Accounting for Earth Curvature in Directional Drilling
- Noel D. Zinn (ExxonMobil Exploration Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Annual Technical Conference and Exhibition, 9-12 October, Dallas, Texas
- Publication Date
- Document Type
- Conference Paper
- 2005. Society of Petroleum Engineers
- 1.6 Drilling Operations, 1.6.6 Directional Drilling, 1.12.1 Measurement While Drilling, 4.3.4 Scale, 1.6.7 Geosteering / Reservoir Navigation, 1.10 Drilling Equipment
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In an important contribution to the geodesy of directional drilling (SPE 56013, "Directional Drilling and Earth Curvature", Ref. 1) authors H. S. Williamson and H. Wilson cite seven errors of Earth curvature often found in current oil field practice. Four of those errors relate to the use of map projections to convert cubical coordinates (e.g., those computed from Minimum Curvature) to geodetic coordinates. This paper offers an algorithm that converts cubical coordinates directly into latitude and longitude without an intervening map projection, thus avoiding these four errors plus one other of the seven cited.
Directional surveys of stations along the path of extended-reach, deviated wells are reported in observations of inclination with respect to the vertical, azimuth with respect to north, and measured depth (actually a length) along the path traversed from the Kelly bushing to that station. The Minimum Curvature algorithm is today's most-common method (among many) of converting inclination, azimuth and measured depth into "cubical" or "flat-Earth" coordinates called displacements or departures in the horizontal plane (N/S and E/W) and true vertical depth (TVD) in the vertical dimension.
In current oil field practice many wells may be drilled from closely spaced surface locations. Several wells from disparately spaced surface locations may be drilled into the same sub-surface target. Because the Earth is not a cube, cubical coordinates are an inadequate Coordinate Reference System (CRS) for important technical and safety issues, such as avoiding well collisions. Cubical coordinates must be converted into geodetic coordinates, an adequate CRS. The usual way to accomplish this is to relate cubical coordinates to projected coordinates (called Northings and Eastings, or X's and Y's) determined with a map projection (such as Transverse Mercator or Lambert Conformal Conic). This does work, but complete geodetic accuracy requires the computation and application of several corrections (e.g., convergence of the meridians, linear scale factor, and depth amplification factor) at every station along the well path. Unfortunately, these corrections are not always computed and applied. Errors result. Also, map projections introduce grid north as an azimuth reference in addition to true and magnetic north. Grid north is a common cause of mistakes in the management of directional well survey data.
This paper offers an algorithm (called LMP) that converts cubical coordinates into geographical coordinates (latitude and longitude) without an intervening map projection. Instead, LMP uses three easily computed radii of the ellipsoidal Earth. Repeated computation of these radii is more efficient than computing the corrections required by the use of a map projection, and the results are equivalent. Grid north is avoided.
A Geodesy Primer
Many geodetic terms used in this paper are explained in the geodesy primer offered in the cited paper by Williamson and Wilson, which is worth a second reading.
Directional Well Survey Data Types and Formats
To better understand the problems that LMP solves, we begin with the following taxonomy of DWS (Directional Well Survey) data types and the formats in which these data are delivered, viz., raw data, intermediate data, cubical coordinates and geodetic coordinates.
Raw DWS data are those acquired by either wireline or MWD (Measurement While Drilling) survey instruments. In the MWD case, for example, three tri-axially-mounted accelerometers sense acceleration, especially the acceleration of gravity to determine the vertical, tool-face angle and inclination. Three tri-axially-mounted magnetometers sense magnetic north to determine horizontal azimuth. MWD gyros are a supplementary - or alternative - azimuth reference. A tally of drilling pipe determines progress (measured depth) along the path of the well.
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