Efficient Simulations of Electromagnetic Induction Tool in a Deviated Borehole for Resistivity Inversions

For the petroleum industry, layered medium subsurface detection plays an important role in discovering reservoirs and drilling wells. In geophysics, resistivity is an essential property for distinguishing formation layers or even small fractures. Well logging with electromagnetic induction tools can measure the subsurface resistivity. This measurement includes two steps: 1) directly measure the low-frequency response signals using the tool and 2) determine the subsurface geometric model and resistivity. The problem is that no formula can directly calculate the resistivity from the measured tool responses. A systematic solution is to combine forward electromagnetic simulations and inversion of the subsurface model. In this dissertation, two categories of inversion are investigated: Determine the proper subsurface model by 1) optimizing the objective function, such as data misfit, and 2) training a surrogate model for the inverse mapping. Many forward simulations are demanded for either estimating the data misfit of new candidate models or collecting data for training. Therefore, efficient electromagnetic simulation is critical for resistivity logging. From complex to simple, three types of simulation are discussed: 1) borehole simulation with real tool configuration, 2) borehole simulation with point sources as the virtual tool, and 3) simplified layered medium simulation with virtual tool. Three optimal methods are implemented, respectively: the domain decomposition method, the finite element boundary integral method, and the analytical method. The tool calibration and the borehole effects are studied in the comparison of these simulations. Ideally, the simplest forward simulation should be used in the inversion, and the additional effects can be extracted as correction terms. The optimization-based inversion of the formation model uses simulations of a virtual tool in the layered medium. The Occam inversion or Monte Carlo Markov chain can minimize the data misfit. Another special simulation for small fractures using the thin dielectric sheet approximate method collects the dataset of fracture models. Fracture parameters such as resistivity, extension, and tilt angle are accurately determined by machine learning methods. The surrogate model also tends to predict fracture properties correctly, even for the complete simulation result.