Modeling of Low-Frequency Downhole Electrical Measurements for Mapping Proppant Distribution in Hydraulic Fractures in Casedhole Wells

A tool concept using downhole electrical measurements for mapping electrically conductive proppant in hydraulic fractures is presented in this paper. The method relies on direct excitation of the casing, which is expected to overcome the severe limitations of induction tools in casedhole wells. An array of insulating gaps is installed and cemented in place as a permanent part of the casing string. The envisioned electrical measurements are performed by imposing a voltage across each insulating gap, one at a time, before and after hydraulic-fracture operations. The voltages across other insulating gaps near the transmitter gap are recorded. The proposed tool's response to the geometry of a single fracture was modeled by solving for the electrical potential with a finite-volume method. Previous simulation results have shown that the electrically conductive proppant alters the path of the electrical current in the formation, and this is recorded as differential signals by the string of insulating gaps surrounding the source gap. The simulated differential signals are highly sensitive to a fracture's location, length, and orientation, and less sensitive to the fracture's aspect ratio. However, to enable the implementation of such a practical system, various aspects of the tool concept must be investigated further through simulations. Following our previous work, this paper focuses on the forward modeling of the tool's response to multiple fractures, which demonstrates the influence of these fractures on the signals, and provides important guidance for inverse modeling. Parametric inversion of fractures from synthetic data, generated by exciting various insulating gaps, is solved with very fast simulated annealing (VFSA). Simulation results show that, when multiple hydraulic fractures are present, the voltages measured at the receiver gaps are determined primarily by the fracture that is in direct contact with the excited section of casing. When two fractures touch the same casing section, they induce voltages very similar to those from a single fracture with the same conductivity and volume. Preliminary inversion results that use synthetic data computed from circular fractures indicate that the proposed VFSA can solve for the multiple fractures' widths and radii at the same time, without requiring numerous forward simulations. Even with noisy synthetic data, VFSA can make good estimates of the fractures' parameters. This indicates that the VFSA technique is a proper and robust inversion technique for the measured voltages at various receiver gaps.