ANALYTICAL CALCULATION OF THE WEDGE SETTING ANGLE WHEN A BOREHOLE DEVIATES FROM THE SPECIFIED TRAJECTORY

G. L. Kalinichenko, S. S. Sudakshin

(Presented by the Department of Engineering and Computational Mathematics)

When calculating the wedge setting angle in a borehole in order to change its azimuth \(\alpha\), one usually proceeds from the assumption that the azimuthal rotation of the wedge \(\Delta \alpha = \angle XOC\) is approximately equal to \(\angle KOC'\) in Fig. 1 (see [1, 2, 3]).

Fig. 1.

This assumption introduces a certain error into the results obtained. Therefore, we shall derive all the formulas necessary for the indicated calculation without any simplifications.

Consider the wedge in the Cartesian coordinate system \(XYZ\) (Fig. 1). The rotation angle of the wedge \(\psi'\), lying in the plane \(ZOX\), is defined as the angle between the axis \(OX\) and the vector \(\overline{BC}\). For this, we first need to know the coordinates of the vector \(\overline{BC}(x,y,z)\), or the coordinates of the points \(B\) and \(C\). From the drawing it is clear that the point \(B\) is defined by the coordinates \(B(l\sin\Theta;\ l\cos\Theta;\ 0)\). We define the point \(C\) by the coordinates \(C(x_C,y_C,z_C)\), where \(x_C=OD;\ y_C=0;\ z_C=CD\). From the drawing we also find that

\[ z_C=x_C\tg\Delta\alpha . \tag{1} \]

Therefore the coordinates \(x,y,z\) of the vector \(\overline{BC}\) will be equal to:

\[ x=x_C-l\sin\Theta;\quad y=0;\quad z=z_C=x_C\tg\Delta\alpha . \tag{2} \]

From this, we further find from the drawing that the vectors \(\overline{AB}\) and \(\overline{AC}\) are defined by the coordinates

\[ \overline{AB}(l\sin\Theta;\ -l\cos\Theta;\ 0), \]

\[ \overline{AC}(x_C;\ -l\cos\Theta;\ x_C\tg\Delta\alpha). \tag{3} \]

The coordinates of the vector \(\overline{BC}\) have been obtained through the unknown abscissa \(x_C\) of the point \(C\), which we shall find from the formula for the angle between the vectors \(\overline{AB}\) and \(\overline{AC}\), since the angle under consideration is the wedge skew angle (a design quantity). We have

\[ \cos(\overline{AB},\widehat{\overline{AC}})=\cos\delta= \]

\[ =\frac{l\sin\Theta\cdot x_C+l^2\cos^2\Theta} {\sqrt{l^2\sin^2\Theta+l^2\cos^2\Theta}\cdot \sqrt{x_C^2+l^2\cos^2\Theta+x_C^2\tg^2\Delta\alpha}}, \]

whence

\[ \cos\delta= \frac{x_C\sin\Theta+l\cos^2\Theta} {\sqrt{x_C^2+l^2\cos^2\Theta+x_C^2\tg^2\Delta\alpha}} . \tag{4} \]

From this formula we find the coordinate \(x_C\) for given \(\delta\), \(\Theta\), and \(\Delta\alpha\). After the corresponding transformations we obtain:

\[ x_{C1,2}=\frac{l\cos\delta}{a} \left[ \sin\Theta\cos\Theta \pm \sqrt{\sin^2\Theta\cos^2\Theta-a(\cos^2\delta-\cos^2\Theta)} \right], \tag{5} \]

where

\[ a=\cos^2\delta\sec^2\Delta\alpha-\sin^2\Theta . \tag{6} \]

The real roots for \(x_C\) will exist under the condition

\[ \sin^2\Theta\cos^2\Theta-a(\cos^2\delta-\cos^2\Theta)\geq 0 \]

or

\[ a\leq\frac{\sin^2\Theta\cdot\cos^2\Theta}{\cos^2\delta-\cos^2\Theta}. \tag{7} \]

In the case of equality, we obtain the condition under which both roots will be equal and the angle \(\psi'\) will be greatest. For the given \(\Theta\) and \(\delta\), we find \(\Delta\alpha\) for which \(\psi'\) will be greatest:

\[ \sec^2\Delta\alpha= \frac{\sin^2\Theta\cdot\cos^2\Theta} {(\cos^2\delta-\cos^2\Theta)\cdot\cos^2\delta} + \frac{\sin^2\Theta}{\cos^2\delta}. \tag{8} \]

After we find \(x_C\), \(\psi'\) can be determined from the formula

\[ \psi'=\arc\cos \frac{x_C-l\sin\Theta} {\sqrt{(x_C-l\sin\Theta)^2+x_C^2\tg^2\Delta\alpha}} . \tag{9} \]

The inclination angle \(\Theta'\) of the deviated borehole is found from the formula for the angle between the vector \(\overline{AC}\) and the \(Oy\) axis

\[ \Theta'=\arccos\left[\frac{l\cos\Theta}{\sqrt{x^2\sec^2\Delta\alpha+l^2\cos^2\Theta}}\right]. \tag{10} \]

Since formula (5) gives two values for \(x\), formulas (9) and (10) will likewise give two values for the angles \(\psi'\) and \(\Theta'\). It should be noted that in [1], [2], [3] the second values of these angles are not mentioned at all.

All calculations according to the formulas presented have been programmed, and tables have been compiled on the Minsk-1 electronic computer for \(\delta=2^\circ\), \(\delta=3^\circ\), and \(\delta=4^\circ\), and for \(l=2,3,4\) meters for \(\Delta\alpha\), beginning with \(\Delta\alpha=\delta\) and up to \(\Delta\alpha_{\max}\), at an interval of 1 degree, and for \(\Theta\), beginning with \(\Theta=2\delta\) up to the possible value, at an interval of \(2^\circ\); in addition, the maximum values of the angle \(\psi'\) have also been found for \(\delta=3\) and \(l=3\).

Comparing the calculations carried out by Kurmashev [1] according to the formula

\[ \sin(\psi'-\Delta\alpha)=\frac{\sin\Theta\sin\Delta\alpha}{\operatorname{tg}\beta}, \]

with the calculations according to formula (9), we see that the error in determining the angle \(\psi'\) reaches 4 degrees and more.

REFERENCES

1. A. M. Kurmashev. Device for calculating deviation in an inclined borehole. Information Bulletin of the Ministry of Geology and Protection of Mineral Resources of the USSR, No. 27, 1961.
2. A. G. Kalinin et al. Orientation of Deflecting Systems in Boreholes. Moscow, 1963.
3. A. G. Kalinin. Curvatures of Drilled Boreholes. Moscow, 1963.