ON THE PROFILING OF MULTILATERAL WELLS IN SHOT DRILLING

V. P. ROZHKOV

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

One of the progressive methods for increasing the technical and economic efficiency of sinking wells is multilateral drilling. With this method, the main bore is used repeatedly; therefore the amount of drilling through the upper unproductive horizons is considerably reduced \[1, 2\].

On the other hand, multilateral drilling entails certain additional expenditures of funds and time (cementing the well, or setting a bridge plug, setting a wedge, drilling off from the main bore, etc.). Therefore, the drilling of a multilateral well must be preceded by a thorough analysis of the technical feasibility and economic expediency of sinking such a well. In advance one can only assert that the economic effect will be the greater, the greater the depth from which the additional bores are drilled off. Guided only by this general proposition, it is impossible to decide the question of the expediency of drilling a particular well. For this it is necessary to have the profile of the designed multilateral well, drawn up so that the well fully solves the geological problem posed, the total length of the well is as small as possible, and the drilling is technically feasible.

The profile of a multilateral well depends on the means used and on the adopted drilling scheme.

Depending on the sequence in which the additional bores are drilled off, two schemes of exploration by multilateral wells are distinguished \[3\]. According to the first scheme (Fig. 1, a), the first additional bore \(C_1B_1\) is cut into the wall of an already drilled main well \(AB\), by which the ore body has been intersected at depth \(H\). After completion of the first additional bore \(C_1B_1\), if the appropriate conditions exist, it is possible to sink additional bores \(C_2B_2\), \(C_3B_3\), drilled off from the main well \(AB\). Thus, under this scheme of multilateral drilling, exploration by additional bores proceeds “from below upward” along the rise of the ore body.

According to the second scheme (Fig. 1, b), the main bore \(AB\) is set at point \(A\) with an initial zenith angle determined by the task of intersecting the ore body at depth \(H\), and is sunk to point \(C_1\). After this, an additional well \(C_1B_1\) is drilled off from the bottom of the main bore. If the ore body is found at depth \(H_1\), the main bore is drilled off to point \(C_2\), and a second additional bore \(C_2B_2\) is made. In such

in the same sequence, all the remaining auxiliary holes can be drilled, and then the main hole can be drilled out to the final depth (point \(B\)). In this case exploration proceeds not along the rise of the ore body, as in the first case, but along its dip, i.e., “from top to bottom.”

As means for directing the hole to a specified point, tools and devices created specially for this purpose may be used, as well as the phenomenon of natural curvature of boreholes. In exploration according to the first scheme, the auxiliary hole is started with the aid of a wedge; when the second scheme is used, the auxiliary hole is drilled off from the main one with the aid of a single-hinge deflector, and subsequent deepening of the main hole (in the zone where the auxiliary is started) is carried out with the aid of a long rigid assembly.

Fig. 1. Scheme of exploration by multilateral boreholes (after B. I. Spiridonov):
\(a\)—first scheme, \(b\)—second scheme.
\(AB\)—main hole; \(C_1B_1,\ C_2B_2,\ C_3B_3\)—auxiliary holes; \(H_1,\ H_2,\ H_3,\ H\)—vertical depths of intersection of the mineral.

Let us consider the profiling of multilateral shot-drilling boreholes when they are drilled according to the schemes described and when natural curvature is used. Then the profile of the auxiliary hole, like the profile of a single naturally curved borehole, is described by the differential equation (4)

\[ K=\frac{d\Theta}{dl}=a+b\Theta, \]

where \(K\) is the curvature of the borehole, representing the increment of the zenith angle over an interval of one meter;

\(\Theta,\ l\)—respectively the zenith angle and the depth of the borehole;

\(a,\ b\)—parameters of the regression line, constant for the given section or for the deposit as a whole.

The solution of this equation leads to the following dependences of the borehole projections on the magnitude of the zenith angle

\[ X=\frac{1}{b}\{\cos A\,[\operatorname{Ci}(A+\Theta)-\operatorname{Ci}(A+\Theta_0)]+ \sin A\,[\operatorname{Si}(A+\Theta)-\operatorname{Si}(A+\Theta_0)]\}, \tag{2} \]

\[ Y=\frac{1}{b}\{\cos A\,[\operatorname{Si}(A+\Theta)-\operatorname{Si}(A+\Theta_0)]- \sin A\,[\operatorname{Ci}(A+\Theta)-\operatorname{Ci}(A+\Theta_0)]\}, \tag{3} \]

where \(X,\ Y\) are the current coordinates of the point under consideration of the curved borehole, corresponding to the zenith angle;

\(\operatorname{Si},\ \operatorname{Ci}\)—respectively the sine integral and the cosine integral;

\[ A=\frac{a}{b} \]

is a quantity constant for the given section or deposit as a whole.

Analytical profiling by these equations is possible only when the integral functions are replaced by simpler ones. By point approximation it was established that, for \(\Theta = 10\text{--}30^\circ\), the relations

\[ \operatorname{Si}(\Theta) \approx \Theta, \tag{4} \]

\[ \operatorname{Ci}(\Theta) \approx -c + d\Theta - f\Theta^2, \tag{5} \]

hold, where \(\Theta\) is the value of the zenith angle in radians; \(c, d, f\) are constants equal, respectively, to \(1.76\), \(4.25\), and \(2.08\).

Substituting these equalities into equations (2) and (3), we obtain approximate equations for the projections of the shot well,

\[ X = \frac{1}{b}\,[M(\Theta-\Theta_0) - f(\Theta^2-\Theta_0^2)], \tag{6} \]

\[ Y \approx \frac{1}{b}\,[N(\Theta-\Theta_0) + Af(\Theta^2-\Theta_0^2)], \tag{7} \]

where

\[ M = d - A(2f-1) = 4.25 - 3.16A; \tag{8} \]

\[ N = 1 - A \cdot d + 2fA^2 = 1 - 4.25A + 4.16A^2; \tag{9} \]

\(\Theta_0, \Theta\) are, respectively, the zenith angle of the kick-off and the zenith angle of the well corresponding to the projections \(X\) and \(Y\), in radians.

The profiling of a multilateral shot well when drilling according to the “bottom-up” scheme (Fig. 2, a) begins with the design of the main bore. The method for profiling the main bore remains the same as when profiling a single directional well [4]; therefore, here we consider only the design of the additional bores.

The principal task in calculating the profile of an additional bore is to determine the point of its kick-off. The slotting point is easily determined if the zenith angle at the point where the ore body is cut, \(\Theta_{\mathrm{g}}\), and the zenith angle of the main bore at the slotting point of the additional bore, \(\Theta_{\mathrm{z}}\), are known. To determine these angles, one may use the conditions following from Fig. 2, a:

\[ H_{\mathrm{v}} = X_{\mathrm{main}} + X_{\mathrm{add}}, \tag{10} \]

\[ H_{\mathrm{g}} = Y_{\mathrm{main}} + Y_{\mathrm{add}}, \tag{11} \]

\[ \Theta_i = \Theta_{\mathrm{z}} + \omega, \tag{12} \]

where \(H_{\mathrm{v}}, H_{\mathrm{g}}\) are the distances from point \(B_1\) to the mouth (point \(O\)), respectively along the vertical and the horizontal;

\(X_{\mathrm{main}}, X_{\mathrm{add}}\) are the vertical projections of the sections \(OA\) of the main bore and \(AB_1\) of the additional bore, respectively;

\(Y_{\mathrm{main}}, Y_{\mathrm{add}}\) are the horizontal projections of the same sections;

\(\Theta_{\mathrm{z}}, \Theta_i\) are the zenith angles at the kick-off point of the additional bore, respectively for the main and additional bores;

\(\omega\) is the wedge deflection angle.

Indeed, expressing in equations (10) and (11) the projections by means of equalities (6), (7), and (12) through the zenith angles, we obtain a system of two equations:

\[ bH_{\mathrm{v}} = 2f\omega\Theta_{\mathrm{z}} + M(\Theta_{\mathrm{g}}-\Theta_0-\omega) - f(\Theta_{\mathrm{g}}^2-\Theta_0^2-\omega^2), \]

\[ bH_{\mathrm{g}} = -2f\omega A\Theta_{\mathrm{z}} + N(\Theta_{\mathrm{g}}-\Theta_0-\omega) - Af(\Theta_{\mathrm{g}}^2-\Theta_0^2-\omega^2), \]

whence it is easy to obtain the required relations:

\[ \Theta_{\mathrm{z}}= \frac{b\left(AH_{\mathrm{v}}+H_{\mathrm{r}}\right)+A(1-f)\left(\Theta_{\mathrm{v}}^{2}+\omega^{2}\right)} {1+A^{2}}, \]

\[ \Theta_{\mathrm{r}}=\Theta_{\mathrm{z}}+\Theta_{0}+\omega, \]

\[ \Theta_{\mathrm{e}}= \frac{bH_{\mathrm{v}}+f\left(\Theta_{\mathrm{r}}^{2}-\Theta_{0}^{2}-\omega^{2}\right)-M\Theta_{\mathrm{z}}} {2\omega f}, \]

Fig. 2. Schemes of multilateral wells:
a — when drilling an additional borehole with the aid of a wedge; b — with the wedgeless method of drilling an additional borehole (with the aid of a single-hinge deflector).
\(H\) — depth of the point of intersection along the vertical; \(X_{\mathrm{main}}, X_{\mathrm{add}}\) and \(y_{\mathrm{main}}, y_{\mathrm{add}}\) — the vertical and horizontal projections of the section \(OA\) of the main borehole and of the additional \(AB\); \(\Theta_{\mathrm{e}}, \Theta_i\) — zenith angles at the kickoff point of the additional borehole, respectively for the main and the additional borehole.

Substituting certain numerical values into these initial relations, we finally obtain:

\[ \Theta_{\mathrm{z}}= \frac{b\left(H_{\mathrm{r}}+bH_{\mathrm{v}}\right)-1.08A\left(\Theta_{0}^{2}+\omega^{2}\right)} {1+A^{2}}, \tag{13} \]

\[ \Theta_{\mathrm{r}}=\Theta_{\mathrm{z}}+\Theta_{0}+\omega, \tag{14} \]

\[ \Theta_{\mathrm{e}}= \frac{bH_{\mathrm{v}}+2.08\left(\Theta_{\mathrm{r}}^{2}-\Theta_{0}^{2}-\omega^{2}\right)-M\cdot\Theta_{\mathrm{z}}} {4.16\omega}, \tag{15} \]

where \(M=4.25-3.16A\),

\(\Theta_{\Sigma}\) — the total deviation of the additional borehole from the initial direction of the main borehole due to natural curvature (without taking into account deviation due to the wedge).

From the scheme of a multilateral borehole (Fig. 2, a), whose additional borehole is spudded by means of a wedge, the following conditions are quite obvious:

1) \(\Theta_3 > \Theta_0\) — spudding of the additional borehole (from the point of view of the trajectories of naturally curved boreholes) is completely impossible;

2) \(\Theta_3 = \Theta_0\) — the additional borehole turns into an independent borehole that has a common mouth with the previously drilled borehole (i.e., the “main” borehole);

3) \(\Theta_3 < \Theta_0\) — the additional borehole turns into a completely independent borehole.

In cases 2 and 3, profiling of the “additional” borehole must be carried out according to the procedure developed for a single borehole [4].

To determine the place for spudding the additional borehole, one can obtain a dependence from differential equation (1) by integrating it and solving with respect to the borehole length. After the indicated operations we shall have

\[ L_3=\frac{1}{b}\ln\left(\frac{A+\Theta_3}{A+\Theta_0}\right) =\frac{2.3}{b}\lg\left(\frac{A+\Theta_3}{A+\Theta_0}\right), \tag{16} \]

where \(L_3\) is the depth of wedge setting along the main borehole, in meters;
\(b\) is the intensity of curvature build-up, in \(1/\text{meter}\);
\(A, \Theta_0, \Theta_3\) are in degrees or radians.

Thus, profiling of the additional borehole reduces to determining, with the aid of a geological section, the vertical and horizontal distances from the intersection point to the mouth of the main borehole (i.e., \(H_v\) and \(H_g\)), determining by formulas (13), (14), and (15) the total deviation \((\Theta_{\Sigma})\), the zenith angle at the intersection point of the ore body \((\Theta_g)\), and the zenith angle of the main borehole at the point where the additional borehole is spudded \((\Theta_3)\), and then, by formula (16), the depth of wedge setting.

Profiling a multilateral shot borehole when drilling according to the “top-down” scheme (Fig. 2, b) can be reduced to the case just considered.

When drilling according to the “top-down” scheme, the additional borehole is spudded with a single-hinge deflector. Drilling practice with this deflector shows that with its aid one can achieve a curvature from 0.2 to 0.45 degrees/m, with an average value \(K = 0.3\) degrees/m, i.e., the borehole curves on average along a radius equal to \(190\) m.

For the case of spudding an additional borehole by means of a single-hinge deflector, it is important to know over what interval this deflector must be applied so that the additional borehole completely departs from the direction of the main borehole. To determine this interval we shall use Fig. 3. Directly from the figure it follows that the central angle \(\alpha\), corresponding to the interval being determined \(l_3\), will be equal to

\[ \alpha=\arccos\left(\frac{R-\dfrac{d}{2}}{R+\dfrac{d}{2}}\right),\ \text{degrees}, \tag{17} \]

where

\[ R=\frac{57.3}{K}=190\ \text{m} \]

is the radius of curvature of the well when drilling with a single-hinged deflector;

\(d\) is the diameter of the additional borehole.

Then the required interval \(l_3\) is determined as

\[ l_3=\frac{2\pi(R+d)\alpha}{360}=\frac{\alpha(R+d)}{57.3}. \tag{18} \]

Suppose that the final diameter of the additional borehole is equal to 112 mm; then, with \(R=190\ \text{m}\), the central angle will be equal to \(1^\circ 58'\), and the required interval for sidetracking the additional borehole is \(l_3=6.5\ \text{m}\).

Fig. 3. Diagram of sidetracking an additional borehole using a single-hinged deflector

Thus, as the calculations show, in most cases the magnitude of the sidetracking interval of the additional borehole is small in comparison with the remaining length of the additional borehole; therefore, with a small error, in profiling it may be assumed that the well also curves along the natural profile over the sidetracking interval, while the sidetracking itself is carried out by means of a wedge with a deflection angle

\[ \omega_{\mathrm{e}}=K\cdot l_3, \tag{19} \]

where \(K\) is the average curvature of the well when drilling with a single-hinged deflector, in degrees/m;

\(l_3\) is the length of the interval for sidetracking the additional borehole using a single-hinged deflector. It may be calculated by formulas (17), (18) or taken from experience.

It should also be noted that the larger the adopted “equivalent wedge deflection angle” \(\omega_{\mathrm{e}}\), the greater the depth from which the sidetracking of the additional borehole can be carried out and, consequently, the more economical multilateral drilling becomes. On the other hand, it must be borne in mind that the greater the curvature of the well over short sections, the more difficult the operating conditions for the drill pipes will be.

After determining the “equivalent skew angle” by the formula, further profiling calculations may be carried out as in the preceding case, i.e., by formulas (13)—(16).

REFERENCES

1. S. S. Sulakshin. The problem of directional drilling of exploratory wells. Izv. vuzov—Geology and Exploration, No. 5, 1962.
2. A. G. Kalinin. Curvature of Boreholes. Gostoptekhizdat, Moscow, 1963.
3. B. I. Spiridonov. On a method for drilling multilateral wells. Izv. vuzov—Geology and Exploration, No. 5, 1962.
4. V. P. Rozhkov. Some methods of profiling wells in shot drilling. Izv. TPI, vol. 151, 1966.