Anisotropic resistivity estimation and uncertainty quantification from borehole triaxial electromagnetic induction measurements: Gradient-based inversion and physics-informed neural network

Abstract

Rapid and accurate petrophysical reservoir description and quantification is important for subsurface energy resource modeling and engineering. Triaxial borehole resistivity measurements enable the estimation of key in-situ rock properties, such as the volumetric concentration of shale and sandstone resistivity. However, traditional approaches for estimating these properties from triaxial, or anisotropic, borehole resistivity measurements rely on analytical solutions that solve a system of equations simultaneously, which can lead to numerical instability and inefficiency. By reformulating the system of anisotropic resistivity equations as an inverse problem, we can achieve a more stable, accurate, and efficient estimation of key petrophysical properties. We propose two methods, namely nonlinear gradient-based and physics-informed neural network (PINN) inversion, to estimate the volumetric concentration of shale and sandstone resistivity from the parallel- and perpendicular-to-bedding-plane resistivity logs, posed as the solution of an inverse problem. Furthermore, we compare the PINN, nonlinear gradient-based inversion and analytical solution methods in terms of accuracy, computational efficiency, and uncertainty quantification using four different datasets of anisotropic resistivity logs, i.e., two synthetic and two field cases. The PINN inversion technique estimates the petrophysical properties within 0.5 CPU milliseconds with an accuracy between 91% and 99%, while nonlinear gradient-based inversion can estimate the petrophysical properties with an accuracy between 98% and 99% but requires several CPU minutes depending on the size of the dataset. The proposed PINN technique is therefore capable of providing fast and accurate estimation and uncertainty quantification of key petrophysical properties from triaxial resistivity logs, with comparable accuracy and approximately $10^6$ speedup compared to nonlinear gradient-based inversion, and can be used for real-time applications such as automatic shale properties estimation, logging-while-drilling measurement interpretation, automated well geosteering, and time-lapse reservoir monitoring.