11-CoordSys

Sperry Drilling Services

2007.1

Directional Drilling

FUNDAMENTAL COORDINATE SYSTEMS

Coordinate Systems Projections

Sperry Drilling Services 2007

2-D Rectangular Coordinate System

Basic Coordinate Systems

+Y • Rectangular (Cartesian) : P(x;y;z) P (x,y)

y

• Polar ------------------------- : P(R;α) • Cylindrical ----------------- : P(R;α;D)

1

0

1

+X

x

Direction from +X to +Y is counter-clockwise

3-D Rectangular Coordinate System

Polar Coordinate System

+Y 0

x

P (R, α)

y

+X

R

z

α

P (x,y,z)

reference direction

+Z

Direction from +X to +Y is counter-clockwise

Directional Drilling – Coordinate Systems

R polar radius polar angle

α

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2007.1

Cylindrical Coordinate System

Measurement of Angles 0°

reference direction

D

P α R

P

P (R;α;D)

CCW

α

P

α

E

X

in surveying ...

calculators do ...

Location Coordinates - Conversion

Location Coordinates - Conversion

Y

N P

y

Math convention

R

x

X

P

n

Surveying convention

α

x = Rcosα y = Rsinα

α 0

CW

N

Y

0

R e

R = (x2 +y2 )1/2 α = atan (y/x)

E

e = Rsinα n = Rcosα R = (e2 +n2 )1/2 α = atan (e/n)

Using a Calculator’s Built-in Rectangular-polar Conversion

Example : Target displacement (R) 3200 ft Direction to target center (α) 53.5° GN

1. Calculate ΔN and ΔE 2. Enter as y and x coordinates 3. Calculate R and α polar angle

x = 3200 x sin(53.5) = 3200 x 0.8039 = 2,572.3 ft y = 3200 x cos(53.5) = 3200 x 0.5948 = 1,903.4 ft

Ngrid

’ The distance is R (ft or m)  The grid Azimuth is 90 - α (deg)

Y

Remember :

α

TC X

Directional Drilling – Coordinate Systems

E

• x - East and y - North ? Check ! • α is measured from X to Y (ccw) • AZ is measured from North to East (cw)

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Direction to a Location or Target

Distance Between 2 Locations or to a Target

GN

S =

Y

Δx

2

+ Δy

2

P2

Δy = 55.95

Example:

P1

P1 → P 2 = a tan

Δx Δy

= a tan

432 . 59 55 .95

Surface coordinates TD coordinates

X

Δx = 432.59

= 82 .6

Horizontal References

à

Earth models (spheroids)

š

Projections

e6.00 e161.97

Δy = 853.97 – 3.00 = 850.97 ft Δx = 161.97 – 6.00 = 155.97 ft

deg rees

S =

Direction P1 Æ P2 = 82.6° (referenced to grid North)

Vertical references

n3.00 n853.97

850.97

2

+ 155.97

2

= 865.15

feet

Turn ri ght

Turn`left Build The Bull’s D ro

VERTICAL REFERENCES

Eye

p

The Solid Earth and the Oceans

Vertical Reference Surfaces Houston

sea level (geoid)

topographic surface

geoid surface

SOLID EARTH ellipsoid surface

topographic surface Mt. Everest

The Geoid is a level mathematical surface (equipotential surface of the gravity field), best fit to MSL Peter H. Dana 9/1/94

Directional Drilling – Coordinate Systems

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Vertical Direction

Vertical (depth) References

(

Deviation of the “vertical”

MSL = Mean Sea Level – a global depth reference MSL Geoid = equipotential surface (equal acceleration of gravity)

Geoid = Mean Sea Level

Ellipsoid

Example : Difference btw MSL Geoid and WGS84 ellipsoid : +/- 40m avg. extremes : + 60m -100m For vertical coordinates :

For horizontal coordinates :

Geoid height Geoid

Geoid

Ellipsoid

Which vertical do we use ?

Examples of Depth References LAT = Lowest Annual Tide AHD = Australian Height Datum AGD84 = Australian Geodetic Datum, 1984 GDA94 = Australian Geodetic Datum, 1994 NAP = Nieuw Amsterdamse Peil • Acronyms : MSL BRT AHRT SS

AMSL AHD MD GL

TVD RF RKB DFE AHORT ODF

Depth References on Land Locations Options : • Ground level • Wellhead • Rotary table • Kelly bushing

Note : MSL or SS depth are vertical depth

Ground Level and Elevation

Directional Drilling – Coordinate Systems

Rotary Table as Depth Reference

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2007.1

Elevations Based on the Geoid

Kelly Bushing as Depth Reference

RKB Elevation

MSL

Ground Elevation

Subsea Depth

Wellhead Elevation RT Elevation

… a land rig

Jack-up Rig – What is the Rotary Table Elevation ?

GlobalSantaFe

Semi-submersible Rig

Semi-submersible Rig

How we measure the depth here ?

GlobalSantaFe

Directional Drilling – Coordinate Systems

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2007.1

Other Vertical Reference Points

Drill Ship

Well Reference Point near or at GL

Drill Datum usually rotary table (RT) or kelly bushing (KB)

Mean Sea Level MSL

subsea depth, target depths are typically given as TVDss = vertical depth below MSL

Well Reference Point at or near mudline

GlobalSantaFe

Ways of Expressing Direction …

’ Quadrant Bearings (e.g. N15E) (in quadrature system) ’ Azimuth

(0 - 360°)

HORIZONTAL REFERENCES ¾ Angle units : degrees or radians (360°=2¶ radian, or 6.28… radian, where ¶ = 3.1415965) R

¾ Definition of radian :

The Quadrature System

R

1 radian

Quadrant Bearings

Cardinal bearings : N-E-S-W Intercardinal bearings : NE-SE-SW-NW N

N NW

NE

NW W

W

E SW

NxxE

NxxW

NE

SE

E

SxxE

SxxW SE

SW

S

S Quadrants : NE-SE-SW-NW

Directional Drilling – Coordinate Systems

The advantage is …

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2007.1

Azimuth Examples :

N

=

0° or 360° AZ

N88E

=

88°

E

=

90°

S23E

=

157°

S

=

180°

S55W =

235°

W

=

270°

N15W =

345°

N

360 0

E

W

Azimuth is 0 - 360°

S

North is 0° or 360°

Azimuth Change Horizontal Direction References N 350°

10°

> True North - direction to the geodetic N pole > Magnetic North - direction to the magnetic N pole > Grid North - aligned with the central meridian(s) on maps

ΔAZ = (360-350) + (10-0) = 20° or ΔAZ = 350-10 = 340°

True & Magnetic North

True & Magnetic North TN

TN

MN

meridians

TN MN MN

Equator MN

Equator agonic line TN

TN

MN

Magnetic Declination

Directional Drilling – Coordinate Systems

polar axis

Magnetic Declination = 0

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2007.1

Azimuth Correction

Magnetic Declination Convention

True North

True North

Magnetic North

Magnetic North

♠ Magnetic Declination (MD) : difference btw TN and MN MD = ƒ (location, time)

West Declination (-)

East Declination (+)

Azimuth correction : AZTRUE=AZMAG + (MD) East Declination : MN is East of TN West Declination: MN is West of TN

Application of Magnetic Declination West Declination

Application of Magnetic Declination East Declination

TN

TN MN

MT MD

T M

measured direction

φ

measured direction

T φ

M

P

P

AZTRUE=AZMAG+ MD

AZTRUE=AZMAG+ MD

Ellipsoids and the Geoid MSL (geoid) globally fitting ellipsoid eg. WGS 84

EARTH MODELS

locally fitting ellipsoid eg. Clarke 1866 area of best fit of ellipsoid to geoid

Directional Drilling – Coordinate Systems

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Earth Models - Spheroids (ellipsoids)

• Spheroids : Everest (1830) Bessel (1841) Airy (1830) Clarke (1866) Clarke (1880) Hayford (1909-1910) International (1924) Krassovsky (1940) GRS80 (1980) Geodetic Reference System WGS84 (World Geodetic System)

Ellipsoid Dimensions (approx.)

Ellipsoid name

Semi-major (metres)

Semi-minor (metres)

Inv. Flattening

Airy, 1848 (UK) Australian Nat. Clarke, 1866 (US) International 1924 WGS 72 WGS 84

6 377 563 6 378 160 6 378 206 6 378 388 6 378 135 6 378 137

6 356 257 6 356 745 6 356 584 6 356 912 6 356 750 6 356 752

299.325 298.25 294.979 297 298.26 298.257

ITRF (Int. Terrestial Reference Frame)

Ellipsoid – WGS84

Best Fitting Ellipsoids USA, Canada, Philippines

Clarke, 1866

Eu,N.Africa, Middle East

International, 1924

UK

Airy, 1848 International, 1924

Chile, Borneo, Indonesia

Bessel, 1841

Africa, France

Clarke, 1880

India, Afghanistan, Pakistan Thailand

Everest, 1830

Peninsular Malaysia

mod. Everest, 1830

Datums

• Sphere flattened by 1 part in 300 • Coincides with solid earth to +/- 10 km • Coincides with MSL (geoid) to +/- 100m

Datum Describes a Location, Height

Definition of datum : • Defines the shape of the reference ellipsoid • Defines position of the ellipsoid relative to the Earth • Defines how a coordinate system is seated on the reference ellipsoid

• At the datum the ellipsoid touches the surface of the earth, the location coordinates are fixed (“reference point”)

• Ideal datum : geocentric, with correct polar and equatorial radii (e.g. NAD83) • Important datums :

ED50 for the GCS (European Datum, 1950) NAD27 (North American Datum, 1927) NAD83 (North American Datum, 1983) WGS84 (World Geodetic System, 1984) Pulkovo (1946) Tokyo (TD) South American (1969) Australian Geodetic (1966) Indian

www.colorado.edu/geography/gcraft/notes/datum/datum.html

Directional Drilling – Coordinate Systems

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Comparison of Coordinates at Different Datums

NAD27 and NAD83

DATUM

OPTION

ELEVATION

LONGITUDE

LATITUDE

• NAD27 (1927) fixed location : origin is at Meade’s Ranch, Kansas USGS stated that the Clarke 1866 ellipsoid is a good approx.

NAD27

CENTRAL

51 ft

95°20’24.2” W

29°56’10.2” N

NAD27

CONUS

66 ft

95°20’24.2” W

29°56’10.2” N

• NAD83 (1983) geocentric, centered on Earth’s center of mass Based on the WGS84 ellipsoid. Natural Resources of Canada adopted in 1990 as its new geodetic reference system

NAD83

n/a

74 ft

95°20’24.6” W

29°56’11.0” N

WGS84

n/a

95 ft

95°20’24.1” W

29°56’10.7” N

GDA

n/a

74 ft

95°20’24.3” W

29°56’10.8” N

GEODETIC

1949

75 ft

95°20’27.5” W

29°56’07.7” N

• WGS84 (1984) geocentric, uses GRS80 ellipsoid, which is almost identical to WGS84 ellipsoid • Conversion between NAD27 and 83 : NADCON software accuracy : +/- 0.5m Note :

GPS uses the Earth’s centre of mass as origin – WGS84 ellipsoid, thus the NAD83 is compatible with

Accuracy : 22 ft Location : Houston, Zone 15

Central Meridian : 93°W

PROJECTIONS Distortions by Projection

1. Shape 2. Bearing 3. Scale 4. Area

Zones are defined in order to minimize distortion and preserve accuracy within ea. projection

Projection Types • Cylindrical

cylinder (Mercator, UTM)

• Conical

cone (Lambert Conformal Conic) ( Albers Equal-Area Conic)

• Azimuthal

plane (Lambert Azimuthal Equal-area)

• Miscellanous

(Unprojected Lat and Long)

Projections Used in the US and Territories • State Plane Coordinate System : – SPC27 based on NAD27 coordinates in feet – SPC83 NAD83 meters !

• • • •

Lambert Conformal Conic (Cal – Tx – La) Transverse Mercator Oblique Mercator Approximate Azimuthal Equidistant

• Alaska : Lambert or Oblique Mercator

Directional Drilling – Coordinate Systems

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2007.1

Geographic Coordinate System - GCS Geographic Coordinate System - GCS ato r

North Pole

Eq u

N 90°

Longitudes (great circles)

P

Latitude (parallels)

Equator

Geodetic Latitude (angle)

0°

Prime Meridian

Latitudes : 0 - 90° N and 0 - 90° S Horizontal Earth Rate : 15.000 deg/hr

Longitudes : 0-180° E or W Latitudes -- : 0-90° N or S

Geographic Coordinate System - GCS

Longitudes : 0-360° or 0-180°E and 0-180°W

S

geometric center of spin axis

Examples Geocentric Latitudes

N 90°

50°N London

P Equator

N

Geocentric Latitude (angle) 0° Equator

Latitudes : 0 - 90° N and 0 - 90° S Longitudes : 0-360° or 0-180°E and 0-180°W

S

6°S Jakarta

35°S Buenos Aires

geometric center of spin axis

Why a Location has Several Latitudes

Examples Longitudes - angle E/W from Greenwich (London) 0° London

45°E Baghdad

Location

A. ellipsoid B. ellipsoid perpendicular to A. ellipsoid

45°E 90°W 107°E

N

107°E Jakarta

Directional Drilling – Coordinate Systems

90°W New Orleans 95°W Houston

φΑ φΒ

Equator perpendicular to B. ellipsoid

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Unprojected Lat / Long Map

Geographic Coordinates - Example

Lat 53° 01’ 30.848” N -

Long 3° 33’ 05.185” E

Conversion to decimal degrees :

Latitudes

Lat /Long = HHH+ MM + SS 60 3600 Longitudes

Example :

Lat = 53.025°

Long = 3.551 ° • Simple rectangular coordinate system • Scale, distance, area and shape are all distorted • Distortion is increasing towards to poles

Note : distance on 1° Latitude change along a meridian is 60 nautical miles distance on 1° Longitude change is 60 nautical miles on the Equator 1’ is 1 nm (1853 m)

Gerard Kremer

Projections

Gerardus Mercator de Rupelmonde

1512 - 1594

• Transverse Mercator (TM) - Gauss-Kruger • Universal Transverse Mercator (UTM) • Lambert (conical)

www.colorado.edu/geography/gcraft/notes/mapproj/mapproj.html

Mercator-projection

Mercator Projection

North N

B Equator

East

central meridian

A S

• •

Directional Drilling – Coordinate Systems

Easy navigation by holding constant direction from A to B Long, Lat and rhomb lines are straight lines on the map

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2007.1

Zones of the UTM Projection

Universal Transverse Mercator (UTM) Projection US Army, 1947

N

Equator

S

Central Meridian

Cylindrical projection to a horizontal cylinder

www.fmnh.helsinki.fi/english/botany/afe/map/utm.htm

UTM Grid Zones

Eastings are distances from the false Easting line :

Central Meridian

84°N

N Origin of false Easting 500,000m

Easting at Central Meridian = 500 000 m Approximate range : 200 000 - 800 000 m

Northings are distances from the Equator : 8°

Northing = 0m Origins of false Northing

Northing at Equator = 0 for the Northern Hemisphere 107 Southern 3°

Equator

3°

E

8° Northing =107m

n=0m S

Equator

n = 10,000,000 m

80°S

appr. 600,000m

UTM Zone Limits

UTM Grid Blocks an Zone Limits

Lat = 84°N Long = CM - 3° 465,003 m E 9,329,292 m N

CM Last block 12° high !

North Zones Lat = 0° Long = CM - 3° 166,008 m E 10,000,000 m N Lat = 0° Long = CM 500,000 m E 10,000,000 m N

Equator

Lat = 0° Long = CM 500,000 m E 0mN

12°

X

8° 8°

D C

84°N

I and O are excluded

Lat = 0° Long = CM + 3° 833,992 m E 0mN

South Zones

Directional Drilling – Coordinate Systems

Lat = 80°S Long = CM + 3° 558,135 m E 1,116,652 m N

Block heights : 8°

80°S

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UTM Zones - Lower 48 States

UTM Standard Grid Zones Numbering

West Longitudes

126° 174°W

120°

114°

108°

102° 96°

90°

84°

78°

72°

66°

International Date Line

180°

19

zone 60

zone 1

10 North Pole 90°W

11

90°E

18 12

13

14

16

15

17

zone 31 zone 30

0° 6°E

Central Meridian

Prime Meridian (Greenwich)

Grid Scale Factor

Grid Scale Factor CM distance on map distance on surface

ace Surf th’s Ear

map surface grid scale factor > 1

map surface

th’s Ear e fac Su r

grid scale factor < 1

Scale factor =

distance on map (grid) true dist. on Ea. surface

Scale Factors in UTM Zones

grid scale factor = 1

Scale factor =

grid scale factor > 1

distance on map (grid) true dist. on Ea. surface

UTM Coordinates - Example East from the CM

F0 = 0.9996 Scale factor =

grid distance true distance

F = 1.0004

UTM coordinates : e 536987.41 - n5875344.05 GCS coordinates : Lat 53° 33’ 05.123” N

Long 5° 12’ 32.453”E Zone 31, central meridian 3°E (location is East from the CM)

block U

Central meridian

Note : Note : distances are true on central meridian only

Directional Drilling – Coordinate Systems

• 31U (std. zone 31 – sector U) ⇒ CM3 (3° E Longitude) • Coordinates in meters !

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UTM Coordinates - Example

Location Coordinates - example :

Geographic - Latitude and Longitude (degrees) e.g. LAT 53° 33’ 05.123”N LONG 5° 12’ 32.453”E

e536987.41 - n5875344.05 31U easting

northing

block

UTM Grid - Northings and Eastings (metres) e 536987.41 n 5875344.05

zone

CM3

Note : Spheroid and datum must be given for both

Note : CM longitude or zone number must be given !

Rectangular Grid within a UTM Zone

Location Coordinates - example : M ⇒ T + 16.076 M ⇒ G + 14.851 T ⇒ G + 1.225

Corrections :

Grid North

North

Position :

Grid East

Distance (R) = 1331.5 ft Direction (α) = 57.7°MN

East Equator Grid South

PMN(1331.5 ; 57.7°) or (711.49N ; 1125.47E) PTN (1331.5 ; 73.78°)

(371.92N ; 1278.50E)

PGN (1331.5 ; 72.55°)

(399.28N ; 1270.22E)

Central Meridian

Grid North vs True North

Grid North vs True North

Southern Hemisphere

Northern Hemisphere

UTM Projection TN

TN GN

UTM Projection

Equator

TN

GN

GN

GN

TN

GN

TN

P TN

K

GN

K

K

P Grid

K

P P

Equator

Meridians

Grid

TN

TN

Meridians

Directional Drilling – Coordinate Systems

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Meridian Convergence

Meridian Convergence N

TN

GN

MC = f ( LAT, Δλ )

TN

GN

TN

Δλ = λ − λ0

GN

λ

LAT

Δλ

LONG

Equator

E

LAT,LONG : geographic coordinates

Angular difference between Grid North and True North If Grid North is East of True North : positive Grid North West True North : negative

CM

λ0 Longitude of the CM

Azimuth correction : AZGRID=AZMAG + MD - (MC) Note : other names used

λ Longitude of the location

- Grid Convergence - Grid Correction

2

3

+

5 Δλ

15

(

)

sin LAT ⋅ cos 2 LAT 1 + 3n2 + 2n 4 +

(

sin LAT ⋅ cos 4 LAT 2 − t 2

)

Approximate meridian convergence (grid correction) value : MC = (LONG

[ Eq. 1 ]

λ0 = Longitude of central meridian e 2 ⋅ cos 2 LAT

λ = λ − λ0

n2 =

t = tan LAT

where : e = 2.7182

Δ

λ0

Meridian Convergence – Approximate Calculation

Meridian Convergence – Exact Solution

MC = Δλ ⋅ sin LAT + Δλ

S

Δ λ distance from CM, degrees

MC LONGCM LATLOC LONGLOC MD

LOC

− LONG

CM ) × sin LAT LOC

[ Eq. 2 ]

meridian convergence (grid correction), degrees Longitude of central meridian, degrees Latitude of the location, degrees Longitude of the location, degrees magnetic declination, degrees

1− e2

Example :

Total Azimuth correction : AZGRID=AZMAG + MD - (MC)

Azimuth Corrections - Example

Lat 54.0046 N - Long 4.9129 E

TM5 projection (!)

Convergence from Eq.1 = - 0.0695 degree Eq.2 = - 0.0705 Rounded to 2 decimals = - 0.07

Corrections :

MD = 16.076 MC = - 1.225

M ⇒ T + 16.076 T ⇒ G + 1.225 M ⇒ G + 14.851

Measured Direction (α) = 57.7° MN Azimuth, from MN = 57.7° 57.7 + 16.076 = 73.776 ~ 73.8° TN

E.g. : AZTN = 37° ⇒ AZGN = AZTN - ( - 0.07) = 37.07°

73.8 + 1.225 = 75.025 ~~ 75.0° GN

Note : the location is W from the central meridian 5° (TM5) thus GN is W from TN ( AZGN > AZTN ) : West convergence

Directional Drilling – Coordinate Systems

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GN

2007.1

Application of Magnetic Declination and Grid Correction (1)

TN

Application of Magnetic Declination and Grid Correction (2)

GN TN

MN MN

measured direction G φ

M