NOMOGRAMS

FOR CALCULATING THE WEDGE SETTING ANGLE

DURING SPATIAL DEFLECTION OF A BOREHOLE

G. L. KALINICHENKO, S. S. SULAKSHIN

The formulas known in the literature for setting a wedge in a borehole in order to change its azimuth are given with certain assumptions, which introduce corresponding errors into the results obtained. In [1] we gave formulas for setting the wedge without these assumptions. Calculations using them, however, are in many respects complicated and present known difficulties. Therefore we compiled a program for all calculations by these formulas on the “Minsk-1” digital computer and prepared tables for \(\delta = 2^\circ\), \(\delta = 3^\circ\), \(\delta = 4^\circ\). In the calculations, in addition to [1], the following three provisions were adopted:

1. The angles \(\psi^1\) and \(\theta^1\)—the wedge rotation angle in the horizontal plane and the new zenith angle, determined by the formulas:

\[ \psi^1=\arccos\left[ \frac{x_c-l\sin\theta} {\sqrt{(x_c-l\sin\theta)^2+x_c^2 \operatorname{tg}^2\Delta\alpha}} \right], \]

\[ \theta^1=\arccos\left[ \frac{l\sin\theta} {\sqrt{x_c^2\sec^2\Delta\alpha+l^2\cos\theta}} \right], \]

do not depend on \(l\). Indeed, if we denote

\[ u_{1,2}=\frac{\cos\alpha}{a} \left[ \sin\theta\cos\theta \pm \sqrt{\sin^2\theta\cos^2\theta-a(\cos^2\delta-\cos^2\theta)} \right] \]

then \(x_c=lu_{1,2}\), and then formulas 1 and 2 take the form

\[ \psi^1=\arccos\left[ \frac{u_c-\sin\theta} {\sqrt{\sec^2\Delta\alpha+\cos^2\theta \operatorname{tg}^2\Delta\alpha}} \right], \]

\[ \theta^1=\arccos\left[ \frac{\cos\theta} {\sqrt{u_c^2(u_c-\sin\theta)^2+u}} \right]. \]

It follows from this that \(\psi^1\) and \(\theta^1\) do not depend on \(l\).

1. Often it is of interest not to know the angle of rotation of the wedge in the plane \(xOy\) (in Fig. 1, the angle \(\angle CBD\)), but the angle \(\angle C^1B^1D^1\), i.e., the angle between the plane \(xOz\) and the plane \(ABC\).

From the figure it is immediately evident that the coordinates of the points will be
\(A(0, 0, l\cos\theta)\); \(B(l\sin\theta, 0, 0)\); \(C(x_c, x_c\operatorname{tg}\Delta\alpha, 0)\). Then the coordinates of the normal to the plane \(ABC\) will be the following:

\[ N\left(x_c\operatorname{tg}\Delta\alpha\cos\theta;\ (x_c-l\sin\theta)\cos\theta;\ x_c\operatorname{tg}\Delta\alpha\sin\theta\right) \]

Hence the angle \(\psi\) will be found from the formula

\[ \psi=\arccos \frac{(x_c-l\sin\theta)\cos\theta} {\sqrt{x_c^{2}\operatorname{ctg}^{2}\Delta\alpha+(x_c-l\sin\theta)^2\cos^2\theta}} . \]

1. Let us show that formulas 4 and 5 for \(\delta=\Delta\alpha\) have one and the same value, i.e., the new zenith angle is equal to the wedge turning angle.

Fig. 1.

Indeed, for \(\delta=\Delta\alpha\)

\[ x_{c_{1,2}}=\frac{l\cos\theta}{a} \left[ \sin\theta\cos\theta \pm \sqrt{\sin^2\theta\cos^2\theta-a(\cos^2\delta-\cos^2\theta)} \right], \]

where \(a=\cos^2\delta\,\sec^2\Delta\alpha-\sin^2\theta\), we shall have

\[ x_{c}=l(\sin\theta\pm\sin\delta). \]

Substituting 6 into the numerator and into the second term of the radicand of formula 5, and dividing the numerator and denominator by \(\sin\delta\), we obtain formula 4.

1. From the compiled tables, three nomograms were constructed for \(\delta=2^\circ\), \(\delta=3^\circ\), and \(\delta=4^\circ\).

In each nomogram, a definite \(\Delta\alpha\) corresponds to two functions: \(\psi=f_1(\theta)\) and \(\psi^{1}=f_2(\theta)\), which, as they increase, tend to coincide; thus, for example, at \(\delta=2^\circ\), already at \(\Delta\alpha=10^\circ\) the curves \(\psi\) and \(\psi^{1}\) practically coincide. Each point of a nomogram curve is a set of three quantities \((\Delta\alpha,\theta,\psi)\) or \((\Delta\alpha,\theta,\psi^{1})\). Thus, by specifying two of the three quantities, we obtain the third quantity from the nomogram.

Three analogous nomograms were constructed for finding the new zenith angle, relating the three quantities \((\theta,\theta^{1},\Delta\alpha)\) (Figs. 2, 3).

Fig. 4. Nomograms for bevel angle \(\delta = 4^\circ\).

Conclusions

1. The angles $\psi^{1}$ and $\theta^{1}$ of the deflecting wedge system do not depend on the length of the wedge.

2. At $\Delta \alpha = \delta$, the new zenith angle is equal to the wedge rotation angle.

3. For all three values of the wedge bevel, $\delta = 2^\circ$, $\delta = 3^\circ$, $\delta = 4^\circ$, at $\Delta \alpha > 10^\circ$ the curves $\psi$ and $\psi^{1}$ practically coincide; hence the assumptions used in works [2, 3, 4] are valid only at $\Delta \alpha > 10^\circ$.

4. At small $\Delta \alpha$, the difference between $\psi$ and $\psi^{1}$ reaches $8^\circ$.

REFERENCES

1. G. L. Kalinichenko, S. S. Sulakshin. Analytical calculation of the wedge setting angle when deflecting a borehole from a prescribed trajectory. Izv. TPI, vol. 154, 1967.

2. A. M. Kurmashev. Instrument for calculating deflections in an inclined borehole. Information Bulletin of the Ministry of Geology and Mineral Resources Protection of the USSR, 27, 1961.

3. A. G. Kalinin et al. Orientation of deflecting systems in boreholes. Moscow, GNTI, 1960.

4. A. G. Kalinin. Curvature of drilled boreholes, Gostoptekhizdat, Moscow, 1963.