ALGORITHM FACILITATES PREPLANNING WELL BORE TRAJECTORIES
John Taft Nicholson
Smith International
Houston
An algorithm has been developed for generating and navigating well bore trajectories with prescribed initial and final coordinates, and angular conditions.
This type of three-dimensional trajectory occurs while planning relief, extended reach, horizontal, and noncollinear multitarget wells.
The selection of a mechanically feasible well bore trajectory is based upon optimizing the drillstring loads and the steerability of the trajectory.
The use of the algorithm is illustrated with detailed examples.
The algorithm can be easily solved on a personal computer.
The basic nature of the algorithm differs from that of conventional directional drilling, planning, and survey techniques' in that no assumption was made as to the shape of the trajectory. The algorithm generates data bases that can be used to compare the accuracy of survey calculation methods.
The algorithm can be generalized in two ways. One or two more derivatives at the terminal ends can be prescribed. This is equivalent to specifying the initial and final curvature and torsion. Such generalizations would increase the order of the polynomial (Equation 1 in equation box) from a third order to a fifth and seventh order polynomial.
PREPLANNING
The purpose of a preplanned well bore trajectory is to serve as a navigational guide for the directional driller.
The drilled trajectory is rarely identical to that of the preplanned trajectory.
The advent of steerable drilling systems has made possible any mechanically feasible well bore trajectory. Well bore trajectories with prescribed terminal coordinate and angular conditions (EWo, NSo, VDo, INCo, AZo, EW, NS, VD, INC, AZ), are applicable for the planning of relief, horizontal, and multiple target wells. The abbreviations are explained in the Nomenclature box.
The relief well usually requires a parallel approach to the blowout well for both detection and kill purposes. A horizontal well can be thought of as a controlled fracture in a prescribed direction. Many multiple target wells' objectives are noncollinear.
For illustration purposes in this article, selection of a well bore trajectory from a set of mechanically feasible trajectories with the same objectives has been limited to optimizing drillstring loads and steerability. In practice, economic optimization would be used .2
THEORY
The frame of reference used in this article is that of a conventional left-handed triad, positive (north, east), and vertical (down). Inclination is positive from vertical, and azimuth is positive clockwise from north. The unit tangent vector (T) at any measured depth (curve length) is:
= sin(AZ) x sin(INC)T + cos(AZ) x sin(INC)l + cos(INC)k
AZ and INC are the azimuth and inclination, respectively. The i, j, and k are unit vectors in the east, north, and vertical directions.
A space curve between two points (NSO, EWO, VDO) and (NS, EW, VD) with prescribed terminal angular conditions (INCO, AZO) and (INC, AZ) can be represented parametrically in terms of a position vector (P) in Equation 1 .3
Curve length, s (measured depth), is a function of the parameter t. The unit tangent vector () is defined by Equations 2 and 3.
The vector expressions for P and dP/dt can be expanded to give scalar Equations 4-9. NS, EW, and VD are constants.
Component values of the unit tangent at t are given by Equations 10-12.
The curve length increment between t, to tn,l is calculated by the integral in Equation 13.
Instantaneous curvature and torsion at any point t are given by the vector Equations 14-15. The represent absolute value.
A family of space curves all satisfying the terminal conditions can be generated by setting to equal to the initial measured depth (ds/dt = 1 at to). Equations 1 and 2 are set equal to the initial prescribed conditions.
Next, the value of t (tto) is arbitrarily chosen (ds/dt = 1 at t) with Equations 1 and 2 set equal to the final prescribed conditions.
This leads to a linear system of 12 equations with 12 unknowns (NS, EW, VD) that can be solved for by standard methods.
Once the constants have been found, the space curve can be represented as a data base of discrete stations along the curve by incrementing t from to to t. At each value of t, Equations 4-15 can be solved for the measured depth, inclination, azimuth, spatial coordinates, instantaneous curvature, and torsion.
This data base completely describes a member of the family. Another member of the family can be generated by choosing a different value for t. Selection of a member for implementation is based upon optimizing the drillstring loads and the steerability of the trajectory.
The size of the family of curves satisfying the prescribed conditions, of course, is infinite. Not all of these members are mechanically feasible. A feasible trajectory is one that will not exceed the mechanical limits of the drillstring while the route is navigable by modern directional drilling techniques.
An optimum trajectory in the feasible set is selected by the tabulation and comparison of the constraints. Ideally, the drillstring loads will be optimum when the pickup load, drilling torque, and dogleg severity (DLS) are a minimum while the slack off load is a maximum (ability to apply weight on bit while drilling oriented).
A trajectory is steerable if the curvature requirements fall within those attainable by conventional steerable drilling systems. A preferred trajectory will be one that requires the least amount of oriented drilling.
The set of constraints chosen for optimization are:
- Pickup and slack off loads
- Off bottom rotating torque
- Average DLS (sum of all DLS in data base/number of stations - 1)
- Maximum DLS observed in the data base.
The average instantaneous torsion/100 ft has been included for observation only.
In general, the feasible set of trajectories can be generated by setting t equal to values of the same order of magnitude as the sum of the initial measured depth plus the cordal distance between the terminals.
The last statement is a conjecture. If the value of t is too small, and the difference between the terminal angular conditions are large, then at least the curvature at one of the terminals will be great.
If the value of t is too large, the curve meanders through space with portions of it having the vertical depth decrease as the measured depth increases. The size of the increment (-t) from t, to tv should not be too large or some of the important aspects of the space curve will not be represented.
If the values of DLS and instantaneous curvature (CUR) are approximately the same, then the size of the increment is usually sufficient. The selection of t and the size of the increment between stations are illustrated in the following three examples.
SIDETRACK
The following well is to be sidetracked. At the target, the inclination should be 90 with an azimuth of 114.53. All distances are in meters. The well was kicked off at the surface.
Tie in:
MD = 1,125
INC = 15.5
AZ = 52.45
VD = 1,094.45
NS = 150.48
EW = 189.55
Target:
VD = 1,962
NS = - 583.6
EW = 1,273
SOLUTION
The value of the initial measured depth plus the distance between the terminals is 2,695. Equations 4-9 are solved for the constants (VD, NS, EW) while setting to = 1, 1 25 and t = 2,000, 2,500, 3,000, 3,500, 4,000.
This generates a family of five space curves all satisfying the prescribed terminal conditions. The overhead view of three members of the family is shown in Fig. 1.
The values of the constants in Equations 4-6 with t = 3,500 are:
Vo = -491.5177627744453
V1 = 1.908161488689422
V2 = -4.895165006124027 -4
V3 = 4.131847661491191 -8
Eo = -82.92925595000243
E1 = 0.3194831786815353
E2 = -1.104243705182529E -4
E3 = 3.709473281938452 -8
No = -571.5405232356044
N1 = 1.203445267045877
N2 = - 5.720164533803783 -4
N3 = 6.491157183406541 -8
Next, the polynomials are used to generate a data base (MD, INC, AZ, VD, NS, EW, CUR, TOR) by incrementing t in steps of 25 from to to t. The data base shown in Table 1 is for t = 3,500 and -t = 125.
The instantaneous curvature (CUR) and torsion (TOR) per 100 ft are compared with the DLS in Table 1. Torsion is a measure of the space curves noncoplanarness.
The bottom hole location (VD, NS, EW), as calculated by the circular arc survey method 4 using the surveys of the 95 (-t = 25) station data base, is 1,962.03, -583.92, and 1,273.03.
The selection of a member trajectory from the family for implementation is now based upon optimizing the torque and drag of the drillstring and the trajectory's steerability. The torque and drag software used in this analysis is based upon the model developed by Johancsik .5 6
For demonstration purposes only, the drillstring chosen for this example consists of 360 ft of 8-in. drill collars, 450 ft of 5-in. heavywall drill pipe, and 5 in. by 19.5-lb/ft drill pipe.
The drilling parameters were taken as an overpull at the bit of 25,000 lb, weight on bit = 1,000 lb, and drilling torque = 2,000 ft-lb. The friction factor was set at 0.4.
The drillstring loads and steerability factors for the family of curves are compared in Table 2 (^t = 25). If the drillstring loads and the trajectory's steerability are the only constraints used for optimization, then the trajectory with t = 3,500 would be selected for this drilling scenario. For further optimization, the above analysis would be repeated in a neighborhood of t = 3,500.4 In practice, many different drillstrings, drilling, and other parameters would be considered during the selection of the optimum trajectory.
RELIEF WELL
This example demonstrates the use of the algorithm for the planning of a relief well. All distances are in feet.
The relief well's present position is:
Tie-in:
MD = 4,250
INC = 29.57
AZ = 136.93
VD = 4,045.67
NS = - 1,070.13
EW = 1,036.76
The tie-in point was reached by kicking the well off at 1,500 ft, with a buildup rate of 1.97/100 ft. The objective of the relief well was to be tangent to the blowout well as follows:
Target:
INC = 26.95
AZ = 217.85
VD = 6,427.48
NS = - 1,070.13
EW = 1,036.76
SOLUTION
The value of the initial measured depth plus the distance between the terminals is 6,703 ft. Equations 4-9 are solved for the constants (VD, NS, EW) while setting to 4,250 and t = 6,000, 7,000, 7,500, 8,000, 9,000.
The increment chosen between tn and tn,1 is 50. This generates a family of five space curves all satisfying the prescribed terminal conditions. The overhead view of three members of the family is represented in Fig. 2.
The data base for t 7,000 and t = 125 is presented in Table 3. The bottom hole location (VD, NS, EW) as calculated by the circular arc survey method using the surveys of the 55 (-t = 50) station data base is (6,427,49, - 1,070.26, 1,036.69).
The drilistring chosen for this example consists of 120 ft of 8-in. drill -collars, 750 ft of 5-in. heavy wall drill pipe, and 5 in. by 19.5-lb/ft drill pipe with the same drilling scenario used in the sidetrack example.
The drillstring loads and steerability factors for the family of curves are compared in Table 4 (-t = 50).
Based upon the optimization of the drillstring loads, and the trajectory's steerability, the trajectory with t = 7,000 would be selected. Further optimization can be performed in a neighborhood of t = 7,000.
NONCOLLINEAR TARGETS
This example of a multiple noncollinear target well is adapted from Sidman.8
The well is to be kicked off at 700 ft. The five targets are defined in Table 5. The inclination and azimuth shown in Table 5 are those needed for the next target.
The trajectory between kickoff and Target 1 was planned by conventional methods with a build rate of 3/l 00 ft. The drilling program calls for a 4.232 left-hand lead (AZ = 269.014) at Target 4 because this trajectory (4-5) is to be drilled with a nonsteerable BHA.
The polynomials were derived between Targets 1-2 (t = 11,836, -t 500), 2-3 (t = 12,590.8, -t 500), 3-4 (t= 13,310.4, -t = 250), and 4-5 (t = 14,293.4, -t = 250) with the prescribed terminal angular conditions shown in Table 5.
The horizontal plot for this trajectory is shown in Fig. 3. The 26-station data base for this trajectory is presented in Table 6 (a right-hand helix has positive torsion, and a left-hand helix has negative torsion).
The difference between the DLS and the CUR presented in Table 6 is quite large at some stations (12,806.06). This indicates that a smaller value for -t should be chosen.
The bottom hole location (VD, NS, EW) as calculated by the circular-arc survey method using the surveys of the data base in Table 6, is (1 O,l 93.76, 3,342.79, - 8,620.36) . The bottom hole location can be recalculated with a 225 (-t = 50) station data base as (10,191.01, 3,348.79, -8,621.23). The algorithm provides a convenient means of comparing the accuracy of the many survey calculation methods.
This trajectory is an example of a mechanically feasible solution.
In practice, other trajectories with different terminal conditions would be reviewed before final selection.
The solution presented in Sidman's paper has a dogleg severity of 5.5/100 ft. The maximum DLS observed in the 255-station data base is 1.77/100 ft.
The algorithm can also be used as a navigational tool while drilling a well. Given the bottom hole location and bit attitude, the directional driller can adjust the tool-face orientation and dogleg of the steerable drilling system to match the survey of the next station in the data base.910
If the actual course is far from the preplanned trajectory, the algorithm can be recalculated based upon the new initial coordinate and angular conditions. This procedure has been implemented on a portable PC and used at the well site while drilling a well.
All arithmetic operations presented in this paper were done in double precision. Only the two first significant digits have been shown.
ACKNOWLEDGMENTS
The author thanks Smith International for allowing the publication of this article. He also appreciates the time Robert Duveau spent proofreading the text.
REFERENCES
- SPE Textbook Series, Applied drilling engineering, 1986.
- Vrielink, H.J., "The optimization of slant-well drilling in the Lindbergh field," SPE Drilling Engineering, December 1989.
- Lipschutz, M.M., Differential Geometry, Schaum's Outline Series.
- McKown, G.K., "Drilistring design optimization for high-angle wells," SPE/IADC cong., SPE paper 18650, New Orleans, 1989.
- Johancsik, C.A., Dawson, R., and Friensen, D.B., "Torque and drag in directional wells-predication and measurement," IADC/SPE conf., SPE paper 11,380, New Orleans, 1983, pp. 201-208,
- Maurer Engineering Inc., "Project to determine the limitations of directional drilling phase II," Report No. TR88-4, January 1988.
- Zaremba, W.A., "Directional surveys by the circular arc method," SPE Journal, February 1973, pp.5-11, Trans., AIME, Vol. 255.
- Sidman, R.D, "Mathematical approach helps plan directional wells," OGJ, June 4, 1979, pp. 142-44.
- Millheim, K.K., "Evaluating and planning directional wells utilizing postanalysis techniques and three dimensional bottomhole assembly program," SPE Annual Technical Conference and Exhibition, SPE paper 8339, Las Vegas, Sept. 2326, 1979.
- Nicholson, J.T., "Calculator program developed for directional drilling," OGJ, Sept. 28, 1981, pp. 213-24.
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