Two-Dimensional High-Accuracy Simulation of Resistivity Logging-While-Drilling (LWD) Measurements Using a Self-Adaptive Goal-Oriented hp Finite Element Method

We simulate electromagnetic (EM) measurements acquired with a logging-while-drilling (LWD) instrument in a borehole environment. The measurements are used to assess electrical properties of rock formations. Logging instruments as well as rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. The simulations are performed with a self-adaptive goal-oriented hp-finite element method that delivers exponential convergence rates in terms of the quantity of interest (for example, the difference in the electrical current measured at two receiver antennas) against the CPU time. Goal-oriented adaptivity allows for accurate approximations of the quantity of interest without the need to obtain an accurate solution in the entire computational domain. In particular, goal-oriented hp-adaptivity becomes essential to simulating LWD instruments, since it reduces the computational cost by several orders of magnitude with respect to the global energy-norm-based hp-adaptivity. Numerical results illustrate the efficiency and high accuracy of the method, and provide physical interpretation of resistivity measurements obtained with LWD instruments. These results also describe the advantages of using magnetic buffers in combination with solenoidal antennas for strengthening the measured EM signal so that the " signal-to-noise " ratio is minimized. 1. Introduction. A plethora of energy-norm-based algorithms intended to generate optimal grids have been developed throughout recent decades (see, for example, [10, 23] and references therein) to accurately solve a large class of engineering problems. However, the energy-norm is a quantity of limited relevance for most engineering applications, especially when a particular objective is pursued, such as simulating the electromagnetic response of geophysical resistivity logging instruments in a borehole environment. In these instruments, the amplitude of the measurement (for example, the electric field) is typically several orders of magnitude smaller at the receiver antennas than at the transmitter antennas. Thus, small relative errors of the solution in the energy-norm do not imply small relative errors of the solution at the receiver