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, hydrocarbon reservoir management, and earthquake hazard assessment. Relevant publications are therefore spread widely through the literature. Although it is recognized 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 exponents and fractal dimensions from observations, although outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these techniques and suggest guidelines for the accurate and objective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after critical appraisal of published studies, to show a wide variation, 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 physical causes of power law scaling and variation in exponents and fractal dimensions are still poorly understood. INTRODUCTIONThe study of fracture systems (terms in italic are defined in the glossary, after the main text) has been an active area of research for the last 25 years motivated to a large extent by the siting of hazardous waste disposal sites in crystalline rocks, by the problems of multiphase flow in fractured hydrocarbon reservoirs, and by earthquake hazards and the possibility of prediction. Here we define a fracture as any discontinuity within a rock mass that developed as a response to stress. This comprises primarily mode I and mode II fractures. In mode I fracturing, fractures are in tensile or opening mode in which displacements are normal to the discontinuity walls (joints and many veins). Faults correspond to mode II fractures, i.e., an in-plane shear mode, in which the displacements are in the plane of the discontinuity. Fractures exist on a wide range of scales from microns to hundreds of kilometers, and it is known that throughout this scale range they have a sign...

[1] Long-term bedrock incision is driven by daily discharge events of variable magnitude and frequency, with ineffective events below an incision threshold. We explore theoretically how this short-term stochastic behavior controls long-term steady state incision rates and bedrock channel profiles, combining a realistic frequency-magnitude distribution of discharge with a deterministic, detachment-limited incision model in which incision rate is a power function of basal shear stress above a critical shear stress. Our model predicts a power law relationship between steady state slope and drainage area consistent with observations. The exponent of this power law is independent of discharge mean and variability, while the amplitude factor, which controls mountain belt relief, is a power law function of mean runoff (with an exponent of À0.5) and a complex function of runoff variability. In accordance with evidence that incision occurs between 6 and 20% of time in rapidly incising rivers (>1 mm/yr) our model predicts that channel steepness is virtually insensitive to runoff variability. Runoff variability can only decrease channel steepness for very slow incision rates and/or weak lithologies. The relationship between channel steepness and incision rate is always a power law whose exponent depends on the channel cross-sectional geometry and runoff variability. This contradicts models neglecting discharge stochasticity in which the steepness-incision scaling is set by the incision law exponent. Our results suggest that changes in climate variability cannot explain an increase in bedrock incision rates during the Late Cenozoic within the context of a detachment limited model.Citation: Lague, D., N. Hovius, and P. Davy (2005), Discharge, discharge variability, and the bedrock channel profile, J. Geophys.

Understanding how water and solutes enter and propagate through freshwater landscapes in the Anthropocene is critical to protecting and restoring aquatic ecosystems and ensuring human water security. However, high hydrochemical variability in headwater streams, where most carbon and nutrients enter river networks, has hindered effective modelling and management. We developed an analytical framework informed by landscape ecology and catchment hydrology to quantify spatiotemporal variability across scales, which we tested in 56 headwater catchments, sampled periodically over 12 years in western France. Unexpectedly, temporal variability in dissolved carbon, nutrients and major ions was preserved moving downstream and spatial patterns of water chemistry were stable on annual to decadal timescales, partly because of synchronous variation in solute concentrations. These findings suggest that while concentration and flux cannot be extrapolated among subcatchments, periodic sampling of headwaters provides valuable information about solute sources and subcatchment resilience to disturbance.