Structural and topological information play a key role in modeling flow and transport through fractured rock in the sub-surface. Discrete fracture Restricting the flowing fracture network to this backbone provides a significant reduction in the network's effective size. However, the particle tracking simulations needed to determine the reduction are computationally intensive. Such methods may be impractical for large systems or for robust uncertainty quantification of fracture networks, where thousands of forward simulations are needed to bound system behavior.In this paper, we develop an alternative network reduction approach to characterizing transport in DFNs, by combining graph theoretical and machine learning methods. We consider a graph representation where nodes signify fractures and edges denote their intersections. Using random forest and support vector machines, we rapidly identify a subnetwork that captures the flow patterns of the full DFN, based primarily on node centrality features in the graph. Our supervised learning techniques train on particle-tracking backbone paths found by dfnWorks, but run in negligible time compared to those simulations. We find that our predictions can reduce the network to approximately 20% of its original size, while still generating breakthrough curves consistent with those of the original network.
Given a planar point set X, we study the convex shells and the convex layers.We prove that when X consists of points independently and uniformly sampled inside a convex polygon with k vertices, the expected number of vertices on the first t convex shells is O(kt log n) for t = O( √ n), and the expected number of vertices on the first t convex layers is O kt 3 log n t 2 . We also show a lower bound of Ω(t log n) for both quantities in the special cases where k = 3, 4. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.
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