An analytic method for 3D well trajectory design with dual-target, Part II

Mathematical model of 3D well trajectory with dual-target is a nonlinear system of equations which is usually solved by numerical iteration method which gives approximate solution, but the iteration method has inherent defects of initial value dependence, slow convergence rate, and being likely to diverge. For a design issue with the first target orientation given, the design system of equations can be decoupled into two systems of equations with few unknowns respectively for the first and the second target. Simplification and elimination by pseudo-analytic method is performed to reduce the system of equations of the first target into a polynomial equation of one unknown with at most 10 degree. All the real roots of this polynomial equation can be solved by separation of real roots, and all the unknowns of the first target’s system of equations can be expressed by the analytical formula of these real roots. The second target’s system of equations results in two quadratic polynomial equations with one unknown, and the analytical solution can be obtained. The new algorithm overcomes the inherent defect of iteration method, and has great advantage in computation speed. It can give all the solutions at the one-run when needed, which is impossible by iteration method. The new method opens a new direction in the algorithm study of 3D design issue, especially in analytical algorithm research. The simplification and elimination techniques provide a new mathematical tool for solving analytical solution of definite-condition problem in other 3D design.