Intensity of Wellbore Curvature in the Aleksandrovsk Oil Exploration Expedition (Middle Ob Region)

INTENSITY OF WELLBORE DEVIATION IN THE ALEXANDROVSK OIL EXPLORATION EXPEDITION

(MIDDLE OB REGION)

L. A. Pukhlyakov

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

By the intensity of wellbore deviation \((i)\) it is customary to understand the change in the direction of the wellbore in space over a given interval. This is precisely how this term is understood by Yu. S. Vasil'ev, N. V. Sivokhina, A. S. Bronzov [1], and certain other authors. A. Lubinskii [2] considers it necessary always to relate this quantity to one and the same interval—10 m—and recommends calling it the degree of wellbore curvature. However, it seems more convenient to relate this quantity to the interval over which, at the given enterprise, it is customary to measure the elements of wellbore curvature. This is sufficient for identifying zones of greatest deviation, and for comparing the intensity of wellbore deviation at different enterprises a recalculation can be made as necessary.

Some authors recommend calling this quantity the absolute intensity, or the intensity of the total deviation of the wellbore, contrasting with it separately the intensity of increase in zenith angle and the intensity of increase in the azimuth of wellbore inclination. Here, however, it should be noted that such a separation only distorts the picture, since with one and the same difference in azimuths, as the zenith angle increases, the absolute intensity will increase.

To derive the formula for absolute intensity, let us adopt the following notation: \(\Theta_1\) and \(\Theta_2\) are the zenith angles of the wellbore at the first and second measurement points, respectively; \(A_1\) and \(A_2\) are the azimuths of the wellbore inclination at these same points; and \(\varphi\) is the difference in azimuths,

\[ \varphi = A_1 - A_2 . \tag{1} \]

Next, in the plane coinciding with the plane of wellbore inclination at the first measurement point, we construct the tangent to the wellbore axis at this point, \(BF\), and the projection onto it of the tangent \(C'F\) at the second measurement point \(C\); after this, we intersect them with the plane \(CG\), perpendicular to the tangent to the wellbore axis at the first measurement point \(BF\) (Fig. 1). Obviously, on the tangent at the second measurement point this plane will cut off a segment that will be related to the corresponding segment on the tangent \(BF\) by the following relation:

\[ FG = CF \cdot \cos i . \tag{2} \]

Its projection, however, will be related to this segment by a somewhat different relation. Namely:

\[ FG = C'F \cdot \cos \angle C'FG \]

or

$$ FG = C'F(\cos \Theta_1 \cdot \cos \angle C'FD) + \sin \Theta_1 \cdot \sin \angle C'FD). \tag{3} $$

On the other hand, the sine and cosine of the angle \(C'FD\) can be expressed in terms of the sides of the right triangle \(C'FD\), in which the line \(FD\) is vertical,

$$ \cos \angle C'FD = \frac{z}{C'F}, $$

$$ \sin \angle C'FD = \frac{x}{C'F}. $$

In turn, the legs of this triangle can be expressed through the zenith angle at the second measurement point \(\Theta_2\) and the difference of azimuths \(\varphi\):

$$ z = CF \cdot \cos \Theta_2, $$

$$ x = CF \cdot \sin \Theta_2 \cdot \cos \varphi. $$

As a result, expression (3) takes the form:

$$ FG = C'F\left(\cos \Theta_1 \cdot \frac{CF \cdot \cos \Theta_2}{C'F} + + \sin \Theta_1 \frac{CF \cdot \sin \Theta_2 \cdot \cos \varphi}{C'F}\right) $$

or, carrying out the corresponding reductions,

$$ FG = CF(\cos \Theta_1 \cdot \cos \Theta_2 + + \sin \Theta_1 \cdot \sin \Theta_2 \cdot \cos \varphi). \tag{4} $$

Fig. 1. Relationship between tangents drawn to the wellbore axis at two adjacent measurement points

Finally, comparing expressions (2) and (4), we find

$$ \cos i = \cos \Theta_1 \cdot \cos \Theta_2 + \sin \Theta_1 \cdot \sin \Theta_2 \cdot \cos \varphi. \tag{5} $$

To determine the intensity of wellbore curvature, various authors propose different means. For example, the instrument of Yu. S. Vasiliev is known [1, pp. 44–45], consisting of two protractors and three rulers with uniform scales. A. Lubinskii [2] recommends a special nomogram for the same purpose. However, in analyzing the intensity of wells of the Aleksandrovsk Oil Exploration Expedition, none of these means was used. The main reason for rejecting them was the unusually high labor input required for their use. To simplify the work, a special table was compiled—a table for determining the intensity of slightly curved wells [3], an excerpt from which is given in Appendix 1. This table consists, as it were, of several independent small tables, each of which corresponds to the zenith angle at the first measurement point \(\Theta_1\) and has 10–12 rows. In turn, each row corresponds to different values of the zenith angles at the second measurement point \(\Theta_2\). Here the second measurement point is understood to be the one at which the value of the zenith angle is greater than at the other, regardless of whether it is located higher or lower along the borehole. The vertical columns of the table correspond to different values of the well-inclination azimuths \(\varphi\).

Appendix 1

Table for determining the intensity of curvature of slightly curved wells

\(\Theta_1 = 0^\circ 30'\)

\(\Theta_1 = 1^\circ 00'\)

\(\Theta_1 = 1^\circ 30'\)

\(\theta_1 = 2^\circ 00'\)

\(\theta_1 = 2^\circ 30'\)

\(\theta_1 = 3^\circ 00'\)

\[ \Theta_1 = 3^\circ 30' \]

\[ \Theta_1 = 4^\circ 00' \]

\[ \Theta_1 = 4^\circ 30' \]

Fig. 2. Intensity of wellbore deviation in wells of the Alexandrovsk oil-exploration area. The vertical straight lines represent the well axes. The numbers to the left of them indicate depth in meters. The intensity values are given in degrees per 25 m of borehole; the common scale is shown in the lower part of the figure.

Visible well labels: Well 17; Well 16; Well 4; Well 19; Well 23; Well 22; Well 2; Well 3; Well 5.

Scale: \(0^\circ,\ 1^\circ,\ 2^\circ,\ 3^\circ,\ 4^\circ\).

\(\theta_1 = 5^\circ 00'\)

The results of determining the intensity of curvature for various wells of the Alexandrovsk oil exploration expedition are given in Fig. 2. Greater intensities here correspond to greater distances of the broken line from the vertical straight line symbolizing the well axis.

As can be seen from this figure, the intensity of curvature of the wells of the Alexandrovsk oil exploration expedition is comparatively small. As a rule, it is no more than one degree per 25 m, although at times it nevertheless exceeds this value. For comparison, let us note that in the wells of the “Artemneft” oil-field administration (Azerbaijan), in the zone where curvature accumulates, the intensity of curvature sometimes reaches \(4^\circ 00'\) per 25 m; and in the wells of the “Chernushkaneft” oil-field administration it is \(2^\circ 30'\) and even \(3^\circ 00'\) per 25 m. It follows from what has been said that, during operation of the wells of the Alexandrovsk oil exploration expedition, they should not exhibit the uneven wear of sucker rods that is observed in the fields of the “Artemneft” oil-field administration.

Furthermore, Fig. 1 makes it possible to say that in some wells of the Alexandrovsk oil exploration expedition the intensity of curvature is distributed along the borehole more or less uniformly—for example, in wells 3 and 17—whereas in others it is greater in some parts and smaller in others. For example, in well 4, in the interval 1000–1600 m it is close to zero, while in the intervals 300–1000 and 1750–2000 m it exceeds \(0^\circ 30'\) per 25 m. Low curvature intensity is observed in well 2 in the intervals 750–825 m and 1275–1575 m; in well 18 in the intervals 475–675 m, 750–800 m, 950–1175 m, and 1750–1825 m; and in well 19 in the interval 1200–1400 m. High intensity is observed in well 17; moreover, as noted above, it is distributed approximately uniformly over the entire borehole.

It does not appear possible to associate an increase in the intensity of well curvature with any suite of the geological section; therefore the cause of well curvature in this area should be sought in the technical conditions of drilling them. Incidentally, the curvatures of the wells of the Alexandrovsk oil exploration expedition fall within the permissible norms—the zenith angles here do not exceed \(5^\circ 00'\).

REFERENCES

1. Yu. S. Vasil’ev, N. B. Sivokhina, A. S. Bronzov. Permissible deviations of wellbores from the design. Gostoptekhizdat, 1963.
2. G. Woods, A. Lubinsky. Well deviation during drilling. Gostoptekhizdat, 1960.
3. L. A. Pukhlyakov. Table for determining the intensity of well curvature. Manuscript, 1966.