  Modeling gas flow through fractures of subsurface rock is a particularly
challenging problem because of the heterogeneous nature of the material.
High-fidelity simulations using discrete fracture network (DFN) models are one
methodology for predicting gas particle breakthrough times at the surface, but
are computationally demanding. We propose a Bayesian machine learning method
that serves as an efficient surrogate model, or emulator, for these
three-dimensional DFN simulations. Our model trains on a small quantity of
simulation data and, using a graph/path-based decomposition of the fracture
network, rapidly predicts quantiles of the breakthrough time distribution. The
approach, based on Gaussian Process Regression (GPR), outputs predictions that
are within 20-30% of high-fidelity DFN simulation results. Unlike previously
proposed methods, it also provides uncertainty quantification, outputting
confidence intervals that are essential given the uncertainty inherent in
subsurface modeling. Our trained model runs within a fraction of a second,
which is considerably faster than other methods with comparable accuracy and
multiple orders of magnitude faster than high-fidelity simulations.
