Fracture distribution plays a significant role in the behavior of subsurface environments, affecting such activities as geothermal production, exploitation and management of groundwater resources, and long-term storage of nuclear waste and carbon dioxide. A key challenge in these and other applications is to estimate the fracture network properties from sparse and noisy observations. We evaluate the utility of cross-borehole thermal experiments for this task, using both physics-based particle-tracking (PBPT) heat-transfer approach and its deep neural network (DNN) surrogates. Synthetic data are provided by the PBPT simulations and used to train and test the DNN surrogates over a full range of the fracture network properties. We propose regionalized and step-by-step training techniques to reduce the computational cost of expensive PBPT forward solves over large ranges of the (to-be-estimated) parameters. Our numerical experiments suggest the feasibility of training a regionalized DNN surrogate over parameter ranges for which the PBPT solves are fast and extrapolating its predictions to parameter ranges with few additional data. We analyze the balance between computational cost and model accuracy, and provide both PBPT and DNN models for applications to others kinds of data.

A two-dimensional particle-based heat transfer model is used to train a deep neural network. The latter provides a highly efficient surrogate that can be used in standard inversion methods, such as grid search algorithms. The resulting inversion strategy is utilized to infer statistical properties of fracture networks (fracture density and fractal dimension) from synthetic thermal experimental data. The (to-be-estimated) fracture density is well constrained by this method, whereas the fractal dimension is harder to determine and requires adding prior information on the fracture network connectivity. The method is tested on several fracture networks and hydraulic conditions.