This paper concerns a model of bed load transport, which describes the advection and dispersion of coarse particles carried by a turbulent water stream. The challenge is to develop a microstructural approach that, on the one hand, yields a parsimonious description of particle transport at the microscopic scale and, on the other hand, leads to averaged equations at the macroscopic scale that can be consistently interpreted in light of the continuum equations used in hydraulics. The cornerstone of the theory is the proper determination of the particle flux fluctuations. Apart from turbulence-induced noise, fluctuations in the particle transport rate are generated by particle exchanges with the bed consisting of particle entrainment and deposition. At the particle scale, the evolution of the number of moving particles can be described probabilistically using a coupled set of reaction-diffusion master equations. Theoretically, this is interesting but impractical, as solving the governing equations is fraught with difficulty. Using the Poisson representation, we show that these multivariate master equations can be converted into Fokker-Planck equations without any simplifying approximations. Thus, in the continuum limit, we end up with a Langevin-like stochastic partial differential equation that governs the time and space variations of the probability density function for the number of moving particles. For steady-state flow conditions and a fixed control volume, the probability distributions of the number of moving particles and the particle flux can be calculated analytically. Taking the average of the microscopic governing equations leads to an average mass conservation equation, which takes the form of the classic Exner equation under certain conditions carefully addressed in the paper. Analysis also highlights the specific part played by a process we refer to as collective entrainment, i.e. a nonlinear feedback process in particle entrainment. In the absence of collective entrainment the fluctuations in the number of moving particles are Poissonian, which implies that at the macroscopic scale they act as white noise that mediates bed evolution. In contrast, when collective entrainment occurs, large non-Poissonian fluctuations arise, with the important consequence that the evolution at the macroscopic scale may depart significantly that predicted by the averaged Exner equation. Comparison with experimental data gives satisfactory results for steady-state flows.

The advection-diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system. Citation:Ancey, C., P. Bohorquez, and J. Heyman (2015), Stochastic interpretation of the advection-diffusion equation and its relevance to bed load transport, p particle volume (m 3 ).Gaussian noise increment (s −1∕2 ). b space and time Gaussian noise increment (m −1∕2 s −1∕2 ) .

Fluid flow in porous media drives the transport, mixing, and reaction of molecules, particles, and microorganisms across a wide spectrum of natural and industrial processes. Current macroscopic models that average pore-scale fluctuations into an effective dispersion coefficient have shown significant limitations in the prediction of many important chemical and biological processes. Yet, it is unclear how three-dimensional flow in porous structures govern the microscale chemical gradients controlling these processes. Here, we obtain high-resolution experimental images of microscale mixing patterns in three-dimensional porous media and uncover an unexpected and general mixing mechanism that strongly enhances concentration gradients at pore-scale. Our experiments reveal that systematic stretching and folding of fluid elements are produced in the pore space by grain contacts, through a mechanism that leads to efficient microscale chaotic mixing. These insights form the basis for a general kinematic model linking chaotic-mixing rates in the fluid phase to the generic structural properties of granular matter. The model successfully predicts the resulting enhancement of pore-scale chemical gradients, which appear to be orders of magnitude larger than predicted by dispersive approaches. These findings offer perspectives for predicting and controlling the vast diversity of reactive transport processes in natural and synthetic porous materials, beyond the current dispersion paradigm.

[1] Steep slope streams show large fluctuations of sediment discharge across several time scales. These fluctuations may be inherent to the internal dynamics of the sediment transport process. A probabilistic framework thus seems appropriate to analyze such a process. In this paper, we present an experimental study of bedload transport over a steep slope flume for small to moderate Shields numbers. The sampling technique allows the acquisition of highresolution time series of the solid discharge. The resolved time scales range from 10 −1 s up to 10 5 s. We show that two distinct time scales can be observed in the probability density function for the waiting time between moving particles. We make the point that the separation of time scales is related to collective dynamics. Proper statistics of a Markov process including collective entrainment are derived. The separation of time scales is recovered theoretically for low entrainment rates. Citation: Heyman, J., F. Mettra, H. B.Ma, and C. Ancey (2013), Statistics of bedload transport over steep slopes: Separation of time scales and collective motion, Geophys.

International audienceIn gravel bed rivers, bed load transport exhibits considerable variability in time and space. Recently, stochastic bed load transport theories have been developed to address the mechanisms and effects of bed load transport fluctuations. Stochastic models involve parameters such as particle diffusivity, entrainment, and deposition rates. The lack of hard information on how these parameters vary with flow conditions is a clear impediment to their application to real-world scenarios. In this paper, we determined the closure equations for the above parameters from laboratory experiments. We focused on shallow supercritical flow on a sloping mobile bed in straight channels, a setting that was representative of flow conditions in mountain rivers. Experiments were run at low sediment transport rates under steady nonuniform flow conditions (i.e., the water discharge was kept constant, but bed forms developed and migrated upstream, making flow nonuniform). Using image processing, we reconstructed particle paths to deduce the particle velocity and its probability distribution, particle diffusivity, and rates of deposition and entrainment. We found that on average, particle acceleration, velocity, and deposition rate were responsive to local flow conditions, whereas entrainment rate depended strongly on local bed activity. Particle diffusivity varied linearly with the depth-averaged flow velocity. The empirical probability distribution of particle velocity was well approximated by a Gaussian distribution when all particle positions were considered together. In contrast, the particles located in close vicinity to the bed had exponentially distributed velocities. Our experimental results provide closure equations for stochastic or deterministic bed load transport models. ©2016. American Geophysical Union. All Rights Reserved

This article examines the spatial dynamics of bed load particles in water. We focus particularly on the fluctuations of particle activity, which is defined as the number of moving particles per unit bed length. Based on a stochastic model recently proposed by Ancey and Heyman (2014), we derive the second moment of particle activity analytically, that is, the spatial correlation functions of particle activity. From these expressions, we show that large moving particle clusters can develop spatially. Also, we provide evidence that fluctuations of particle activity are scale dependent. Two characteristic lengths emerge from the model: a saturation length sat describing the length needed for a perturbation in particle activity to relax to the homogeneous solution and a correlation length c describing the typical size of moving particle clusters. A dimensionless Péclet number can also be defined according to the transport model. Three different experimental data sets are used to test the theoretical results. We show that the stochastic model describes spatial patterns of particle activity well at all scales. In particular, we show that c and sat may be relatively large compared to typical scales encountered in bed load experiments (grain diameter, water depth, bed form wavelength, flume length, etc.) suggesting that the spatial fluctuations of particle activity have a nonnegligible impact on the average transport process.

This paper describes the relationship between the statistics of bed load transport flux and the timescale over which it is sampled. A stochastic formulation is developed for the probability distribution function of bed load transport flux, based on the Ancey et al. (2008) theory. An analytical solution for the variance of bed load transport flux over differing sampling timescales is presented. The solution demonstrates that the timescale dependence of the variance of bed load transport flux reduces to a three-regime relation demarcated by an intermittency timescale (t I ) and a memory timescale (t c ). As the sampling timescale increases, this variance passes through an intermittent stage (≪t I ), an invariant stage (t I < t < t c ), and a memoryless stage (≫ t c ). We propose a dimensionless number (Ra) to represent the relative strength of fluctuation, which provides a common ground for comparison of fluctuation strength among different experiments, as well as different sampling timescales for each experiment. Our analysis indicates that correlated motion and the discrete nature of bed load particles are responsible for this three-regime behavior. We use the data from three experiments with high temporal resolution of bed load transport flux to validate the proposed three-regime behavior. The theoretical solution for the variance agrees well with all three sets of experimental data. Our findings contribute to the understanding of the observed fluctuations of bed load transport flux over monosize/multiple-size grain beds, to the characterization of an inherent connection between short-term measurements and long-term statistics, and to the design of appropriate sampling strategies for bed load transport flux.

The µ(I)-rheology was recently proposed as a potential candidate to model the incompressible flow of frictional grains in the dense inertial regime. However, this rheology was shown to be ill-posed in the mathematical sense for a large range of parameters, notably in the low and large inertial number limits (Barker et al. 2015). In this rapid communication, we extend the stability analysis of Barker et al. (2015) to compressible flows. We show that compressibility regularizes mostly the equations, making the problem well-posed for all parameters, with the condition that sufficient dissipation be associated with volume changes. In addition to the usual Coulomb shear friction coefficient µ, we introduce a bulk friction coefficient µ b , associated with volume changes and show that the problem is well-posed if µ b > 1 − 7µ/6. Moreover, we show that the ill-posed domain defined by Barker et al. (2015) transforms into a domain where the flow is unstable but remains well-posed when compressibility is taken into account. These results suggest the importance of taking into account dynamic compressibility for the modelling of dense granular flows and open new perspectives to investigate the emission and propagation of acoustic waves inside these flows.