A Fast Forward and Inversion Method for Long Distance Cross-well Electromagnetic Imaging Logging System

Long distance cross-well electromagnetic imaging method can explain the formation profile more than 500 m. The emphasis and difficulty of formation data inversion focus on the accuracy and speed for inversion imaging. Inversion imaging uses iterative method to gradually reduce the difference between the forward model and the measured results to approximate the real formation. Forward calculation includes Green's function method, finite element method and finite difference method. The calculation process of Green's function method in axisymmetric coordinates is simple, but it needs to know the background resistivity. The calculation accuracy decreases with the increase of distance, which affects the inversion imaging accuracy. Finite element method and finite difference method establish large linear equations in Cartesian coordinates to solve the singular matrix, which is inefficient. The advantage of inversion imaging is that the Jacobian matrix can be solved by the reciprocity theorem. In this paper, numerical mode matching method is proposed for forward calculation for the first time, which has the characteristics of longitudinal analysis and lateral approximation, and takes into account both accuracy and efficiency. However, in axisymmetric coordinates, the reciprocity theorem cannot be used to solve Jacobian matrix, and the inversion efficiency is too low through difference approximated Jacob matrix calculation. In this paper, a fast algorithm for computing Jacobian matrix by using Green function method instead of difference approximation method is proposed under axisymmetric coordinates. Jacobian matrix can be obtained only by the product of one forward and one Green function. Then the method is verified by contrast with the result of difference approximation method. It improves the calculation speed significantly. The simulation results under formation model show that the Gauss-Newton inverse algorithm with modified Green function Jacob calculation applied in long distance cross-well electromagnetic imaging logging system is feasible for axial symmetry 2D non-uniform formation. The parameters and bounders can be determined for 2D formation. The speed is dozens of times faster than difference approximation method.