WELL PROFILING WITH DIFFERENT, BUT INTERVAL-WISE CONSTANT, CURVATURE

G. L. Kalinichenko, S. S. Sulakshin

(Presented by the scientific seminar of the Department of Engineering and Computational Mathematics)

Let us first consider the case of a well with constant curvature (intensity of curvature) \(K\). If the origin of coordinates is placed at the point of intersection of the well with the mineral deposit, then the equation of the trajectory of a well of constant curvature \(K\), that is, the equation of a circle of radius \(R=\dfrac{1}{K}\) (Fig. 1), will have the form

\[ (x-x_0)^2+(y-y_0)^2=R^2, \tag{1} \]

where \(x_0, y_0\) are the coordinates of the center of the circle,
\(x, y\) are the current coordinates of the circle.

We find the coordinates \(x_0, y_0\) of the center of the circle from the condition that the well passes through the origin of coordinates (the point of intersection of the bed with the well) and intersects the \(OX\) axis at an angle \(\eta_n\) at a certain point \(A_n\). Therefore,

\[ x_0=-R\sin\eta_n, \]

\[ y_0=R\cos\eta_n. \tag{2} \]

Substituting the found values \(x_0\) and \(y_0\) into equation (1), we obtain the final form of the equation of the trajectory of a well with constant curvature (intensity of curvature) \(K=\dfrac{1}{R}\):

\[ (x+R\sin\eta_n)^2+(y-R\cos\eta_n)^2=R^2. \tag{3} \]

Using this equation, we can construct the planned (or actual) profile of the well at the required scale.

To determine the coordinates \(x_1, y_1\) of the point \(A_1\) of well spudding, we use the condition that the ordinate \(y_1\) is equal to the vertical depth \(H\) of the well, i.e., \(y_1=H\). Then from equation (3) it is seen that

\[ x_1=\sqrt{R^2\sin^2\eta_n+2RH\cos\eta_n-H^2}-R\sin\eta_n. \tag{4} \]

The initial angle of well spudding \(\eta_1\) is determined from the condition that the angular coefficient of the tangent to the well at point \(A_1\) must

be equal to the value of the derivative \(\dfrac{dy}{dx}\) of function (3), evaluated at this point. Carrying out the corresponding calculations, we obtain

\[ \eta_1=\operatorname{arctg}\frac{\sqrt{R^2\sin^2\eta_n+2RH\cos\eta_n-H^2}}{R\cos\eta_n-H}. \tag{5} \]

Fig. 1.

Knowing the initial well-setting angle \(\eta_1\), the length \(L\) of the wellbore in meters is found as follows:

\[ L=R(\eta_1-\eta_n)=R\arcsin\left[\frac{\sqrt{R^2\sin^2\eta_n+2RH\cos\eta_n-H^2}}{R}-\eta_n\right]. \tag{6} \]

In the case when the initial inclination angle \(\eta_1\) of the well at point \(A_1\) has already been calculated by formula (5), the arc length \(L\) can be found from the simpler expression

\[ L=0.01745R|(\eta_1-\eta_n)|^\circ, \tag{7} \]

where \(\eta_1\) is the well inclination angle at the beginning (degrees);
\(\eta_n\) is the inclination angle at the end of the well (degrees).

The calculation of the well profile with different, but constant on separate intervals \(h_1, h_2, h_3, \ldots, h_{n-2}, h_{n-1}\), curvature intensities \(K_1, K_2, K_3, \ldots, K_{n-2}, K_{n-1}\), is carried out, using the formulas derived earlier, interval by interval—from bottom to top (Fig. 2).

Fig. 2.

The coordinates of the point \(A_1\)—the wellhead—will be \(x_1=S\) and \(y_1=H\), i.e. \(A_1(S,H)\). The abscissa \(x=S\) is determined as the sum of the projections of the intervals onto the \(y\)-axis

\[ S = S_1 + S_2 + S_3 + \cdots + S_{n-2} + S_{n-1} \tag{8} \]

or

\[ S = \sum_{i=1}^{n-1} S_i = \sum_{i=1}^{n-1} \left(\sqrt{R_i^2 \sin^2 \eta_i + 2R_i h_i \cos \eta_i} - R_i \sin \eta_i\right). \tag{9} \]

The length of the wellbore \(L\) along the axis is determined as the sum of the lengths of the intervals \(L_i\) by formula (6)

\[ L = L_1 + L_2 + L_3 + \cdots + L_{n-2} + L_{n-1} \tag{10} \]

or

\[ L = \sum_{i=1}^{n-1} L_i = \sum_{i=1}^{n-1} R \left[ \arcsin \frac{\sqrt{R_i^2 \sin^2 \eta_i + 2R_i h_i \cos \eta_i - h_i^2}}{R_i} - \eta_i \right]. \tag{11} \]

The initial inclination angle of the well \(\eta_1\) in this case is found by successively determining the angles at each point of the beginning and end of the intervals by formulas (5) and

\[ \eta_n = 180^\circ - (\gamma+\beta), \]

\[ \eta_{n-1}=\operatorname{arc\,tg}\left( \frac{\sqrt{R_{n-1}^{2}\sin^{2}\eta_n+2R_{n-1}h_{n-1}\cos\eta_n-h_{n-1}^{2}}} {R_{n-1}\cos\eta_n-h_{n-1}} \right) \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ \eta_2=\operatorname{arc\,tg}\left( \frac{\sqrt{R_2^{2}\sin^{2}\eta_3+2R_2h_2\cos\eta_3-h_2^{2}}} {R_2\cos\eta_3-h_2} \right), \]

\[ \eta_1=\operatorname{arc\,tg}\left( \frac{\sqrt{R_1^{2}\sin\eta_2+2R_1h_1\cos\eta_2-h_1^{2}}} {R_1\cos\eta_2-h_1} \right). \]

If, conversely, the initial inclination angle \(\eta_1\) is specified, then, solving the problem in the reverse sequence, one can find the inclination angle \(\eta_n\) of the well at the final depth \(H_n\).

When profiling a well with different curvature intensity \(K_i\) on different intervals \(h_i\), equations are compiled for each interval, having its own coordinate system, with the axes arranged parallel to the original system \(x_n, y_n\).

Then for the points \(A_n, A_{n-1}\) we shall have the coordinates \(A_n(0,0)\), \(A_{n-1}(S_n,h_n)\), where \(h_n=H_n-H_{n-1}\), and therefore \(A_{n-1}(S_n,H_n-H_{n-1})\).

The coordinates of the point \(A_{n-2}\) relative to the coordinate system \(x_{n-1}, y_{n-1}\) with origin at the point \(A_{n-1}\) will be \(A_{n-2}(S_{n-1},h_{n-1})\), where \(h_{n-1}=H_{n-1}-H_{n-2}\), and therefore \(A_{n-2}(S_{n-1},H_{n-1}-H_{n-2})\). The coordinates of this same point in the original system \(x_n, y_n\) will be \(A_{n-2}(S_n+S_{n-1};\, H_n-H_{n-2})\); and so on.

The coordinates of the point \(A_2\) relative to the system \(x_2,y_2\) will be \(A_2(S_1,H_1)\), and in the original system \(x_n,y_n\), respectively,

\[ A_2(S_n+S_{n-1}+S_{n-2}+\cdots+S_2;\, H_n-H_1), \]

or otherwise

\[ A_2\left(\sum_{i=2}^{n} S_i;\, H_n-H_1\right). \]

Finally, the coordinates of the point \(A_1\) in the original system will be

\[ A_1(S_1,H_n), \]

or otherwise

\[ A_1\left(\sum_{i=n}^{n} S_i;\, H_n\right). \]

Obviously, on any \(p\)-th section of the well with the given curvature intensity

\[ K_p=\frac{1}{R_p} \]

the trajectory equation in the \(p\)-th local system will have the form:

\[ (x+R_p\sin\eta_p)^2+(y-R_p\cos\eta_p)^2=R_p^2. \]

In the original coordinate system this equation is written as:

\[ \left(x+R_p\sin\eta_p-\sum_{i=1}^{p}S_i\right)^2+ \left(y-R_p\cos\eta_p-H_n+H_p\right)^2=R_p^2; \tag{12} \]

or, explicitly,

\[ y_p=\pm\sqrt{R_p^2-\left(x+R_p\sin\eta_p-\sum_{i=1}^{p}S_i\right)} +R_p\cos\eta_p+H_n-H_p. \tag{13} \]

Then the trajectory of the entire borehole, consisting of \(n\) intervals, is determined by a system of \(n\) equations of the form

\[ y = \begin{cases} R_n \cos \eta_n - \sqrt{R_n^2 - (x + R_n \sin \eta_n)^2}, & \text{if } 0 \leqslant x < S_n; \\[6pt] R_{n-1}\cos \eta_{n-1} + H_n - H_{n-1} - \sqrt{R_{n-1}^{2} - (x + R_{n-1}\sin \eta_{n-1} - S_n)^2}, & \\[-2pt] \qquad \text{if } S_{n-1} \leqslant x < S_n + S_{n-1}; \\[6pt] R_{n-2}\cos \eta_{n-2} + H_n - H_{n-2} - \sqrt{R_{n-2}^{2} - (x + R_{n-2}\sin \eta_{n-2} - S_n - S_{n-1})^2}, & \\[-2pt] \qquad \text{if } S_n + S_{n-1} < S_n + S_{n-1} + S_{n-2}; \\[6pt] \ldots \\[-2pt] \ldots \\[6pt] R_2\cos \eta_2 + H_n - H_2 - \sqrt{R_2^2 - (x + R_2\sin \eta_2 - S + S_1 + S_2)^2}, & \\[-2pt] \qquad \text{if } S - S_1 - S_2 \leqslant x < S - S_1; \\[6pt] R_1\cos \eta_1 + H_n - H_1 - \sqrt{R_1^2 - (x + R_1\sin \eta_1 - S + S_1)^2}, & \\[-2pt] \qquad \text{if } S - S_1 \leqslant x < S. \end{cases} \]

With the aid of these equations, the coordinates of any number of points on the borehole axis are found, from which its profile can be constructed on a geological cross-section.

Such a problem is readily carried out by means of a calculating machine using a floating-point program.