Olivier Bour

SCALING OF FRACTURE SYSTEMS IN GEOLOGICAL MEDIA

Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

Connectivity of random fault networks following a power law fault length distribution

We present a theoretical and numerical study of the connectivity of fault networks following power law fault length distributions, n(l) ∼ αl−a, as expected for natural fault networks. Different regimes of connectivity are identified depending on a. For a > 3, faults smaller than the system size rule the network connectivity and classical laws of percolation theory apply. On the opposite, for a < 1, the connectivity is ruled by the largest fault in the system. For 1 < a < 3, both small and large faults control the connectivity in a ratio which depends on a. The geometrical properties of the fault network and of its connected parts (density, scaling properties) are established at the percolation threshold. Finally, implications are discussed in the case of fault networks with constant density. In particular, we predict the existence of a critical scale at which fault networks are always connected, whatever a smaller than 3, and whatever their fault density.

Hydraulic properties of two‐dimensional random fracture networks following a power law length distribution: 1. Effective connectivity

Natural fracture networks involve a very broad range of fractures of variable lengths and apertures, modeled, in general, by a power law length distribution and a lognormal aperture distribution. The objective of this two-part paper is to characterize the permeability variations as well as the relevant flow structure of two-dimensional isotropic models of fracture networks as determined by the fracture length and aperture distributions and by the other parameters of the model (such as density and scale). In this paper we study the sole influence of the fracture length distribution on permeability by assigning the same aperture to all fractures. In the following paper [de Dreuzy et al., this issue] we study the more general case of networks in which fractures have both length and aperture distributions. Theoretical and numerical studies show that the hydraulic properties of power law length fracture networks can be classified into three types of simplified model. If a power law length distribution n (l) ∼ l−a is used in the network design, the classical percolation model based on a population of small fractures is applicable for a power law exponent a higher than 3. For a lower than 2, on the contrary, the applicable model is the one made up of the largest fractures of the network. Between these two limits, i.e., for a in the range 2–3, neither of the previous simplified models can be applied so that a simplified two-scale structure is proposed. For this latter model the crossover scale is the classical correlation length, defined in the percolation theory, above which networks can be homogenized and below which networks have a multipath, multisegment structure. Moreover, the determination of the effective fracture length range, within which fractures significantly contribute to flow, corroborates the relevance of the previous models and clarifies their geometrical characteristics. Finally, whatever the exponent a, the sole significant scale effect is a decrease of the equivalent permeability for networks below or at percolation threshold.

Hydraulic properties of two-dimensional random fracture networks following a power law length distribution: 2. Permeability of networks based on lognormal distribution of apertures

The broad length and aperture distributions are two characteristics of the heterogeneity of fractured media that make difficult, and even theoretically irrelevant, the application of homogenization techniques. We propose a numerical and theoretical study of the consequences of these two properties on the permeability of bidimensional synthetic fracture networks. We use a power law for the model of length distribution and a lognormal model for aperture distribution. We have especially studied the two endmost models for which length and aperture are (1) independent and (2) perfectly positively correlated. For the model without correlation between length and aperture we show that the permeability can be adequately characterized by a power-averaging function whose parameters are detailed in the text. In contrast, for the model with correlation we show that the prevailing parameter is the correlation when the power law length exponent a is lower than 3, whereas the random structure of the network is a second-order parameter. We also determine the permeability scaling and the scale dependence of the flow pattern structure. Three types of scale effects are found, depending exclusively on the geometrical properties of the network, i.e., on the length distribution parameter a. For a larger than 3, permeability decreases for scales below a definite correlation length and becomes constant above. We show in this case that a correlation between length and aperture does not fundamentally change the permeability model. In all other cases the correlation entails much larger-scale effects. For a in the range 1-3 in the case of an absence of correlation and for a in the range 2-3 in the case of correlation, permeability increases and tends to a limit, whereas the flow structure is channeled when permeability increases and tends to homogenize when permeability tends to its limit. We note that this permeability model is consistent with natural observations of permeability scaling. For a in the range 1-2, in the case of correlation, permeability increases with scale with no apparent limit. We characterize the channeled flow pattern, and we show that permeability may increase even when flow is distributed in several independent structures.

On the connectivity of three‐dimensional fault networks

Natural fault networks involve a very broad range of fault lengths, modeled in general by a power law length distribution, n(l ) ; al. Such a scaling law does not allow to define any a priori pertinent scale of observation for hydraulic field experiments in fractured media. To investigate the relative effects of faults depending on their length, we undertake in the spirit of percolation theory a theoretical and numerical study of the connectivity of three-dimensional fault networks following power law length distributions. We first establish the correct analytical expression of a percolation parameter p, which describes the connectivity of the system. The parameter p is found to be dependent on the third moment of the length distribution for fault planes. It allows us to identify different regimes of connectivity depending on a, the exponent of the fault length distribution. The geometrical properties of the infinite cluster, which partly control transport properties, are also established at the percolation threshold. For natural fault networks, our theoretical analysis suggests that faults larger than a critical length scale may form a well-connected network, while smaller faults may be not connected on average. This result, which implies an increase of the connectivity with scale, is consistent with scaling effects observed on permeability measurements.

Clustering and size distributions of fault patterns: Theory and measurements

The fractal geometry of fault systems has been mainly characterized by two scaling-laws describing either their spatial distribution (clustering) or their size distribution. However, the relationships between the exponents of both scaling-laws has been poorly investigated. We show theoretically and numerically that the fractal dimension D and the exponent a of the frequency length distribution of fault networks, are related through the relation x=(a−1)/D, where x is the exponent of a new scaling law involving the average distance from a fault to its nearest neighbor of larger length. Measurements of the relevant exponents on the San Andreas fault pattern are in agreement with the theoretical analysis and allows us to test the fragmentation models proposed in the literature. We also found a correlation between the position of a fault and its length so that large faults have their nearest neighbor located at greater distances than small faults.

Scaling of fracture connectivity in geological formations

A new method to quantify fracture network connectivity is developed and applied to analyze two classical examples of fault and joint networks in natural geological formations. The connectivity measure accounts for the scaling properties of fracture networks, which are controlled by the power law length distribution exponent a, the fractal dimension D and the fracture density. The connectivity behavior of fracture patterns depends on the scale of measurement, for a D + 1. Analysis of the San Andreas fault system shows that a < D+1 and that the connectivity threshold is reached only at a critical length scale. In contrast, for a typical sandstone joint pattern, a ≈ D + 1, which is on the cusp where the connectivity threshold is highly sensitive to the minimum fracture length in the system.

Inferring transport characteristics in a fractured rock aquifer by combining single‐hole ground‐penetrating radar reflection monitoring and tracer test data

[1] Investigations of solute transport in fractured rock aquifers often rely on tracer test data acquired at a limited number of observation points. Such data do not, by themselves, allow detailed assessments of the spreading of the injected tracer plume. To better understand the transport behavior in a granitic aquifer, we combine tracer test data with single-hole ground-penetrating radar (GPR) reflection monitoring data. Five successful tracer tests were performed under various experimental conditions between two boreholes 6 m apart. For each experiment, saline tracer was injected into a previously identified packed-off transmissive fracture while repeatedly acquiring single-hole GPR reflection profiles together with electrical conductivity logs in the pumping borehole. By analyzing depth-migrated GPR difference images together with tracer breakthrough curves and associated simplified flow and transport modeling, we estimate (1) the number, the connectivity, and the geometry of fractures that contribute to tracer transport, (2) the velocity and the mass of tracer that was carried along each flow path, and (3) the effective transport parameters of the identified flow paths. We find a qualitative agreement when comparing the time evolution of GPR reflectivity strengths at strategic locations in the formation with those arising from simulated transport. The discrepancies are on the same order as those between observed and simulated breakthrough curves at the outflow locations. The rather subtle and repeatable GPR signals provide useful and complementary information to tracer test data acquired at the outflow locations and may help us to characterize transport phenomena in fractured rock aquifers.

Impact of velocity correlation and distribution on transport in fractured media: Field evidence and theoretical model

Flow and transport through fractured geologic media often leads to anomalous (non-Fickian) transport behavior, the origin of which remains a matter of debate: whether it arises from variability in fracture permeability (velocity distribution), connectedness in the flow paths through fractures (velocity correlation), or interaction between fractures and matrix. Here we show that this uncertainty of distribution- versus correlation-controlled transport can be resolved by combining convergent and push-pull tracer tests because flow reversibility is strongly dependent on velocity correlation, whereas late-time scaling of breakthrough curves is mainly controlled by velocity distribution. We build on this insight, and propose a Lagrangian statistical model that takes the form of a continuous time random walk (CTRW) with correlated particle velocities. In this framework, velocity distribution and velocity correlation are quantified by a Markov process of particle transition times that is characterized by a distribution function and a transition probability. Our transport model accurately captures the anomalous behavior in the breakthrough curves for both push-pull and convergent flow geometries, with the same set of parameters. Thus, the proposed correlated CTRW modeling approach provides a simple yet powerful framework for characterizing the impact of velocity distribution and correlation on transport in fractured media.

Flow in multiscale fractal fracture networks

Abstract The paper aims at defining the flow models, including equivalent permeability, that are appropriate for multiscale fracture networks. As a prerequisite of the flow analysis, we define the scaling nature of fracture networks that is likely quantified by power-law length distributions whose exponent fixes the contribution of large fractures versus small ones. Despite the absence of any characteristic length scale of the power-law model, the flow structure appears to contain three length scales at the very maximum: the connecting scale, the channelling scale, and the homogenization scale, above which the equivalent permeability tends to a constant value. These scales, including their existence, depend on the fracture length distribution and on the transmissivity distribution per fracture. They are basic in defining the flow properties of fracture networks.