Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The transport of materials is of fundamental importance, with studies on diffusion being at the forefront. Diffusion in a simple matrix is typically considered Fickian. However, anomalous diffusion in various media is a dominant process. It is well documented that anomalous diffusion results from medium heterogeneity, extreme events, phase transitions, medium surface dynamics, and other factors. The present work demonstrated that the diffusion of a cationic dye (crystal violet, CV) in an agar gel medium was anomalous, as shown by spatial and time-series data of dye movement. We estimated the classical diffusion coefficients using the Einstein-Smoluchowski solution to Fick's law, but these did not yield consistent Gaussianity, stationarity, or non-seasonality. However, anomalous diffusion was confirmed by modelling the experimental data. The exponent values (α) ranged from 0.468 ± 0.027 to 0.883 ± 0.107, indicating anomalous sub-diffusion that depended on the concentration of the CV dye in the same medium. We then evaluated time- and ensemble-averaged mean squared displacement using the dye-spreading data via image processing. The discrepancy in the distribution functions over a long experimental period highlighted the non-ergodic nature of the stochastic process. The dimensionless ergodicity-breaking parameter was evaluated, confirming the non-ergodic nature of the process. We found that the continuous time random walk (CTRW) model was well-suited for describing the anomalous diffusion in this system. We attributed the sub-diffusion of the dye to violations of the assumptions of Brownian motion of the particles and the nature of the diffusing medium.

The purpose of the current study was to evaluate the performance of a continuous-time random-walk (CTRW) diffusion model for differentiating malignant and benign breast lesions and to consider the potential association between CTRW parameters and the Ki-67 expression. Sixty-four patients (46.2 ± 11.4 years) with breast lesions (29 malignant and 35 benign) were evaluated with the CTRW model, intravoxel incoherent motion model, and diffusion-weighted imaging. Echo planar diffusion-weighted imaging was conducted using 13 b-values (0-3000 s/mm2 ). Three CTRW model parameters, including an anomalous diffusion coefficient Dm , and two parameters related to temporal and spatial diffusion heterogeneity, α and β, respectively, were obtained, and had MRI b-values of 0-3000 s/mm2 . Receiver operating characteristic (ROC) analysis was conducted to determine the sensitivity, specificity, and diagnostic accuracy of CTRW parameters for differentiating malignant from benign breast lesions. In malignant breast lesions, the CTRW parameters Dm , α, and β were significantly lower than the corresponding parameters of benign breast lesions. In the malignant breast lesion group, the CTRW parameter Dm was significantly lower in high Ki-67 expression than in low Ki-67 expression. In ROC analysis, the combination of CTRW parameters (Dm , α, β) demonstrated the highest area under the curve value (0.985) and diagnostic accuracy (94.23%) in differentiating malignant and benign breast lesions. The CTRW model effectively differentiated malignant from benign breast lesions. The CTRW diffusion model offers a new way for noninvasive assessment of breast malignancy and better understanding of the proliferation of malignant lesions.

Fluid flow and solute transport simulations in tight geologic formations: Discrete fracture network and continuous time random walk analyses

In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

The dispersal of individuals of a species is the key driving force of various spatiotemporal phenomena which occur on geographical scales. It can synchronise populations of interacting species, stabilise them, and diversify gene pools [1-3]. The geographic spread of human infectious diseases such as influenza, measles and the recent severe acute respiratory syndrome (SARS) is essentially promoted by human travel which occurs on many length scales and is sustained by a variety of means of transportation [4-8]. In the light of increasing international trade, intensified human traffic, and an imminent influenza A pandemic the knowledge of dynamical and statistical properties of human dispersal is of fundamental importance and acute [7,9,10]. A quantitative statistical theory for human travel and concomitant reliable forecasts would substantially improve and extend existing prevention strategies. Despite its crucial role, a quantitative assessment of human dispersal remains elusive and the opinion that humans disperse diffusively still prevails in many models [11]. In this chapter I will report on a recently developed technique which permits a solid and quantitative assessment of human dispersal on geographical scales [11]. The key idea is to infer the statistical properties of human travel by analysing the geographic circulation of individual bank notes for which comprehensive datasets are collected at the online bill-tracking website www.wheresgeorge.com. The analysis shows that the distribution of travelling distances decays as a power law, indicating that the movement of bank notes is reminiscent of superdiffusive, scale free random walks known as Lèvy flights [13]. Secondly, the probability of remaining in a small, spatially confined region for a time T is dominated by heavy tails which attenuate superdiffusive dispersal. I will show that the dispersal of bank notes can be described on many spatiotemporal scales by a two parameter continuous time random walk (CTRW) model to a surprising accuracy. To this end, I will provide a brief introduction to continuous time random walk theory [14] and will show that human dispersal is an ambivalent, effectively superdiffusive process.

Continuous time random walks for non-local radial solute transport

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

In Chapter 4, we obtained the governing equations of continuous-time random processes by taking a suitable joint limit Δ -→0 and τ -→0 of a lattice random walk, where Δ is the lattice spacing and τ the constant time between successive steps. The continuous-time processes so obtained are governed by stable densities (§4.3), with the Gaussian density representing the most typical case. In taking the continuum limit the lattice structure and the identity of individual steps are lost, and the analysis of many properties of the processes so obtained requires rather deep and subtle aspects of advanced probability theory, in contrast to the elegant conceptual simplicity of the original random walk. There is a less drastic way to obtain a continuous time process related to the ordinary random walk, without losing either the lattice structure or the notion of well-defined individual steps. This approach, the continuous time random walk model, owes its popularity in contemporary physics to Montroll and Weiss (1965). From a mathematical standpoint it is a particularly useful special case of the semi-Markov process (Levy 1954, Smith 1955, Pyke 1961, Feller 1964, Qinlar 1969, 1975).

On fractional diffusion and continuous time random walks

Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition directionand waiting time for every state and2) detailed balance among the transition probabilities. Togetherthey are also sufficient condition. For a CTRW in an open systemwith explicit chemical energy input 1) still holds while 2)breaks down. Hence, CTRW models not satisfying 1) are eitherinconsistent with thermodynamics or cannot attain equilibriumdue to hidden dissipation in non-Markovian states.Each CTRW defines a unique corresponding Markov process (cMP).The steady-state distribution of a CTRW equals that of the corresponding Markov process,and the two systems have the same steady-state flux,the same exit probabilities and thesame mean trapping times. Mechanicity isdiscussed; a paradox observed by Kolomeisky and Fisher(J. Chem. Phys., 113 (2000) 10867) is resolved.

We study the upscaling of pore-scale solute transport in partially saturated porous media at different saturation degrees. The interaction between structural heterogeneity, phases distribution and small-scale flow dynamics induces complex flow patterns and broad probability distributions of flow. In turn, this spatial distribution of flow velocities, at the pore scale, induces irregular (non-Fickian) transport of dissolved substances (e.g., contaminants), causing an earlier arrival and longer tailing, which may have grave consequences in underestimating risk assessments and prolonged cleanup times of contaminated sites.Here, we suggest an integrated continuous time random walk (CTRW) modeling framework, which accounts for also the entrapping of particles in zones of low flow velocities, to estimate the resident times of solutes in the media. Furthermore, comparing the results of the CTRW model to a well-established numerical simulation method allows a phenomenological evaluation of the model's physical parameters for different conditions (i.e., volume of entrapped air, mean water flow rate, or solute molecular diffusion coefficient).   In this study we show that entrapped air promotes preferential solute transport and solute trapping in low flow regions. Moreover, we demonstrate that the trapping frequency and trapping time depend on the interaction between advection and diffusion (i.e., the Péclet number). An integrative CTRW model captures the effects of trapping in stagnant regions and preferential transport on non-Fickian dispersion of solutes.

Non-Fickian transport ubiquitously occurs across all scales within fractured geological media. Detailed characterization of non-Fickian transport through single fractures is thus critical for predicting the fate of solutes and other fluid-borne entities through fractured media. Our direct numerical simulations of solute transport through two-dimensional rough-walled fractures showed early arrival and heavy tailing in breakthrough curves (BTCs), which are salient characteristics of non-Fickian transport. Analyses for dispersion coefficients (DADE) using the standard advection-dispersion equation (ADE) led to errors which increased linearly with fracture heterogeneity. Estimated Taylor dispersion coefficients deviated from estimated DADE even at higher Peclet numbers. Alternatively, we used continuous time random walk (CTRW) model with truncated power law transition rate probability to characterize the non-Fickian transport. CTRW modeling markedly and consistently improved fits to the BTCs relative to those fitted with ADE solutions. The degree of deviation of transport from Fickian to non-Fickian is captured by the parameter β of the truncated power law. We found that β is proportional to fracture heterogeneity. We also found that the CTRW transport velocity can be predicted based on the flow velocity. Along with the ability to predict β, this is a major step toward prediction of transport through CTRW using measurable physical properties.

Superdiffusion in decoupled continuous time random walks

Continuous time random walk model as a model of anomalous relaxation

The Portevin–Le Chatelier effect in the continuous time random walk framework