EPSG

SPE paper 96813 "Accounting for Earth Curvature in Directional Drilling", 2005.

2019-02-17

For wellbore survey.

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided. LMP requires that the input local horizontal coordinates are referenced to true north, and that depths (D) are referenced to the reference surface of the vertical CRS. Depths are required as input but are not changed by the algorithm. They are used to compute the depth correction factor and therefore a depth of zero must be close to Mean Sea Level (strictly speaking to the reference ellipsoid). Required inputs are: Latitude of the Well Reference Point (WRP) in the target 2D geographic CRS; Longitude of the Well Reference Point in the target 2D geographic CRS; dcf flag: 0 = no depth correction is applied; 2 = a variable depth correction factor along the path is applied; n[]: Array of length num_points containing local northing coordinates, oriented to true north; e[]: Array of length num_points containing local easting coordinates, oriented to true east; D[]: Array of length num_points containing depths below the Vertical Reference Surface at survey stations down the wellbore (referred to as TVD in the SPE paper). The output of LMP is ellipsoidal coordinates of the wellbore trajectory in a geographic 2D CRS. If required, projected coordinates of the wellbore may be calculated from these. The following equations assume that the local coordinates (n,e) and depth D are in metres. If these use a different unit of measure, then as a first step they should be converted to metres. When dcf flag = 2, output arrays LAT, LON (in radians) are computed as: for i=1:num_points ρ = a (1 – e^2) / (1 – e^2 sin^2(LAT[i-1]))^3/2 ν = a / (1 – e^2 sin^2(LAT[i-1]))^1/2 LAT[i] = LAT[i-1] + (n[i] – n[i-1]) / (ρ – D[i]) LON[i] = LON[i-1] + (e[i] – e[i-1]) / (ν – D[i]) / cos(LAT[i-1]) where for i=1, for the first station: LAT[0] and LON[0] are the geographic coordinates of the WRP in radians; n[0] = 0, e[0] = 0. and a is the semi-major axis of the reference ellipsoid; e is the eccentricity of the reference ellipsoid (not to be confused with local easting); ρ is the radius of curvature in the meridian; ν is the radius of curvature in the prime vertical. When dcf flag = 0, the equations above are modified to: LAT[i] = LAT[i-1] + (n[i] – n[i-1]) / ρ LON[i] = LON[i-1] + (e[i] – e[i-1]) / {ν cos(LAT[i-1])} Reverse case: To compute (n,e) from (LAT,LON,D): For i=1:num_points ρ = a (1 – e^2) / (1 – e^2 sin^2(LAT[i-1]))^3/2 ν = a / (1 – e^2 sin^2(LAT[i-1]))^1/2 n[i] = n[i-1] + (LAT[i] – LAT[i-1]) * (ρ – D[i]) e[i] = e[i-1] + (LON[i] – LON[i-1]) * (ν – D[i]) * cos(LAT[i-1]) where for the first station i=0 variables are initialised as in the forward case. Note that LAT[i-1], the latitude of the previous point, is used in the reverse formulas to compute the radii of curvature in order to be internally consistent in round trip calculations. When dcf flag = 0, the equations above are modified to: n[i] = n[i-1] + (LAT[i] – LAT[i-1]) * ρ e[i] = e[i-1] + (LON[i] – LON[i-1]) * ν * cos(LAT[i-1])

An example computation for the LMP method is provided below. This is not a realistic wellbore survey but demonstrates the computation steps. A straight path is modelled to a bottom hole at 5 km depth, approximately 14 km away from the surface location, using a constant inclination 70.54002°, azimuth 51.3178° from true north, and survey stations every 30.0167 metre. Parameters: Ellipsoid: WGS 84, a = 6378137 metres, 1/f = 298.257223563 then e = 0.0818191908 and e^2 = 0. 00669437999 Forward calculation for a well with surface coordinates (origin of the local coordinates) Latitude of Well Reference Point 9.0364081° = 0.1577150739 rad Longitude of Well Reference Point -30.2713073° = -0.5283339813 rad Depth Correction dcf flag = 2 Zero Depth Point MSL height H = 25 m (not used in equations) Local northing (n) and easting (e) given in the table are the unscaled output of an algorithm such as minimum curvature as commonly applied in wellbore survey (described as cubical coordinates in the SPE paper). Axes n and e are aligned with true north and true east, respectively. Initialisation: LAT[0] = 0.1577150739 rad; LON[0] = -0.5283339813 rad. i D(m) n(m) e(m) ρ(m) ν(m) LAT(rad) LON(rad) 1 0.00 0.00 0.00 6337009.005 6378663.709 0.1577150739 -0.5283339813 2 10.00 15.00 24.00 6337009.005 6378663.709 0.1577174409 -0.5283301714 3 20.00 30.00 48.00 6337009.052 6378663.724 0.1577198080 -0.5283263616 4 30.00 45.00 72.00 6337009.099 6378663.740 0.1577221751 -0.5283225517 5 40.00 60.00 96.00 6337009.145 6378663.756 0.1577245421 -0.5283187419 ... 497 4960.00 7440.00 11904.00 6337032.231 6378,671.502 0.1588895878 -0.5264433944 498 4970.00 7455.00 11928.00 6337032.278 6378,671.517 0.1588919567 -0.5264395808 499 4980.00 7470.00 11952.00 6337032.325 6378,671.533 0.1588943256 -0.5264357673 500 4990.00 7485.00 11976.00 6337032.373 6378,671.549 0.1588966945 -0.5264319538 501 5000.00 7500.00 12000.00 6337032.420 6378,671.565 0.1588990634 -0.5264281403