Abstract
In some geological formations, borehole resistivity measurements can be simulated using a sequence of 1D models. By considering a 1D layered media, we can reduce the dimensionality of the problem from 3D to 1.5D via a Hankel transform. The resulting formulation is often solved via a semi-analytic method, mainly due to its high performance. However, semi-analytic methods have important limitations such as, for example, their inability to model piecewise linear variations on the resistivity. Herein, we develop a multi-scale finite element method (FEM) to solve the secondary field formulation. This numerical scheme overcomes the limitations of semi-analytic methods while still delivering high performance. We illustrate the performance of the method with numerical synthetic examples based on two symmetric logging-while-drilling (LWD) induction devices operating at 2 MHz and 500 KHz, respectively.
Highlights
There exist a wide variety of geophysical prospection methods
In a 1D model, we reduce the dimension of the problem via a Hankel or a 2D Fourier transform along the directions over which we assume the material properties to be invariant [15,16,17]
If a common factorization of the system matrix based on a lower and an upper triangular matrix would be precomputed for all source positions, the situation would only worsen: one would need a refined grid to properly model all sources, and since the cost of backward substitution is proportional to the discretization size, the large computational cost required to perform backward substitution would significantly increase the total cost of the solver
Summary
There exist a wide variety of geophysical prospection methods. In this work, we focus on resistivity methods. We seek a method with the following features: (a) we can consider arbitrary resistivity distributions along the 1D direction, and (b) we can and rapidly construct derivatives with respect to the material properties and position of the bed boundaries by using an adjoint formulation, which allows us to compute numerically the derivatives forming the Jacobian matrix needed by the Gauss-Newton inversion method at (almost) no additional cost. Despite these advances, presently our proposed multi-scale method is slower than the semi-analytic one.
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