A hybrid method for inversion of 3D AC resistivity logging measurements

• Hybrid strategy hp-HGS for solving inverse problems.
• Global step solved by Hierarchical Genetic Search (HGS) coupled with self-adaptive hp-Finite Element Method (hp-FEM).
• Local search performed with Broyden–Fletcher–Goldfar–Shanno (BFGS).
• hp-HGS algorithm adapts the length of the genetic code and the accuracy of the direct solver.
• Application to 3D AC resistivity logging measurement simulations in deviated wells.

In this paper, we propose the use of a hybrid algorithm for the inversion of 3D Alternate Current (AC) resistivity logging measurements. The forward problem is solved using a goal-oriented self-adaptive hp-Finite Element Method (hp-FEM) that provides exponential convergence of the numerical error with respect to the mesh size. The inverse problem is solved using a Hierarchical Genetic Search (HGS) coupled with a Broyden–Fletcher–Goldfar–Shanno (BFGS) method. Individuals from the genetic populations represent the resistivity of the formation layers. The fitness function is estimated based on hp-FEM results. The hybrid method controls the accuracy of evaluation of particular individuals, as well as the accuracy of the genetic coding. After finding those regions where the fitness function has small values, the local search method by means of BFGS algorithm is executed. The paper is concluded with numerical results for the hybrid algorithm.