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

Abstract

We present a method combining multilevel Monte Carlo (MLMC) and a graph‐based primary subnetwork identification algorithm to provide estimates of the mean and variance of the distribution of first passage times in fracture media at significantly lower computational cost than standard Monte Carlo (MC) methods. Simulations of solute transport are performed using a discrete fracture network (DFN), and instead of using various grid resolutions for levels in the MLMC, which is standard practice in MLMC, we identify a hierarchy of subnetworks in the DFN based on the shortest topological paths through the network using a graph‐based method. While the mean of these ensembles is of critical importance, the variance is also essential in fractured media where uncertainty is an overarching theme, and understanding variability across an ensemble is a requirement for safety assessments. The method provides good estimates of the mean and variance at two orders of magnitude lower computational cost than MC.

Graph-based flow modeling approach adapted to multiscale discrete-fracture-network models

“…If the flow is adequately estimated by the graph-based models, they can be used to reduce the cost of DFN modeling with little loss of information, thus making possible the calculation of flow in DFN models with fracture sizes spanning multiple orders of magnitude. The practical applications with graph-based models are to simplify the DFN structure before calculating flow or transport properties (e.g., selecting the network backbone or important paths), or to consider very large ensembles of DFN models with both high-and low-fidelity descriptions to quantify the variability and uncertainty in these systems (O'Malley et al [33] and Berrone et al [34]).…”

“…They have been successfully used to identify the DFN backbone, i.e., fractures participating to shortest travel times or carrying the major flows [27][28][29][30][31]. They have also been used to directly solve flow and transport [32] in order to reduce the cost associated with uncertainty quantifications (e.g., multifidelity Monte Carlo, O'Malley et al [33], and multilevel Monte Carlo, Berrone et al [34]). In these recent studies, simple analytical definitions were given as edges hydraulic attributes to preserve the rapidity (quasi-immediate) of the graph computations.…”

From Fluid Flow to Coupled Processes in Fractured Rock: Recent Advances and New Frontiers

“…This dimension reduction drastically reduces the number of degrees of freedom in the solution matrix increasing the computational efficiency. There has been a recent re‐interest in these CN models, which are now commonly referred to as graph‐based emulators (see Section 5), as they allow for significantly more fractures to be included in networks compared to high‐fidelity DFN models, which in turn facilitates and allows for many more simulations to be performed at low cost to bound uncertainty (Berrone et al., 2020; Doolaeghe et al., 2020; Hyman et al., 2018; Karra et al., 2018; O’Malley et al., 2018; Osthus et al., 2020), a topic which is discussed further in the next section. The CN approach is attractive when there are a large number of fractures or when the fracture intersections or solution enhanced permeability are a dominant factor in determining where flow and transport occur within the network.…”

The numerical simulation of rock mass grouting: a literature review

“…These additional features increase the cost of a DFN model run. Berrone et al (2018Berrone et al ( , 2020 used multilevel Monte Carlo and graph theory to solve the problems of time-consuming and excessive assumptions in DFN modeling and used this model to simulate the grouting process of the rock mass. Mohajerani et al (2017Mohajerani et al ( , 2018 proposed calculating the grout permeability of rock joints and fissures based on the pressure before grouting, extended this method to 2D and 3D random fissure networks, and solved the actual diffusion path of grout in fissures by recursion.…”