A bi‐fidelity model for hydraulic fracturing

Abstract

AbstractHydraulic fracture, whether a natural phenomenon or an engineered process, consists of a highly pressurized flow evolving within a rock and, when necessary, fracturing such a medium. Particularly, in the context of oil and gas exploitation, a mixture of water and proppant is injected through a well to improve the oil extraction conditions by artificially increasing the local permeability. From a modeling standpoint, it is fundamental to acknowledge that this complex flow involves, among other disciplines, fracture mechanics, fluid dynamics, non‐Newtonian flow, solid mechanics, and flow in porous media. Therefore, explicitly or not, a model bears multiple scales and multiple physics, leading to a non‐linear moving boundary mathematical problem that is hard to solve through analytical techniques. Amongst them, the Pseudo‐3D, which computes fracture evolution in reservoirs confined by symmetric stress barriers, is frequently employed in the oil and gas industry domain. The different assumptions made to obtain the P3D model lead to inaccurate predictions, like the overestimation of fracture height for important operating conditions. We propose a model discrepancy term in a multi‐fidelity approach to correct such drawbacks. The main difficulty associated is constructing a mapping between the inputs and outputs that faithfully represents the error between high and low fidelity models. In this paper, we investigated the performance of a nonlinear multi‐fidelity approach relying on Gaussian Processes. We present a careful analysis to assess the good performance of such a proposition and also present results involving Uncertainty Quantification, which is made feasible by using the bi‐fidelity surrogate.