We study the intermittency of fluid velocities in porous media and its relation to anomalous dispersion. Lagrangian velocities measured at equidistant points along streamlines are shown to form a ...spatial Markov process. As a consequence of this remarkable property, the dispersion of fluid particles can be described by a continuous time random walk with correlated temporal increments. This new dynamical picture of intermittency provides a direct link between the microscale flow, its intermittent properties, and non-Fickian dispersion.
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.
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 −0.5) 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.
► Non-Fickian mixing is defined as the anomalous scaling of the scalar dissipation rate. ► Porous media heterogeneity implies non-Fickian mixing and non Fickian spreading.► Non-Fickian mixing ...quantifies the concentration heterogeneity in the mixing zone. ► Non-Fickian spreading quantifies the anomalous growth of the mixing zone. ► The two processes are related through an analytical expression at large travel times.
We investigate the temporal scaling properties of mixing in heterogeneous permeability fields with variances ranging from very small
(
σ
ln
K
2
=
0.01
)
to very large
(
σ
ln
K
2
=
9
)
. We quantify mixing by the scalar dissipation rate, which we estimate over a large range of temporal scales. For an initial pulse line injection, we find that moderate and strong heterogeneity induce anomalous temporal scaling of the scalar dissipation rate, which we call non-Fickian mixing. This effect is particularly relevant for upscaling reactive transport as it implies a non-Fickian scaling of reactive transport. Although spreading and mixing are intimately coupled, we find that their scaling properties are not directly related in general. In the non-Fickian mixing regime, the temporal scaling of the scalar dissipation rate depends on the complex spatial distribution of the concentration field that generates transverse mixing. For times larger than the characteristic diffusion time associated with one permeability field correlation length, the heterogeneity of concentration in the plume is attenuated and progressively erased by diffusion. Thus, at large times, the temporal scaling of mixing and spreading can be related through a simple analytical expression.
The realism of Discrete Fracture Network (DFN) models relies on the spatial organization of fractures, which is not issued by purely stochastic DFN models. In this study, we introduce correlations ...between fractures by enhancing the genetic model (UFM) of Davy et al. 1 based on simplified concepts of nucleation, growth and arrest with hierarchical rules. To do so, the nucleation of new fractures is correlated with the elastic strain energy of distortion stored in the matrix, which is a function of preexisting fractures. Discrete Fracture Networks so generated show multi-scale clustering effects with fractal dimensions below the topological dimension over a broad range of scales. The fractal dimension depends on the way one correlates the nucleation occurrence to the strain energy. Fracture clustering entails a spatial variability of the fracture density, which increases with the intensity of the coupling between stress and nucleation. The analysis of connected clusters density and of fracture intersections also highlights the differences between the UFM models and its equivalent Poisson model. We show that our stress-dependent nucleation model introduces some new fracture size-positions correlations, with small fractures tending to connect to the largest ones.
We present the results of an experimental study of topography dynamics under conditions of constant precipitation and uplift rate. The experiment is designed to develop a complete drainage network by ...the growth and propagation of erosion instabilities in response to tectonic perturbations. The quantitative analysis of topographic evolution is made possible by using telemetric lasers that perform elevation measurements at an excellent level of precision. We focus our study on the effect of initial surface organization and of uplift rate on both the transient dynamics and the steady state forms of topography. We show that the transient phase is strongly dependent on the initial internally drained area, which is found to decrease exponentially with time. The topography always reaches a steady state whose mean elevation depends linearly on uplift rate with a strictly positive value when uplift is zero. Steady state surfaces are characterized by a well‐defined slope–area power law with a constant exponent of −0.12 and an amplitude that depends linearly on uplift rate with a strictly positive value when uplift is zero. These results are consistent with a stream power law erosion model that includes a nonnegligible threshold for particle detachment. Uncertainty regarding the sediment transport length is resolved by calibrating the transient dynamics with a surface process model. Reappraising published results on the linear dependency between mean elevation, or relief, and denudation rate, we suggest that an erosion threshold is worth considering for large‐scale systems.
The large database of topographic form and uplift rates that exists for the Siwaliks Hills (central Nepal) makes possible a thorough analysis of the long‐term erosion model. The study especially ...focuses on drainage areas larger than 5.10−3 km2, fixed by the database resolution, and smaller than 1 km2 above which a fluvial signature is recorded. This area range corresponds to colluvial valleys in which the dominant erosion process is likely debris flow. We evaluate a phenomenological model wherein erosion is considered to depend on drainage area and slope. We test this model by assuming that the uplift rate is in approximate equilibrium with erosion. The stream power law model, formulated by analogy to river incision and transport problems, is found to be consistent with data since an inverse power law relationship between slope and drainage area is systematically observed between 7.10−3 and ∼1 km2, with little variability on the exponent ∼−0.24. Thanks to the range of uplift rates, we obtain constraints on the slope dependency of erosion law, which appears linear and which predicts a significant erosion threshold. The linear dependence on slope in the debris‐flow zone is consistent with findings by Kirby and Whipple 2001 in the fluvial downstream zone and with the linear relationship between local relief and uplift rate documented by Hurtrez et al. 1999. The transition between this colluvial‐channel regime and the fluvial regime appears quite sharp in contrast with recent studies, but the latter regime is not sufficiently documented to derive definite conclusions.
Because of the competition between brittle and ductile rheologies and their interplay with tectonic and buoyancy forces, lithospheric deformation results in very contrasting styles. In continental ...collision, especially with unconfined boundaries, deformation can be either homogeneously distributed or localized on complex fault patterns, and different deformation modes such as contraction, extension, and strike‐slip interfere. Using scaled lithospheric analog experiments made of dry sand, silicone putties, and dense honey, we investigate the mechanical parameters that control the deformation style in colliding systems, with a particular focus on the roles of buoyancy and brittle‐ductile coupling. The analysis of tens of experiments shows that the principal deformation features depend on two main parameters: a brittle‐to‐ductile strength ratio Γ, which controls deformation localization at the largest scale, and a buoyancy‐to‐strength ratio Ar, which fixes the relative amount of contractional, extensional, and strike‐slip structures. Strain localization occurs only for Γ larger than a critical value (∼0.5), and the range of Γ values, over which the transition from nonlocalized to localized deformation occurs, is small. The three main deformation regimes (contraction, strike‐slip, and extension), which coexist in most of the collision experiments, occur in relative proportions that depend mainly on Ar and on the nature of the boundary conditions.
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.
PDF
