6 b shows an expansion of the ellipses carried out in the second method of calculating the expansion factor
  • FIGS. 7 a - 9 b show dual and single sided expansion of various configurations of ellipses
  • FIG. 10 shows a wellbore positioning system according to the present invention
  • FIG. 11 shows an example of a planned wellbore in simplified collision avoidance plot
  • FIG. 12 shows the steps taken in determining the relative position of a wellbore according to the present invention
  • FIG. 13 shows a schematic diagram of a wellbore being drilled into a formation.
  • an ellipse In directional work an ellipse is generally defined by its centre (x 0 , y 0 ), the lengths of its semi-major and semi-minor axes a and b and the orientation ⁇ of the major axis direction a relative to some reference direction.
  • each ellipse represents some confidence interval that the wellbore lies within its boundary
  • the equal expansion or contraction of both ellipses until they touch is a measure of a potential collision between the wells. Since the point at which two ellipses touch is a function of both their sizes and orientations, conceptually, the available space can also be calculated by expanding only one ellipse with the other one fixed. Therefore both dual sided and single sided expansion can be applied to calculate a relevant expansion factor k.
  • the quadratic may be represented in matrix form, as shown by Eq. 12, where E is the 3 ⁇ 3 symmetric matrix. It is noted that the expansion factor appears in only one of the matrix elements as its square k 2 .
  • ZPM The following calculation, referred to hereinafter as the “ZPM” method, can be used to calculate the expansion factor using dual sided expansion, where each of two ellipses is expanded equally.
  • step S 501 the elliptical parameters a 1 , b 1 , ⁇ 1 , x 0,1 , y 0,1 of ellipse E 1 and a 2 , b 2 , ⁇ 2 , x 0,2 , y 0,2 of ellipse E 2 are input into a wellbore positioning system, as described below with respect to FIG. 10 .
  • step S 502 a determination is made as to whether the centres of the ellipses are separated by a distance greater than ⁇ min , as explained above in relation to Eq. 3.
  • step S 503 it is determined in step S 503 that the wellbores physically interfere and the calculation is stopped in step S 504 .
  • the distance of closest approach ⁇ cr is calculated in step S 505 , as explained below.
  • the distance of closest approach ⁇ cr of two arbitrary hard ellipses in 2D can be determined using the method disclosed in Zheng, X., Palffy-Muhoray, P.: “Distance of Closest Approach of Two Arbitrary Hard Ellipses in 2D”.
  • the ellipse E 2 is translated towards E 1 in the direction joining their centres until it reaches the position E 2 * when the ellipses touch externally.
  • the orientations of the two ellipses are maintained throughout.
  • the ellipse E 1 is then transformed into a circle C 1 and the same mathematical transformation used to obtain the circle is applied to the ellipse E 2 ( FIG. 5 c ).
  • Zheng and Palffy-Muhoray also describe a method for calculating the contact point and provided computer code for both of the closest approach and contact point calculations.
  • Knowledge of the contact point may be used to verify the expansion factor results, checking for each ellipse that
  • a further method referred to hereinafter as the “YKC” method, can be used to calculate an expansion factor using dual sided expansion, where each of the two ellipses is expanded equally, or single sided expansion, where only one ellipse is expanded while the other remains fixed.
  • the ZPM approach is preferred. Tests show that it is more stable computationally, particularly for similarly sized ellipses with centres that are close together.
  • step S 601 the elliptical parameters a 1 , b 1 , ⁇ 1 , x 0,1 , y 0,1 of ellipse E 1 and a 2 , b 2 , ⁇ 2 , x 0,2 , y 0,2 of ellipse E 2 are input into a wellbore positioning system, as described below with respect to FIG. 10 .
  • step S 602 a determination is made as to whether dual sided or single sided expansion is preferred. Single sided expansion may be preferred in some cases because of the greater area of space obtained about the expanded wellbore.
  • step S 602 For single sided expansion (output “Y” at step S 602 ), the symmetry present in the dual sided expansion is broken and a different approach must be used.
  • the centre (x 0,2 , y 0,2 ) of the second ellipse E 2 must lie outside its boundary (step S 603 ).
  • this requires the condition that E 1 (x 0,2 , y 0,2 , 1)>0.
  • step S 604 if the centre of E 2 does not lie outside of E 1 , the system determines that no solution is possible, and the calculation is stopped at step S 605 .
  • the cubic's discriminant vanishes when the ellipses touch, leaving a quartic equation in k 2 , as shown by Eq.13. Taking the square root gives the expansion factor k.
  • ⁇ ⁇ 4 a 2 4 ⁇ b 2 4 ⁇ ( z 12 2 - 4 ⁇ a 1 2 ⁇ a 2 2 ⁇ b 1 2 ⁇ b 2 2 ) ( 14 )
  • ⁇ 3 2 ⁇ a 2 2 ⁇ b 2 2 ⁇ [ 6 ⁇ a 1 2 ⁇ a 2 2 ⁇ b 1 2 ⁇ b 2 2 ⁇ ( a 2 2 ⁇ p 1 + b 2 2 ⁇ q 1 ) + 9 ⁇ a 1 2 ⁇ a 2 2 ⁇ b 1 2 ⁇ b 2 2 ⁇ z 12 ⁇ a 2 2 ⁇ b 2 2 ⁇ r 1 ⁇ z 12 - z 12 2 ⁇ ( a 2 2 ⁇ p 1 + b 2 2 ⁇ q 1 ) - 2 ⁇ z 12 3 ] ( 15 )
  • ⁇ 2 - 27 ⁇ a 1 4 ⁇ b 1 4 ⁇ a 2 4 ⁇ b 2
  • this calculation of the closest distance of a point (which may represent an object) to an ellipse is equivalent to the single sided expansion of a unit circle (which is a special case of an ellipse) centred on the point against the ellipse, as shown in FIG. 6 b .
  • This distance is equal to the expansion factor k (step S 607 ).
  • the expansion factor is output at step S 608 .
  • FIGS. 7 a to 9 b Some examples of elliptical configurations are shown in FIGS. 7 a to 9 b .
  • the configurations of two ellipses on FIGS. 7 a , 8 a and 9 a correspond to the configurations in FIGS. 7 b , 8 b and 9 b , respectively.
  • the dashed ellipses represent the expanded, osculating ellipses when a dual expansion method is used.
  • FIGS. 7 b , 8 b and 9 b the dashed ellipse represents the expansion of one of the ellipses in a single sided expansion.
  • Table 1 shows a comparison of the CVM and PCM separation factors (k CVM and k PCM , respectively) with the dual sided expansion factor (k ZPM ) for the three elliptical configurations.
  • wellbore drilling systems generally comprise drilling equipment 4 arranged to drill a wellbore 5 into the one or more hydrocarbon-bearing reservoirs in a formation 6 .
  • the drilling system typically comprises a controller 7 arranged to control the drilling equipment.
  • An existing wellbore 8 is also shown.
  • the wellbore positioning system 100 comprises suitable computer-implemented models, software tools and hardware, as shown in FIG. 10 .
  • a reservoir model 121 may be employed.
  • a reservoir model is a conceptual 3-dimensional construction of a reservoir that is constructed from incomplete data with much of the inter-well space estimated from data obtained from nearby wells or from seismic data.
  • a trajectory model 123 that is, a computer model that constructs 2D and/or 3D representations of the geographical locations and/or trajectories of wellbores may be employed.
  • the trajectory model may comprise or make use of a collision avoidance plot, also known as a travelling cylinder or normal plane diagram.
  • An expansion factor calculation tool 111 can calculate the expansion factor as explained above.
  • the trajectory model 123 can use information such as the volume and shape of the reservoir 3 (including the arrangement of overlying rock formations and the locations of any faults or fractures in the rock formations and sub-surface hazards), the porosity of the oil-bearing rock formations, the location of existing production well(s) and injection well(s), in combination with the results of the expansion factor calculation tool 111 , to provide an indication as to the possible trajectory of a planned wellbore.
  • the expansion factor calculation tool 111 and optionally the reservoir model 121 , the trajectory model 123 and an optimisation tool 125 are executed by the wellbore positioning system 100 .
  • the wellbore positioning system 100 which is for example a control system on a platform, can comprise conventional operating system and storage components such as a system bus connecting a central processing unit (CPU) 105 , a hard disk 103 , a random access memory (RAM) 101 , and I/O and network adaptors 107 facilitating connection to user input/output devices and interconnection with other devices on a network N 1 .
  • CPU central processing unit
  • RAM random access memory
  • I/O and network adaptors 107 facilitating connection to user input/output devices and interconnection with other devices on a network N 1 .
  • the Random Access Memory (RAM) 101 contains operating system software 131 which controls, in a known manner, low-level operation of the wellbore positioning system 100 .
  • the server RAM 101 contains the software tools and models 111 , 121 , 123 and 125 during execution thereof.
  • Each item of software is configurable with measurement and/or predetermined data stored in a database or other storage component which is operatively coupled or connected to the wellbore positioning system 100 ; in the system of FIG. 2 , storage component DB 1 stores all such data relating to the expansion factor calculation tool 111 and is accessible thereby, while storage component DB 2 stores all other data for use by the other components of the system 100 .
  • Input data received by receiving means of the system 100 comprise the elliptical parameter values and are based on a measured position of an existing wellbore or an estimated (i.e. modelled or simulated) position of a planned wellbore.
  • estimated input data can be modelled or estimated upon planning a wellbore, for example upon an initial assessment or appraisal of a reservoir when developing a new field.
  • the input data includes measurement data relating to the position of the object.
  • the measurement data may comprise specific measured values as directly measured by suitably positioned measurement equipment such as survey instruments 12 , or may comprise values derived from a number of separate positional measurements. Therefore, the raw measured data may, if necessary or preferred, be manipulated by appropriate software and executed by the CPU 105 of the system 100 , in order to generate measurement or estimated position data that are suitable for inputting into the expansion factor calculation tool 111 . Such manipulation may comprise using the reservoir and/or trajectory models to determine the parameter values of the two ellipses.
  • the expansion factor calculation tool 111 may comprise a software program such as Mathematica. This program can be used in a number of ways during the calculation of the expansion factor. Firstly by making use of its symbolic manipulation, the substitutions, for example, for A, B, C, D, F, G (which is equivalent to H ⁇ a 2 b 2 k 2 —see Appendix A), H can be made. The determinants can then be expanded and the equations simplified using this program. Additionally, MathematicaTM is preferably employed to program the resulting quartic coefficients and solve the quartic equation. Alternatively, the expressions can be programmed in, for example, Visual BasicTM within an EXCELTM spreadsheet.
  • An optimisation tool 125 may be provided to assist in the planning and drilling of wellbores.
  • the optimisation tool may be used in conjunction with the trajectory model 123 to compute an optimal position for the wellbore in 2D or an optimal trajectory in 3D, based on input data including the calculated expansion factor and the measured or estimated input data that relates to the position of one or more existing wellbores or objects.
  • the optimisation tool 125 may be programmed with rules that take into account additional data representing, for example, threshold values representing practical limits to the degree of curvature of the wellbore trajectory. In this way, the optimisation tool 125 can determine an optimum alignment of the trajectory, as explained further below with reference to FIG. 11 .
  • FIG. 11 shows a simplified collision avoidance plot which may be produced by the trajectory model 123 upon calculation of the expansion factor; the x and y axes represent length in metres.
  • the dashed ellipse represents the tolerable errors, including an acceptable operational margin, for a planned wellbore at some point in space.
  • the solid ellipses represent the tolerable errors surrounding three adjacent, drilled wellbores.
  • the centre vector method is generally excluded in such a scenario as it is overly optimistic.
  • the use of the expansion factor in the wellbore positioning method and system of the invention is advantageous in the planning and drilling of wellbores, as it provides more space in which to plan and optimise the trajectories of wellbores.
  • a planner concluded that it was not possible to drill through the gap of FIG. 11 , then the wellbore would have to be planned around the existing wellbores.
  • Such activities add to the tortuosity of the wellbore's trajectory, which increases torque and drag forces, and/or may be difficult to achieve with the available tools.
  • the detour may not be possible. In subsurface terms, the detour may make it difficult to achieve optimum alignment to a target. If so, oil and gas reserves and production may be adversely affected.
  • the wellbore positioning system 100 is preferably operatively connected to a controller 133 of the wellbore drilling system, for example via the network N 1 .
  • the controller 133 of the wellbore drilling system is automatically configured with the one or more operating modes determined by the system 100 , the controller 133 being arranged to apply the one or more operating modes.
  • FIG. 12 the steps involved in a first embodiment of a computer-implemented method for determining one or more operating modes for the wellbore drilling system are shown.
  • step S 1201 the input data is received by the wellbore positioning system 100 .
  • the input data are input into the expansion factor software tool 111 , the calculations of which are described above in relation to FIGS. 5 a - 5 e , 6 a and 6 b .
  • the expansion factor calculation tool is then run in step S 1203 , and generates, at step S 1204 , position data indicative of a relative position or proximity of the planned wellbore to the existing or simulated wellbore or object.
  • This data may be output in various forms, for example, as coordinates of a 2D or 3D simulation of a reservoir, or as a collision avoidance plot.
  • the generated position data are used to determine one or more operating modes of the wellbore drilling system.
  • the operating mode can represent an instruction or suggested setting for the drilling system, which can subsequently be applied to the drilling system.
  • the determination can include the step of comparing, in accordance with a predetermined set of rules (which can be set using a collision avoidance plot implemented by the trajectory model 123 ), the calculated position data to predetermined known or threshold position data that is accessible from the database DB 2 .
  • the determination may be based on a known position of an existing wellbore or a sub-surface hazard.
  • Software executed by the CPU 105 of the system 100 determines, on the basis of the determined position data, the one or more operating modes of the wellbore drilling system.
  • the expansion factor calculation tool 111 , the reservoir model 121 and/or the trajectory model 123 may be configured to determine the operating mode(s) upon generation of the position data, or a separate software component may be provided. Additional technical and physical constraints determined by the reservoir model 121 or the trajectory model 123 may be taken into account in order to determine the operating mode, and can be stored and accessed from the databases DB 1 and DB 2 as necessary.
  • the operating mode can comprise an instruction to go ahead with the drilling of a planned wellbore or not, this determination being based on a determination by the trajectory model 123 that the trajectory of the planned wellbore under consideration is drillable.
  • the operating mode can comprise one or more specific configuration settings for the wellbore drilling system, such as a drilling speed or trajectory.
  • the software component used to determine the operating mode is configured to use a predetermined set of rules in conjunction with input data such as the calculated expansion factor, in order to determine the operating mode. These rules are stored in and accessible from the database DB 1 and DB 2 as necessary.
  • the computer-implemented method can further include an optional step, S 1206 , of applying or inputting the determined operating mode into a controller of the wellbore drilling system.
  • the derivations using the YKC conditions depend on the ability to translate freely between the ellipse representations.
  • the first and second quadratic forms are mathematically equivalent.
  • E _ 0 [ b 2 0 0 0 a 2 0 0 0 - a 2 ⁇ b 2 ] ( A ⁇ - ⁇ 2 )
  • the matrix T translates a point on the ellipse by an amount x 0 in the x direction and y 0 in the y direction.
  • the rotation matrix R rotates a point by an amount ⁇ clockwise about the origin.
  • the scaling matrix S scales a point by a factor k relative to the origin.
  • T _ [ 1 0 0 0 1 0 - x 0 - y 0 1 ] ( A ⁇ - ⁇ 3 )
  • R _ [ cos ⁇ ⁇ ⁇ - sin ⁇ ⁇ ⁇ 0 sin ⁇ ⁇ ⁇ cos ⁇ ⁇ ⁇ 0 0 0 1 ] ( A ⁇ - ⁇ 4 )
  • S _ [ k - 1 0 0 0 k - 1 0 0 0 1 ] ( A ⁇ - ⁇ 5 )
  • the coordinates of the ellipse's centre, semi-major and semi-minor axes and orientation may be recovered from the first quadratic form using the inverse transform, Eqs. A-7 to A-11.
  • the inverse transform is not used in either the separation or expansion factor calculations it provides an effective means of testing the correctness of the transform. Note that the constant G is equivalent to H ⁇ a 2 b 2 k 2 .
  • x 0 CD - BF B 2 - AC ( A ⁇ - ⁇ 7 )
  • y 0 AF - BD B 2 - AC ( A ⁇ - ⁇ 8 )
  • a 2 ⁇ ( AF 2 + CD 2 + GB 2 - 2 ⁇ BDF - ACG ) ( B 2 - AC ) ⁇ [ ( A - C ) 2 + 4 ⁇ B 2 - ( A + C ) ] ( A ⁇ - ⁇ 9 )
  • the ellipse can also be represented so the symmetric matrix is independent of the ellipse's origin, Eq. A-12 and A-13, (Zheng and Palffy-Muhoray, 2010).
  • k 1 throughout.
  • u 0 - a 1 4 ⁇ b 1 4 ( B ⁇ - ⁇ 5 )
  • u 1 a 1 2 ⁇ b 1 2 ⁇ ⁇ - a 2 2 ⁇ b 1 2 ⁇ cos 2 ⁇ ( ⁇ 2 - ⁇ 1 ) - a 1 2 ⁇ ⁇ a 2 2 ⁇ sin 2 ⁇ ( ⁇ 2 - ⁇ 1 ) + b 2 2 ⁇ cos 2 ⁇ ( ⁇ 2 - ⁇ 1 ) ⁇ + b 2 2 ⁇ ⁇ - b 1 2 ⁇ sin 2 ⁇ ( ⁇ 2 - ⁇ 1 ) + ( ⁇ ⁇ ⁇ x 0 ⁇ cos ⁇ ⁇ ⁇ 2 + ⁇ ⁇ ⁇ y 0 ⁇ sin ⁇ ⁇ ⁇ 2 ) 2 ] + a 2 2 ⁇ ( ⁇ ⁇ ⁇ y 0 ⁇ cos ⁇ ⁇ ⁇ 2 - ⁇ ⁇ x x ⁇
  • the coefficients ⁇ i of the quartic equation may be calculated as Eq. B-12 to B-24.
  • ⁇ ⁇ 1 ⁇ ⁇ ⁇ y ⁇ ⁇ cos ⁇ ⁇ ⁇ 1 - ⁇ ⁇ ⁇ x ⁇ ⁇ sin ⁇ ⁇ ⁇ 1 ( B ⁇ - ⁇ 12 )
  • ⁇ ⁇ 1 ⁇ ⁇ ⁇ x ⁇ ⁇ cos ⁇ ⁇ ⁇ 1 + ⁇ ⁇ ⁇ y ⁇ ⁇ sin ⁇ ⁇ ⁇ 1 ( B ⁇ - ⁇ 13 )
  • ⁇ ⁇ 2 ⁇ ⁇ ⁇ y ⁇ ⁇ cos ⁇ ⁇ ⁇ 2 - ⁇ ⁇ ⁇ x ⁇ ⁇ sin ⁇ ⁇ ⁇ 2 ( B ⁇ - ⁇ 14 )
  • ⁇ ⁇ 2 ⁇ ⁇ ⁇ x ⁇ ⁇ cos ⁇ ⁇ ⁇ 2 + ⁇ ⁇ ⁇ y ⁇ ⁇ sin ⁇ ⁇ ⁇ 2 ( B ⁇ - ⁇ 15 )
  • ⁇ r 1 ⁇ 1 2 ⁇

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