`
`Applied Drilling Engineering
`
`Downloaded from http://onepetro.org/books/book-pdf/2794205/9781555630010.pdf by Robert Durham on 26 January 2023
`
`Adam T. Bourgoyne Jr.
`
`
`Professor of Petroleum Engineering
`Louisiana
`
`State U.
`
`Keith K. Millheim
`
`
`Manager-Critical Drilling Facility
`
`Amoco Production Co.
`Martin E. Chenevert
`
`
`Senior Lecturer of Petroleum Engineering
`
`U. of Texas
`
`F.S. Young Jr.
`President
`Woodway Energy Co.
`
`SPE Textbook Series, Volume 2
`Henry L. Doherty Memorial Fund of AIME
`
`Society of Petro\eum Engineers
`
`
`
`Richardson, TX USA
`
`LIBERTY EXHIBIT 1045, Page 1
`
`

`

`Downloaded from http://onepetro.org/books/book-pdf/2794205/9781555630010.pdf by Robert Durham on 26 January 2023
`
`Dedication
`This book is dedicated to the many students who were forced to study from the trial drafts of this work.
`
`Disclaimer
`
`This book was prepared by members of the Society of Petroleum Engineers and their well-qualified colleagues from
`material published in the recognized technical literature and from their own individual experience and expertise.
`While the material presented is believed to be based on sound technical knowledge, neither the Society of Petroleum
`Engineers nor any of the authors or editors herein provide a warranty either expressed or implied in its application.
`Correspondingly, the discussion of materials, methods, or techniques that may be covered by letters patents implies
`no freedom to use such materials, methods, or techniques without permission through appropriate licensing.
`Nothing described within this book should be construed to lessen the need to apply sound engineering judgment
`nor to carefully apply accepted engineering practices in the design, implementation, or application of the techniques
`described herein.
`
`© Copyright 1986 Society of Petroleum Engineers
`
`All rights reserved. No portion of this book may be reproduced in any form or by any means, including electronic
`storage and retrieval systems, except by explicit, prior written permission of the publisher except for brief passages
`excerpted for review and critical purposes.
`
`Manufactured in the United States of America.
`
`ISBN 978-1-55563-001-0
`ISBN 978-1-61399-159-6 (Digital)
`
`Society of Petroleum Engineers
`222 Palisades Creek Drive
`Richardson, TX 75080-2040 USA
`http://store.spe.org
`books@spe.org
`1.972.952.9393
`
`LIBERTY EXHIBIT 1045, Page 2
`
`

`

`Chapter 8
`Directional Drilling
`and Deviation Control
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`deflected into the X-Y planes
`(see Fig. 8 .1 ) The X plane is defined as
`and the Y plane is
`sociated with the departures in the X and Y planes are
`
`and thus there may be environmental restrictions that pro­
`
`
`
`
`
`hibit the use of power vessels and equipment such as off­
`
`
`shore drilling rigs and production facilities.
`The early drilling of directional wells was clearly moti­
`
`
`
`Directional drilling is the process of directing the well­
`
`
`
`vated by economics . The oil fields offshore California
`
`bore along some trajectory to a predetermined target.
`
`
`
`were the spawning ground for directional drilling prac­
`
`
`
`
`Deviation control is the process of keeping the well bore
`
`tices and equipment, and for a special group of people
`
`
`
`contained within some prescribed limits relative to incli­
`
`
`
`
`called "directional drillers. " Later discoveries of oil and
`
`
`nation angle, horizontal excursion from the vertical, or
`
`gas in the Gulf of Mexico and in other countries promot­
`
`
`both. This chapter discusses the principles and mecha­
`
`
`
`
`ed the expanded application of directional drilling. Off­
`
`
`
`nisms associated with directional drilling and deviation
`
`
`
`shore field development has accounted for the majority
`control.
`
`
`
`of directional drilling activities. Fig. 8 . 3 shows a typical
`
`
`
`The preceding chapters deal with the one-dimensional
`process of penetrating the earth with the bit to some ver­
`
`
`
`offshore platform development. In a number of cases,
`
`
`tical depth. However, drilling is a three-dimensional proc­
`
`
`fields have been discovered beneath population centers,
`
`
`ess. The bit not only penetrates vertically but is either
`
`
`and the only way to develop the fields economically has
`
`purposely or unintentionally
`
`
`been to use a drilling pad and to drill directionally (see
`
`the direction plane
`
`
`
`
`Fig. 8 .4). Natural obstructions such as mountains or other
`the inclination plane. The angles as­
`
`
`
`
`severe topographical features frequently prohibit build­
`
`
`
`
`ing a surface location and drilling a near-vertical well (Fig.
`called "direction" and "inclination" angles, respectively.
`
`
`
`
`
`
`
`8 . 5 ) . Sidetracking out of an existing wellbore is another
`
`
`Fig. 8 . 2 presents a typical example of the trajectory­
`
`
`
`
`application of directional drilling. This sidetracking may
`
`
`control situation. Here a structure is located almost en­
`
`
`be done to bypass an obstruction (a " fish" ) in the origi­
`
`
`tirely under a lake. Well 1 , drilled on a part ofthe struc­
`
`
`nal wellbore (see Fig. 8 . 6) or to explore for additional
`
`ture that is not under the lake, could be treated simply
`
`
`
`producing horizons in adjacent sectors of the field (see
`
`as a deviation-control well drilled on the shore. To de­
`Fig. 8 . 7 ) .
`velop the rest of the field, however, will necessitate drill­
`Strong economic and environmental pressures have in­
`
`
`
`
`ing directional wells. The only way vertical wells could
`
`
`
`creased the use of directional drilling. In some areas it
`
`
`
`
`be drilled would be from a floating drilling vessel or plat­
`
`is no longer possible to develop a field by making roads
`
`form, with the wells being completed on the lake bed (sub­
`
`
`
`
`to each surface location and drilling a near-vertical well.
`
`
`lake completions), or from a floating or fixed production
`
`
`
`
`Instead, as in offshore installations, drilling pads must be
`
`platform; and the economics of those approaches would
`
`built from which a number of wells can be drilled. Not
`
`
`
`be far less attractive than drilling directional wells from
`
`
`
`only is directional drilling increasing, but trajectory pro­
`
`
`
`some convenient land-based site where a standard land
`
`
`grams are becoming more complicated and directional
`rig can be used. In some situations, there is no alterna-­
`
`drilling is being applied in situations and areas where
`
`
`tive to drilling a directional well. For example, the lake
`
`
`
`directional drilling has not been common. In hot-rock de­
`
`may be the only source for drinking water in the area,
`
`velopments, for example, directional wells are being
`
`8.1 Definitions
`
`and Reasons for
`Directional Drilling
`
`LIBERTY EXHIBIT 1045, Page 3
`
`

`

`352
`
`APPLIED D R ILLING E N G I N E E R I N G
`
`N
`
`- - -
`
`- -
`
`BOTTOM HOLE LOCATIOt,
`
`� c>,,' """'''''
`SURFACE �CATION
`FOR WELL NO 1
`/' -
`I
`I
`<>
`--tl
`FOR WELL 2
`I
`
`I Z AXIS (TRUE VERTICAL
`t DEPTH)
`
`NORT H
`
`P L A N E I
`
`D I R E C T I O N
`
`- -
`
`II -
`I
`
`I I I I I I
`
`L __
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`,--_1
`___ J
`SCALE
`
`Fig. 8 . 1- l nclination and direction planes as a wel lbore proceeds
`
`Fig. 8 . 2-Plan view of a typical oil and gas structure under a
`
`
`in the depth plane.
`
`lake showing how d irectional wells could be used to
`
`develop it.
`
`
`
`DRILLING R I G INSIDE BUI LDING
`
`
`wells.
`
`Fig. 8.4-Developing a field under a city using d i rectionally
`
`Fig. 8.3-Typical offshore development platform with directional
`drilled wells.
`
`LIBERTY EXHIBIT 1045, Page 4
`
`

`

`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`
`
`D I R ECTIONAL D R I L L I N G A N D DEVIATION CONTROL
`
`
`
`353
`
`
`
`SIDETRACKED HOLE AROUND FISH
`
`wel l s where the reservoi r is Fig. 8.S-Dri l l i n g of d irectional
`
`
`
`beneath a major surface obstructi o n .
`
`Fig. 8.S-Sidetracking aro u n d a fish.
`
`and metamorph­
`drilled in hard granites and other igneous
`
`
`
`
`ic rocks. Geothermal projects have been developed with
`
`
`
`directional wells. Wells with extended horizontal reaches
`
`of 1 4 ,000 ft are being drilled, with goals of going even
`
`farther. As the costs of field development increase-in
`
`deeper waters, remote locations, hostile environments,
`
`
`and deeper producing zones-the use of directional drill­
`ing will also increase.
`
`8.2 Planning the Directional Well
`
`Trajectory
`
`The first step in planning any directional well is to de­
`
`
`
`sign the wellbore path, or trajectory, to intersect a given
`
`
`
`
`target. The initial design should propose the various types
`
`
`
`of paths that can be drilled economically. The second,
`
`or refined, plan should include the effects of geology on
`
`the bottomhole assemblies (BHA' s) that will be used and
`
`
`other factors that could influence the final well bore trajec­
`
`
`
`tory. This section explains how to plan the initial trajec­
`
`tory for most common directional wells.
`
`
`Fig. 8.8 depicts three types of trajectories that could
`
`be drilled to hit the target. Path A is a build-and-hold
`
`
`trajectory: the well bore penetrates the target at an angle
`
`equal to the maximum buildup angle. Path B is a
`" modified-S" and C is an " S " trajectory. With the S­
`
`
`shape trajectory the wellbore penetrates the target verti­
`
`cally, and with the modified-S trajectory the wellbore
`
`
`penetrates the target at some inclination angle less than
`the maximum inclination angle in the hold section. For
`cking Fig. 8.7-Using an old well to explore for new oil by sidetra
`
`
`
`
`
`
`Path D, a " continuous-build trajectory, " the inclination
`
`out of the cas i n g and d r i l l i n g d i rectionally.
`
`
`keeps increasing right up to or through the target. The
`
`
`
`build-and-hold path requires the lowest inclination angle
`
`to hit the target; the modified-S requires more inclina­
`
`tion; and the S-shape requires still more. The continuous­
`
`
`
`
`build path requires the highest inclination of all the trajec­
`tory types to hit the target.
`
`LIBERTY EXHIBIT 1045, Page 5
`
`

`

`354
`
`APPLIED DRILLING ENGINEERING
`
`AND DROP ("S· TYPE)
`
`DROP AND/OR HOLD (MODIFIED
`oS· TYPE)
`
`CONTINUOUS
`
`\ \ \. '\ C BUILD-HOLD
`\ ',yV j I
`BUILD AND HOLD TYPE;' A \ \ :
`r0(', B /b<.. BUILD-HOLD
`0\\ \
`. \ \ \1
`BUILD " \ \ I /-
`,' \1// / ��/ I. /��'l
`/,// / /
`/ / / /
`<..... c. - ./ --
`/ / / /
`
`
`
`Fig . 8.8-Major types of wellbore trajectories.
`
`8.2 . 1 Build-and�Hold Trajectory
`
`Fig. 8 . 9 depicts a simple build-and-hold wellbore trajec­
`
`
`
`
`
`tory intersecting a target at a true vertical depth (TVD)
`
`
`
`of D3 and at a horizontal departure of X3 (Point B). The
`kickoff point is at a TVD of depth D \ , where the rate
`and
`
`of inclination angle buildup
`per unit length.
`The radius of curvature, r \, is found thus:
`
`.
`
`where
`
`r \
`
`DB
`
`
`
`. . . . . . . . (8.4)
`
`1 80 1
`
`
`Substituting DB into Eq. 8 . 4 gives
`. . . . . . . . . . . . . . . . . . .
`(8. 1 )
`
`in
`
`SIll 0 =- . . . . . . . . . . . . . . . . . . . .
`is q in degrees
`7r q
`r \ =-X-. .. . . . . . .
`To find the maximum inclination angle, 0, consider
`
`The maximum inclination angle, (), for the build-and-hold
`
`
`case, is not limited
`
`Fig. 8 . 9 that
`
`or
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`to X 3 < rt. It is also valid for X 3 � rt.
`( r \ -X3 )
`
`
`
`
`
`0 = 0 - 7. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . (8. 2)
`
`The angle 7 can be found by considering Triangle DAB,
`
`where
`
`- arc tan
`D 3 -D \
`
`
`
`. . . . . . . . . . . . . . . . . . . (8. 6)
`
`BA r \-X3
`tan 7 =- =
`, . . . . . . . . . . . . . . . . . (8.3a)
`
`
`AD D 3-D \
`
`The length of the arc, Section DC, is
`
`and
`
`7r
`LDC= - X r \ xO,
`
`1 80
`
`r \-X3
`or
`
`7 =arc tan . . . . . . . . . . . . . . . . . . . . (8.3b)
`D 3 -D \
`
`
`
`
`
`Angle 0 can be found by considering Triangle DBC,
`
`
`
`
`
`LDC=-' . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . (8. 7)
`
`o
`
`q
`
`LIBERTY EXHIBIT 1045, Page 6
`
`

`

`
`
`D I R ECTIONAL D R I L L I N G A N D DEVIATION CONTROL
`
`355
`
`D1
`D
`
`0' r1
`
`Df
`
`DN
`
`r1
`
`0
`
`A
`
`Fig. 8 . 10-Geometry for the build sectio n .
`
`a I I Tf-*' I
`I "
`I J_ ,,,
`I I I I I I 0
`p'; i I I
`-C'I I I I I I
`I
`M B x3H
`I
`r1 �
`XN�
`t
`x �
`ld-type well path for X3 <fl'
`
`Fig. 8.S-Geometry of build-and-ho
`
`inclination angle a t C ' (Fig. 8 .1 0) , which will yield a new
`
`
`
`horizontal departure, XN. The distance dN can be deter­
`
`mined considering Triangle D ' OC ' , where
`The length of the trajectory path, CB, at a constant in­
`
`clination angle can be determined from Triangle BCO as
`
`
`
`DN=D ) + r ) sin () ' , . . . . . . . . . . . . . . . . . . . . (8.1 0)
`
`
`
`and the horizontal displacement, X N , is
`
`
`
`
`
`XN= r ) - r ) cos (} ' =r ) ( l- cos ()) .. . . . . . . . (8.1 1)
`
`The TVD at the end of the build section is D2, which
`
`
`
`can be derived from Triangle D ' OC (Fig. 8 . 9 ) :
`
`CO r )
`= --
`LCB LCB
`
`and
`
`r )
`
`tan !l
`
`D2= D) + rl s i n () .. . . . . . . . . . . . . . . . . . . . .
`(8.12)
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`.. . . . . . . . . . . . . . .
`
`tan!l =--
`LCB=-- '
`.. . . . . . . . . .
`D M = D) + - +--
`q tan!l
`X2 = r ) - r ) cos (} = r ) ( l- cos ()). . . . . . . . . . . .
`CP=-­
`
`
`The total measured depth, D M , for a TVD of D 3 is
`
`
`
`
`
`The new measured depth for any part of the buildup is
`
`() r )
`
`
`, . . . . . . . . .(8. 8)
`
`()'
`D MN =D ) + -. .. . . . . . .
`
`(8.1 3)
`
`
`
`
`where D M equals the vertical section to kickoff plus build
`mined from Triangle PP' C :
`
`
`
`section plus constant inclination section (Fig. 8 . 9 ) .
`
`The horizontal departure E C (X2) a t the end o f the
`
`
`build can be determined by considering Triangle D ' OC,
`where
`
`()
`
`
`DMP=D ) + -+ CP , . . . . . . . . . . . . . . . . . . . . . ( 8 .14)
`
`The new measured depth at a TVD of D ' can be deter­
`
`To find the measured depth and horizontal departure
`
`
`along any part of the build before reaching maximum an­
`
`
`
`gle (), consider the intermediate inclination angle () ' , the
`
`CP'
`
`cos ()
`
`( 8 . 9)
`
`where
`
`q
`q
`
`LIBERTY EXHIBIT 1045, Page 7
`
`

`

`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`356
`
`and
`
`CP'= D' -D2 = (D '-D1 - r , sin () .
`
`Therefore,
`
`(D' -D 1 - rl sin ()
`CP=
`
`cos ()
`
`Since X3 < r" () = D - r. From Eq. 8 .3a,
`
`
`
`APPLIED DRILLING E N G I N E E R I N G
`
`r\ - X3 2 1 0 ft
`tanr =
`= 0 . 026 1,
`D 3 - D, 8 ,050 ft
`
`r = arc tan 0 . 0261= 1 .5 ° .
`
`
`. . . . . . . . . . . . . . . (8.1 5)
`
`q
`
`
`
`Substituting Eq. 8 .1 5 into Eq. 8 .1 4 ,
`
`From Eq. 8 . 5 ,
`
`
`() D ' - D 1 -r , sin ()
`DMP=D, +-+
`
`
`cos ()
`
`. . . . . . . (8.1 6)
`
`Eq. 8 .1 6 also can be used instead of Eq. 8 .14 to calcu­
`
`
`late the measured depth by making D ' =D 3.
`
`The horizontal departure at Point P is
`
`---
`
`2 ,865 ft
`= 0 . 35 5 8 ,
`8 ,053 ft
`
`X ' = X2+ P' P , . . . . . . . . . . . . . . . . . . . . . . . . . . (8. 17)
`
`D = arc sin 0 .356= 20 . 84 ° .
`
`The maximum inclination angle is
`
`where P ' P= CP ' tan () .
`
`Combining Eq. 8 .1 7 , E q . 8 . 9 , and CP' yields
`
`() = 20 . 84 ° - 1 .5 ° = 1 9 . 34 ° .
`
`X '= r , ( l- cos ()+ (D ' - D ,- r \ sin ()tan () . . ( 8 .1 8)
`
`The preceding derivation is valid only when X 3 < r \ .
`
`
`
`Another way of expressing the maximum inclination
`() = arc tan
`angle, (), in terms of r \, D \, D 3, and X3 for X3 < r\ is
`2 , 865 ft - 2, 65 5 ft
`
`() = arc tan
`
`-arc cos
`D3 -D,
`
`- arc cos
`9 , 650 ft -l, 600 ft
`
`(8.1 9)
`
`x sin arc tan
`2 ,865 ft - 2, 655 ft
`
`D M= I,600 ft +--
`
`1 9 . 34 °
`x l00 ft = 2, 565 ft ,
`2 °
`
`Using Eq. 8 .1 9 , (9 , 650 ft -l, 600 ft)
`r, -X3
`( D 3 - D, ) [( r l )
`( 2 , 865 ft )
`[ (9 , 650 ft -l, 600 ft)]
`x sin [arc tan ( �; �;3' ) ] J. . . . . . . . . . . .
`= 1 9 . 34 ° .
`( 1 ) the radius of curvature, R 1 ; (2) the
`ft, using Eq. 8 .1 4 , is R,
`
`The measured depth to the end of the build at an incli­
`
`nation of 1 9 . 34 ° is
`Example 8.1 . It is desired to drill under the lake to the
`
`
`location designated for Well 2 . For this well, a build-and­
`
`hold traj ectory will be used. Horizontal departure to the
`
`
`target is 2 , 655 ft at a TVD of9 , 650 ft. The recommended
`rate of build is 2 . 0 ° 11 00 ft. The kickoff depth is
`1 ,600
`
`ft . Determine
`
`
`maximum inclination angle, () ; (3) the measured depth to
`
`the end ofthe build; (4) the total measured depth; (5) the
`
`
`horizontal departure to the end of the build; (6) the meas­
`
`ured depth at a TVD of 1 ,9 1 5 ft; (7) the horizontal dis­
`
`
`placement at a TVD of 1 ,9 1 5 ft; (8) the measured depth
`at a TVD of 7 , 6 1 4 ft; and (9) the horizontal departure
`= 1 0 , 09 1 ft .
`at a TVD of 7 , 6 1 4 ft .
`
`
`
`and the total measured depth t o the target TVD o f 9 , 650
`
`Dtar= 2, 565 ft +-- = 2 , 565 ft +----
`tan 0
`tan(20. 84 0)
`
`2 , 865 ft
`
`From Eq. 8 .1
`Solution.
`
`The horizontal departure t o the end o f the build, from
`
`
`Eq. 8 . 9 , is
`
`1 80 1
`r , = -
`
`7r 2 ° 1 1 00 ft
`
`= 2 , 865 ft .
`
`X 2 = r 1 ( I- cos () = 2 , 865 ft [1 - cos(I 9 . 34)] = 1 6 1 ft.
`
`LIBERTY EXHIBIT 1045, Page 8
`
`

`

`
`
`D I R ECTIONAL DRILLING A N D DEVIATION CONTROL
`
`
`
`357
`
`8.2.2 Build-Hold-and-Drop
`
`1 ,9 1 5 ft =I, 600 ft +2 , 865 ft sin ()
`
`("S") Trajectory
`At a TVD of 1 ,9 1 5 ft, the measured depth at a rate of
`
`
`
`build of 2 ° / 1 00 ft can be determined by first calculating
`The second type of trajectory is the build, hold, and
`
`the inclination at 1 ,9 1 5 ft using Eq. 8 .10:
`
`
`drop-or S-shape curve-which is depicted by Fig. 8 .1 1
`for the cases were r l < X3 and r l + r2 > X4 , and in Fig.
`8 .1 2 for the cases where r l < X3 and r l + r2 < X4 . In
`
`both cases, the maximum inclination is reduced to zero
`
`in the same at D4 with drop radius r2, which is derived
`
`
`manner as the build radius, r l . The following equations
`
`
`are used to calculate the maximum inclination angles for
`
`r [ + r2 > X4 and for r l + r2 < X4·
`using
`
`= 6 . 3 1 0 .
`8 = arc sin
`2 , 865 ft
`
`6 . 3 1 ° can be calculated
`
`( 3 1 5 ft )
`
`The arc length of the build to
`Eq.8 . 7 :
`
`LDc=--
`
`6 .3 1 °
`x loo ft = 3 15 . 5 ft .
`2 . 0 °
`
`
`
`The measured depth for a TVD o f 1 ,9 1 5 ft is
`
`DM= 31 5 . 5 ft+ 1 ,600 ft= 1 ,9 1 5 . 5 ft,
`
`which is only 0 . 5 ft more than the TVD .
`
`
`The horizontal departure at a TVD o f 1 ,9 1 5 ft i s found
`fromEq.8 .1 1 :
`
`X 1,91S = 2 , 865 ft ( l. O - cos 6 . 31 ) =1 7 . 36 ft .
`
`
`
`The measured depth a t a TVD o f 7 , 6 1 4 ft i s
`
`X Sin[arctan( D4- D1 )J]
`. . . . . . . . . .
`
`r l + r2 -X4
`
`(8. 2 1 )
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`X Sin[arc tan [ D 4-D 1 ]]) .
`. • . . . . . (8. 22)
`8.2.3 Build, Hold, Partial
`+------------------------------
`LCA=- . q
`8'
`X'7,614 =2,865 ft ( I- cos 19.34)
`L CB = r2 sin 8' . . . .. . .. . . ... . . . . . . .. . . .
`S BA = r2 - r2 cos 8' = r2( I- cos 8 ' ) . . . . . . . .
`
`X4 - (r l + r2)
`
`Drop, and Hold
`(Modified "S") Trajectory
`
`The build, hold, partial drop, and hold (Fig. 8 .1 3) is the
`
`
`modified S type of wellbore path. Consider that the arc
`length
`
`7 , 6 1 4 ft - 1 , 600 ft -2 , 865 ft sin(1 9 . 34 ° )
`
`= 7,934 ft .
`
`The horizontal departure at a TVD of7 , 6 1 4 ft i s calcu­
`
`lated with Eq. 8 .1 8 :
`
`From the Right Triangle CO'B, the following relation­
`
`ships can be written.
`
`+ (7 , 6 1 4 ft -l , 600 ft -2 , 865 ft sin 1 9 . 34 ° )
`
`X tan 1 9 . 34 = 1 ,935. 5 ft .
`
`and
`
`(S .23a)
`
`(8.23b)
`
`Eqs. S . 2 1 and 8 . 22 can be rewritten by substituting
`
`Ds + r2 sin 8 ' for D4 and Xs + r2( I- cos 8 ' ) for X4.
`
`
`
`
`The preceding derivation and example calculation is for
`
`
`For any of the S-shape curves, the measured depths and
`
`the case where r [ > X3 for a simple build-and-hold trajec­
`
`
`
`horizontal departures can be calculated in the same way
`tory. For the case where r l < X3 , the maximum angle,
`
`they are calculated for the build-and-hold trajectory by
`
`
`8, can be calculated by
`
`
`
`
`
`deriving the appropriate relationships for the various ge­
`ometries .
`
`x sin [arc tan ( :: ��ll ) ]]. . . . . . . . . . . .
`
`Targets
`
`
`When a directional well is being planned, the depth and
`
`
`
`horizontal departure of the target are given, as well as
`
`
`
`its dimensions. Targets may be rectangular, square, or
`(8.20)
`
`circular. If the target is a circle, a radius is designated.
`
`8.2.4 Multiple
`
`LIBERTY EXHIBIT 1045, Page 9
`
`

`

`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`358
`
`APPLIED D R ILLING E N G I N E E R I N G
`
`-j0 START OF B U I L D U P
`
`-I--I _ _ --'r 1 __
`
`_ -;70 K I CKOFF
`
`03
`
`04
`
`04 03
`
`.
`
`DROP OFF
`
`t-_ �r�1 ___
`I
`I I I
`I I
`-
`- 1------
`I
`--� -T --- O· -I
`..l.- __ I_ -- ------ _
`I
`,'l- I
`o --
`I TARGET
`I
`I I I TARGET
`X2� I I I I
`,
`r-----X3 -------.,� I
`I I
`X3 --1 I
`Fig. 8 . 12-Build-hold-and-drop for the case where (1 < X 3 and
`-and-drop for the case where (1 < X 3 and
`X. -I
`�X 4 �
`(1 + (2 >X 4·
`
`Fig. 8 . 11-Build-hold
`
`(1 + (2 < X4·
`
`Sometimes there are multiple targets, as shown by Figs.
`
`
`
`8 .14a and 8 .1 4b. If they are favorably positioned, multiple
`
`
`
`targets can be economically penetrated with one of the
`Example 8.2. What are the directions , in the alternative
`
`
`aforementioned types of traj ectories (Fig. 8 .1 4a). Some­
`
`format, of each of the following wells?
`
`
`
`times, however, they are unfavorably aligned (Fig. 8. 14b)
`
`
`
`and expensive traj ectory alterations are required. The
`
`trajectory in Fig. 8. 14b could be difficult and expensive
`
`
`to drill even though the vertical section appears the same
`
`as that in Fig. 8 .14a. The direction change to hit Target
`
`
`
`3 would in most situations be extremely difficult to
`execute.
`
`Well A
`Well B
`Well C
`
`N l SE
`225°
`NOE
`
`and is I S 0; Well
`X-Y Trajectory
`dimensional Y-Z trajectory (Fig. 8 .1 ) . The next step is
`
`Well A is in the first quadrant
`Solution.
`
`
`
`B is in the third quadrant and should be read as S4SW ;
`8.2.5 Direction Quadrant and Compass Schemes
`
`
`and Well C represents 0 ° o r north.
`
`
`
`
`In the previous discussions all the trajectory planning has
`
`
`been reduced to a two-dimensional problem, considering
`
`
`only depth and horizontal departure. All directional wells
`8.2.6 Planning the
`
`
`also have an X component that is associated with direction.
`
`
`
`For example, Well 2 in Fig. 8 . 2 has a target direction
`of 1 00 ° east of north by a normal compass reading. In
`to account for the X component of the trajectory that
`
`
`
`
`
`directional drilling, a 90° quadrant scheme is used to cite
`
`
`
`
`departs from the vertical plane section between the surface
`
`
`directions and the degrees are always read from north to
`
`
`
`location and the bottornhole target. Fig. 8 .1 6 is a plan
`east or west, and from south to east or west. For example,
`
`
`view, looking down on the straight line projected path
`
`
`
`the direction angle in Fig. 8 .1 Sa by compass (always read
`
`
`
`from Well 2 ' s surface location to the bull's-eye of a tar­
`
`clockwise from due north) is 1 8 0 , and by the quadrant
`
`get with a 1 00-ft radius . The dashed line indicates a pos­
`
`scheme it is N 1 8E . The well in the second quadrant (Fig.
`
`sible path the bit could follow because of certain influences
`
`
`8 . I Sb) at l S 7 ° is read S23E . In quadrant three (Fig.
`
`
`8 . I Sc), the well is S20W , for a measured angle of 200° .
`
`
`
`exerted by the bit, the BHA configuration, the geology,
`
`
`
`general hole conditions , and other factors that are covered
`
`In quadrant four (Fig. 8 .1 Sd), the compass angle of 30S 0
`later in this chapter.
`
`is read NSSW.
`
`The first step in planning a well is to determine the two­
`
`LIBERTY EXHIBIT 1045, Page 10
`
`

`

`
`
`D I R ECTIONAL D R I L L I N G AND DEVIATION CONTROL
`
`
`
`359
`
`
`
`1-----'----70 START OF BUILDUP
`
`D, D3
`
`
`
`Fig. 8 . 13-Build-hold-and-drop and hold (modified-S) where
`
`point i s D 1 .
`
`The target area provides a zone of tolerance for the well­
`
`
`bore trajectory to pass through. The size and dimensions
`
`
`
`of the target are usually based on factors pertaining to the
`
`
`drainage of a reservoir, geological criteria, and lease
`
`
`boundary constraints.
`When a well is kicked off, the practice is to orient the
`
`
`
`
`
`
`trajectory to some specific direction angle called' 'lead. "
`
`
`
`
`This lead usually is to the left of the target departure line
`
`and ranges from 5 to 250 • The value used is generally
`
`based on local experience or some rule of thumb. More
`
`
`
`
`recent research on direction variation (or, to use an older
`term, "bit walk") indicates that the lead can be selected
`
`on the basis of analysis of offset wells and of factors that
`
`might cause bit walk.
`
`
`As the drilling progresses after the lead is set, the trajec­
`tory varies
`in
`
`Figs. 8 . 1 7 and 8 .1 8 are vertical and horizontal
`
`
`
`(elevation and plan) views of a typical trajectory path. Past
`the lead angle, the trajectory shows a clockwise, or right­
`
`
`hand, tendency or bit walk. A counter-clockwise curva­
`
`
`ture is called left-hand tendency or bit walk.
`
`
`
`The initial trajectory design did not account for the ex­
`
`
`cursion of the bit away from the vertical plane that goes
`
`
`through the surface location and the target' s bull' s-eye.
`
`
`
`There are many ways to calculate the three-dimensional
`. 1-3 The most common method used
`path of the wellbore
`
`in the field is " angle averaging, " which can be performed
`
`on a hand calculator with trigonometric functions.
`
`
`
`Consider the vertical section as depicted by Fig. 8 .1 7 .
`
`The distance from the surface to the kickoff
`
`kicked off and drilled to A2. The in­
`
`clination angle at the kickoff is zero. Fig. 8 .1 8 shows the
`
`in the X and Y planes as the bit penetrates
`the Z plane.
`I I
`I
`0
`� -+- �\STOPDROPOFF
`," I
`1 L __ � ___
`o
`I HOLD
`.. -1----�
`_____
`I I \ TO TARGET
`IO��,
`--
`� l-. lr-TARGET
`0'
`I
`I I
`-I 1
`-1-�---�
`--
`�----i--1-- - - -0' -I--it
`X.
`1--- ----
`X5---- --I
`X,-!
`X,
`f--------
`At A 1 the well is
`_ I
`r 1 < X 3 and r 1 + r 2 < X 4 .
`
`(-/. 3
`1 2 V_'
`-I
`-- f-0 l.:ECTED TRAJECTORY
`
`� _
`
`_
`
`-
`
`D1
`
`WITH LEFT TURN TO HIT TARGETS
`
`f---
`f4----
`�---
`
`X1 "I 1
`X2
`X3 ----j
`-I I
`(a)
`
`TARGET 3
`
`(b)
`
`
`
`
`
`m u ltiple targets. Fig. 8 . 14-Directional well used to i ntersect
`
`
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`LIBERTY EXHIBIT 1045, Page 11
`
`

`

`360
`
`
`
`APPLIED DRILLING E N G INEERING
`
`90 E
`II
`
`IV
`W 90
`III
`S
`(a)
`N
`0
`
`IV
`
`III
`
`(c)
`
`N
`
`IV
`
`11\
`
`(b)
`N o
`
`II
`
`III
`o s
`(�)
`
`
`
`Fig. 8 . 15-Directional quadrants and compass measurements.
`
`top, or plan, view of the trajectory; Point A 1 on the ver­
`
`
`
`tical section corresponds to the starting point, A I, on the
`
`
`plan view. Using the angle-averaging method, the follow­
`
`
`
`ing equations can be derived for the north/south (L) and
`
`east/west (M) coordinates .
`
`/
`1
`
`N
`
`III
`
`TARGET AT
`A TVO OF
`9650 Ft
`
`LAKE
`
`Fig. 8 . 16-Plan view.
`
`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`. (CJ. A + CJ. A- l) (f A + f A - l )
`. (CJ. A + CJ. A- l) . (f A + fA-1)
`(CJ. A + CJ. A -l)
`
`L = tJ)M sm
`
`cos
`
`2
`2
`
`
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8 . 24)
`
`and
`
`M = tJ) M sm
`
`2
`
`sm
`
`2
`
`.
`
`
`
`. . . . . . . . . . . . . . . . . . . . . . . . . . . (8. 25)
`
`
`
`
`
`The TVD can be calculated by
`
`D = tJ) M cos
`
`
`
`, . . . . . . . . . . . . ( 8 . 26)
`
`2
`
`
`where tJ) M is the measured depth increment.
`
`LIBERTY EXHIBIT 1045, Page 12
`
`

`

`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`
`
`D I R ECTIONAL D R I L L I N G A N D D EVIATION CONTROL
`
`361
`
`NORTH
`
`BULLSEYE
`
`
`
`
`
`TARGET
`i ..o;,l"------ SURFACE
`"' � b
`-------
`"'� &
`TARGET Q:-�
`TARGET
`RADIUS
`l�-----
`CLOSURE
`KICK OFF
`(InClination At Kickoff = 0°)
`N,
`I
`A, M,..."-I:-----":I ___ ----'-;-___
`
`LEAD ANGLE
`
`I
`SURFACE K, K3
`
`LOCATION
`
`K4
`
`EAST
`
`_
`
`_
`
`__
`
`�_
`
`
`
`Fig. 8 . 18-Horizontal calcu l ation.
`
`and (1 +0)
`
`- = 99 . 996 ft ,
`D 2 = 1 00 ft cos -
`2
`
`D = 8,000 ft + 99 . 996 ft = 8,099. 996 ft .
`
`From 8 ,1 00 t o 8 , 200 ft: ( I + 2 ) ( 20 + 20 )
`
`L3 = 1 00 ft sin -
`- cos
`2
`2
`
`= 2.46 ft .
`
`Total north =0 . 82 +2 .46 = 3 .28 ft .
`
`
`
`Fig. 8 . 17-Vertical calcu l ati o n .
`
`Example 8.3. Calculate the trajectory for the well from
`
`8 ,000 to 8 ,400 ft , where the kickoff is at 8 , 000 ft and
`the rate of build is 1 °1 1 00 ft, using a lead of 1 0 ° and a
`
`
`
`
`right -hand walk rate of I ° 1 1 00 ft . Direction to the bull's­
`eye is N30E. Assume that the first 200 ft is to set the lead,
`
`
`where the direction is held constant to 8 , 200 ft and then
`
`
`turns right at a rate of 1 ° / 1 00 ft.
`
`The north and east coordinates are calculated
`
`Solution.
`
`using Eqs. 8 . 24 and 8 . 2 5 , and the TVD from 8 , 000 to
`8 ,1 00 ft is calculated from Eq. 8 .26.
`
`Total east = 0.30 + 0.90 = 1 .20 ft.
`D3 = 1 00 ft cose � 2) = 99.966 ft .
`D = 8,099.996 + 99.966 = 8 ,1 99.962 ft .
`ft from N20E t o N21 E . (2 + 3) (20+ 2 1)
`
`
`
`
`
`From 8,200 to 8 , 300 ft, the direction changes by 1 0/ 1 00
`
`L4 = 1 00 ft sin -
`- cos
`2
`2
`
`= 4 . 09 ft.
`
`(1 ° + 0)
`(1 0 +0)
`
`cos(20) * = 0 . 82 ft,
`
`L 2 = 1 00 ft sin -
`
`2-
`
`sin(20)= 0 . 30 ft ,
`
`M 2 = 1 00 ft sin -
`
`2-
`
`' For the first point the direction should not b e averaged.
`
`Total north = 3. 2 8+ 4 .09 = 7. 3 7 ft.
`
`LIBERTY EXHIBIT 1045, Page 13
`
`

`

`Downloaded from http://onepetro.org/books/book/chapter-pdf/2794220/chapter08.pdf by Robert Durham on 26 January 2023
`
`362
`
`1 .5 3 ft.
`
`Total east = 1 .20 + 1 .5 3 = 2.73 ft.
`
`DM TVD N North N East Departure Angle'
`
`APPLIED D R ILLING E N G I N E E R I N G
`
`TABLE 8 . 1-DAT A F O R EXAMPLE 8 . 3
`
`Departure
`
`(tt) (tt) (tt) (tt) (tt) (degrees)
`8,000 8,000.00 0.00 0.00
`0.00
`8,100 8,099.99 0.82 0.30
`0.87
`8,200 8,199.96 3.28 1.20
`3.49
`8,300 8,299.86 7.37 2.73
`7.86
`8,400 8,399.67 13.05 4.97
`13.97
`
`• Note that the statement
`
`20.1
`20.1
`20.33
`20.85
`
`angle to be 20° to 8,200
`
`of the problem requ ires the departure
`
`
`ft. Roundoff error in the very small early-departure distances can cause the calculated
`departure angle to be different.
`From 8 ,300 to 8 ,400 ft, the direction further changes
`
`
`
`D4 = 1 00 ft cose ; 3) = 99.90 ft.
`D = 8, 1 99.962 + 99.90 = 8,299 .862 ft.
`to N22E. (3 +4) (2 1+ 22)
`M 5 = 1 00 ftsine ; 4)Sine1 ; 22) = 2 . 24 ft.
`Total east = 2. 7 3+ 2 .24= 4 . 97 ft . (3 +4)
`
`- cos
`L s = 1 00 ft sin -
`2
`2
`
`= 5. 68 ft .
`
`Total north = 7. 37 +5 .68= 1 3. 05 ft .
`
`D s = 1 00 ft cos -
`- = 99 . 8 1 ft,
`2
`
`
`
`D = 8,299.862 + 99. 8 1 = 8,399.672 ft.
`
`1 A 2 K 2, A 2 A3K3, A3A 4 K4 , and
`
`proper sign convention is observed. In planning a trajec­
`
`
`
`
`tory that is near 0 0 (first quadrant) and 360 ° (fourth quad­
`
`rant), special care must be tak