This work presents a mathematical method to design complex trajectories for three-dimensional (3D) wells. Three-dimensional cubic trajectories are obtained for various end conditions: free end, set end, free inclination/set azimuth, and set inclination/free azimuth. The resulting trajectories are smooth continuous functions, which better suit the expected performance of modern rotary steerable deviation tools, in particular point-the-bit and push-the-bit systems. A continuous and gradual change in path curvature and tool face results in the smoothest trajectory for 3D wells, that in turn results in lower torque, drag, and equipment wear. The degree of freedom and the associated parameters of the 3D curves express the commitment between the average curvature to the final length of the path, which can be adjusted to fit the design requirements and to optimize the trajectory. Several numerical examples illustrate the various end conditions. The paper also presents the full mathematical results (expressions for the 3D path, actual curvature, and actual tool face). The method is directly applicable to the well planning cycle as well as to automatic and manual hole steering.

1.
Planeix
,
M. Y.
, and
Fox
,
R. C.
, 1979, “
Use of an Exact Mathematical Formulation to Plan Three Dimensional Directional Wells
,” Paper No. 8338, September.
2.
Mitchell
,
B.
, 2005,
,
10th ed.
.
SPE-AIME
, Dallas, Chap. 3, pp.
355
370
.
3.
Guo
,
B.
,
Miska
,
S.
, and
Lee
,
R.
, 1992, “
Constant Curvature Method for Planning a 3-D Directional Well
,” Paper No. 24381, May.
4.
Ebrahim
,
A. A.
, 1995, General Three-Dimensional Well Trajectory Planning for Single and Multiple Directional Wells, MS thesis No. T4708, December, Colorado School of Mines, Golden, CO.
5.
Liu
,
X.
, and
Shi
,
Z.
, 2001, “
Improved Method Makes a Soft Landing of Well Path
,”
Oil & Gas J.
0030-1388,
99
(
43
), pp.
47
51
.
6.
Sawaryn
,
S. J.
, and
Thorogood
,
J. L.
, 2003, “
A Compendium of Directional Calculations Based on Minimum Curvature Method
,” Paper No. 84246, October.
7.
Schweikert
,
D.
, 1966, “
An Interpolation Curve Using Splines in Tension
,”
J. Math. Phys.
0022-2488,
45
, pp.
312
347
.
8.
Stoker
,
J. J.
, 1969,
Differential Geometry
,
Wiley
, New York, pp.
12
48
.
9.
Press
,
W. H.
,
Teukolsky
,
S. A.
,
Vetterling
,
W. T.
, and
Flannery
,
B. P.
, 1992,
Numerical Recipes in Fortran: The Art of Scientific Computing
,
2nd ed.
,
Cambridge University Press
, New York, pp.
390
398
.