ON THE MATHEMATICAL DESCRIPTION OF TRAJECTORIES OF EXPLORATORY SPATIALLY CURVED BOREHOLES

V. P. ROZHKOV

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

All boreholes are, to one degree or another, subject to curvature even when special precautionary measures have been taken during drilling. Therefore, in many exploration organizations, along with the use of measures to reduce curvature, boreholes have begun to be profiled with allowance for their future natural trajectories. The empirical methods used for this purpose in most cases do not sufficiently take into account the regularities of curvature and contain assumptions not confirmed by practice. In this connection, several works have appeared in the specialized literature [3, 5, 6, 11] devoted to the mathematical description of curved boreholes. In some works the problem posed is solved by analytical methods, in others by statistical methods. Since the methods of analysis determine the final result to a known extent, it is necessary to examine these approaches in greater detail and to determine the areas in which they may be used in studying borehole curvature.

Let us first note that borehole curvature is influenced by a large number of diverse factors. According to the nature of their action, as was shown earlier [4, 7], they are divided into two groups. The first includes factors that regularly change the overall curvature (twisting) of boreholes, while the second comprises factors that distort the regular curvature.

In the work of B. Z. Sultanov [11] an attempt is made to determine the trajectory of boreholes analytically. It is quite clear that, with such an approach, it is impossible to take all curvature factors into account. Therefore, the work considers only part of the factors of the first group and completely disregards the factors of the second group. Using this work as an example, one can see how difficult it is, in an analytical approach, to relate curvature factors to the trajectory of a borehole. It is not accidental that, in deriving formulas for the borehole axis, the work uses the assumption that it bends in zenith and azimuth in the form of a parabola. Meanwhile, the subject of this assumption is the objective of the entire work.

We shall note in passing that in the work of B. Z. Sultanov [11] errors of another kind, no less significant, are also made. Thus, in deriving the formulas, it is essentially assumed that the axis of the drilling assembly is constantly located in one and the same plane in the maximally tilted state, while the cause of borehole curvature in all cases is the contacts of hard and soft rocks (moreover, ideally even ones). Similar

idealization are admissible only for approximate qualitative analysis; however, in the work we find that the derived “formulas will make it possible to calculate the deviation of a borehole from a specified direction.” Let us cite one of these formulas. The horizontal displacement of a borehole in shot drilling is recommended to be determined from the relation

\[ I=\frac{k f \pi \cdot \sigma_{\mathrm{comp}}\cdot h_{\min}\cdot \delta \cdot R\,(D_{\mathrm{b}}-D_{\mathrm{n}})\cdot H^2} {2(EI)\cdot \tan(\gamma-\Theta_0)\cdot n\cdot L}, \tag{1} \]

where \(k\) is a correction coefficient (its value is not determined in the work); \(f\) is the coefficient of interbedding of the rocks of the geological section; \(\sigma_{\mathrm{comp}}\) is the temporary compressive strength of the rock, averaged for the geological section; \(h_{\min}\) is the averaged minute drilling rate; \(\delta\) is the diameter of the drilling shot; \(R\) is the distance between the reactive forces acting on the bit from the side of the hard and soft rocks when rocks of different hardness meet at the contact; \(D_{\mathrm{b}}, D_{\mathrm{n}}\) are the diameters of the borehole and of the core barrel; \(H\) is the interval for which the displacement of the borehole relative to the vertical is determined; \((EI)\) is the rigidity of the drill-string column; \(\gamma\) is the dip angle of the rocks; \(\Theta_0\) is the zenith angle of the borehole collar; \(n\) is the number of revolutions of the drilling assembly; \(L\) is the half-wavelength of the bent column.

According to the author of the work being analyzed, “the area of practical application of formula (15)* is very extensive.” However, it is easy to see that some of the parameters of this formula are very difficult to determine quantitatively (\(L, R, k, f, \sigma_{\mathrm{comp}}\)) or have a variable value (\(L, R, n, D_{\mathrm{b}}, D_{\mathrm{n}}, EI\)), and it is unknown how average values are to be determined. If one also recalls the assumptions made in deriving it, one may conclude that relations such as (1) have no practical significance.

In assessing the possibilities of the analytical approach, it may be noted that analytical relations cannot take into account distorting factors (since the latter in most cases do not lend themselves to quantitative evaluation), and, unless one resorts to considerable cumbersomeness of the final relations, part of the factors of the first group will not be taken into account. In addition, some factors of this group are also difficult to determine quantitatively. The choice of the most essential factors presents significant difficulties, since under some conditions they may be one set, and under other conditions another.

Thus, the analytical approach may serve as one of the principal means for the qualitative study of the process of borehole curvature [2, 7, 9], but not as a means of determining borehole trajectories for quantitative calculations.

The first attempt to determine the trajectory of boreholes quantitatively solely by methods of statistical analysis apparently belongs to N. O. Yakobi [13]. Because of the complexity of the spatial trajectory, he analyzed only the zenith curvature. To determine the borehole profile from measurements of zenith curvature, horizontal \((X)\) and vertical \((Y)\) projections of the borehole were constructed. Such projections (in contrast to projections constructed with account of azimuthal curvature) were called by him modified projections. Studying the modified projections, N. O. Yakobi came to the conclusion that, owing to zenith curvature, “...inclined shot boreholes have a tendency to form lines of the hyperbola type” and can be described by the equation

\[ X=\frac{Y}{a+bY}, \tag{2} \]

* In the present article, this is formula (1).

where \(a\) and \(b\) are empirical coefficients determined for each borehole being analyzed.

Formula (2) at present is of scientific significance only. Its practical value is small, since it does not take into account the causes of curvature and was calculated on the basis of an analysis of individual boreholes.

For profiling purposes it is important to determine not the profile of several specific boreholes, but, from a large number of measurements of curvature, to identify an averaged trajectory characteristic of a particular deposit (section). In this connection, at the Department of Exploration Engineering of the Tomsk Polytechnic Institute, work began several years ago on developing a methodology for the statistical study of the causes and regularities of borehole curvature. S. S. Sulakshin proposed a general statistical analysis [8, 9, 10] of curved boreholes, which includes the study of the frequency of deviation of intervals or of the borehole as a whole (in zenith and azimuth), determination of borehole deviations (in zenith and azimuth) as a function of depth and zenith angle, determination of average curvature over individual intervals, etc. Individual types of statistical relationships from this analysis may serve as the basis for determining the averaged trajectory of boreholes in a deposit.

A. A. Sorokin [5, 6], in determining the averaged trajectory, took as the basis the statistical dependence of the increment of zenith and azimuthal angles on depth. Later this dependence (though only for zenith angles) was used by A. M. Kurmashev [3]. As a result of the mathematical processing of the aforementioned statistical relationship, they obtained dependences in the form of a polynomial

\[ X = \sum C_n L^n \quad \text{for } n = 1, 2, 3; \tag{3} \]

where \(X\) is the increment of the magnitude of the zenith (or azimuthal) angle from the initial one; \(L\) is the borehole depth; \(C_n\) are empirical coefficients.

Since the magnitude of the increment according to formulas of type (3) does not depend on the setting angle, the authors recommend dividing all boreholes in a deposit into groups with similar zenith collaring angles and determining their own empirical coefficients \((C_n)\) for each group.

Dependences of type (3) are difficult to use for practical purposes without resorting to cumbersome calculations or simplifications. Thus, if one begins profiling the borehole from the point of intersection and determines the collaring location and the zenith collaring angle (and this is much more correct than profiling from the collaring point), then the problem can be solved either by trial calculations for several groups of coefficients \((C_n)\), or by selecting the most suitable standard profile, which must be constructed for each group of boreholes. In the second case, only the collaring location is determined unambiguously, while the zenith collaring angle has to be taken within those limits of angles for which the profile used was constructed.

Thus, the choice of the initial dependence in works [3, 5, 6] was made unsuccessfully.

On the whole, the statistical approach is the only correct path toward determining the profile of curved boreholes, since it makes it possible to obtain fairly simple finite dependences that take into account most factors of both groups. In addition, with the aid of statistical dependences one can determine the limits of applicability of directional drilling, based on the regularities of curvature, in the following way. Boreholes specified in one direction and drilled under similar conditions follow not one and the same trajectory, but form—

form a curved cone of dispersion [11, 12]. The cross sections of this cone are easily determined by taking into account the root-mean-square deviations of the empirical coefficients in the final relations. Comparison of the cone sections with the exploration grid makes the aforementioned assessment possible. It should also be noted that, in order to increase the effectiveness of the statistical method, it is necessary to make the proper choice of parameters as functions of which the curvature is to be considered.

Obviously, the more complete our knowledge of the causes and regularities of borehole curvature, and the more extensively curved boreholes are studied (cavernometry, oriented core, etc.), the more objectively these parameters can be chosen. Therefore, as the natural curvature of boreholes is studied, they will change, and it is impossible to recommend one parameter or another for permanent use. We shall dwell only on general considerations for their selection.

Each parameter must be chosen so that it includes, as far as possible, a larger number of unit factors of curvature (i.e., is the result of the action of, or conditions the appearance of, such factors) and at the same time has a simple quantitative characteristic. Such a parameter will be called a complex one. By a unit factor is meant a factor that takes into account some narrow area of phenomena (for example, the diameter of the drill bit, the diameter of the shot, etc.). In this interpretation, a complex parameter is an intermediate link between the sum of all the causes (factors) of curvature and the result of their action.

Curvature is ultimately conditioned by the skewing of the drill-string assembly in the borehole. The direction and intensity of curvature depend on the angle between the axis of the drill-string assembly and the axis of the near-bottom section of the borehole, as well as on the position of the plane passing through these axes. These, in essence, are the most complex factors of borehole curvature, but they cannot be adopted for practical purposes, since the assembly in the borehole constantly vibrates, changes its position and the magnitude of its skew; as a result, it is not possible to predict the average skew and the plane in which it will occupy the predominant position.

With modern methods for measuring curved boreholes and with the existing degree of study of the causes of curvature, the zenith angle or depth may be adopted as a complex factor. The zenith angle should be considered the most general, since the magnitude of the zenith angle of the near-bottom section determines the forces that deflect the drilling assembly toward the lower wall, and partly the orientation of the near-bottom section relative to the line of least resistance of the anisotropic rock being drilled, whereas only the result of curvature is associated with depth.

With further study of the form of the trajectories of exploratory boreholes, the complex factors may be taken to be, for example, simultaneously the borehole diameter and the zenith angle, or the borehole diameter, the zenith angle, and the angle between the strike of the main structures of the deposit and the azimuthal direction of the borehole, etc.

The forms of statistical analysis may be the most varied. Thus, for example, by means of correlation analysis one may determine the quantitative interrelationship between the curvature parameters and the adopted complex factors of curvature, or use the theory of stochastic processes. In principle, it is possible to apply the theory of Markov chains [1], since the drilling assembly drills the borehole, which then becomes a guide channel for it. In other words, the future position of the borehole is determined mainly

the present position of its near-bottom section and does not depend on the position of this section in the past.

The statistical approach proves still more effective if the dependences of the borehole-curvature parameters on the selected composite factors are determined in differential form. Integration of these dependences and the use of boundary conditions make it possible to obtain particular formulas for various borehole-drilling conditions with the smallest amount of computational work.

As a result of statistical study, dependences are obtained between the curvature parameters and the adopted composite factors, as well as constants entering into these dependences. The magnitudes of the constants and their probable deviations from the mean values will take into account all factors not organically included in the adopted composite factors. It is quite obvious that the more fully the individual factors are taken into account by the selected composite factors, the more accurately the averaged trajectory of curved boreholes is described and the smaller the mean-square deviations of the constants; as a result, the theoretical cone of dispersion approaches the natural one. The latter is due to the fact that, in the statistical approach to studying borehole trajectories, the theoretical cone of dispersion takes into account not only the most probable actual dispersion of borehole shafts, but also, in part, the influence of individual factors not accounted for by the selected composite factors.

Further refinement of borehole trajectories can be achieved by increasing the number of composite factors or by seeking more general factors. The second path is obviously more promising, since it will make it possible to obtain simpler dependences. Let us consider the determination of the trajectory of an averaged naturally curved borehole by determining the correlation relation between the zenith angle and the curvature of boreholes drilled in one and the same field. First, let us make a remark on terminology and on the method of separately determining borehole deviations by zenith and azimuth.

Any averaged trajectory of boreholes can be represented by a line in a vertical surface curved in accordance with the horizontal projection. We shall call such a surface apsidal, by analogy with the apsidal plane [7, 9]. The apsidal plane will always touch the apsidal surface along a vertical line. The proposed representation of an averaged borehole trajectory promotes a simpler approach to their mathematical description. In this case, the study of the spatial curvature of boreholes is completely replaced by the study of their horizontal projections and by the study of curvature in the apsidal surface.

The mathematical description of spatially curved boreholes should begin with finding the law of curvature in the apsidal surface, since the length of the horizontal projections (and, consequently, the curvature) depends on the curvature in the apsidal surface.

The mathematical description of the averaged trajectory is obtained most simply if the overwhelming majority of the boreholes analyzed do not have characteristic bends of the shaft confined to definite horizons, but rather increase (or decrease) their zenith angle sufficiently smoothly.

Having, for such boreholes, data on the magnitudes of the zenith angles, one can find the empirical dependence

\[ K_a=\frac{\Theta_{i+1}-\Theta_i}{l_{i+1}-l_i} =\frac{\Delta\Theta_i}{\Delta l_i} =f_1(\Theta_i), \tag{4} \]

where \(l_i,\, l_{i+1},\, \Delta l_i\) are, respectively, the depth of the \(i\)-th and \((i+1)\)-th points of the borehole and the distance between these points; \(\Theta_i,\, \Theta_{i+1},\, \Delta\Theta_i\) are the zenith angle

at the \(i\) and \(i+1\) points of the borehole and the increment of the zenith angle between these points; \(K_a\) is the average curvature of the borehole between the measurement points.

If the increment \(\Delta l\) is taken to be small (in comparison with the length of the borehole), and if, from the empirical tabular dependence (4), by known methods one constructs an empirical formula \(F_1(\Theta)\) that would best describe the empirical data, then dependence (4) can be represented in the form of the differential equation

\[ K_a=\frac{d\Theta}{dl}=F_1(\Theta). \tag{5} \]

By integrating equation (5), it is easy to obtain the relationship between the borehole depth and the zenith angle. The vertical and horizontal projections of the borehole can be determined from the equations

\[ dz=dl\cdot\cos\Theta, \]

\[ ds=dl\cdot\sin\Theta, \]

known from differential geometry, if in them the value of the borehole length element is substituted from expression (5). After the corresponding substitution we obtain the differential equations

\[ dz=\frac{\cos\Theta}{F_1(\Theta)}\,d\Theta, \tag{6} \]

\[ ds=\frac{\sin\Theta}{F_1(\Theta)}\,d\Theta, \tag{7} \]

and, after their integration, the desired projections (Fig. 1, a):

\[ Z=\int_{\Theta_0}^{\Theta_i}\frac{\cos\Theta}{F_1(\Theta)}\,d\Theta, \tag{8} \]

\[ S=\int_{\Theta_0}^{\Theta_i}\frac{\sin\Theta}{F_1(\Theta)}\,d\Theta. \tag{9} \]

where \(\Theta_0,\ \Theta_i\) are the zenith angle of the spud-in point and the current zenith angle of the borehole.

If the boreholes have no curvature in the horizontal plane, then their trajectories are completely described by equations (8) and (9). In the case of spatial curvature, the horizontal projection of the borehole is curvilinear, and it can be resolved into two mutually perpendicular axes located in the horizontal plane (Fig. 1, b). The vertical projection of the borehole depends only on curvature in the apsidal surface; therefore expression (8) is valid simultaneously for plane-curved and spatially curved boreholes.

The differential law of azimuthal curvature must contain the differential of the azimuthal direction, the function of variation of this differential as a function of the zenith angle, and the differential of the zenith angle. The relation between these quantities can be expressed by the following simple relation

\[ \frac{d\alpha}{d\Theta}=F_2(\Theta). \tag{10} \]

The empirical analog of equation (10) can be written in the form

\[ \frac{\Delta x_i}{\Delta \Theta_i} = \frac{a_{i+1}-a_i}{\Theta_{i+1}-\Theta_i} = f_2(\Theta). \tag{11} \]

Imposing analogous conditions on this dependence and performing the same operations as with expression (4), we obtain the differential equation (10).

The law of the azimuthal turn of the borehole can be determined by solving the integral

\[ \alpha=\int F(\Theta)\,d\Theta, \tag{12} \]

and the magnitude of the total azimuthal turn by

\[ \alpha_{\mathrm{п}}=\int_{\Theta_0}^{\Theta_i} F_3(\Theta)\,d\Theta . \tag{13} \]

Fig. 1. Diagram of the projections of a curved borehole onto coordinate planes: a — for a plane-curved borehole; b — for a spatially curved borehole.

The projections of the borehole onto the horizontal axes can be determined analogously to expressions (8) and (9). For this purpose we shall use the same relations from differential geometry. As applied to the case under consideration, they are written in the form

\[ dx = ds\cdot \cos \alpha, \]

\[ dy = ds\cdot \sin \alpha. \]

Substituting into these initial expressions the value of \(ds\) from expression (7), and the law of the azimuthal turn of the borehole from equation (12), and integrating them, we obtain the projections of the borehole onto the horizontal axes

\[ X=\int_{\Theta_0}^{\Theta_i} \frac{\sin \Theta}{F_1(\Theta)} \cos \left[\int F_3(\Theta)\,d\Theta\right]\,d\Theta, \tag{14} \]

\[ Y=\int_{\Theta_0}^{\Theta_i} \frac{\sin \Theta}{F_1(\Theta)} \sin \left[\int F_3(\Theta)\,d\Theta\right]\,d\Theta. \tag{15} \]

In the event that the integrands in relations (8), (9), (14), and (15) prove to be very cumbersome and integration is very difficult or altogether impossible, then in the integrands the functions \(F_1(\Theta)\) and \(\int F_3(\Theta)\,d\Theta\) must be replaced by tabular empirical dependences \(f_1(\Theta)\) and \(f_4(\Theta)=(a_i-a_0)\), and from them the tabular values of the integrand should be computed. After this, from the tabular values an empirical formula can be constructed so that it describes the empirical data in the best possible way. This formula must be substituted in place of the integrand, and the integral solved. The empirical formula should be chosen in such a form that it is easy to integrate. As experience in processing empirical data has shown, dependence (5) most often represents the equation of a straight line or a parabola, while dependence (10) is an equation of exponential form. In such cases no difficulties arise in integrating expressions (8), (9), (12), and (13). To solve equations (14) and (15), the above-described substitution should be carried out.

Thus, in order to obtain the projections of a spatially curved borehole, it is sufficient to determine the empirical dependences by expressions (4) and (11), and then, from them, to proceed to the empirical formulas (5) and (10). Substituting these empirical formulas into relations (8), (14), (15) and integrating over the indicated limits, we obtain the required projections. In the case of processing plane-curved boreholes, the calculations are limited to relations (4), (5), (8), and (9). The advantages of the dependences obtained by this method have been described by us earlier [4].

The dispersion cone of naturally curved boreholes can be determined by taking proper account of the possible deviations of the constants entering equations (8), (9), (14), and (15). Comparison of the transverse dimensions of the cone with the dimensions of the exploration grid will make it possible to determine the area of effective application of directional drilling using natural curvature at the deposits of interest.

Conclusions

1. For the purposes of a mathematical description of naturally curved boreholes, analytical (“dynamic,” “deterministic”) dependences are unsuitable for many reasons. A sufficiently accurate description can be achieved by identifying complex curvature factors and studying borehole trajectories statistically.

2. As a first approximation, the zenith angle or the zenith angle of the borehole and the depth may be taken as complex parameters simultaneously; however, in that case priority should be given to the zenith angle of the borehole.

1. In order to determine more general equations for borehole trajectories and to reduce the volume of computational work, the dependence between the curvature parameters and the complex factors should be sought in differential form.

2. The developed methods for describing the trajectories of naturally curved boreholes make it possible to determine with sufficient accuracy the profiles of averaged boreholes in deposits, which in turn may serve as a basis for more precise design of plane- and spatially curved boreholes.

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