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Exploratory data analysis (EDA) for functional data—data objects where observations are entire functions—is a difficult problem that has seen significant attention in recent literature. This surge in interest is motivated by the ubiquitous nature of functional data, which are prevalent in applications across fields such as meteorology, biology, medicine and engineering. Empirical probability density functions (PDFs) can be viewed as constrained functional data objects that must integrate to one and be nonnegative. They show up in contexts such as yearly income distributions, zooplankton size structure in oceanography and in connectivity patterns in the brain, among others. While PDF data are certainly common in modern research, little attention has been given to EDA specifically for PDFs. In this paper, we extend several methods for EDA on functional data for PDFs and compare them on simulated data that exhibit different types of variation, designed to mimic that seen in real‐world applications. We then use our new methods to perform EDA on the breakthrough curves observed in gas transport simulations for underground fracture networks. [ABSTRACT FROM AUTHOR]

Understanding mechanistic causes of non‐Fickian transport in fractured media is important for many hydrogeologic processes and subsurface applications. This study elucidates the effects of dead‐end fractures on non‐Fickian transport in three‐dimensional (3D) fracture networks. Although dead‐end fractures have been identified as low‐velocity regions that could delay solute transport, the direct relation between dead‐end fractures and non‐Fickian transport has been elusive. We systematically generate a large number of 3D discrete fracture networks with different fracture length distributions and fracture densities. We then identify dead‐end fractures using a novel graph‐based method. The effect of dead‐end fractures on solute residence time maximizes at the critical fracture density of the percolation threshold, leading to strong late‐time tailing. As fracture density increases beyond the percolation threshold, the network connectivity increases, and dead‐end fractures diminish. Consequently, the increase in network connectivity leads to a reduction in the degree of late‐time tailing. We also show that dead‐end fractures can inform about main transport paths, such as the mean tortuosity of particle trajectories. This study advances our mechanistic understanding of solute transport in 3D fracture networks. Plain Language Summary: A critical phenomenon often observed in fractured rocks is an anomalously slow decay of solute concentration, which is often called solute tailing. Long retention of solute contaminants poses severe threats to human health, and predicting solute tailing is critical for remediation purposes. However, solute tailing is challenging to predict, and understanding the mechanisms inducing solute tailing is important for improving our predictive capability. Recent studies showed that dead‐end fractures form slow‐flow regions that could cause solute tailing. However, no study has explicitly quantified the effects of dead‐end fractures on solute tailing. By simulating flow and transport in three‐dimensional fracture networks over more than 200 realizations, we find that the relationship between fracture density and dead‐end fracture formation is non‐monotonic. Specifically, we observe that there exists a critical level of fracture density where the presence of dead‐end fractures is maximized. At the optimum level, solute particles' residence time in dead‐end fractures significantly increases, leading to strong solute tailing. As fracture density increases beyond the optimum, dead‐end fractures diminish due to the improved network connectivity, reducing solute tailing. This study improves our mechanistic understanding of how dead‐end fractures control solute tailing in fracture networks. Key Points: A new graph‐based measure that identifies dead‐end fractures is proposedLate‐time solute tailing is maximized at the percolation thresholdThe density of dead‐end fractures is a good predictor of non‐Fickian transport [ABSTRACT FROM AUTHOR]

We present a graph-based method to identify primary flow and transport subnetworks in three-dimensional discrete fracture networks (DFNs). The structure of a DFN lends itself to the use of graphs as a coarse-scale representation that retains the multiscale nature of flow and transport through fracture media. We develop a bipartite graph representation that integrates fracture network topology, fracture geometry, and hydraulic properties. We show that the two most common graph-representations of DFNs, vertices representing intersections and vertices representing fractures, are projections of this bipartite graph thereby providing a generalization of previous DFN-graph frameworks. The primary subnetworks in each DFN are identified by running a heuristic algorithm that determines the edge-disjoint shortest paths through the graph which correspond to the regions where the fastest transport occurs. The method does not have any user-defined parameters and terminates in a finite number of steps. The quality of the method is demonstrated by comparing transport simulations on the identified primary subnetwork and full network, which are in good agreement for early and middle times. These estimates of the first passage times can be achieved with close to an order of magnitude reduction of computational expense using the proposed method. [ABSTRACT FROM AUTHOR]

Motivated by a triaxial coreflood experiment with a sample of Utica shale where an abrupt jump in permeability was observed, possibly due to the creation of a percolating fracture network through the sample, we perform numerical simulations based on the experiment to characterize how the effective permeability of otherwise low-permeability porous media depends on fracture formation, connectivity, and the contrast between the fracture and matrix permeabilities. While a change in effective permeability due to fracture formation is expected, the dependence of its magnitude upon the contrast between the matrix permeability and fracture permeability and the fracture network structure is poorly characterized. We use two different high-fidelity fracture network models to characterize how effective permeability changes as percolation occurs. The first is a dynamic two-dimensional fracture propagation model designed to mimic the laboratory settings of the experiment. The second is a static three-dimensional discrete fracture network (DFN) model, whose fracture and network statistics are based on the fractured sample of Utica shale. Once the network connects the inflow and outflow boundaries, the effective permeability increases non-linearly with network density. In most networks considered, a jump in the effective permeability was observed when the embedded fracture network percolated. We characterize how the magnitude of the jump, should it occur, depends on the contrast between the fracture and matrix permeabilities. For small contrasts between the matrix and fracture permeabilities the change is insignificant. However, for larger contrasts, there is a substantial jump whose magnitude depends non-linearly on the difference between matrix and fracture permeabilities. A power-law relationship between the size of the jump and the difference between the matrix and fracture permeabilities is observed. The presented results underscore the importance of fracture network topology on the upscaled properties of the porous medium in which it is embedded. [ABSTRACT FROM AUTHOR]

The apertures of natural fractures in fractured rock are highly heterogeneous. However, in-fracture aperture variability is often neglected in flow and transport modeling and individual fractures are assumed to have uniform aperture distribution. The relative importance of in-fracture variability in flow and transport modeling within kilometer-scale field–scale fracture networks has been under a matter of debate for a long time because the flow in each single fracture is controlled not only by in-fracture variability but also by boundary conditions. Computational limitations have previously prohibited researchers from investigating the relative importance of in-fracture variability in flow and transport modeling within large-scale fracture networks. We address this question by incorporating internal heterogeneity of individual fractures into flow simulations within kilometer scale three-dimensional fracture networks, where fracture intensity, P 32 (ratio between total fracture area and domain volume) is between 0.027 and 0.031 [1/m]. A recently developed discrete fracture network (DFN) simulation capability, dfnWorks , is used to generate DFNs that include in-fracture aperture variability represented by a stationary log-normal stochastic field with various correlation lengths and variances. The Lagrangian transport parameters, non-reacting travel time and cumulative retention, are calculated along particles streamlines. It is observed that due to local flow channeling early particle travel times are more sensitive to in-fracture variability than the tails of travel time distributions, where no significant effect of the in-fracture transmissivity variations and spatial correlation length is observed. [ABSTRACT FROM AUTHOR]

dfn W orks is a parallelized computational suite to generate three-dimensional discrete fracture networks (DFN) and simulate flow and transport. Developed at Los Alamos National Laboratory over the past five years, it has been used to study flow and transport in fractured media at scales ranging from millimeters to kilometers. The networks are created and meshed using dfn G en , which combines fram (the feature rejection algorithm for meshing) methodology to stochastically generate three-dimensional DFNs with the L a G ri T meshing toolbox to create a high-quality computational mesh representation. The representation produces a conforming Delaunay triangulation suitable for high performance computing finite volume solvers in an intrinsically parallel fashion. Flow through the network is simulated in dfn F low , which utilizes the massively parallel subsurface flow and reactive transport finite volume code pflotran . A Lagrangian approach to simulating transport through the DFN is adopted within dfn T rans to determine pathlines and solute transport through the DFN. Example applications of this suite in the areas of nuclear waste repository science, hydraulic fracturing and CO 2 sequestration are also included. [ABSTRACT FROM AUTHOR]

In this work, we compare hydrocarbon production curves obtained from a graph-based reduced-order model with the high-fidelity Discrete Fracture Network (DFN) predictions for a fracture network created using data from a real shale site. We observe that the bounds for the high fidelity DFN model lie within the bounds for the reduced order model, implying that the reduced-order model provides a conservative estimate. Moreover, we found that except for first-passage times and late arriving mass, the production curves from the reduced-order model predict transport accurately. However, it is to be noted that the results are inspite of trading a three-dimensional geometry for a reduced system in the form of a graph, one that is 500–1000 times faster in terms of computational efficiency (for this particular application). In addition, we also compare the production curves for large drawdown and small drawdown using our graph approach. The reduced-order model is successful in showing that the long term productivity is higher in case of small drawdown although the initial productivity is higher for large drawdown. Thus, this reduced-order model offers great potential in uncertainty quantification for production, as well as in providing operators with information to make real-time decisions for optimal production. • Flow and transport simulation on DFN mesh is replaced with flow and transport simulation on a graph representation of the DFN. • Graph theory provides a much faster and elegant way for model order reduction of DFNs. • Production curves for the reduced-order model match well with the high fidelity DFN model except early times and late times. [ABSTRACT FROM AUTHOR]

A quantitative assessment of how faults control the migration of geofluids is critical in many areas of geosciences. We integrated geological fieldwork, quantitative analysis of the fractures distribution and numerical modeling to build a geometrical representation of a fault zone and to characterize its hydraulic properties. Our target is a fault located in the Majella Mountain (Italy). We collected 21 scan lines across the fault profile in order to characterize its architecture. The numerical modeling of the fracture network of the damage zones and their hydraulic parameters was performed using both commercial (Move®) and open source software (dfnWorks and PFLOTRAN). Move® was used to build a representative model of the fault zone using fracture spacing as a proxy, and to model the hydraulic parameters of the different fault domains. dfnWorks and PFLOTRAN were employed to infer the hydraulic parameters of the damage zones of the fault and then upscale these properties to an equivalent continuum domain, suitable for fluid flow simulations through the whole fault zone. Our findings show how even in a relatively small area it is possible to describe changes in terms of hydraulic properties of a fault zone and to build models capable to represent these variations. • We integrate geological fieldwork, quantitative analysis of fractures and numerical modeling to infer the hydraulic properties of a fault zone. • Our results stress the value of detailed field investigations in the hydraulic characterization of fault zones. • The use of open source software like dfnWorks and PFLOTRAN allows for a wide control of the parameters used to build a numerical model of the fault. • The proposed workflow has potential applications in several fields, especially when is crucial to assess the impact of faults on fluid migration. [ABSTRACT FROM AUTHOR]