A goal-oriented framework for rapid integral-equation-based simulation of borehole resistivity measurements of 3D hydraulic fractures

Abstract

We have developed a goal-oriented framework for fast integral-equation-based simulation of low-frequency borehole resistivity measurements of 3D arbitrarily shaped hydraulic fractures. The framework explores the possibility of applying various approximate integral-equation methods to simulate borehole electromagnetic (EM) measurements acquired in the vicinity of 3D hydraulic fractures generated with electrically conductive proppant. It includes four approximate methods that are progressively more accurate, costly, and rigorous. Each method is used to approximate the method-of-moments solution of the integral equation and to evaluate/extract quantities of interest, e.g., bucked signals detected at receivers. When compared with rigorous fast Fourier transform (FFT)-accelerated method-of-moments solutions, the numerical results obtained with the four methods indicate the following (1) All of the approximate methods capture the main features of the quantities of interest, e.g., the shape of detected signals. (2) Different approximate methods exhibit different accuracies and efficiencies in the simulation of EM scattering from various 3D fractures. (3) The identified approximate method achieves accurate results (error [Formula: see text]) while reducing the simulation time by a factor of 2–1000 compared with the FFT-accelerated rigorous method. Thus, our approximate simulation framework is a promising candidate for evaluating the Jacobian matrix in the fast inversion of borehole EM measurements to detect and assess the geometry of 3D hydraulic fractures generated with electrically conductive proppant.