Machine learning for flux regression in discrete fracture networks

Abstract

In several applications concerning underground flow simulations in fractured media, the fractured rock matrix is modeled by means of the Discrete Fracture Network (DFN) model. The fractures are typically described through stochastic parameters sampled from known distributions. In this framework, it is worth considering the application of suitable complexity reduction techniques, also in view of possible uncertainty quantification analyses or other applications requiring a fast approximation of the flow through the network. Herein, we propose the application of Neural Networks to flux regression problems in a DFN characterized by stochastic trasmissivities as an approach to predict fluxes.

Graph-Informed Neural Networks for Regressions on Graph-Structured Data

“…Given the stochastic flow networks G 1 and G 2 , the corresponding maximum-flow regression problems consist of the approximation of the functions F 1 : R n → R m , n = 126, m = 15, and F 2 : R n → R m , n = 269, m = 15, respectively, where F 1 and F 2 are defined as in (16). For each i = 1, 2, we build the dataset D i of G i made of 10,000 pairs…”

“…In recent literature, several new methods have been proposed to reduce the computational costs of the DFN flow simulations (e.g., see [55][56][57]); nonetheless, they are still computationally expensive in many situations and the UQ analyses can involve thousands of these simulations. Therefore, it is fundamental to take into account the techniques for the complexity reduction, such as machine learning-based techniques (e.g., see [16,17,58]). In particular, in [16,17], NN models are trained on datasets built using DFN simulations to provide surrogate models; finally, in a negligible amount of time, the NN models are used to generate a large set of approximated DFN flow simulation results, which are particularly useful to speed up the UQ analyses.…”

“…Therefore, it is fundamental to take into account the techniques for the complexity reduction, such as machine learning-based techniques (e.g., see [16,17,58]). In particular, in [16,17], NN models are trained on datasets built using DFN simulations to provide surrogate models; finally, in a negligible amount of time, the NN models are used to generate a large set of approximated DFN flow simulation results, which are particularly useful to speed up the UQ analyses.…”

“…Given a graph G with n nodes, a regression task on graphstructured data based on G consists of approximating a function F : Ω ⊆R n → R m , m ≤ n, depending on the adjacency matrix of G, and that returns the m values related to a fixed subset of m nodes for each set of values assigned to the nodes of G. This type of regression task has applications in many interesting fields, such as circulation with demand (CwD) problems (see [chap. 7.7] in [14]), network interdiction models (NIMs) [15], and flux regression problems in underground fractured media [16,17]. A classic multi-layer perceptron (MLP), or its suitable variants, can perform this regression task on the graph data with a good performance [16,17], implicitly learning the node relationships during the training (see [18,19]).…”

“…7.7] in [14]), network interdiction models (NIMs) [15], and flux regression problems in underground fractured media [16,17]. A classic multi-layer perceptron (MLP), or its suitable variants, can perform this regression task on the graph data with a good performance [16,17], implicitly learning the node relationships during the training (see [18,19]). On the other hand, the current GCNs in the literature are not comparable to MLPs for such a regression task; indeed, as mentioned above, they are designed mainly for other kinds of tasks and, in practice, they cannot exploit deep architectures.…”