Olivier Bour

Bio: Olivier Bour is an academic researcher from University of Rennes. The author has contributed to research in topics: Aquifer & Hydrogeology. The author has an hindex of 44, co-authored 120 publications receiving 6213 citations. Previous affiliations of Olivier Bour include University of Rennes 1 & Centre national de la recherche scientifique.

Scaling of fracture systems in geological media

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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.

VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

Scaling of fracture systems in geological media

Abstract: 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.