Characterizing flow and transport in fractured geological media: A review

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.

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Scaling of fracture systems in geological media

A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones

Among the current problems that hydrogeologists face, perhaps there is none as challenging as the characterization of fractured rock (Faybishenko and Benson 2000). This paper discusses issues associated with the quantification of flow and transport through fractured rocks on scales not exceeding those typically associated with single- and multi-well pressure (or flow) and tracer tests. As much of the corresponding literature has focused on fractured crystalline rocks and hard sedimentary rocks such as sandstones, limestones (karst is excluded) and chalk, so by default does this paper. Direct quantification of flow and transport in such rocks is commonly done on the basis of fracture geometric data coupled with pressure (or flow) and tracer tests, which therefore form the main focus. Geological, geophysical and geochemical (including isotope) data are critical for the qualitative conceptualization of flow and transport in fractured rocks, and are being gradually incorporated in quantitative flow and transport models, in ways that this paper unfortunately cannot describe but in passing. The hydrogeology of fractured aquifers and other earth science aspects of fractured rock hydrology merit separate treatments. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. One way is to depict the rock as a network of discrete fractures (with permeable or impermeable matrix blocks) and another as a nonuniform (single, dual or multiple) continuum. A third way is to combine these into a hybrid model of a nonuniform continuum containing a relatively small number of discrete dominant features. In either case the description can be deterministic or stochastic. The paper contains a brief assessment of these trends in light of recent experimental and theoretical findings, ending with a short list of prospects and challenges for the future.

Trends, prospects and challenges in quantifying flow and transport through fractured rocks

The concept of tortuosity is used to characterize the structure of porous media, to estimate their electrical and hydraulic conductivity, and to study the travel time and length for tracer dispersion, but different types of tortuosity—geometric, hydraulic, electrical, and diffusive—have been used essentially interchangeably in the literature. Here, we critically review the tortuosity models developed empirically, analytically, and numerically for flow in both saturated and unsaturated porous media. We emphasize that the proposed tortuosity models are distinct and thus may not be used interchangeably. Given the variety of models that have been developed, and the sharp differences between some of them, no consensus has emerged unifying the models in a coherent way. Related treatments of tortuosity are found in the literature on porous catalysts. In such materials, nonlinear reactions ordinarily accompany transport, and the effective diffusivity within the pore space in the presence of the reactions is distinct from the one in their absence. Thus, because tortuosity may be defined as the ratio of the effective diffusivities in the bulk material and within the pore space, a careful treatment of tortuosity may need to distinguish between transport with and without reactions. This complication is ultimately relevant to soils as well, because bioremediation and biodegradation in soils are always accompanied by nonlinear reactions. Common models of tortuosity include both logarithmic functions and power laws. In many cases, the differences between the logarithmic and power-law phenomenologies are not great, but power laws can usually be reconciled with percolation concepts. Invoking percolation theory provides both insight into the origin of the power functions and a framework for understanding differences between tortuosity models.

Tortuosity in Porous Media: A Critical Review

Scaling and connectivity of joint systems in sandstones from western Norway

[1] Fracture patterns are characterized by a complex geometry which involves a large length distribution and nonhomogeneous density distributions. Here we address the issue of the modeling of this complex geometry over scales through a first-order model providing the characterization of the number of fractures of a given length and at a given scale. We propose that the simplest model is a power law in both space and fracture length, with three main parameters: the exponent a of a power law length distribution, the fractal dimension D which fixes the scale dependence of the number of fractures, and the fracture density α. We verify the applicability of this model on seven fracture patterns mapped from the metric scale up to almost the kilometric scale in the Hornelen basin. The model efficiently describes fracture network properties at all scales with a single set of parameters. For the Hornelen fracture networks we found a simple relationship between basic exponents, a = D + 1, implying that the fracture network is self-similar. Through this analysis, we present different methodological developments for deriving the basic exponents a and D and for verifying the consistency of the model. Overall, the main methodological development is about the normalization of the measurements made from different scales of observation. Finally, we discuss both the limitations and the uses of such a model for analyzing the hydraulic and mechanical properties of fracture networks.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001JB000176

A statistical scaling model for fracture network geometry, with validation on a multiscale mapping of a joint network (Hornelen Basin, Norway)

The influence that fractures exert on the permeability of a fractured rock is, to a large extent, controlled by the nature of the network formed by the fracture system. Here, the network properties of a two-dimensional natural pattern, mapped from the surface of a sandstone layer, are investigated and compared to those of realizations of spatially randomly distributed line segments with similar orientation and length distributions and line segment density (line length per unit area) to the natural pattern. These patterns are composed of clusters of varying size and shape, made up of interconnected fracture traces or line segments. Comparing the natural pattern with the realizations, the natural pattern was found to contain roughly half the number of clusters while the mass (total line length) of the largest cluster is approximately double that of the realizations. The size of the largest cluster controls the connectivity of the patterns, as can be seen by comparing the largest cluster of the natural pattern, which connects all four sides of the region, with those of the realizations, which are unconnected or connect only two sides. Cluster scaling characteristics were found to be similar in the natural pattern and the realizations and show a crossover from a dimension of one (their topological dimension) to two (the dimension of the embedding medium) at a point that corresponds to the fracture spacing. An investigation of the self-similarity dimension, using the box-counting method, showed similar characteristics with a broad transition zone between one- and two-dimensional behaviour at smaller box sizes. The patterns are therefore found to be non-fractal. The effect of the spatial distribution shown by the natural pattern is thus to modify the manner in which fractures are distributed among clusters, increasing connectivity (and permeability in the case of open fractures), but does not affect the cluster scaling characteristics or the self-similarity dimension of the fracture patterns.

Noelle E. Odling

Papers

Scaling of fracture systems in geological media

Scaling and connectivity of joint systems in sandstones from western Norway

A statistical scaling model for fracture network geometry, with validation on a multiscale mapping of a joint network (Hornelen Basin, Norway)

Network properties of a two-dimensional natural fracture pattern

Permeability scaling properties of fault damage zones in siliclastic rocks