Deep learning (DL) is a numerical method that approximates functions. Recently, its use has become attractive for the simulation and inversion of multiple problems in computational mechanics, including the inversion of borehole logging measurements for oil and gas applications. In this context, DL methods exhibit two key attractive features: (a) once trained, they enable to solve an inverse problem in a fraction of a second, which is convenient for borehole geosteering operations as well as in other real-time inversion applications. (b) DL methods exhibit a superior capability for approximating highly complex functions across different areas of knowledge. Nevertheless, as it occurs with most numerical methods, DL also relies on expert design decisions that are problem specific to achieve reliable and robust results. Herein, we investigate two key aspects of deep neural networks (DNNs) when applied to the inversion of borehole resistivity measurements: error control and adequate selection of the loss function. As we illustrate via theoretical considerations and extensive numerical experiments, these interrelated aspects are critical to recover accurate inversion results. K E Y W O R D S deep learning, deep neural networks, error estimation, geophysical applications, real-time inversion

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.

Borehole resistivity measurements are routinely employed to measure the electrical properties of rocks penetrated by a well and to quantify the hydrocarbon pore volume of a reservoir. Depending on the degree of geometrical complexity, inversion techniques are often used to estimate layer-by-layer electrical properties from measurements. When used for well geosteering purposes, it becomes essential to invert the measurements into layer-by-layer values of electrical resistivity in real time. We explore the possibility of using deep neural networks (DNNs) to perform rapid inversion of borehole resistivity measurements. Accordingly, we construct a DNN that approximates the following inverse problem:given a set of borehole resistivity measurements, the DNN is designed to deliver a physically reliable and data-consistent piecewise one-dimensional layered model of the surrounding subsurface. Once the DNN is constructed, we can invert borehole measurements in real time. We illustrate the performance of the DNN for inverting logging-while-drilling (LWD) measurements acquired in high-angle wells via synthetic examples. Numerical results are promising, although further work is needed to achieve the accuracy and reliability required by petrophysicists and drillers.