simplified equations for planning directional and horizontals

7/28/2019 Simplified Equations for Planning Directional and Horizontals

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SPESociety of Petroleum Engineers

SPE 21261Simplified Equations for Planning Directional andHorizontal WellsM.L. Wiggins and H.C. Juvkam-Wold, Texas A&M U.SPE Members

Copyright 1990, Society of Petroleum Engineers, Inc.This paper was prepared for presentat ion at the SPE Eastern Regional Meeting held in Columbus, Ohio, October 31-November 2, 1990.This paper was selected for presentat ion by an SPE Program Committee following review of infonmation contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been revIewed by the Society of Petroleum Engmeers and are sUbject to correct ion by the author(s). The material, as presented, does not necessarily reflectanyposItIOn of the Society of Petroleum Engineers, ,ts officers, or members. Papers presented at SPE meetings are subjectto publication review by Editorial Committees of the Societyof Petroleum Engmeers. PermIssIon to copy IS restricted to an abstract of notmore than 300words. Illustrationsmay not be copied. The abstractshould contain conspicuous acknowledgmento f where and by whom the paper IS presented. Wrote PublicatIOns Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.

ABSTRACfDirectional drilling techniques have been used for manyyears to reach subsurface objectives that have had inaccessiblesurface locations. Economic considerations and increasedenvironmental concerns have increased the number of directionalwells drilled in recent years. Directional drilling techniques havealso been applied to horizontal drilling with interest in this areaincreasing greatly over the last several years.In the past, there have been two major methods utilized inplanning the direct ional well. These methods are the use ofbuildup or composite buildup charts and the use of severaldirectional well planning equations, each method depending on the

partic.ular wellbore geometry desired. The use of buildup charts isa tedIOUS process that often yields inaccurate results since itrequires using preplotted graphs that require interpretation andinterpolation. The equation approach is often confusing to usedue to the similarity of the various equations with the selection ofthe proper equation dependent on the desired wellbore geometry.This paper presents the derivation of a single e q u ~ t i o n forplanning the trajectory of any directional well and a similarequation for any horizontal well. Several examples are used todemonstrate the application of the equations to various wellboregeometries.

INTRODUCTIONDirectional drilling is the purposeful deflection of a wellbore fromthe vertical along some preplanned trajectory to a predeterminedtarget. The necessity of directional drilling is usually dictated byeconomic and envi ronmental concerns. Appl ica tions fordirectional drilling are varied with the most common application ofdirectional drilling techniques probably being in offshore waters.Here, several wells are drilled from a single platform to differentbottomhole locations, allowing the optimization of developmentcosts. Similarly, directional wells have been drilled in remoteareas from artificial islands and dril ling pads. This type of

References and figures at end of paper.17

operation is common in places like Alaska, the Canadian Arcticand the jungle areas of South America. Drilling multiple wellsfrom a single location can greatly simplify the installation ofgathering and production facilities.Inaccessible surface locations also require the use ofdirectional drilling techniques. The inaccessibility may be theresult of natural barriers, such as rivers, lakes and mountainousterrain, or man-made barriers, like highways and populated areas.In these situations, single wells may be directionally drilled toreach the objective target. Directional drilling techniques are alsoused in drilling relief wells, drilling in saltdome areas and in somefishing operations.Another and rapidly growing area requiring directionaldrilling techniques is horizontal and extended reach drilling.Generally, horizontal wells are drilled for economic reasons.Applications include consolidated, naturally fractured reservoirswhere the wellbore may intersect multiple fracture systems. Theyhave also been drilled to reduce coning problems in reservoirs thathave large gas caps or strong water drives. In some reservoirs thehorizontal well may improve drainage by increasing the area of thewellbore in contact with the reservoir.Deflecting a wellbore involves many factors which mustbe considered individually. As a result, careful planning is thekey to successful directional drilling. One of the first steps inplanning the directional well should be the design of the wellboretrajectory. Wellbore trajectories can be categorized into twoclasses, the directional well and the horizontal well.In the directional well class, there are three basic types ofwellbore geometries: Type I, Type II and Type III.I Fig. I showsthese major types of wellbore configurations. A Type I well isbasically a "build and hold" trajectory where the wellbore isdeflected from the vertical at some kickoff point and the angle builtuntil a maximum angle is reached and then held until the target isintercepted by the wellbore. A Type II well is a "build, hold anddrop" or "S" trajectory where the wellbore is deflected to someangle, the angle is held and then dropped in a manner such that thetarget is penetrated vertically. A modified Type II well differs inthat the wellbore is not returned to the vertical in the drop portion.A Type III well has a continuous build trajectory where the


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2 SIMPLIFIED EQUATIONS FOR PLANNING DIRECTIONAL AND HORIZONTAl W R l . T . ~inclination continues to increase right through the target.

The second class of wellbore trajectories is the horizontalwell. This includes the horizontal or near horizontal well (anglesgreater than about 80 degrees) and the extended reach well. Fig. 2shows these wellbore configurations.In the past, there have been two major methods utilized inplanning the direct ional well. These methods are the use ofbuildup or composi te buildup char ts and the use of several

directional well planning equations, each method depending on theparticular wellbore geometry desired.1.2 The use of buildupchart s is tedious and of ten yields inaccurate results since i trequires using preplotted graphs that require interpretation andinterpolation. The equation approach is often confusing to usedue to the similarity of the various equations where the selectionof the proper equation is dependent on the desired wellboregeometry.This paper presents the derivation of an equation forplanning the trajectory of any directional well. A similar equationfor planning any horizontal well is also presented. Severalexamples are used to demonstrate the application of the equationsto various wellbore geometries.

DIRECTIONAL WEll , CLASSDerivation of Equation

Fig. 3 presents a typical geometry for a directional wellthat can be modified to fit any of the directional well geometries.The derivation of the equation from this figure is straightforwardand depends on basic geometric relationships. The equation,which can be derived in several ways, is based on the Type II or"S" shape geometry and can be modified for the other wellgeometries.It is desired to determine the maximum angle of inclinationfrom Fig. 3. This angle is related to the hold portion of thewellbore bytan 9 =,::,:X:,L3_-

OJ -D2 (1)These five variables are unknown and must be related tothe known parameters. The known variables usually includekickoff point, total true vertical depth, maximum lateraldisplacement and the radius of curvature in the build and dropsegments of the wellbore. These radii can be determined from thedesired rate of build and drop by the relationr=l8.Q. x l1t q (2)

These variables can be related by the following equations.X2 = rl - r l cos 9 (3)X3 = ~ - r2 + r2 cos 9 (4)D2 = DI + rl sin 9 (5)03 = 04 - r2 sin 8 (6)

Subtracting Eq. 3 from Eq. 4 and Eq. 5 from Eq. 6 results inEqs. 7 and 8.X3 - X2 = ~ - rl - r2 + (ri + r2) cos 9 (7)

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03 - 02 = 04 - 0 1 - (r i + r2) sin 9Substituting Eqs. 7 and 8 into Eq. 1 yields

9 X4 - r l - r2 + (ri + r2) cos 9tan = - - " - ~ - = - - - - - ' ' - - " - - - - ' = - - - - -04 - 01 - (ri + r2) sin 8

which can be rearranged astan 8 (D4 - DI - (ri + r2) sin 8) = ~ - rl - r2 + (ri + r2)...............................................................

Multiplying Eq. 10 by cos 8 and collecting terms results insin 9 (D4 -OIl = cos 9 (X4 - r l - r2) + (ri + r2)

Advantage can now be taken of the following trigonoidentities.cos 2cjl = 2 cos2 cjl- 1sin 2cjl=2sincjlcos cjlsec2 cjl = 1 + tan2 cjl

By letting

in Eq. 11 and dividing bycos2

the equation becomes2 t a n ~ ( 0 4 - 0 1)=X4 + tan2ft(2 (ri + r2) - ~ )2 2

Eq. 12 can be written in the form of a second degree polynas(2 (ri + r2) - ~ ) tan2ji - 2 (D4 - DIl tan ji + ~ = 02 2

Solving Eq. 13 for the angle 9 yields Eq. 14.8 =2 tan-I [(D4 -OI l - -../(D4 - Dif - 2 (ri + r2) ~ +2 (ri + r2) - X4. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. .It should be noted that the positive root of the discrimin solving Eq. 13 is ignored since it yields invalid resultsapplication. This equation can be used to plan any tydirectional well. It can be used as written for a Type II wefor a TypeI and III well where r2 =O.For a modified Type II well, Eq. 14 must be modFig. 4 represents the geometry of a modified Type II weorder to use Eq. 14, D4 and ~ must be determined. Fromgeometric relations


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SPE 21261 M. L. WIGGINS &H. C. JUVKAM-WOLD 3

Inserting these values into Eq. 14 yields

Type II Well. A Type II well is to be drilled inside the city limitsof a heavily populated area. The desired true vertical depth is12,850 ft. The well needs to be vertical at 8200 ft with a 3250 fthorizontal displacement prior to drilling into a shale interval. Thedesired build rate is 2.0 degrees per 100 ft while the drop rate is2.5 degrees per 100 ft. The well will bekicked offat 300 ft.For this particular well,

04 = 05 + r2 sin, (15)X4 = Xs + r2(1- cos,) (16)

where , is the desired angle in the final hold portion of thewellbore. For the modified Type II well, Eqs. 15 and 16 shouldbe used to determine 04 and X4 which are then be used with Eq.14to yield the maximum angle in the hold portion of the wellbore.It should be noted that 05 and Xs in Eqs. 15 and 16represent the vertical and horizontal distances to the beginning ofthe final hold portion of the wellbore. This corresponds to the endof the drop portion of the wellbore and not to the vertical andhorizontal displacement to the bottomhole objective.

01 = 300.0rl =2864.8X4 = 3250.0

04= 8200.0r2 = 2291.8

There are times when one may know the maximum angleallowed in the hold portion of the well and desires to determine thekickoffpoint. This can be done by solving Eq. 14 for 0 1 whichresults in Eq. 17.9 = 2 tan,l {[(8200.0 - 300.0)

- "'!(8200.0 - 3oo.of- 2 (2864.8 + 2291.8) 3250.0 + 3250.02 ]+ [2 (2864.8 + 2291.8) - 3250.0] }

or

e= 25.8Themaximum angle obtained in the hold portion of the well willbe 25.8 degrees.Modified Type II Well. It has been decided that an offshore wellwill be drilled using a modifiedType II trajectory. The final holdportion of the wellbore should begin at a true vertical depth of11,000 ft and a horizontal displacement of 6000 ft. It is desiredthat the well be deviated at 2500 ft and the target formation bepenetrated at an angle of 25 degrees. The buildup rate is 3 degreesper 100 ft while the droprate is 2 degrees per 100 ft.

For a modified Type I I well, Eq. 14 must be modified byusing Eqs. 15 and 16. For this well,

tan (2 (rl + r2) - X4)2 (17)

It should be noted that Eqs. 14 and 17 can be used,without modification, for the near horizontal well and the extendedreach well if there is only one build sect ion in the wellboretrajectory, i.e., r2 = O.Examples

Several examples are presented that show the applicationofEqs. 14 and 17 for the applicable wellbore trajectories.Type I Well. It is desired to drill a Type I well to a true verticaldepth of 8600 ft under a local golf course. The requiredhorizontal displacement is 1200 ft and the maximum angle desiredat total depth is 20.0 degrees. A build rate of 1.5 degrees per 100ft will be used.

rl = 1909.901 =2500.0

rz = 2864.8

Since the maximum angle allowed is known, one wouldbe interested in determining the deepest kickoffpoint which couldbe utilized. Eq. 17 can be used to make this determination. Forthis example,

04 = 11000.0 + 2864.8 sin 25.0 = 12210.7X4 = 6000.0 + 2864.8 (1.0 - cos 25) = 6268.4

01 = 8600.0 + 1200.02 - 2 (3819.7 + 0.0) 1200.02 tan (10.0) (2 (3819.7 + 0.0) - 1200.0)

180 - 381971t (1.5/100) - .04 = 8600.0

Inserting these values intoEq. 17 yields,

r2 =0.0X4 = 1200.0

Substituting these values into Eq. 14 will allow the determinationof the maximum angle in the well bore. That angle is determinedto be 37.8 degrees.Type III Well. It is desired to drill a Type III well to a true verticaldepth of 10,800 ft with a total horizontal displacement of 425 ftand a build rate of 1.5 degrees per 100 ft. It is planned to deviatethe wellbore at 8000 ft.

In planning the well, one needs to determine themaximumangle the wellbore will obtain at i l l . For this well,

Utilizing Eq. 14 to determine the maximum angle yields 9.8degrees at total depth.

tan (10.0) (2 (3819.7 + 0.0) - 1200.0)2

or01 = 4629.5

Thus a kickoff point of 4629.5 ft will penetrate the target at anangle of 20.0 degrees.19

01 = 8000.0rl = 3819.7X4=425.0

04= 10800.0r2 =0.0


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4 SIMPLIFIED EOUATIONS FOR PLANNING DIRECTIONAL AND HORIZONTAL WELLS ~ P F . ?l?hHORIZONTALWELLCLASSDerivation ofEQuation

Fig. 5 presents a typical geometry for a horizontal well thatcan be modified to fit an extended reach or nearly horizontal well.The derivation of the equation from this figure is very similar tothe derivation ofEq. 14 for the directional well class.It is desired to determine the angle of inclination, 8, fromFig. 5. This angle is related to the hold portion of thewellbore byEq.1.tan 9 = X3 - Xz

D:3 -Dz (1)The five variables in this relationship are unknown and must berelated to the known parameters. Xz and Oz can be. determinedfrom Eqs. 3 and 5.

X2 = rl - rl cos 9 (3)D2 = Dl + rl sin 9 (5)

However, since there is a second build section in the horizontalwell, new relationships must be determined for X3and 0 3, Thesevariables can be related by the following equations.X3 = X4 - rz cos 9 (18)D3 = D4 - rz + rz sin 9 (19)

Sub tracting Eq. 3 from Eq. 18 and Eq. 5 from Eq. 19 andutilizing the relationship of Eq. 1 yields9 X4- rl - (rz - r l) cos 8tan = - - - = - - - - = - - - ' : . . = . . . - - - = ~ - -D4 - DI - rz + (rz - rI) sin 9 (20)

This equation can be rearranged astan 9 (D4- DI - rz + (rz - rl) sin 9) = X4 - rl - (rz - rI) cos 8

Eq. 23 can then be solved by the quadratic formula for 9, thangle in the hold portion of the wellbore. This yields Eq. 24.9= 2 tan-I [(01-04+ (2) + ../(04- 01 -f2'T -(f2 - 2 fl + X4) (f2 -X4)

f2-2fl+X4. (2

Eq. 24 can be used as is for a horizontal well with one or twbuild sections. For a horizontal well with only one build sectioone should let rz equal O.

Eq. 24 must bemodified to allow its use for planning thextended reach or near horizontal well that contain two buisections. Fig. 6 depicts the geometry of these types of wells. Fothese cases, 04 and X4 should be calculated as follows:D4 = 05 + r2 (1 - sin ep ) (2X4 = Xs + rz cos ep ..................................... (2

The angle ep represents the desired angle in the final hold portioof the wellbore. Once 04 and X4 are determined, Eq. 24 can bused as written. As in the directional case, 05 and Xs represethe displacement to the beginning of the last hold portion of thwellbore and not the total displacement to the bottomhoobjective.Examples

The use of Eq. 24 is presented in the following examplefor a horizontal well, an extended reach well and a near horizontwell.Horizontal well. It is desired to drill a horizontal well infractured chalk reservoir . True vertical depth of the targformation is 9800 ft with 4000 ft of horizontal displacement to thend of the second build section of the wellbore. A horizontsection of 1800 ft is desired. The wellbore will be deviated2500 ft with an initial build rate of 4 degreesper 100 ft and a fmabuild rate of 3 degrees per 100 fL

The determination of the angle in the hold portion of thwell is as follows:................................................................. (21)

Multiplying Eq. 21 by cos 9 and collecting terms results in

By taking advantage of the trigonometric identities used earlier,letting

01 =2500.0rl = 1432.4X4 =4000.0

Using Eq. 24,8 = 20.9

04=9800.0r2 = 1909.9

dividing byc o s z ( ~ )

and collecting terms, Eq. 22 becomes

Extended Reach Well with Two Build Sections. It is desired t~ an extended ~ e a c h . well from an offshore platform. Thhonzontal and vertlcal displacement to the beginning of the fmhold portion of the wellbore is 15800 ft and 8800 ft, respectivelyThe well will be kicked off at 500 ft with initial and final buirates of 5 and 4.5 degrees per 100 ft, respectively, with a finhold angle of 70 degrees.

For this well,(r2 - 2 rl + X4) tanZa + 2 (04 - 0 1- rz) tan.a+ rz - X4 = 02 2.................................................................(23)

20

rl = 1145.901 =500.0

rz = 1273.2


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SPE 21261 M. 1. WIGGINS & H. C. JUVKAM WOLD 5Eqs. 25 and 26 should be used to detennine D4 andX4

04 =8800.0 + 1273.2(1.0 - sin 70) =8876.8X4 =15800.0 + 1273.2 cos 70.0 =16235.5

Substituting the values into Eq. 24 yields 64.4 degrees as therequired angle in the hold portion of the well.Near Horizontal Well with Two Build Sections. A near horizontalwell is to be drilled. The target depth to the final hold portion ofthe wellbore is 8400 ft TVD. The horizontal displacement is 3200ft. The target fonnation is to be penetrated at an angle of 85degrees. The well will be deviated from the vertical at 300 ft withinitial and final build rates of 5 and 6 degrees per 100 ft

In this case,

REFERENCES1. "Introduction to Directional Drilling," Eastman Whipstock,Houston, Texas, 4-6.2. Bourgoyne, Jr., A.T., Millheim, K.K., Chenevert, M.E. andYoung, Jr., F .S.: Applied D r i l I i n ~ E n ~ i n e e r i n ~ , SPETextbook Series, SPE, Richardson, TX (1986) 2,354-357.

rl = 1145.90 1 = 300.0

r2 = 954.9

Using Eqs. 25 and 26 once again,04 =8400.0 + 954.9 (1.0 - sin 85)= 8403.6X4 = 3200.0 + 954.9 cos 85.0 =3283.2

For this well, the hole angle required in the hold ponion of thewellbore will be 18.1 degrees.CONCLUSIONS

This paper presents two simplified equations for planningdirectional (Eq. 14) and horizontal (Eq. 24) wells. Forplanning amodified Type I I well, Eq. 14 should be used with Eqs. 15 and16. In planning an extended reach or near horizontal well withtwo build sections, Eqs. 25 and 26 should be used in conjunctionwith Eq. 24. These equations can be used for any directionalwellbore trajectory desired and simplifies well planning.NOMENCLAllJRE

01 = Kickoff point, ft02 = True vertical depth to end of first build section, ftOJ = True vertical depth to end of first hold section, ft04 = True vertical depth to target, ftOs = True venical depth to end of second drop sectionfor directional wells or build section for horizontalwells, ftX2 = Horizontal displacement to end of first buildsection, ftX3 = Horizontal displacement to end of first holdsection, ftX4 = Horizontal displacement to target, ftXs = Horizontal displacement to end of drop section fordirectional wells or second bui ld section forhorizontal wells, ftrl = Radius of curvature in first build section, ftrz = Radius of curvature in drop section for directionalwells or second build section for horizontal wells,ftq = rate of angle build, degrees per 100 fte = Maximum angle in hold section of well, degreesIII = Desired final angle in drop section for directionalwells or second build section for horizontal wells,degrees

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Xl '1I: x. 1------x. .1

, I ., i Start of Buildup

~ ~ IXl II. x. 1 I1

.1 1 7 Start of BUildup8

D"

D.

Target, (V r !, _-..i I "

Fig. 3. Geometty of a Type IT directional well. Fig. 4. Geometty of a modified Type IT directionalwell.


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Target

D,

- - - - . ! : : . - \ -7S ta r t of Buildup0//

////L - - - t - - ~ I E n d of Build

10II

/l/1/"-i

/ /90 - 0/

////////L--- --+__+- ~ B u i l d u p

IIIiIID:.

D.I

I+I.------x.----------IFig. 5. Geometry of a horizontalwellwith two buildup sections.

D, ---c, - - - 7 Start of Buildup0//

///. I-_..---: ,_----:: ! \( End of Build

/11/"-LI90-0 / l/ / I/ /'-...1/ /9 0 -

/ I/ / I/ / I

/ / 1/ / 1 r2

// / I. I - - - - - + - - - f - - - - - ~ Buildup / 1/ 1

.I - -+__+- -+-__ ~ .... End of Build 1...._ I---------- J - -------------l--I1f-x,11-- - - Xo. . , - - -+ lII------x.

TargetFig. 6. GeometIy of a near horizontal well or an extended reach well with two buildup sections.

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simplified equations for planning directional and horizontals

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