Romain Le Goc

Fracture networks are preferential flow paths playing a critical role in a wide range of environmental and industrial problems. Their complex multiscale structure leads to broad distributions of fluid travel times, affecting many biogeochemical processes. Yet, the relationship between the fracture network structures, their hydrodynamic properties, and the resulting anomalous transport dynamics remains unclear. We use a large database of fracture network models to investigate the factors controlling fluid velocity and travel-time distributions across a wide range of networks, from synthetic to field-calibrated models, with aperture variability at both fracture and network scales. Analysis reveals that transport statistics have generic properties across investigated networks, including notably heavy-tailed travel time distributions. Networks of increasing complexity and heterogeneity lead to broader velocity distributions and more channeled velocity fields, where flow concentrates in a narrow channel web in the three-dimensional (3D) fracture structure. While heterogeneity in point-velocity statistics increases travel-time variability, channeling tends to reduce it. This counterintuitive phenomenon challenges current theories, which assume that long travel time power law exponents are determined solely by point-velocity statistics. By analyzing velocity and travel time statistics for different flow structures, we develop a coupled Continuous Time Random Walk framework capturing the unexpected control of the velocity field’s spatial structure on anomalous transport in fracture networks. This leads to a unique class of random walk models capturing the respective roles of velocity heterogeneity and spatial structure on transport in networks. These findings open a prospective for characterizing, modeling, and predicting transport dynamics in complex networks, with potential applications to geological, biological, and engineered networks.

The objective of the paper is to better understand and quantify the flow structure in fractured rocks from flow logs, and to propose relevant indicators for validating, calibrating or even rejecting hydrogeological models. We first studied what the inflow distribution tells us about the permeability structure from a series of analyses: distribution of transmissivities as a function of depth, proportion of flowing sections as a function of section scale, and scaling of the arithmetically-averaged and geometrically-averaged permeability. We then define three indicators that describe few fundamental characteristics of the flow/permeability, whatever the scale: a percolation scale l s , the way permeability increases with scale above l s , and the variability of permeability. A 4th indicator on the representative elemental volume could in principle be defined but the data show that this volume/scale is beyond the 300 m investigated. We tested a series of numerical models built in three steps: the geo-DFN based on the observed fracture network, the open-DFN which is the part of the geo-DFN where fractures are open, and a transmissivity model applying on each fracture of the open-DFN (Discrete Fracture Network). The analysis of the models showed that the percolation scale is controlled by the open-DFN structure and that the percolation scale can be predicted from a scale analysis of the percolation parameter (basically, the third moment of the fracture size distribution that provides a measure of the network connectivity). The way permeability increases with scale above the percolation threshold is controlled by the transmissivity model and in particular by the dependence of the fracture transmissivity on either the orientation of the fractures via a stress-controlled transmissivity or their size or both. The comparison with data on the first two indicators shows that a model that matches the characteristics of the geo-DFN with an open fraction of 15% as measured adequately fits the data provided that the large fractures remain open and that the fracture transmissivity model is well selected. Most of the other models show unacceptable differences with data but other models or model combinations has still to be explored before rejecting them. The third indicator on model variability is still problematic since the natural data show a higher variability than the models but the open fraction is also much more variable in the data than in the models.

We derive the relationships that link the general elastic properties of rock masses to the geometrical properties of fracture networks, with a special emphasis to the case of frictional crack surfaces. We extend the well‐known elastic solutions for free‐slipping cracks to fractures whose plane resistance is defined by an elastic fracture (shear) stiffness k_s and a stick‐slip Coulomb threshold. A complete set of analytical solutions have been derived for (i) the shear displacement in the fracture plane for stresses below the slip threshold and above, (ii) the partitioning between the resistances of the fracture plane on the one hand and of the elastic matrix on the other hand, and (iii) the stress conditions to trigger slip. All the expressions have been checked with numerical simulations. The Young's modulus and Poisson's ratio were also derived for a population of fractures. They are controlled both by the total fracture surface for fractures larger than the stiffness length l_S (defined by k_s and the intact matrix elastic properties) and by the percolation parameter of smaller fractures. These results were applied to power law fracture size distributions, which are likely relevant to geological cases. We show that if the fracture size exponent is in the range −3 to −4, which corresponds to a wide range of geological fracture networks, the elastic properties of the bulk rock are almost exclusively controlled by k_s and the stiffness length, meaning that the fractures of size l_S play a major role in the definition of the elastic properties.

We derive the relationships that link the general elastic properties of rock masses to the geometrical properties of fracture networks, with a special emphasis to the case of frictional crack surfaces. We extend the well-known elastic solutions for free-slipping cracks to fractures whose plane resistance is defined by an elastic fracture (shear) stiffness k_s and a stick-slip Coulomb threshold. A complete set of analytical solutions have been derived for i) the shear displacement in the fracture plane for stresses below the slip threshold and above, ii) the partitioning between the resistances of the fracture plane in the one hand, and of the elastic matrix in the other hand, and iii) the stress conditions to trigger slip. All the expressions have been checked with numerical simulations. The Young’s modulus and Poisson’s ratio were also derived for a population of fractures. They are controlled both by the total fracture surface for fractures larger than the mechanical length l_M (defined by k_s and the intact matrix elastic properties), and by the percolation parameter of smaller fractures. These results were applied to power-law fracture size distributions, which are likely relevant to geological cases. We show that, if the fracture size exponent is in the range -3 to -4, which corresponds to a wide range of geological fracture networks, the elastic properties of the bulk rock are almost exclusively controlled by k_s and the mechanical length, meaning that the fractures of size l_M play a major role in the definition of the elastic properties.

A major use of DFN models for industrial applications is to evaluate permeability and flow structure in hardrock aquifers from geological observations of fracture networks. The relationship between the statistical fracture density distributions and permeability has been extensively studied, but there has been little interest in the spatial structure of DFN models, which is generally assumed to be spatially random (i.e., Poisson). In this paper, we compare the predictions of Poisson DFNs to new DFN models where fractures result from a growth process defined by simplified kinematic rules for nucleation, growth, and fracture arrest. This so‐called “kinematic fracture model” is characterized by a large proportion of T intersections, and a smaller number of intersections per fracture. Several kinematic models were tested and compared with Poisson DFN models with the same density, length, and orientation distributions. Connectivity, permeability, and flow distribution were calculated for 3‐D networks with a self‐similar power law fracture length distribution. For the same statistical properties in orientation and density, the permeability is systematically and significantly smaller by a factor of 1.5–10 for kinematic than for Poisson models. In both cases, the permeability is well described by a linear relationship with the areal density , but the threshold of kinematic models is 50% larger than of Poisson models. Flow channeling is also enhanced in kinematic DFN models. This analysis demonstrates the importance of choosing an appropriate DFN organization for predicting flow properties from fracture network parameters.

The topology of two discrete fracture network models is compared to investigate the impact of constrained fracture growth. In the Poissonian discrete fracture network model the fractures are assigned length, position and orientation independent of all other fractures, while in the mechanical discrete fracture network model the fractures grow and the growth can be limited by the presence of other fractures. The topology is found to be impacted by both the choice of model, as well as the choice of rules for the mechanical model. A significant difference is the degree mixing. In two dimensions the Poissonian model results in assortative networks, while the mechanical model results in disassortative networks. In three dimensions both models produce disassortative networks, but the disassortative mixing is strongest for the mechanical model.

In order to improve discrete fracture network (DFN) models, which are increasingly required into groundwater and rock mechanics applications, we propose a new DFN modeling based on the evolution of fracture network formation—nucleation, growth, and arrest—with simplified mechanical rules. The central idea of the model relies on the mechanical role played by large fractures in stopping the growth of smaller ones. The modeling framework combines, in a time‐wise approach, fracture nucleation, growth, and arrest. It yields two main regimes. Below a certain critical scale, the density distribution of fracture sizes is a power law with a scaling exponent directly derived from the growth law and nuclei properties; above the critical scale, a quasi‐universal self‐similar regime establishes with a self‐similar scaling. The density term of the dense regime is related to the details of arrest rule and to the orientation distribution of the fractures. The DFN model, so defined, is fully consistent with field cases former studied. Unlike more usual stochastic DFN models, ours is based on a simplified description of fracture interactions, which eventually reproduces the multiscale self‐similar fracture size distribution often observed and reported in the literature. The model is a potential significant step forward for further applications to groundwater flow and rock mechanical issues.

We propose a mapping from fracture systems consisting of intersecting fracture sheets in three dimensions to an abstract network consisting of nodes and links. This makes it possible to analyze fracture systems with the methods developed within modern network theory. We test the mapping for two-dimensional geological fracture outcrops and find that the equivalent networks are small-world and dissasortative. By analyzing the Discrete Fracture Network model, which is used to generate artificial fracture outcrop networks, we also find small world networks. However, the networks turn out to be assortative.