Modeling oil spill trajectory in coastal waters based on fractional Brownian motion

This paper proposes a numerical method to simulate oil spill trajectories, which are affected by the combination of advection, turbulent diffusion and mechanical spreading process, based on a particle tracking algorithm. Recent studies have shown that the trajectories of drifters on the ocean surface have a fractal structure that is far from being described using ordinary Brownian motion. Thus, in modeling the diffusion process, a discrete method has been employed for the generation of fractional Brownian motion (fBm) to illustrate superdiffusive transport. The algorithm is implemented to predict oil slick trajectories following the “Arteaga” oil spill accident that occurred near the Dalian coastal region in 2005. When compared with the observed data and the results of traditional diffusion modeling, the numerical results based on the fBm model are encouraging.     

Fractional calculus in hydrologic modeling: A numerical perspective

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.     

Stochastic particle based models for suspended particle movement in surface flows

Modeling of suspended sediment particle movement in surface water can be achieved by stochastic particle tracking model approaches.In this paper,different mathematical forms of particle tracking models are introduced to describe particle movement under various flow conditions,i.e.,the stochastic diffusion process,stochastic jump process,and stochastic jump diffusion process.While the stochastic diffusion process can be used to represent the stochastic movement of suspended particles in turbulent flows,the stochastic jump and the stochastic jump diffusion processes can be used to describe suspended particle movement in the occurrences of a sequence of extreme flows.An extreme flow herein is defined as a hydrologic flow event or a hydrodynamic flow phenomenon with a low probability of occurrence and a high impact on its ambient flow environment.In this paper,the suspended sediment particle is assumed to immediately follow the extreme flows in the jump process（i.e.the time lag between the flow particle and the sediment particle in extreme flows is considered negligible）.In the proposed particle tracking models,a random term mainly caused by fluid eddy motions is modeled as a Wiener process,while the random occurrences of a sequence of extreme flows can be modeled as a Poisson process.The frequency of occurrence of the extreme flows in the proposed particle tracking model can be explicitly accounted for by the Poisson process when evaluating particle movement.The ensemble mean and variance of particle trajectory can be obtained from the proposed stochastic models via simulations.The ensemble mean and variance of particle velocity are verified with available data.Applicability of the proposed stochastic particle tracking models for sediment transport modeling is also discussed.     

Multiscale flow and transport model in three-dimensional fractal porous media

This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levy-stable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant long-tailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.     

Is fractal dispersion subdiffusive or superdiffusive? A theoretical investigation

A probabilistic approach is used to simulate particle tracking along a fractal path. The particle tracking is modelled as the sum of elementary steps with independent random variables. An exponential distribution is obtained for each elementary step and a Gamma distribution or probability density function is then deduced. The relationship between fractal dispersivity and the elementary step is given. It is theoretically demonstrated that the fractal dispersion is subdiffusive and that the linear dispersion coefficient and the linear dispersivity decrease with the mean travel distance. A review of some fractal models showing an increase of the linear dispersivity is given and an explanation that shows why these models may be not correct is proposed. It is shown that the results presented are in agreement with other studies relating to the application of the fractional calculus to diffusion transport. Lastly, a relation between the fractal dimension and the order of the fractional Langevin equation is proposed. Copyright © 2007 John Wiley & Sons, Ltd.     

Quantification of longitudinal dispersion by upscaling Brownian motion of tracer displacement in a 3D pore-scale network model

We present a 3D network model with particle tracking to upscale 3D Brownian motion of non-reactive tracer particles subjected to a velocity field in the network bonds, representing both local diffusion and convection. At the intersections of the bonds (nodes) various jump conditions are implemented. Within the bonds, two different velocity profiles are used. At the network scale the longitudinal dispersion of the particles is quantified through the coefficient D_L, for which we evaluate a number of methods already known in the literature. Additionally, we introduce a new method for derivation of D_L based on the first-arrival times distribution (FTD). To validate our particle tracking method, we simulate Taylor’s classical experiments in a single tube. Subsequently, we carry out network simulations for a wide range of the characteristic Péclet number Pe_? to assess the various methods for obtaining D_L. Using the new method, additional simulations have been carried out to evaluate the choice of nodal jump conditions and velocity profile, in combination with varying network heterogeneity. In general, we conclude that the presented network model with particle tracking is a robust tool to obtain the macroscopic longitudinal dispersion coefficient. The new method to determine D_L from the FTD statistics works for the full range of Pe_?, provided that for large Pe_? a sufficiently large number of particles is used. Nodal jump conditions should include molecular diffusion and allow jumps in the upstream direction, and a parabolic velocity profile in the tubes must be implemented. Then, good agreement with experimental evidence is found for the full range of Pe_?, including increased D_L for increased porous medium heterogeneity.     

Numerical simulation of flow and aquaculture organic waste dispersion in a curved channel

The dispersion and deposition of particulate organic matter from a fish cage located in an idealized curved channel with a 90° bend are studied for different horizontal grid resolutions. The model system consists of a three-dimensional, random-walk particle tracking model coupled to a terrain-following ocean model. The particle tracking model is a Lagrangian particle tracking simulator which uses the local flow field, simulated by the ocean model, for advection of the particles and random walk to simulate the turbulent diffusion. The sinking of particles is modeled by imposing an individual particle settling velocity. As the homogeneous water flows through the bend in the channel, the results show that a cross-channel secondary circulation is developed. The motion of this flow is similar to a helical motion where the water in the upper layers moves towards the outer bank and towards the inner bank in the lower layers. The intensity of the secondary circulation will depend on the viscosity scheme and increases as the horizontal grid resolution decreases which significantly affects the distribution of the particles on the seabed. The presence of the secondary circulation leads to that most of the particles that settle, settle close to the inner bank of the channel.     

Prediction uncertainty for tracer migration in random heterogeneities with multifractal character

Travel-time statistics for non-reacting tracers in fractal and multifractal media are addressed through numerical simulations. The logarithm of hydraulic conductivity is modeled using fractional Brownian motion (fBm) and more recently developed multifractal model based on bounded fractional Levy motion (bfLm). These models have been shown previously to accurately reproduce statistical properties of large conductivity datasets. The ensemble-mean travel time increases nearly linearly with travel distance and the variance in the travel time increases nearly parabolically with travel distance. This is consistent with near-field analytical approximations developed for non-fractal media and suggests that these analytical results may have some degree of robustness to non-ideal features in the random-field models. The magnitudes of the travel-time moments are dependent on the system size. For fBm media, this size dependence can be explained using an effective variance that increases with increasing size of the flow system. However, the magnitudes of the travel-time moments are also sensitive to other non-ideal effects such as deviations from Gaussian behavior. This sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations requiring accurate estimates of uncertainty in the plume position.     

A universal field equation for dispersive processes in heterogeneous media

When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing nonstationary increments. The limit process has a mean square displacement that increases as t^α+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient proportional to t^α. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance. With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the general non-local balance law.     

A discussion on the application of b-value to the prediction of seismic tendency

It has been recognized for a long time that the b-value in the Gutenberg-Richters fre-quency-magnitude relation (Gutenberg, Richter, 1954) tends to decrease before some of the earth-quakes (LI, et al, 1979, and the references therein). Since the 1980s, study on b-value has caused much attention among seismologists and physicists when b-value was related to the fractal dimen-sion of an earthquake fault (Aki, 1981; King, 1983) and/or the scaling constant in the model of self-organized critical…     

Magnetic resonance microscopy of biofouling induced scale dependent transport in porous media

Non-invasive magnetic resonance microscopy (MRM) methods are applied to study biofouling of a homogeneous model porous media. MRM of the biofilm biomass using magnetic relaxation weighting shows the heterogeneous nature of the spatial distribution of the biomass as a function of growth. Spatially resolved MRM velocity maps indicate a strong variation in the pore scale velocity as a function of biofilm growth. The hydrodynamic dispersion dynamics for flow through the porous media is quantitatively characterized using a pulsed gradient spin echo technique to measure the propagator of the motion. The propagator indicates a transition in transport dynamics from a Gaussian normal diffusion process following a normal advection diffusion equation to anomalous transport as a function of biofilm growth. Continuous time random walk models resulting in a time fractional advection diffusion equation are shown to model the transition from normal to anomalous transport in the context of a conceptual model for the biofouling. The initially homogeneous porous media is transformed into a more complex heterogeneous porous media by the biofilm growth.     

Permissibility of fractal exponents and models of band-limited two-point functions for fGn and fBm random fields

The fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) random field models have many applications in the environmental sciences. An issue of practical interest is the permissible range and the relations between different fractal exponents used to characterize these processes. Here we derive the bounds of the covariance exponent for fGn and the Hurst exponent for fBm based on the permissibility theorem by Bochner. We exploit the theoretical constraints on the spectral density to construct explicit two-point (covariance and structure) functions that are band-limited fractals with smooth cutoffs. Such functions are useful for modeling a gradual cutoff of power-law correlations. We also point out certain peculiarities of the relations between fractal exponents imposed by the mathematical bounds. Reliable estimation of the correlation and Hurst exponents typically requires measurements over a large range of scales (more than 3 orders of magnitude). For isotropic fractals and partially isotropic self-affine processes the dimensionality curse is partially lifted by estimating the exponent from measurements along fixed directions. We derive relations between the fractal exponents and the one-dimensional spectral density exponents, and we illustrate the relations using measurements of paper roughness.The author would like to acknowledge helpful comments from two anonymous referees.     

A stochastic model for anomalous diffusion in confined nano-films near a strain-induced critical point

A diffusive process is said to be anomalous if in any given direction the average square of the separation a particle experiences from its origin grows nonlinearly with time. Any diffusive process is anomalous if viewed on a short enough time scale, but interestingly, many diffusive processes remain anomalous over longer times. As a canonical example we study one such process here, diffusion in a laterally-confined nano-film as a function of the strain induced critical point. For this example we motivate and illustrate how a simple but novel process, Brownian motion run with a nonlinear clock (Bm-nlc), statistically mimics trajectories generated via Newton’s force law. The model is easily generalized to more complicated random processes and has application in many fields, including but not limited to, random conductivity field or terrain generation, Richardson turbulence in the atmosphere, and time dependent dispersion in hydrology.     

Eddy diffusivity derived from drifter data for dispersion model applications

Ocean transport and dispersion processes are at the present time simulated using Lagrangian stochastic models coupled with Eulerian circulation models that are supplying analyses and forecasts of the ocean currents at unprecedented time and space resolution. Using the Lagrangian approach, each particle displacement is described by an average motion and a fluctuating part. The first one represents the advection associated with the Eulerian current field of the circulation models while the second one describes the sub-grid scale diffusion. The focus of this study is to quantify the sub-grid scale diffusion of the Lagrangian models written in terms of a horizontal eddy diffusivity. Using a large database of drifters released in different regions of the Mediterranean Sea, the Lagrangian sub-grid scale diffusion has been computed, by considering different regimes when averaging statistical quantities. In addition, the real drifters have been simulated using a trajectory model forced by OGCM currents, focusing on how the Lagrangian properties are reproduced by the simulated trajectories.     

Impact of fractional probability distributions on statistics of hydraulic conductivity,dynamics of groundwater flow and solute transport at a low-permeability site

The fractional Brownian motion (fBm) and fractional Lévy motion (fLm) can easily describe the geometry and the statistical structure of hydraulic conductivity (K) for real-world. However, the fBm and fLm models have not been systematically evaluated when building the K field for a low-permeability site. In this study, both the fBm and fLm are used to simulate the low-K field at NingCheGu (NCG), Tianjin, China. Groundwater flow and solute transport are then computed using MODFLOW and MT3DMS, respectively, and the influence of the fBm/fLm models for K on groundwater flow and solute transport is discussed. Results show that the fLm fits better the statistics of the low-K medium than fBm, and the random logarithmic K (LnK) field generated by fLm is more stable because the resultant LnK field captures more of the measured properties at the field site than that generated by fBm. In contrast, the LnK generated by fBm is more likely to form both high-K channels and low-K barriers. The fBm therefore predicts more extreme behaviours in flow and transport, including the preferential flow, low-concentration blocks and solute retention. The overall groundwater renewal period and solute travel time for the fLm simulation are slightly shorter than those for fBm. The impacts of the fLm and fBm models on the statistics of the resultant LnK fields and the dynamics of groundwater flow and solute transport revealed by this study shed light on the selection and evaluation of the fractional probability distribution models in capturing the K fields for low-K media.     

Numerical modelling of organic waste dispersion from fjord located fish farms

In this study, a three-dimensional particle tracking model coupled to a terrain following ocean model is used to investigate the dispersion and the deposition of fish farm particulate matter (uneaten food and fish faeces) on the seabed due to tidal currents. The particle tracking model uses the computed local flow field for advection of the particles and random movement to simulate the turbulent diffusion. Each particle is given a settling velocity which may be drawn from a probability distribution according to settling velocity measurements of faecal and feed pellets. The results show that the maximum concentration of organic waste for fast sinking particles is found under the fish cage and continue monotonically decreasing away from the cage area. The maximum can split into two maximum peaks located at both sides of the centre of the fish cage area in the current direction. This process depends on the sinking time (time needed for a particle to settle at the bottom), the tidal velocity and the fish cage size. If the sinking time is close to a multiple of the tidal period, the maximum concentration point will be under the fish cage irrespective of the tide strength. This is due to the nature of the tidal current first propagating the particles away and then bringing them back when the tide reverses. Increasing the cage size increases the likelihood for a maximum waste accumulation beneath the fish farm, and larger farms usually means larger biomasses which can make the local pollution even more severe. The model is validated by using an analytical model which uses an exact harmonic representation of the tidal current, and the results show an excellent agreement. This study shows that the coupled ocean and particle model can be used in more realistic applications to help estimating the local environmental impact due to fish farms.     

Developing a Lagrangian sediment transport model for open channel flows

A three-dimensional stochastic Lagrangian particle tracking sediment transport model is developed to solve the discrete advection-dispersion equation using a combination of empirical dispersion equations.The performance of three widely-used longitudinal dispersion coefficient equations was examined to select one of them as the primary dispersion equation term in the developed model. Also, a conditional empirical equation was used to consider the effect of vertical dispersion term in top layers n...     

Statistical features of the seismic noise-field

We present a preliminary study of the dependence of the statistical features of the soil motion due to seismic noise on the near-surface geology in the frequency range from 1 Hz to ∼ 40 Hz. In detail, we have investigated the 3D average squared soil displacement 〈r²〉 and the distribution function of the displacement flctuations at different geological sites. The anomalous scaling of the average squared soil displacement 〈r²(τ)〉~τ_α, and the Gaussian shape of the probability distribution function of its fluctuations suggest that the soil motion under the influence of the seismic noise is consistent with a persistent fractional Brownian motion (fBm) characterized by a scaling exponent 1.5 < α < 2. Therefore, the seismic noise-field, thought as a stochastic process, shows a markovian character with a memory longer than a pure Brownian motion (α = 1/2). Moreover, a dependence of such persistent behavior of the noise-field dynamics on the near-surface local geology has been found and it is discussed.