Uncertainty Quantification in Discrete Fracture Network Models: Stochastic Geometry

Abstract

We consider the problem of uncertainty quantification analysis of the output of underground flow simulations. We consider in particular fractured media described via the discrete fracture network model; within this framework, we address the relevant case of networks in which the geometry of the fractures is described by stochastic parameters. In this context, due to a possible lack of smoothness in the quantity of interest with respect to the stochastic parameters, well assessed techniques such as stochastic collocation may fail in providing reliable estimates of first‐order moments of the quantity of interest. In this paper, we overcome this issue by applying the Multilevel Monte Carlo method, using as underlying solver an extremely robust method.

Multilevel Monte Carlo Predictions of First Passage Times in Three‐Dimensional Discrete Fracture Networks: A Graph‐Based Approach

“…This estimates are refined using a few more accurate samples that are obtained with more costly estimators. In the framework of numerical solutions of partial differential equations, these levels may correspond to different meshes resolutions, referred to as geometric MLMC (see Berrone et al, 2018, for an example of use in the framework of fracture networks), or they may correspond to different models (see O'Malley et al, 2018, for an example of this methodology applied to fracture networks).…”

“…There are currently no methods available to overcome the computational burdens associated with producing an ensemble of large‐scale DFN model runs that are required for standard MC thus rendering its application to be relatively impracticable. However, there are a number of variants that outperform standard MC, namely, multifidelity MC (Canuto et al, 2019; Ng & Willcox, 2014; O'Malley et al, 2018; Peherstorfer et al, 2016) and multilevel MC (Berrone et al, 2018; Giles, 2015; Lu et al, 2016), that could be modified to overcome these issues. The general idea of these variants is to combine estimates with different levels of accuracy where the quality of the estimate depends on the computational cost; low cost simulations/models produce more data but with lower accuracy.…”

Machine learning for flux regression in discrete fracture networks

“…An efficient parallel implementation has been produced and tested (Berrone et al 2019(Berrone et al , 2015. The method has also been effectively used in massive computations for uncertainty quantification analyses in a geometric Multi Level Monte Carlo framework (Berrone et al 2018), taking advantage of the possibility of using very coarse meshes along all the network, even in presence of critical geometrical configurations.…”

Uncertainty Quantification for a Hydraulic Fracture Geometry: Application to Woodford Shale Data

“…Quantifying the effects of spatial variability in formation properties [8,9] on the reliability of hydraulic fracture simulations has been studied [6,[10][11][12][13] but is restricted by simplified deterministic solutions or computational timescales of numerical solutions. The uncertainty quantification for the simple linear elastic model given by [12] calculates the range of possible fracture apertures using Monte-Carlo simulations where Young's modulus and the confining stress are random parameters, and they consequently reformulate the governing equations as stochastic partial differential equations (SPDEs).…”

“…Furthermore, [12] were inspired by [8] who developed the stochastic framework to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales and present the solution as a polynomial approximation. In a recent publication, [13] address variability in discrete fracture network geometry using the very robust multilevel Monte-Carlo methods, rather than the stochastic methodology claiming that the classic stochastic collocation fails to provide reliable estimates of first-order moments of the output solution due to the lack of smoothness. In another recent publication, [14] proposes the (MLMC) methods in conjunction with a wellassessed underlying solver to perform DFN flow simulations using Darcy flow and prove that the method is robust enough to tackle complex geometrical configurations, even with very coarse meshes.…”