Extrapolation of the Zenith Curvature of Exploratory Wells

V. P. Rozhkov, D. A. Ucharov

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

In the practice of exploratory drilling, cases are not uncommon in which the exact position of the upper part of a well is known, while the position of the lower part is unknown, since it is not possible to measure the curvature parameters because of the small diameter of the well or because of an accident that has occurred at the lower horizon. In such cases, geological exploration parties (GEPs) use various, sometimes entirely unfounded, methods for extrapolating the zenith curvature at lower horizons.

When problems of this kind arise, some geological exploration parties assume that in the lower part the well is not curved at all and that its direction coincides with the direction of the very lowest measured interval of the well; others assume that the well is curved with an intensity equal to the average intensity of curvature over the area or over the deposit as a whole (the Tashtagol GEP of the West Siberian Geological Administration); still others compute the average intensity of curvature from the measured part of the borehole and then assume that, in the unmeasured part, the borehole is curved on average with this intensity (the Karbolikha GEP of the West Siberian Geological Administration).

All the methods listed above cannot be regarded as satisfactory, since they either do not use at all the regularities governing well curvature, or use them only partially. Thus, the second and third methods do not take into account that the intensity of zenith curvature of core-drilling wells varies with depth. Moreover, in the second method the adopted well curvature is highly averaged and does not take into account local causes of curvature (primarily geological ones), but reflects only features characteristic of the entire deposit (or area) as a whole. The third method statistically takes these local geological causes into account, but it still does not account for the variability of curvature intensity.

Thus, there arises a need to develop a method for determining the most probable trajectory of the unmeasured part of the borehole, taking into account the general regularities of well curvature, as well as the shape and numerical characteristics of the measured part of the well being extrapolated.

The direction and intensity of curvature of core-drilling wells, in general terms, can be determined as functions of a number of geological and technical causes that bring about curvature. A well—

wells drilled in hard rocks, as a rule, flatten out, tending to assume a position perpendicular to the plane of bedding. The intensity of curvature is greater the more frequent the alternation of hard and soft rocks and the higher their anisotropy. In addition, the intensity depends on the type of core drilling.

Analysis of the intensity of well curvature by numerical methods has made it possible to establish that the dependence of curvature on the zenith angle is very close to linear and, for shot drilling, can be expressed by the equation

\[ \kappa=\frac{d\Theta}{dL}=a+b\Theta, \tag{1} \]

while for hard-alloy drilling it remains practically constant at all depths and for all values of the zenith angle [1], i.e.,

\[ \kappa=\frac{d\Theta}{dL}=\eta, \tag{2} \]

where \(\kappa\) is the curvature of the well, representing the increment of the zenith angle over an interval of one meter;
\(\Theta, L\) are, respectively, the zenith angle and the depth of the well;
\(a, b, \eta\) are constant coefficients.

Since the radius of curvature of a well is inversely proportional to the curvature, it follows from equations (1) and (2) that wells of hard-alloy drilling curve, on average, along the arc of a circle, whereas wells of shot drilling curve along trajectories whose radii of curvature decrease with increasing depth; in other words, they flatten out with increasing intensity.

Thus, from the general relationships of zenith curvature (1) and (2), it follows that extrapolation of hard-alloy-drilled wells is performed correctly only when the third method is applied, and that not a single method is applicable to wells drilled by shot drilling.

The most correct extrapolation procedure may be considered to be one based on relationships (1) and (2), which reflect the form of the well in general terms depending on the type of drilling, while the specific intensity of curvature would be taken into account by the values of the coefficients \(a, b, \eta\), computed from the measured part of the well and reflecting the local drilling conditions. Of course, this is justified only in the case when the measured and unmeasured parts of the well intersect approximately analogous structures, and the length of the measured part is several times greater than the length of the extrapolated part.

For direct use of the general regularities of curvature of wells drilled by hard-alloy drilling, one may use the equation obtained by integrating expression (2):

\[ \Theta_i=\Theta_0+cL_i, \tag{3} \]

where \(\Theta\) is the zenith angle of the well collar.

Integration of expression (1) leads to the comparatively complex relation

\[ \Theta_i=\left(\frac{a}{b}+\Theta_0\right)e^{bL}-\frac{a}{b}, \tag{4} \]

for the use of which it is necessary to have special tables; therefore, in direct calculations of the trajectories of shot-drilled wells of small depth, one may use the modified original equation

\[ \Delta \Theta_i=(a+b\Theta_0)\cdot \Delta L_i, \tag{5} \]

and the calculations are to be carried out according to the scheme

\[ \begin{gathered} \Theta_0;\\ \Theta_1=(a+b\Theta_0)\Delta L_1+\Theta_0;\\ \Theta_2=\Theta_1+(a+b\Theta_0)\Delta L_2;\\ \Theta_i=\Theta_{i-1}+(a+b\Theta_0)\Delta L_i. \end{gathered} \]

If a diamond-drilled borehole has a great depth, it is desirable to use equation (4) or to use nomograms intended for profiling diamond-drilled boreholes.

It follows from the foregoing that the success of extrapolation depends on the correctness of finding the corresponding coefficients in equations (3)—(5). For hard-alloy drilling, the coefficient \(\eta\) can be determined by direct division of the increment of the zenith angle by the interval over which this increment occurred. The coefficients \(a\) and \(b\) are determined somewhat more complicatedly. Practical calculations for several dozen boreholes have established that, for their determination, the following methods may be used: 1) the taut-string method, 2) the method of averages, and 3) the least-squares method.

For any of these methods it is necessary to process the measured part of the borehole in the form given in Table 1, and then, from the results of the processing, it is desirable to construct a graph of the dependence of curvature on the zenith angle.

1. The taut-string method is based on fitting a straight line by eye. Having constructed a graph of the dependence of curvature on the zenith angle, it is necessary graphically to choose the straight line that best fits the observed points. Having selected two arbitrary points on this straight line, their coordinates \((\Theta_1, K_1)\) and \((\Theta_2, K_2)\) are determined. Then, for determining the coefficients \(a\) and \(b\), we shall have two linear equations

\[ \left. \begin{aligned} a+b\Theta_1&=K_1,\\ a+b\Theta_2&=K_2. \end{aligned} \right\} \tag{6} \]

To determine the approximate accuracy of the proposed methods, we shall use data from measurements of the curvature of actual boreholes and shall assume that the lower part of the boreholes is as yet unknown to us, and extrapolate by these methods, and then compare the extrapolation results with the actual measurements.

Thus, Table 1 gives actual measurements of the zenith curvature of a diamond-drilled borehole (borehole No. 597 of the Salair ore field) and the results of extrapolation over the interval 160—200 m when using the taut-string method. The values of the coefficients turned out to be \(a=-0.028\), \(b=+0.00915\). It is seen from the table that the calculated data are close to the actual ones. However, the success of extrapolation by this method depends to a large extent on the ability to draw correctly the mean line on graphs of the dependence of curvature on the zenith angle, and therefore it is best applied to boreholes with smooth curvature.

1. The method of averages consists in the following. The actual curvature of a borehole always contains errors of curvature measurements and, in addition, takes into account the nonuniformity of curvature. The deviation of the actual curvature from the mean on each interval can be calculated by the formula \(\Delta_i=K_i-(a+b\Theta_i)\), for \(i=1, 2, 3\ldots n\). \(\tag{7}\)

The parameters \(a\) and \(b\) in this method must be calculated so that the sum of the errors with their signs is equal to zero, i.e. \(\sum_{i=1}^{n}\Delta i=0\).

From this condition only one equation can be composed, whereas two coefficients must be found. Therefore it is advisable to perform the equation—

Table 1

Extrapolation of the zenith curvature of well No. 597
of the Salair ore field

Discrepancy between the actual and extrapolated bottom holes:

a) by the proposed method — 0.02 m,
b) by the method of the Karbolikhinskaya GRP — 0.86 m.

—not for all curvature values as a whole, but for two groups, each containing approximately half of the observations. This gives a system of equations

\[ \sum_{i=1}^{m} \kappa_i-(a+b\Theta_i)=0,\qquad \sum_{i=m}^{n} \kappa_i-(a+b\Theta_i)=0, \]

where

\(m\) — number of measurements in the first group;
\(n\) — number of curvature measurements over the whole well.

Directly for determining the coefficients, the system may be written as follows:

\[ ma+b\sum_{i=1}^{m}\Theta_i=\sum_{i=1}^{m}\kappa_i, \]

\[ (n-m)a+b\sum_{i=m+1}^{n}\Theta_i=\sum_{i=m+1}^{n}\kappa_i. \]

Table 2

Extrapolation of the zenith curvature of well No. 189 of the Tashtagol deposit with determination of curvature parameters by the mean method

As an example, Table 2 gives the extrapolation of well No. 189 of the Tashtatol’skoye field in the interval 500–640 meters. The values of the curvature parameters determined by the method of averages proved to be \(a=-0.0404\) and \(\varepsilon=+0.00426\).

It is seen from Table 2 that, when the curvature parameters are determined by the method of averages, the assumed trajectory of the unmeasured part passes much closer to the actual one than by the method of Karbolikhinskaya GRP, and the proposed method is quite simple.

1. The method of least squares is based on the fact that the sum of the squares of the deviations of the actual curvature values from the mean values (in absolute magnitude) must be a minimum. This method is widely known [2], [3], and therefore is not set out here. We note only that this method gives the most accurate values of all the proposed methods for determining the curvature parameters of wells, but it requires a somewhat larger number of computations; therefore this method should expediently be applied to wells with considerable variability of zenith curvature.

Conclusions

1. The methods of extrapolating zenith curvature currently used by exploration organizations, as a rule, do not reflect the regularities of curvature of exploratory wells.

2. The proposed extrapolation methods are based on the general regularities of curvature of exploratory wells and take account of local geological drilling conditions. The method of averages should be regarded as the most acceptable in terms of simplicity and accuracy. The taut-string method is expediently applied when the curvature of the well is sufficiently smooth, and the method of least squares, conversely, when there is considerable instability of zenith curvature.

3. The developed methods are applicable if the well intersects approximately analogous structures and the length of the measured part is several times greater than the length of the part being extrapolated.

References

1. S. S. Sulakshin. Curvature of Wells and Methods of Its Measurement. 2nd ed. Gostoptekhizdat, Moscow, 1960.

2. R. S. Guter and B. V. Ovchinskii. Elements of Numerical Analysis and Mathematical Processing of Experimental Results. Fizmatgiz, Moscow, 1961.

3. B. P. Demidovich, I. A. Maron, and E. Z. Shuvalova. Numerical Methods of Analysis. 2nd ed. Fizmatgiz, Moscow, 1963.