NOMOGRAM FOR DETERMINING THE INTENSITY OF CURVATURE OF OIL WELLS

L. A. Pukhlyakov

(Presented by the scientific seminar of the Department of Exploration Technology)

The intensity of curvature of a well within a given interval is understood as the angle enclosed between the directions of its borehole at the ends of this interval, which can be determined from the relation

\[ \cos i = \cos \Theta_1 \cdot \cos \Theta_2 + \sin \Theta_1 \cdot \sin \Theta_2 \cdot \cos \varphi, \tag{1} \]

where \(i\) is the intensity of curvature in the given interval,
\(\Theta_1\) is the zenith angle of the well at the first measurement point,
\(\Theta_2\) is the zenith angle of the well at the second measurement point, and
\(\varphi\) is the difference between the azimuths of the well inclination at the ends of the measurement interval. This is precisely how Yu. S. Vasil’ev, N. B. Sivokhina [1], and several other authors understand the given term. A. Lubinskii [2] considers it necessary always to refer this quantity to one interval—10 m—and recommends calling it the degree of well deflection. Finally, a third group of authors recommends calling this quantity the intensity of total curvature, contrasting it with the intensity of increase in zenith angle and the intensity of azimuthal curvature separately [3].

Here it must first of all be noted that analyzing the intensity of increase in the zenith angle and the intensity of azimuthal curvature of a well separately gives nothing from the point of view of identifying zones of possible casing failure or rapid wear of pump rods during well operation. Moreover, very often such an analysis may lead to entirely incorrect results. Let us demonstrate this by an example.

Suppose that, over some interval of a certain well, the zenith angle was equal to \(1^\circ 30'\) at one end and \(2^\circ 15'\) at the other; and that in a second interval of the same well the zenith angle increased from \(4^\circ 15'\) to \(5^\circ 00'\). In other words, we have assumed that the increase in zenith angle in both intervals was the same. Suppose further that the increase in the azimuth of the well inclination in both intervals was also the same and equal to \(40^\circ\). Thus, when analyzing the intensity of curvature by zenith angle and azimuth separately, for both intervals we obtain identical characteristics. If, however, for this purpose we use formula (1), then for the first interval we obtain the value \(1^\circ 28'\), and for the second, \(3^\circ 14'\). It follows from what has been said that there is no point in analyzing the increase in zenith angle and the azimuth of a well separately, and therefore there is no need to complicate the terminology on this matter. Thus, within the limits of the present article, the quantity,

Fig. 1. Nomogram for determining the intensity of curvature of oil wells (above) and a scheme for using it.
Notation: \(\Theta_1\) is the zenith angle of the well at the first measurement point; \(\varphi\) is the difference between the azimuths of the well inclination at the measurement points; \(\Theta_2\) is the zenith angle of the well at the second measurement point; \(i\) is the intensity of curvature over the given interval.

obtained by calculation from formula (1), will be called the intensity of curvature of the well.

To determine the intensity of curvature of wells, various authors propose different methods and means. For example, the instrument of Yu. S. Vasil’ev is known [1, pp. 44–45], consisting of two protractors and three rulers with uniform scales. This instrument, however, gives a systematic error that is the greater, the greater the difference between the zenith angles and azimuths. This can be verified by solving several examples with Yu. Vasil’ev’s instrument and analytically by formula (1). A second drawback of Yu. S. Vasil’ev’s instrument is that its manufacture involves certain technical difficulties. In any case, this instrument did not come into wide use, and Yu. S. Vasil’ev himself, along with this instrument, also recommends using A. Lubinsky’s nomogram [1, pp. 36–39]. As for this nomogram, the author himself acknowledges that it also gives a systematic error if the difference of zenith angles exceeds \(4^\circ\) [2, p. 156].

The proposed nomogram for determining the intensity of curvature of wells is a plane projection of a hemisphere, more precisely the part of it adjacent to the horizontal diameter. The distances between the horizontal lines of this nomogram express the sines of angles plotted, on a certain scale, upward from the horizontal diameter, and are constant along the entire length of these lines. The distances between the vertical lines are not constant. On the horizontal diameter they are the same as between the horizontal straight lines; on all the remaining horizontal lines they express products of sines plotted from the vertical diameter (the \(0^\circ\)–\(90^\circ\) lines). The arcs are spaced from the zero point at distances equal to the sines of the angles inscribed next to them.

Working with this nomogram amounts to the following. The zenith angle at the first measurement point \(\Theta_1\) is plotted along the horizontal diameter to the left of zero (Fig. 1). From the point obtained, an arc is drawn upward (clockwise), whose angular magnitude must correspond to the difference of the azimuths of the well inclination at the compared points—\(\varphi\) (Fig. 1). To simplify work on the nomogram, some of its arcs have degree graduations. From the second point in the system of vertical lines, the zenith angle of the well at the second measurement point \(\Theta_2\) is counted off to the right. In doing so, the count is made along a straight line directed strictly parallel to the horizontal diameter. As a result, a third point is obtained. Its distance from the zero point of the nomogram corresponds to the magnitude of the desired angle \(i\). With a compass this distance is transferred to the scale located on the left, where each radius is divided into six divisions.

The accuracy of the results obtained on the nomogram under consideration will be the higher, the larger its dimensions.

REFERENCES

1. Yu. S. Vasil’ev, N. B. Sivokhina, A. S. Bronzov. Permissible deviations of wellbores from the design. Gostoptekhizdat, 1963.

2. G. Vuds, A. Lubinsky. Curvature of wells during drilling. Gostoptekhizdat, 1960.

3. Yu. L. Boyarko, V. P. Rozhkov, I. G. Leshchenko. New schemes of inclinometers for magnetic media. Proceedings of the Tomsk Polytechnic Institute, vol. 127, no. 2, 1965.