Fast Simulation of 2.5D LWD Resistivity Tools

We are developing a general one-dimensional (1D), 1.5D, 2D, 2.5D, and 3D Finite Element (FE) library for the fast simulation of borehole resistivity measurements. The library enables to combine problems with different spatial dimensionality and solve them using a single software. As a first step towards the fast inversion of geophysical data, in this work we focus on the rapid simulations of 2.5D logging-while-drilling (LWD) borehole resistivity measurements. Given a commercial logging instrument configuration, we calibrate the FE method offline with respect to (i) the element sizes via non-uniform tensor product grids; (ii) the arbitrary polynomial order of approximation on each element; and (iii) the interpolation of certain Fourier modes. This leads to the design of proper FE discretizations to simulate measurements acquired in an arbitrary 2D formation.Numerical results show that we accurately simulate on a sequential computer any field component at a rate faster than one second per logging position. In parallel, it is expected that the CPU time can be further reduced to below 0.1 seconds per logging position in a modest 128-core for all components of a tri-axial logging instrument. Derivatives (sensitivity functions) needed by gradient-based inverse methods are easily and rapidly computed using an adjoint-based formulation.