Natural parameter method accurately calculates well bore trajectory

Liu Xiushan, Shi Zaihong, Fan Sen
Daqing Petroleum Institute
Anda, China

• Equations [80823 bytes]

This model, the natural parameter method, has several advantages over other models:

• The model is simple and relatively reasonable.

• The calculated results are stable.

• It is convenient for oil field application.

Describing and calculating a well bore trajectory is fundamental, necessary work for vertical, directional, and horizontal wells. Inclinometers can only give parameters from the separate survey points and cannot give the real profile of the trajectory.

There are more than 20 methods for describing and calculating a drilled well bore trajectory. These methods can be summed up into nine categories: tangential method, balanced tangential method, average angle method, rectified average angle method, radius of curvature method (or cylinder helix method), minimum curvature method, chord step method, numerical integral method, and curve and structure method.^1-5

All these methods, except the tangential method which is no longer used because it shows considerable error, are used routinely by drilling engineers. The numerical integral method and curve structure method yield the most accurate results.^{4 5}

The radius of curvature method is more suitable for describing a well bore trajectory drilled in the rotary mode. The minimum curvature method is more suitable for describing the sections drilled in directional drilling mode.⁶

In drilling engineering, interpolation calculation is frequently used for monitoring and controlling the trajectory. The most convenient way to calculate the inclination and azimuth at any depth by interpolation and extrapolation is to assume the rates of inclination change and azimuth change in each course are individually constant.

Based on the same assumption, a new method of survey calculation for simulating an actual well bore trajectory is proposed. This method, the natural parameter method, is suitable for both rotary drilling and directional drilling, compared to the radius of curvature method and minimum curvature method.

Model description

The natural parameter method uses the inclinations and azimuths at the two survey stations of the course and assumes the rates of inclination change and azimuth change remain individually constant over the course length. Because the inclination and direction at various depths (stations) can be measured by using some type of surveying instrument, from the fundamental assumption above, the rates of inclination change and azimuth change can be calculated by Equations 1 and 2.

According to the definitions of rate of inclination change and rate of azimuth change, the inclination function and azimuth function vs. measured depth, L, can be expressed by Equations 3 and 4.

If a vertical plot and a horizontal plot are used to show the well bore trajectory described by Equations 3 and 4 in a vertical expansion plane and a horizontal plane, respectively, the vertical plot will be a linear section or an arc section, but the horizontal plot will be an arc only if the well bore trajectory is a holding section.

In mathematical theory of curves, the parameter standing for the length of a curve is defined as natural parameter. Thus, this method of calculating trajectory is termed the natural parameter method.

Course coordinates

From the well bore trajectory differential model, it is not difficult to derive formulas which can be used to calculate the course coordinates of a survey interval (Equations 5-8).^7-9

Based on the fundamental assumption of the natural parameter method, a(L) and c(L), which are determined by Equations 3 and 4, are linear functions vs. measured depth, L. Substituting Equations 3 and 4 into Equations 5-8 yields Equations 9-12, which can be used through integral calculation to determine the course coordinates.

Equations 9 and 10 require that |Ka| is not equal to |Kc|, and Equations 11 and 12 require that Ka fi 0. So, three special cases should be considered in application:

• If Ka = Kc = 0, then DX and DY are defined by Equations 13 and 14.

• If |Ka| = |Kc| fi 0, then DX and DY are defined by Equations 15 and 16.

• If Ka = 0, then DZ and DS are defined by Equations 17 and 18.

In computer programs, the unit for angles is radian. In field terms, angles are usually reported in degrees. The value of C in these equations will depend on the units used (C = 90/p if the angle is in degrees, or C = 1/2 if the angle is in radians).

This model and its corresponding formulas show that the inclination, azimuth, and coordinates are all functions vs. measured depth. Thus, the method is very convenient for interpolation calculation of well bore trajectory. In fact, the equations are the general formulas for interpolation calculation, and it is only a special case to calculate the course coordinates, in which L is substituted by L2 into the corresponding formulas.

Verification

The model and formulas were programmed into computer software, and the results of its application to eight horizontal wells in the Daqing oil field and Liaohe oil field proved satisfactory.

Two typical theoretical curves, a cylinder helix and an arc curve in an inclined plane, were chosen to verify the calculated results of the natural parameter method and to compare the results with those from the radius of curvature method and minimum curvature method, the most commonly used methods in the oil field.

In the calculations for the following two examples, the parameters at the upper survey station are the same: (1 = 30°, c1 = 150°, and DL = 30 m, but a2 and c2 need to be determined according to the given curves.

In the first example, the course is assumed to be a cylinder helix curve. Its curvature in a vertical plot is KV = 8°/30 m, and its curvature in a horizontal plot is KH = 12°/30 m. Based on the characteristics of a cylinder helix, the inclination and azimuth at the lower survey station can be obtained as follows: a2 = a1 + KVDL = 38.0000°, and c2 = c1 + (KH/KV)[cos(a₁) - cos(a₂)] = 156.7049°.9

Table 1 [24914 bytes] shows the detailed calculated results from the different methods.

In the second example, the course is assumed to be an arc in an inclined plane. Its curvature is K = 8°/30 m, and its initial tool-face angle is v = -60°. So, the following parameters can be obtained: e = KDL = 8.0000°, a2 = cos^-1[cos(a₁)cos(e) - sin(a1)

sin(e)cos(v)] = 34.6335°, and c₂ = c₁ + sin^-1[sin(e)sin(v)/sin(a₂)] = 137.7560°.

Table 2 [25530 bytes] shows the detailed calculated results from the different methods.

With respect to the cylinder helix, the calculated results of the radius of curvature method are exact. With respect to the arc curve in an inclined plane, the calculated results of the minimum curvature method are exact. The example results indicate that the natural parameter method yields correct, reliable, and precise values.

Results

From the assumption that within a survey interval, the rates of inclination change and azimuth change remain individually constant, a new model was derived for calculating the course coordinates and drilled well bore trajectory.

This natural parameter method is a simple and reasonable model. It is convenient for use in the oil field, and its results are accurate.

Compared with the radius of curvature method and minimum curvature method, the calculated results of the natural parameter method are relatively stable.

It is typical in many drilling operations for the rotary method and directional drilling method to be used alternately. In such cases, it is convenient for the engineer to adopt two or more trajectory calculation methods corresponding to different drilling operations in use at the time in one well. In this respect, it would be better and simpler to use just one calculation method, the natural parameter method, which is applicable for both rotary drilling and directional drilling.

All the parameters of a well bore trajectory in the natural parameter method are functions vs. measured depth, so it is convenient for interpolation calculation.

References

1. Craig, J.T. Jr., and Randall, B.V., "Directional Survey Calculations," Petroleum Engineer International, March 1976, pp. 38-45.

2. Callas, N.P., Novak, P.C., and Henderson, J.R., "Directional survey calculation methods compared and programmed," OGJ, Jan. 22, 1979, pp. 53-58.

3. Fuqi, L., "Chord Step Method for Calculating the Real Trace of Well Bore," Natural Gas Industry, Chengdu, China, Dec. 28, 1986, No. 4, pp. 40-46.

4. Xiushan, L., et al., "How to Simulate Actual Well Trajectories with Spline Function," Journal of Daqing Petroleum Institute, Anda, China, March 1991, No. 1, pp. 45-51.

5. Xiushan, L., et al., "The Curve Structure Method of Borehole Trajectory Calculation," Acta Petrolei Sinica, Beijing, China, July 1994, No. 3, pp. 126-33.

6. Zhiyong, H., "On Problem of Selecting Wellbore Survey Calculation Methods," Petroleum Drilling Techniques, Dezhou, China, March 1989, No. 1, pp. 14-17.

7. Planeix, M.Y., and Fox, R.C., "Use of an Exact Mathematical Formulation to Plan Three Dimensional Directional Wells," Society of Petroleum Engineers paper No. 8338, 1979.

8. Wilson, G.J., "An Improved Method for Computing Directional Surveys," Journal of Petroleum Technology, August 1968, pp. 871-76.

9. Xiushan, L., Designing Theory and Describing Method for Wellbore Trajectory, Heilongjiang Science and Technology Press, Harbin, China, 1993.

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