Boyun Guo, Robert L. Lee
New Mexico Institute of Mining & Technology
Socorro, N.M.
Stefan Miska
University of Tulsa
Tulsa
Three dimensional (3-D) drilling trajectory equations derived from the concept of constant curvature can improve the design of economical well trajectories and determine the tool face deflection in directional drilling.
The constant-curvature method developed is based on drilling tendencies of the bottom hole assembly and the technical advancement in directional drilling. It yields a trajectory with less maximum dogleg severity, less drag, and less torque in the drillstring compared to the conventional radius-of-curvature and constant-turn-rate methods.
The equations for the constant-curvature method can easily be used during directional drilling and are especially useful for horizontal drilling where a constant build rate is desired.
In some directional drilling operations, a target in the reservoir cannot be reached by drilling along a planar well path because of underground obstructions, such as faults or existing well bores. In such extreme conditions, a 3-D well trajectory is usually designed. Even though a well trajectory can be designed in a 2-D plane (or a 1-D line for a vertical well), the actual drilled well path is always in a 3-D space.
When a borehole substantially deviates from the original planned trajectory, the engineer usually redesigns the trajectory of the section to be drilled starting from the existing well bore instead of trying to correct the well path to the originally planned trajectory. In this process, a 3-D trajectory is usually used to hit the predetermined target. More generally, 3-D well path sections are always needed during the course of the trajectory correction. Therefore, 3D trajectory calculations are important in directional drilling.
The use of modem bent housing motors and measurement-while-drilling or steering equipment has made the hole inclination angle easy to be monitored and controlled at the surface. Therefore, it is more convenient to use a constant build rate than a varying build rate while drilling.
Based on this operational practice and the performance of deflection tools, Guo, et al., proposed the constant-curvature method for 3-D well trajectory design.1 This method yields a well trajectory with constant curvature in 3-D space; thus, a well can be drilled using constant tool face deflection while a constant build rate is kept.
This article presents three examples of using the constant-curvature method to demonstrate the practical usefulness of the concept. The results reveal that the well trajectory with constant curvature is consistent with directional performance of deflection tools and has less dogleg severity. These improvements, in turn, should increase the fatigue life of drillstring elements passing through the curved segments of the hole. Because of less dogleg severity, the drag and torque are expected to be less in holes designed to have a constant well path curvature.
CONSTANT CURVATURE
Fig. 1 shows the geometrical parameters of a well trajectory. Equation 1 gives the relationship among tool face deflection angle -y, build rate B, and dogleg severity (hole curvature) D. Equation 2 expresses the inclination I at the section length L. The azimuth A at length L is shown in Equation 3. Combining Equations 1 and 3 yields Equation 4. In these equations, the subscript "O" denotes the initial point of the section.
The orthogonal coordinates E, N, and Z for a point on a well path can be calculated with Equations 5-7.
TOOL FACE DEFLECTION
Equations 1, 3, and 4 relate hole inclination angle, azimuth angle, tool face deflection angle, vertical build rate, and tool curvature. These equations are useful for constant tool face deflection calculations in directional drilling, especially for horizontal wells in which a constant build rate is preferred. In such cases, the minimum curvature equations proposed by Taylor and Mason cannot be used.'
For horizontal well applications, the primary goal is to fix the vertical build rate to hit the target . 3 In these wells, it is useful to define an intermediate point (I = Iin, A = Ain) at which the tool face angle can be changed from right to left to negate some of the change in azimuth while continuing the desired constant vertical build rate. The required constant tool face angle can be first determined from Equation 1.
Then, the inclination angle at the intermediate point can be determined as follows:Between the initial point (11, Al) and the intermediate point (Iin, Ain), Equation 4 becomes Equation 8. Between the intermediate (Iin, Ain) and the end point (I2, A2), Equation 4 becomes Equation 9. (Note that 12 is about 90' for a horizontal well.) Adding Equations 8 and 9 and solving for 10 yields Equation 10, which is an additional equation for constant tool face deflection in drilling horizontal wells. The determination Of Iin is important because drillers need to know at what inclination angle they will change the tool face angle from right to left to negate some of the change in azimuth while continuing the desired constant vertical build rate.
The length from I, to Iin is then calculated using Equation 11. The azimuth change after drilling to Iin with tool face angle -y is determined using Equation 12. The azimuth change after drilling from Iin to 12 with a tool face angle -y is calculated with Equation 13. Equation 14 gives the total length of the deflection interval.
AVOIDING AN EXISTING WELL
This first example covers the design of a 3-D well path that needs to be drilled between two specific points to avoid running into an existing well bore. The initial point of the 3-D section is E=100 ft, N=100 ft, and Z= 1,000 ft.-The end point of the section is E=445 ft, N = 205 ft, and Z = 1,890 ft.
To avoid the existing well bore safely, the planned well path section should have an offset, approximately 90 ft in this case, from the vertical plane passing through the two given points. However, the length of the section is limited to within 1,000 ft to reduce the drilling cost. To align with the target of the directional well, the azimuth at the lower point of the 3-D section needs to be about 105.
Although the simple minimum-curvature method yields trajectories with constant curvature, its application is limited because it gives a trajectory which lies in a single plane. In fact, the minimum-curvature method does not yield a trajectory that meets all of the requirements and neither does the McConalogue method proposed by Scholes.4 Therefore, the 3-D well path section in this example is planned using the radius-of-curvature method, the constant-turn-rate method, and the constant-curvature method.
Equations 2, 3, 5, 6, and 7 are used for the well path design. The build rate B, dogleg severity D, turn rate T, and turn rate in horizontal plane H were properly determined by a trial-and-error procedure for the three methods. Tables 1-3 show the results of the computations (well path data). A comparison of the data in these tables indicates that all three plans successfully meet the requirements.
The horizontal trajectory projections (top view from surface) of well path sections obtained by the three methods are plotted in Fig. 2. An examination of Fig. 2 shows that, in terms of azimuth change (turn in horizontal plane), the well path given by the constant-curvature method turns fast in the upper part of the section compared to the well paths given by the radius-of-curvature method and the constant-turn-rate method. Using the constant-curvature method is advantageous because usually with an increase in the inclination angle, the change in azimuth becomes more difficult.
Fig. 3 is a comparison of the well paths for the three trajectories in a vertical spread plane. There is not much difference among the trajectories projected on such a vertical spread plane because the same value of build rate (B=4.13/100 ft) has been used for each method.
Fig. 4 shows the measured depth vs. the dogleg severity of the hole. A comparison of the three designs reveals that the maximum dogleg severity of the well path section planned is greatest for the radius-of-curvature method and smallest for constant-curvature method. The relative difference between the two is up to 84%. Therefore, it is expected that the well path planned using the constant-curvature method will result in less fatigue damage of the drillstring elements as they pass through the curved segments of the hole.
WELL DEFLECTION
This second example covers the determination of the deflection of a directional well. For this example, the deflection tool has a curvature rate of 3.4/100 ft (D). The equations will be used to determine the constant tool face deflection needed to increase the hole inclination from 30 (I1) to 50 (I2) and to change the azimuth from 65 (A1) to 30 (A2).
Equation 4 is used to calculate the tool face angle y as -47.8. The build rate B, 2.28, is calculated with Equation 1. The length of the deflection interval, 875.55 ft, is then calculated with Equation 14. The total tool face deflection dogleg DL can be calculated, based on Equation 1, as 29.77.
If the minimum-curvature method is used, the total dogleg for this deflection is Dm = 29.49. The excess dogleg required to use the constant tool face deflection is ADL = DL - Dm, which equals 0.280. This value is equivalent to an additional well path length of 8.24 ft.
HORIZONTAL WELL
The third example covers the deflection of a 90 horizontal well; the target azimuth is 40. To intersect the target, the current vertical build rate of 8.5/100 ft (B) is kept using a tool with a build of 10/100 ft (D) from a starting inclination of 50 (11) and azimuth of 30 (A1). 12 = 90. The equations will be used to determine the constant tool face deflection and the intermediate point at which the tool face angle is to be changed from right to left.
Equation 1 gives the required tool face angle as 31.8. The inclination angle at the intermediate point (Iin) is calculated with Equation 10 as 76.3. The length from 11 to Iin, according to Equation 11, is 309 ft.
The azimuth change after drilling to Iin with a 31.8 tool face angle is calculated with Equation 12 as 18.53. The azimuth change after drilling from Iin to 90 with a 31.8 tool face angle is calculated with Equation 13 as -8,53. The final azimuth change is 18.53 - 8.53, or 10. Equation 14 is used to calculate the total length of the deflection interval, AL, as 471 ft.
DRAG AND TORQUE
An important step in a 3-D well path design is to estimate the drag and torque of the drillstring in the designed borehole. If the drillstring drag or torque is too large for the available tools and equipment, the well path should be redesigned using less curvature (dogleg severity) until both the drag and torque are within acceptable levels.
The causes of high drag and torque include sliding friction, tight hole conditions, sloughing shale, key-seats (dogleg severity), differential pressure sticking, and cuttings buildup from poor hole cleaning. Although an accurate prediction of the drag and torque is very difficult (if not impossible) at the design stage of a well, an estimate can be made for the drag and torque associated with the sliding friction.
Guo, et al., solved the partial differential equation presented by Sheppard, et al., for constant curvature strings.15 The equations for the effective tension F, and torque Tk, are given by Equations 15 and 16. Equations 17-19 define the quantities C0, C1, and C2, respectively. While Equations 15 and 16 are valid for strings with constant curvature, at the present stage of this study it is assumed that they can also be used piecewise for strings with variable curvatures. In other words, it is assumed that the string behaves (piecewise) like a rope for the purpose of drag and torque calculations.
The following example uses Equations 15 and 16 to compare the drag and torque calculations from the radius-of-curvature method, the constant-turn-rate method, and the constant-curvature method.
The three 3-D well path sections designed in the first example are drilled with a 5-in. OD, 19.5 lb/ft drillstring with a mud weight of 10 ppg. The coefficient of friction between the well bore and drillstring is 0.3. The effective tension and torque in the drillstring at the bottom of the 3-D section are 200,000 lb and 15,000 ft-lb, respectively. The drag and torque in the drillstring in the 3-D sections designed using the radius-of-curvature method and the constant-turn-rate method are obtained by repeated applications of Equations 15 and 16.
The well paths have been divided into 10 segments with piecewise constant curvature for the calculations for the two methods. Of course, the required computations for the constant-curvature method involve direct application of Equations 15 and 16. The results of calculations are summarized in Table 4.
A comparison of the results shows that if the drillstring is modeled as a rope, the drag and torque of the string for the well path designed using the constant-curvature method are slightly lower for these drilling conditions.
Equations 15 and 16 are presented graphically for easier use in Figs. 5 and 6 in terms of drag and torque increment per 100 ft of drillstring. Sensitivity analysis indicates that if the effective tension in the drillstring is greater than 10,000 lb, the drag and torque are very insensitive to the hole inclination angle and build rate. Although these two figures are plotted using a hole inclination of 45 and B/D ratio of 0.75, they give satisfactory results (with error of less than 1%) if the hole inclination varies between 10- and 90 and the B/D ratio ranges between 0.1 and 1.
For an example of using Fig. 5, consider a 1,000-ft drill pipe section just above the bottom hole assembly. If the effective tension at the bottom of the pipe section is 50,000 lb (estimated based on the drill collar size, mud weight, hole inclination, etc.) and the hole curvature is 12/100 ft, the drag (increment in effective tension) can be found in Fig. 5 as 5,000 lb/100 ft. Therefore, the effective tension at the top of this drill pipe section should be 100,000 lb.
DOGLEG SEVERITY
A knowledge of dogleg severity is important for computing the pipe curvature and subsequently the bending stress in the pipe for a given tensile load. Lubinski developed an equation for calculating the dogleg severity between two successive directional survey stations in a closed form.6 It is assumed that the change in the overall angle is a continuous linear function of the length of the hole. For example, if the overall angle change is 5/100 ft of hole, the expected angle change per 1 ft is 0.05.
When the radius-of-curvature, constant-turn-rate, and minimum curvature methods were proposed, approximate formulations for calculating the dogleg severity corresponding to these methods were also proposed. If the well trajectory between two survey stations has a constant build rate B and constant curvature D, then the equation of calculating the dogleg severity of the hole is given by Equation 20.
In drilling practice, all the methods yield almost the same dogleg severity. Consequently, a recommendation is that either Lubinski's equation or Equation 20 be used for actual dogleg severity evaluations because they have closed forms. The equation for calculating hole dogleg severity using the information obtained from the successive directional surveys has a closed form only if the well trajectory assumes a constant curvature.
REFERENCES
- Guo, B., Miska, S., and Lee, R.L., "Constant-Curvature Method for Planning a 3-D Directional Well," SPE paper 24381, presented at the SPE Rocky Mountain Regional Meeting, Casper, wyo,, 1992.
- Taylor, H.L., and Mason, C.M., "A Systematic Approach to Well Surveying Calculations,' Society of Petroleum Engineers Journal, December 1972.
- Schuh, F.J., "Horizontal Well Planning-Build Curve Design," NMT paper 890008, presented at the Centennial Symposium Petroleum Technology into the Second Century, Socorro, N.M., 1989.
- Scholes, H., "A Three Dimensional Well Planning Method for HDR Geothermal Wells," SPE paper 12101, presented at the 58th Annual-Technical Conference and Exhibition, San Francisco, 1983.
- Sheppard, M.C., Wick, C., and Burgess, T., "Designing Well Paths To Reduce Drag and Torque," SPE Drilling Engineering, 1987, pp. 344-50.
- Lubinski, A., Developments in Petroleum Engineering, Vol. 1. Gulf Publishing Co., Houston.
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