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Accurately analyzing the flow and transport behavior in a large discrete fracture network is computationally expensive. Fortunately, recent research shows that most of the flow and transport occurs within a small backbone in the network, and identifying the backbone to replace the original network can greatly reduce computational consumption. However, the existing machine learning based methods mainly focus on the features of the fracture itself to evaluate the importance of the fracture, the local structural information of the fracture network is not fully utilized. More importantly, these machine learning methods can neither control the identified backbone's size nor ensure the backbone's connectivity. To solve these problems, a deep learning model named multi-aggregator graph neural network (MA-GNN) is proposed for identifying the backbone of discrete fracture networks. Briefly, MA-GNN uses multiple aggregators to aggregate neighbors' structural features and thus generates an inductive embedding to evaluate the criticality score of each node in the entire fracture network. Then, a greedy algorithm, which can control the backbone's size and connectivity, is proposed to identify the backbone based on the criticality score. Experimental results demonstrate that the backbone identified by MA-GNN can recover the transport characteristics of the original network, outperforming state-of-the-art baselines. In addition, MA-GNN can identify influential fractures with higher Kendall's τ correlation coefficient and Jaccard similarity coefficient. With the ability of size control, our proposed MA-GNN can provide an effective balance between accuracy and computational efficiency by choosing a suitable backbone size. [ABSTRACT FROM AUTHOR]

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]

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The greatest challenges of rigorously modeling coupled hydro-mechanical processes in fractured rocks at different scales are associated with computational geometry. In addition, selections of continuous or discontinuous models, physical laws, and coupling priorities at different scales based on different geometric features determine the applicability of a numerical model for a certain type of problem. In this study, we present our multi-scale modeling capabilities that have been developed based on the numerical manifold method for analyzing coupled hydro-mechanical processes in fractured rocks. Based on their geometric features, the fractures are modeled as continua—finite-thickness porous zones, and discontinua—discontinuous interfaces and microscale asperities and granular systems. Different governing equations, physical laws, coupling priorities, and approaches for addressing fracture intersections and shearing are then applied to describe these. We applied these models to simulate coupled processes in fractured rocks using realistic geometry obtained from rock images at different scales. We first calculated shearing of a single fracture with different models and demonstrated the impacts of asperities on shearing. We then applied the continuous and discontinuous models to simulate a network of rough fractures, demonstrating that contact dynamics contribute significantly to the geometric, multi-physical evolution of systems where rough fractures are not mineral filled. For a discrete fracture network, our coupled processes modeling demonstrates that shearing of the discrete fractures can have a major impact on stress and pore pressure distribution. Lastly, we applied the discontinuous granular model to simulate evolution of a complex granular system with a deformation band, demonstrating that the deformation band can dominate contact dynamics, the structural and the stress evolution of the granular system.

The greatest challenges of rigorously modeling coupled hydro-mechanical processes in fractured rocks at different scales are associated with computational geometry. In addition, selections of continuous or discontinuous models, physical laws, and coupling priorities at different scales based on different geometric features determine the applicability of a numerical model for a certain type of problem. In this study, we present our multi-scale modeling capabilities that have been developed based on the numerical manifold method for analyzing coupled hydro-mechanical processes in fractured rocks. Based on their geometric features, the fractures are modeled as continua—finite-thickness porous zones, and discontinua—discontinuous interfaces and microscale asperities and granular systems. Different governing equations, physical laws, coupling priorities, and approaches for addressing fracture intersections and shearing are then applied to describe these. We applied these models to simulate coupled processes in fractured rocks using realistic geometry obtained from rock images at different scales. We first calculated shearing of a single fracture with different models and demonstrated the impacts of asperities on shearing. We then applied the continuous and discontinuous models to simulate a network of rough fractures, demonstrating that contact dynamics contribute significantly to the geometric, multi-physical evolution of systems where rough fractures are not mineral filled. For a discrete fracture network, our coupled processes modeling demonstrates that shearing of the discrete fractures can have a major impact on stress and pore pressure distribution. Lastly, we applied the discontinuous granular model to simulate evolution of a complex granular system with a deformation band, demonstrating that the deformation band can dominate contact dynamics, the structural and the stress evolution of the granular system.

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 formulate a finite‐size scaling hypothesis to predict the global permeability of fracture networks. To validate the hypothesis, numerous discrete fracture networks are generated, and the permeability is numerically calculated. Results suggest that the dimensionless permeability, scaled by moments of local conductivity and fracture sizes and corrected by two stereological ratios, can capture variations in fracture attributes (orientations, sizes, and apertures). The universal form obtained in this study can also explain the contradictory observations where the permeability decreases or increases with the domain size of fracture networks. We show that how a clear transition point, where the permeability does not change with the domain size, is obtained from this universal form. This study provides a solid theoretical foundation to understand the connection between fracture attributes and field‐scale hydraulic properties. Plain Language Summary: Knowledge of the permeability of fractured rocks is critical for many geo‐engineering applications (such as deep geological disposal of radioactive wastes, mining and dam engineering) and environmental problems (such as ground water and pollutant transport). The influence of fracture attributes (e.g., orientations, sizes, and apertures) on hydraulic properties of rock masses is significant, but the task of correlating the permeability to individual fracture data is still a challenging one with remarkable knowledge gaps. In this letter, instead of using analytical solutions, we develop a finite‐size scaling (FSS) function, inspired by the percolation theory in statistic physics, to describe the permeability of fracture networks. The universality of the scaling function is validated by massive simulation data which cover a wide range of fracture attributes. Results demonstrate that, regardless of how domain sizes and fracture attributes change, the scaling for the permeability is of an invariant form, after considering the finite‐size effect. Our findings motivate further studies of applying the FSS function at the field scale and improve on the information that earth scientists can obtain from fracture statistics. Key Points: A finite‐size scaling hypothesis for the permeability of discrete fracture networks (DFNs) is formulated with several dimensionless quantitiesNumerous DFN representations and flow simulations are implemented to validate the scaling hypothesisThe scaling is found universal for different DFNs with an invariant form [ABSTRACT FROM AUTHOR]

Abstract We formulate a finite‐size scaling hypothesis to predict the global permeability of fracture networks. To validate the hypothesis, numerous discrete fracture networks are generated, and the permeability is numerically calculated. Results suggest that the dimensionless permeability, scaled by moments of local conductivity and fracture sizes and corrected by two stereological ratios, can capture variations in fracture attributes (orientations, sizes, and apertures). The universal form obtained in this study can also explain the contradictory observations where the permeability decreases or increases with the domain size of fracture networks. We show that how a clear transition point, where the permeability does not change with the domain size, is obtained from this universal form. This study provides a solid theoretical foundation to understand the connection between fracture attributes and field‐scale hydraulic properties.

Heat transport in fractured aquifers is determined by the combined effects of flow velocity heterogeneity in the fracture system, and diffusive exchange between the fluid in the fractures and the rock matrix, which can be assumed as impervious. We analyze the impacts of this diffusive exchange on the response to heat transport, as opposite to the pure advective displacement, which governs solute transport. We focus on the post‐peak behavior where we observe pre‐asymptotic regimes with slopes that differ from the signature of matrix diffusion, which exhibits a decay rate of −3/2. This deviation is driven by the variability of both velocity field and fracture aperture field. We derive theoretical models that predict these pre‐asymptotic tails under three extreme cases that can be related with specific network structures, that is, networks dominated by large or small fractures, networks with highly or poorly channelized flow. These theoretical predictions are compared with results from numerical simulations in different sets of three‐dimensional discrete fracture networks. We determine that the combined observation of solute and heat transport responses allows classifying the network in terms of connectivity structure, and partially characterizing the fracture aperture variability in terms of upscaled parameters. Key Points: Heat transport in heterogeneous fractured aquifers exhibits post‐peak tails that express the characteristics of the fracture networkDFN simulations show that heat BTC transition into diffusive regime depends whether the system is dominated by large or small fracturesThe combined observation of solute and heat decay rates allows characterizing connectivity structure and fracture aperture variability [ABSTRACT FROM AUTHOR]

Since natural fractures are often non-equidimensional, the circular disc model still has great limitations. By contrast, the elliptical disc model is more applicable to representing natural fractures, especially for slender ones. This paper developed a universal elliptical disc (UED) model by incorporating the center point, size, and azimuth of fractures as variables. Specifically, with respect to the azimuth of elliptical fractures in three-dimensional (3D) space, we proposed a paradigm to construct its probability density function (PDF) by coupling the orientation and rotation angle of long axis based on three coordinate transformations. To illustrate the construction process of the PDF of the fracture azimuth, we took the orientation following the Fisher distribution and the rotation angle following Von Mises distribution as an example. A rock slope is used to show the use of the developed UED model, and the 3D DFNs for the slope rock mass are generated by Monte Carlo simulation. In addition, the DFNs for the rock mass are also generated based on the existing circular disc model and non-universal elliptical disc model. The comparison results from the three models clearly illustrate the superiority of the UED model over the existing circular and non-universal elliptical disc models.

Modeling gas flow through fractures of subsurface rock is a particularly challenging problem because of the heterogeneous nature of the material. High-fidelity simulations using discrete fracture network (DFN) models are one methodology for predicting gas particle breakthrough times at the surface but are computationally demanding. We propose a Bayesian machine learning method that serves as an efficient surrogate model, or emulator, for these three-dimensional DFN simulations. Our model trains on a small quantity of simulation data with given statistical properties and, using a graph/path-based decomposition of the fracture network, rapidly predicts quantiles of the breakthrough time distribution on DFNs with those statistical properties. The approach, based on Gaussian Process Regression (GPR), outputs predictions that are within 20%–30% of high-fidelity DFN simulation results. Unlike previously proposed methods, it also provides uncertainty quantification, outputting confidence intervals that are essential given the uncertainty inherent in subsurface modeling. Our trained model runs within a fraction of a second, considerably faster than reduced-order models yielding comparable accuracy (Hyman et al., 2017; Karra et al., 2018) and multiple orders of magnitude faster than high-fidelity simulations. • We predict gas flow in 3D fracture networks using Gaussian process regression. • Our novel machine learning approach trains on high-fidelity simulation data. • Predictions of particle arrival times are within 20%–30% of high-fidelity results. • The method is orders of magnitude faster than high-fidelity simulations. • We provide confidence intervals, crucial given uncertainties in subsurface modeling. [ABSTRACT FROM AUTHOR]

The representative elementary volume (REV) of a fractured rock mass is crucial for evaluating the equivalent continuum approach. An innovative computational framework and a harmony dimension method were proposed to estimate and predict the REV, respectively. These methods were applied to a slope along a road. Initially, a high-fidelity 3D discrete fracture network (DFN) was generated using data from unmanned aerial vehicle photogrammetry. Then, the Möller–Trumbore algorithm and Stokes' theorem were extended for fracture intersection analysis and intensity (P 32) calculation. Subsequently, an equivalent porous medium model was developed. These components were integrated into a framework to calculate the P 32 and equivalent permeability of DFNs of varying sizes, thus determining the optimal REV. Additionally, the harmony dimension method, based on the Levenberg–Marquardt algorithm, was used to predict the relationship between two-dimensional (2D) and 3D DFN properties. This method underwent validation with 10 Poisson processes and 570 percolation simulations. The results show that REV sizes vary with different hydraulic gradients, highlighting the anisotropic nature of 3D fractured media. REV predictions can be made using the variability of 2D parameters. The proposed framework accurately captures geometric and hydraulic behaviors of fractured rock masses with reduced computational cost, while the harmony dimension method simplifies and accelerates prediction. The novel finding of the 2D 3D parameter relationship can streamline DFN modeling and analysis. • The Möller–Trumbore algorithm and Stokes' theorem are extended. • A secondary development EPM model is implemented. • An advanced computational framework for estimating REVs is proposed. • The link between 2D parameters and 3D properties of DFNs is explored. • A harmony dimension method with the Levenberg–Marquardt algorithm is introduced to predict REVs. [ABSTRACT FROM AUTHOR]

Fractures greatly impact hydrocarbon exploration as they modify fluid flow properties within reservoir rocks, creating an interconnected network. The hydrocarbon reservoirs are often difficult to assess, and the methods employed in acquiring information from these locations offer too sparse data or have a low spatial resolution. Otherwise, outcrops allow fracture characterization directly in the field or using 2D and 3D digital representations of outcrops. These fracture networks, usually related to fractal propagation and power-law distribution parameters, can be used as data sources providing useful information when properly adjusted to the reservoir simulation scale. In this sense, attribute estimators, like the Maximum Likelihood Estimator (MLE) and algorithms using MLE, have been widely used for their robustness when compared to linear regression estimators. However, due to the challenges in the power-law characterization, such as the large fluctuations that occur in the tail of the distribution, non-optimum values can be obtained despite the effectiveness of the MLE. Our work proposes the use of an optimization algorithm based on Stochastic Gradient Descent (SGD) with momentum to obtain best-fitting parameters for power-law distributions. The proposed method was first evaluated with synthetic data and several goodness-of-fitness metrics and later using empirical data obtained from fracture characterization in the Digital Outcrop Model (DOM) of a reservoir analogue outcrop. Stochastic DFN sampling based on empirical data was also used to simulate censoring effects. The results showed that the SGD method provided better distribution fitting than other methods based on the MLE when using empirical data while presenting reduced bias when using synthetic data. The estimation of power-law parameters in stochastic DFN data also presented the best-fitting results when applying the proposed method. In conclusion, the proposed optimization method proved a valuable alternative to estimate power-law distributions. • Optimization algorithms like the SGD can achieve better results than the MLE. • KS hypothesis test proved a robust cost function and with rapid convergence. • The SGD presents lower bias and values closer to the expected in the synthetic data. • The SGD presents a better fit to unknown power-law distributions in DFN's length data. • Parallel computing allowed a speedup of the SGD. [ABSTRACT FROM AUTHOR]

• Fracture orientation, density and aperture control solute transport. • Preferential flow paths linked to aperture variability, not network connectivity. • Mega column fractures enhance fluid flow and facilitate fast breakthrough times. • Fracture aperture variability decreases arrival times of the center of plume mass. Numerical simulations explore for the first time the role of mega columns and aperture variability on particle transport through mature volumetric contraction networks as informed by a unique synthesis of network propagation and maturity. Columnar fracture patterns are generated by updating a series of Voronoi centers to the midpoint of a generated polygon over many iterations, creating 250 network realizations. A DFN simulator solves for fluid flow and tracks conservative particle trajectories within each network. Dominant fracture attributes impacting fluid flow and solute transport in volumetric contraction networks are fracture orientation, density, and aperture/transmissivity. Ensemble plume snapshots generated by networks with equal fracture transmissivity define a baseline-level of dispersion that is solely attributed to network structure and connectivity. Longitudinal and transverse dispersion increase and the center of plume mass becomes delayed relative to the baseline case when fracture transmissivity is varied according to a lognormal distribution. The incorporation of highly-transmissive, large-aperture mega column fractures leads to plume snapshots with a more pronounced leading edge and an order of magnitude faster average breakthrough times. The breakthrough curves contain three peaks reflecting contrasting transport pathways in which particles are: (i) initially placed in mega column fractures and remain in these features until exiting the model domain, (ii) initially placed into small column fractures, incur additional time to migrate and enter a mega column fracture, and remain within those mega columns, and (iii) initially placed in small column fractures and remain in these fractures. Incorporating variability in fracture transmissivity for both small column and mega column fractures disrupts the binary distinction between small column and mega column fracture velocities and leads to dispersed breakthroughs over long time scales with a single peak. These results demonstrate that preferential flow paths emerge in volumetric contraction networks due to contracts in fracture transmissivity, not fracture connectivity as observed in tectonic networks. [ABSTRACT FROM AUTHOR]

In this study we use numerical experiments to analyze reactive flow and transport behavior in discrete intersecting fracture networks, focusing on (a) how reaction‐induced changes in physical and chemical properties affect flow connectivity and (b) how fracture networks developed in the Earth's critical zone contribute to carbon sequestration via mineral weathering reactions. In the first part of the study, we used two‐dimensional reactive flow and transport simulations to analyze the impacts of mixing in a natural discrete fracture network. We concluded that reaction‐induced changes can substantially alter the flow connectivity, especially at fracture intersections. The second set of simulations considered the problem of natural weathering of fractured mafic and ultramafic rocks in the partially saturated Earth's critical zone as a function of infiltration rates, fracture permeability, and partially saturated flow parameters. As a model system, we considered an incongruent reaction network with dissolution of forsterite and precipitation of magnesite. The behavior is complex in terms of the rate‐controlling processes because of the multicomponent nature of the system as shown by the grid Peclet number: the CO2 behavior is gas diffusion‐controlled in the partially saturated zone, while the rate of water flow via the Damkӧhler number controls Mg2+ transport through the fracture network. The amounts of carbon that can be trapped are modest, but the naturally fractured domain considered here provides a useful "base case" against which various engineered solutions can be compared. Plain Language Summary: In this study we use numerical experiments to analyze reactive flow and transport behavior in discrete intersecting fracture networks, focusing on (a) how the interplay between intersections of rough fractures and advection, diffusion, and chemical reaction affects flow connectivity via changes in geometry and physical and chemical properties and (b) how fracture networks developed in the Earth's critical zone contribute to carbon sequestration via mineral weathering reactions. In the first part of the study, we used reactive flow and transport simulations to analyze the impacts of mixing in a natural discrete fracture network. We concluded that reaction‐induced changes can substantially alter the flow connectivity, especially at fracture intersections. The second set of simulations considered the problem of natural weathering of fractured mafic and ultramafic rocks in the partially saturated Earth's critical zone as a function of infiltration rates, fracture permeability, and partially saturated flow parameters. The behavior is complex in terms of the rate‐controlling processes because of the multicomponent nature of the system. The amounts of carbon that can be trapped are modest, but the naturally fractured domain considered here provides a useful "base case" for the future design of efficient strategies for carbon trapping and mitigation. Key Points: Modeling of the interplay between intersecting rough fracture networks and advection, gas diffusion, and mineral dissolution/precipitationMineral precipitation due to mixing at fracture intersections can alter the flow connectivity of a fracture networkModeling demonstrates CO2 trapping in naturally fractured rocks within the critical zone, a base case for engineered technologies [ABSTRACT FROM AUTHOR]

A large percentage of subsurface hydrocarbon reservoirs are characterized as very complex due to presence of small to large scale fractures. Modeling of multi-physics processes like, environmental flow, CO2 sequestration, hydrocarbon flows, and so on, through such geologically complex reservoirs is challenging. Main challenges comes from the large scale variation in the fracture scales. Use of traditional modeling approaches, based on dual-porosity/dual permeability medium, to model such complex systems is complicated and results in incorrect flow patterns. Precise and efficient modeling of the fracture networks requires fractures to be represented as lower dimensional objects, which requires an efficient gridding technique. In last decade alone modeling of flow through discrete fracture systems has attracted attention from a number of researchers. As a result few new gridding and discretization techniques have been proposed to model flow through discrete fracture network systems (DFNs). DFN's usually involve very high or very low angle fracture-fracture intersections and sometime presence of small to very large length scale fracture networks. In this article an efficient workflow for meshing of large scale complex DFN's networks as lower dimensional objects is presented supported by quantitative results with aim of developing a software tool box, which could be coupled to a subsurface multi-phase flow simulator. [ABSTRACT FROM AUTHOR]

We investigate the computational performance of hybrid high-order methods applied to flow simulations in extremely large discrete fracture networks (over one million of fractures). We study the choice of basis functions, the trade-off between increasing the polynomial order and refining the mesh, and how to take advantage of polygonal cells to reduce the number of degrees of freedom. [ABSTRACT FROM AUTHOR]

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time. [ABSTRACT FROM AUTHOR]

Tunnels commonly go through fracture zones that used to be analyzed as an equivalent porous medium with homogeneous permeability. However, it is a rough simplification that overlooks the connection triggered by underground works in fractured massifs. This study introduces the use of synthetic discrete fracture networks (DFN) to analyze groundwater inflows through tunnel excavation in a fractured zone considering the daily advance of the drilling front. First, a hypothetical case with six different settings varying the fracture density, the fracture length, and the aperture distribution is analyzed. Each setting has about 100 iterations. DFN hydraulic properties were estimated and compared with previous DFN studies, displaying the same behavior even though the magnitude of the estimated parameters differs. As an application example, structural measurements of the Alaska fault zone in the La Linea massif (Colombia) are used to obtain the statistical parameters of the fracture length and aperture distributions to generate the DFN. Five settings varying the fracture density are built, obtaining measured and simulated groundwater inflows of the same order of magnitude. These results highlight the potential of the synthetic DFN to analyze tunnels’ effects on groundwater flow.

• A two-phase flow model coupled with geomechanics to investigate the transient flow behaviour. • Using dynamic fracture properties improves the model accuracy at early-time period. • Physical constraints in a gas reservoir are considered to capture its actual flow dynamics. • Triangular/rectangular fracture geometries exhibit the lowest/largest deformation, respectively. • Effects of fracture properties/capillary pressure dominate flow behaviour at early stage. A robust and efficient two-phase flow model integrated with geomechanics has been developed, validated, and subsequently utilized to investigate the transient flow characteristics in an unconventional reservoir with consideration of fracture geometry and capillary pressure. As for a fracture subsystem, various hydraulic fracture (HF) geometries are applied to capture a loss in the fracture conductivity due to an alteration in the effective normal stress on the fracture surface during the flowback period. Different from the traditional treatment by assuming it as a constant, a function of interfacial tension between gas and water as well as fracture aperture is employed to obtain the capillary pressure within a fracture, during which the gas/water saturation and the fracture aperture in each fracture segment with an equal length can be iteratively obtained and updated. Not only does the proposed model offer the flexibility to investigate the effects of dynamic fracture properties (i.e., aperture and conductivity) and capillary pressure on the two-phase flow behaviour in the fractures with various fracture geometries, but also its precision has been validated through numerical simulations and subsequently applied in the field cases with outstanding performance. Dynamic fracture properties and capillary pressure are found to exert considerable influences on the gas/water daily production rates and cumulative production at the early time flow. It is worthwhile noting that capillary pressure shows a more pronounced effect on the transient flow behaviour due to a reduction in fracture aperture during production. Moreover, the fracture with a triangular shape exhibits the lowest gas/water production rate, whereas a rectangular fracture results in the largest gas/water production rate. On the basis of fracture geometry and geomechanics, the former shows the smallest deformation induced by the effective stress, and the latter suffers the largest deformation under the same stress condition. In addition, a poorly propped HF exhibits a large aperture reduction, resulting in a reduction in gas/water production rate. As for field cases, the gas/water production rate obtained from the two-phase flow model demonstrate a strong agreement with the observed data, providing further confirmation of the dependability and robustness of the proposed model. [ABSTRACT FROM AUTHOR]

The upscaling of mechanical properties of fractured media requires the definition of an appropriate size for the Representative Elementary Volume (REV). Because of the stochastic nature of the fracture networks, the REV size is not deterministic and should be defined based on the variability of the equivalent properties. This work presents a new general methodology to define the size of the REV for the geometrical and elastic moduli of fractured media. Following previous works on heterogeneous materials, the decision criterion is based on the precision error that arises from the statistical theory of samples. The proposed methodology also relies on the use of the Central Limit Theorem (CLT) to assess the REV of fractured rocks. The CLT is shown to theoretically apply to both the geometrical and the elastic equivalent properties. From that observation, a general equation is drawn to predict the variance of an equivalent property for any REV candidate size, provided that the variance for one size only is known. These concepts are tested using numerous finite element simulations to obtain the distribution of the equivalent elastic moduli of two-dimensional samples containing two fracture networks previously studied for their elastic properties. These properties are confirmed to tend to a normal distribution, as stated by the CLT. Also, the standard deviations associated with the tested REV sizes were predicted with accuracy from the standard deviation obtained in the numerical simulations of only one proper reference volume. The mechanical REV was compared with the geometrical REV, which is based on the first invariant of the fracture tensor. In addition, to reduce computational costs, a procedure to reduce the number of simulations of the reference volume was proposed. A preliminary verification of the applicability of the methodology to non-elastic problems was made. Proper predictions were obtained for the standard deviation of the compression strength calculated in two studies that considered, altogether, both two-dimensional and three-dimensional samples, as well as plastic and damage models. [ABSTRACT FROM AUTHOR]

A new numerical scheme is proposed for flow computation in complex discrete fracture networks. The method is based on a three-field domain decomposition framework in which independent variables are introduced at the interfaces generated in the process of decoupling the original problem on the whole network into a set of fracture-local problems. A PDE-constrained formulation is then used to enforce compatibility conditions at the interfaces. The combination of the three-field domain decomposition and of the optimization-based coupling strategy results in a novel method which can handle nonconforming meshes, independently built on each geometrical ob ject of the computational domain, and ensures a local mass conservation property at fracture intersections, which is of paramount importance for hydrogeological applications. An iterative solver is devised for the method, suitable for parallel implementation on parallel computing architectures. [ABSTRACT FROM AUTHOR]

Well placement in a given geological setting for a fractured geothermal reservoir is necessary for enhanced geothermal operations. High computational cost associated with the framework of fully coupled thermo-hydraulic-mechanical (THM) processes in a fractured reservoir simulation makes the well positioning a missing point in developing a field-scale investigation. To enhance the knowledge of well placement for different working fluids, we present the importance of this topic by examining different injection-production well (doublet) positions in a given fracture network using coupled THM numerical simulations. Results of this study are examined through the thermal breakthrough time, mass flux, and the energy extraction potential to assess the impact of well position in a two-dimensional reservoir framework. Almost ten times the difference between the final amount of heat extraction is observed for different well positions but with the same well spacing and geological characteristics. Furthermore, the stress field is a strong function of well position that is important concerning the possibility of high-stress development. The objective of this work is to exemplify the importance of fracture connectivity and density near the wellbores, and from the simulated cases, it is sufficient to understand this for both the working fluids. Based on the result, the production well position search in the future will be reduced to the high-density fracture area, and it will make the optimization process according to the THM mechanism computationally efficient and economical.

The significance of fracture roughness for hydraulic characterization of sparsely fractured (crystalline) rock using flow logs under steady‐state pumping conditions is investigated by numerical simulations in three‐dimensional fracture networks. A new solver for simulating flow in three‐dimensional discrete fracture networks is implemented using a hybrid finite volume and finite element method. The analysis is focused on different scales of heterogeneity in three‐dimensions: the network scale heterogeneity, the fracture‐to‐fracture scale heterogeneity, and the individual fracture scale heterogeneity (fracture roughness). The results show that the individual fracture scale heterogeneity due to fracture roughness is potentially an additional source of uncertainty for hydraulic tests in fractured rocks using flow logs. Specifically, it enhances the variation in measured flowrates and inferred transmissivity in addition to the network scale heterogeneity and the fracture‐to‐fracture scale heterogeneity. However, the individual fracture scale heterogeneity has relatively small impact on the inferred transmissivity distributions compared to the fracture‐to‐fracture scale heterogeneity. Moreover, fracture roughness and fracture‐to‐fracture heterogeneity have limited influence on the median values of the inferred transmissivity such that the median transmissivity is mostly reliable when inferred from flow logs. Comparison between the inferred and underlying input transmissivity distributions shows that interpreting hydraulic tests in crystalline rock using flow logs and the Thiem equation may underestimate the variation range of the underlying transmissivity. The results are helpful for understanding the uncertainty in hydraulic characterization of sparsely fractured rocks using flow logs that are relevant for various geo‐engineering and water resources applications. Plain Language Summary: We study the impact of fracture roughness on inferred transmissivity from flow log information, in three‐dimensional discrete fracture networks of crystalline rocks. For simulation of water flow, a new solver for three‐dimensional discrete fracture networks using a hybrid finite volume and finite element method is implemented in this work to include borehole hydraulics. We consider three scales of heterogeneity in discrete fracture networks: the network scale heterogeneity, the fracture‐to‐fracture heterogeneity, and the individual fracture roughness. We find that the fracture roughness slightly increases the variability of measured flowrates and inferred transmissivity. The median values of the inferred transmissivity are close to that of the actual underlying input transmissivity, indicating that the fracture roughness has limited impact on the median values. In other words, median values of transmissivity inferred from flow logs under steady‐state pumping are generally representative for the median values of the underlying transmissivity distributions. The method presented in this study is useful for modeling flow in sparsely fractured crystalline rock with conducting boreholes and rough fractures, while the presented results are helpful for understanding and quantifying uncertainty in transmissivity distributions inferred from flow log measurements. Key Points: Three scales of hydraulic heterogeneity affect variability of measured flowrate and inferred transmissivityFracture roughness has relatively small impact on inferred transmissivity compared to the fracture‐to‐fracture scale heterogeneityThe median value of inferred transmissivity from flow logs is representative of the underlying transmissivity distribution [ABSTRACT FROM AUTHOR]

The phenomenon of flow-channeling, or the existence of preferential pathways for flow in fracture networks, is well known. Identification of the channels ("backbone") allows for system reduction and computational efficiency in simulation of flow and transport through fracture networks. However, the purpose of machine learning techniques for backbone identification in fractured media is two-pronged system reduction for computational efficiency in simulation of flow and transport as well as physical insight into the phenomenon of flow channeling. The most critical aspect of this problem is the need to have a truly "physics-informed" technique that respects the constraint of connectivity. We present a method that views a network as a union of connected paths with each path comprising a sequence of fractures. Thus, the fundamental unit of selection becomes a sequence of fractures, classified based on attributes that characterize the sequence. In summary, this method represents a parametrization of the sample space that ensures every selected sample sub-network (which is the union of all selected sequences of fractures) always respects the constraint of connectivity, demonstrating that it is a truly physics-informed method. The algorithm has a user-defined parameter which allows control of the backbone size when using the random forest or logistic regression classifier. Even when the backbones are less than 30% in size (leading to computational savings), the backbones still capture the behavior of the breakthrough curve of the full network. Moreover, there is no need to check for path connectedness in the backbones unlike previous methods since the backbones are guaranteed to be connected. [ABSTRACT FROM AUTHOR]

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. Key Points: Multilevel Monte Carlo is used to predict first passage times in flows through discrete fracture networksThe method provides estimates of the mean and variance at several orders of magnitude lower computational cost than standard Monte CarloWe identify a hierarchy of subnetworks based on the shortest topological paths through the network using a graph‐based method [ABSTRACT FROM AUTHOR]

We present a set of multiphase flow simulations where supercritical CO 2 (scCO 2 ) displaces water at hydrostatic conditions within three-dimensional discrete fracture networks that represent paths for potential leakage through caprock above CO 2 storage reservoirs. The simulations are performed to characterize and compare the relative impact of hydraulic and structural heterogeneity in fractured media on the initial movement of scCO 2 through these caprock formations. In one scenario, intrinsic fracture permeabilities are varied stochastically within a fixed network structure. In another scenario, we generate multiple independent, identically distributed network realizations with varying fracture network densities to explore a wide range of geometric and topological configurations. Analysis of the simulations indicates that network structure, specifically connectivity and the presence of hanging fractures, plays a larger role in controlling the displacement of water by scCO 2 than variations in local hydraulic properties. We identify active surface area of the network as a single-phase feature that could provide a lower bound on the percentage of the network surface area reached by scCO 2 . [ABSTRACT FROM AUTHOR]

The construction of tunnels and underground engineering in China has developed rapidly in recent years in both the number and the length of tunnels. However, with the development of tunnel construction technology, risk assessment of the tunnels has become increasingly important. Water inrush is one of the most important causes of engineering accidents worldwide, resulting in considerable economic and environmental losses. Accordingly, water inrush prediction is important for ensuring the safety of tunnel construction. Therefore, in this study, we constructed a three-dimensional discrete network fracture model using the Monte Carlo method first with the basic data from the engineering geological map of the Longmen Mountain area, the location of the Longmen Mountain tunnel. Subsequently, we transformed the discrete fracture networks into a pipe network model. Next, the DEM of the study area was analysed and a submerged analysis was conducted to determine the water storage area. Finally, we attempted to predict the water inrush along the Longmen Mountain tunnel based on the Darcy flow equation. Based on the contrast of water inrush between the proposed approach, groundwater dynamics and precipitation infiltration method, we conclude the following: the water inflow determined using the groundwater dynamics simulation results are basically consistent with those in the D2K91+020 to D2K110+150 mileage. Specifically, in the D2K91+020 to D2K94+060, D2K96+440 to D2K98+100 and other sections of the tunnel, the simulated and measured results are in close agreement and show that this method is effective. In general, we can predict the water inflow in the area of the Longmen Mountain tunnel based on the existing fracture joint parameters and the hydrogeological data of the Longmen Mountain area, providing a water inrush simulation and guiding the tunnel excavation and construction stages.

In this work, we investigate the sensitivity of a family of multi-task Deep Neural Networks (DNN) trained to predict fluxes through given Discrete Fracture Networks (DFNs), stochastically varying the fracture transmissivities. In particular, detailed performance and reliability analyses of more than two hundred Neural Networks (NN) are performed, training the models on sets of an increasing number of numerical simulations made on several DFNs with two fixed geometries (158 fractures and 385 fractures) and different transmissibility configurations. A quantitative evaluation of the trained NN predictions is proposed, and rules fitting the observed behavior are provided to predict the number of training simulations that are required for a given accuracy with respect to the variability in the stochastic distribution of the fracture transmissivities. A rule for estimating the cardinality of the training dataset for different configurations is proposed. From the analysis performed, an interesting regularity of the NN behaviors is observed, despite the stochasticity that imbues the whole training process. The proposed approach can be relevant for the use of deep learning models as model reduction methods in the framework of uncertainty quantification analysis for fracture networks and can be extended to similar geological problems (for example, to the more complex discrete fracture matrix models). The results of this study have the potential to grant concrete advantages to real underground flow characterization problems, making computational costs less expensive through the use of NNs.

In the present work a message passing interface (MPI) parallel implementation of an optimization-based approach for the simulation of underground flows in large discrete fracture networks is proposed. The software is capable of parallel execution of meshing, discretization, resolution, and postprocessing of the solution. We describe how optimal scalability performances are achieved combining high efficiency in computations with an optimized use of MPI communication protocols. Also, a novel graph-topology for communications, called the multi-Master approach, is tested, allowing for high scalability performances on massive fracture networks. Strong scalability and weak scalability simulations on random networks counting order of 105 fractures are reported. [ABSTRACT FROM AUTHOR]

Microseismic monitoring is widely applied to detect microseisms touched off by shear slippage on bedding planes or natural fractures in the vicinity of the hydraulic fracture stages being stimulated, characterize full fracture geometry and orientation, fracture complexity and also measure the stimulated reservoir volume (SRV). Microseismic events are recorded by geophones at the surface, near-surface or in monitoring boreholes and their hypocenters are mapped via P and S wave arrival picking and hodogram analysis or seismic imaging technology. Fracturing treatment parameters and events attributes, such as seismic moment, rock rigidity, fluid efficiency, injected fluid volumes, displacement along the slip plane and hypocenter focal mechanism, can be used to measure the fracture geometry and orientation for 3D discrete fracture networks (DFNs) modeling. In the DFNs, every modeled fracture plane shaped by uniformly-spaced point arrays is centered on a microseismic event. All these pointsets are classified to different cells in a geocellular model. Then geocellular cubes with uniform dimension are created in the SRV and event-based fractures are mapped onto each cube. Based on the domain classification methodology, the total intersected fracture polygon area in each SRV cube was calculated for fracture properties (such as fracture intensity, fracture porosity) characterization. Fracture permeability scalar was proposed to capture the unscaled permeability enhancement in 3D geocellular framework. 3D visualization plots would be used to determine the high conductivity zone in the productive SRV, which further helps selecting the optimal position for the infill wells and further enhances the oil and gas recovery. • This microseismic-based model helps determine the optimal location of the infill wells. • Approaches of calculating the intersected fracture area within SRV cube were discussed. • Sensitivity analysis of pointset interval and SRV cube size were conducted. • Fracture permeability scalar can quantify the fracture flow capacity. [ABSTRACT FROM AUTHOR]