Quasi-Analytic Solution Theory for Arc Type Well Trajectory Design

To solve the system of multivariate nonlinear equations in well trajectory design quickly and reliably,the analytical solving method for the system of equations has been studied.Based on the theory and techniques of the mathematics and mechanics,a complete theoretical system of quasi-analytical solutions was created through complex mathematical formula derivation.It was proved theoretically that a characteristic polynomial containing only one unknown can be derived from the design equations,and all unknowns in designed equations can be calculated in sequence by all real roots of the characteristic polynomial and a group of analytical formula.The theoretical analysis and practical calculation showed that the quasi-analytical method can determine whether the system of equations are solvable,and calculate the solutions in the cases of the system of equations that have a unique solution or more.The quasi-analytical solution method completely overcomes inherent defects of numerical iteration methods such as initial dependence,convergence,and inability to solve multiple solutions etc.Its accuracy only depends on all real roots of characteristic polynomial,being an exact algorithm.The results showed that proposed method is a fast,reliable and accurate computing technique to solve the system of design equations.It is not only a theoretical innovation in algorithm research,but also has an important practical value in drilling software development.