|Publisher||Society of Petroleum Engineers||Language||English|
|Content Type||Journal Paper|
|Title||Directional Drilling and Earth Curvature|
|Authors||H.S. Williamson, BP Amoco; H.F. Wilson, Baker Hughes INTEQ|
|Journal||SPE Drilling & Completion|
|Volume||Volume 15, Number 1||Pages||37-43|
|Copyright||Copyright 2000. Society of Petroleum Engineers|
This paper provides a review of current practices for calculating directional drilling placement in the light of modern extended-reach applications. The review highlights the potential for gross errors in the application of geodetic reference information and errors inherent in the calculation method. Both types of error are quantified theoretically and illustrated with a real example. The authors borrow established land surveying calculation methods to develop a revised best practice for directional drilling. For the elimination of gross errors they prescribe increased awareness and a more disciplined approach to the handling of positional data.
When calculating well position, directional drillers currently take no account of Earth curvature. In effect, the well is planned and drilled using a “Flat Earth” model. The errors inherent in the Flat Earth assumption were until recently justifiably ignored as insignificant. However, the advent of longer wells aimed at smaller targets has prompted this more detailed analysis. The analysis shows that the errors can no longer be assumed insignificant. This suggests that in the future, oil companies will demand that the directional drilling software used for extended-reach applications incorporates more precise well positioning calculations.
In defining the form of these calculations, a balance must be struck between computational complexity and real requirements. That said, the ubiquity of computers at all stages of the drilling process has virtually eliminated the need for calculations ever to be performed by hand.
A Geodesy Primer
Geodesy is the name given to the study of the size and shape of the Earth. The branch of land surveying which properly takes account of this shape is known as geodetic surveying. To accurately describe the effects of Earth curvature on well positioning, it is necessary to use some geodetic terminology. The standard textbook, which contains full definitions of all the terms which follow, is by Bomford.1
The surface that is everywhere perpendicular to the direction of gravity (an “equipotential surface”) and that on average coincides with mean sea level in the oceans is called the geoid.
The geoid is much smoother than the physical surface of the Earth, but is still too irregular to be used as a reference for spatial coordinates. As an alternative, we use the geometrical shape which most closely approximates the shape of the Earth—an ellipsoid, which in this context is an ellipse rotated about its minor axis. The term spheroid is sometimes used in place of ellipsoid.
To be useful as a coordinate reference, a relationship between the position of the ellipsoid and the solid Earth must be defined. Although sometimes used to refer to just this relationship, the term geodetic datum is more correctly used to include the definition of the ellipsoid as well. When combined with an axes definition, a geodetic datum defines a three-dimensional (3D) geographic coordinate system, the dimensions being (geodetic) latitude and longitude and ellipsoidal height (height above the ellipsoid).
It is possible to define a geodetic datum which approximates the shape of the Earth over the entire globe. WGS 84, used by the Global Positioning System (GPS), is an example. In practice, most geodetic datums used for mapping have been defined to give a more precise fit over a restricted geographical area. As an example, coordinates of points in the North Sea are conventionally quoted with respect to European Datum 1950 (ED50), which incorporates the International 1924 ellipsoid. The proliferation of such regional datums over time has meant that their areas of application frequently overlap. The same set of latitude and longitude coordinates, referenced to different geodetic datums, will refer to different points on the Earth. The coordinates alone, contrary to common belief, do not adequately define a particular location.
Lines of constant latitude and longitude are called parallels and meridians, respectively. These lines are curved in three dimensions, but may be represented on a plane by means of a projection. The rectangular coordinate system on the plane is called a grid.
It is impossible to devise a projection which represents all true directions and distances correctly on the plane. However, it is possible to control this distortion so that the shapes of small areas are preserved. Projections with this property are called orthomorphic or conformal, and include the Transverse Mercator Projection and most others used for oilfield mapping.
For any orthomorphic projection, the amount of distortion to directions at a point on the grid is defined by grid convergence, the angle clockwise between the meridian passing through the point (i.e., true north) and grid north. Likewise, the amount of distortion to scale at a point is defined by the point scale factor. (Not to be confused with the scale factor at the natural origin, which is a fixed parameter used in the definition of many projections. For UTM zones, its value is 0.9996). The point scale factor changes with geographical position, which results in distances calculated from grid coordinates differing from distances measured on (or through) the ground. In this paper, we shall call the ratio of map grid distance to true distance the grid scale factor. Since the arc length of a degree of latitude or longitude decreases with increasing depth, the grid scale factor (Fig. 1 ) must increase to compensate.
For nearby points A and B:
Since grid convergence varies from place to place, the first of these rules is only an approximation—the error will increase as the distance between A and V grows. The second rule is always valid by our definition of grid scale factor.
Horizontal position being defined by ellipsoidal coordinates (latitude and longitude), it seems natural to define vertical position by height above the ellipsoid. This is not done in practice, the main reason being that the surface of the ellipsoid offers no physical reference point for measurement.
The geoid (roughly speaking, mean sea level) is a much more convenient surface to use as a height reference. Surveyors working on land can measure the difference in height above the geoid at two locations by spirit leveling. The reference level used as a zero datum is defined by mean sea level at a selected coastal location, or an average value of mean sea level at several locations, over a specified period of time. Elevations on land should include a reference to this vertical datum. In the U.S., it is termed the North American Vertical Datum of 1988 (NAVD88), which also covers southern Canada. In Britain it is Ordnance Datum Newlyn (ODN).
Surveyors working offshore can measure elevations relative to sea level directly and, by reference to tidal predictions, correct these to mean sea level.
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