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Author: Tongbi Tu Publisher: ISBN: 9780438930575 Category : Languages : en Pages :
Book Description
Solute transport in natural flows is a complex process, which is affected by various factors, such as channel roughness, river geometry, upstream inflow, lateral inflow, solute loadings. These factors are often spatiotemporally heterogeneous and uncertain. As a result, the solute transport process by rivers and stream flows in natural environment is full of uncertainties and has been approached as a stochastic process. Traditional transport governing equations are at point scale and cannot appropriately represent the stochastic dynamics of the transport process when applied in a reach scale or beyond. Upscaled governing equations of solute transport process in open channel flows are proposed in this dissertation to account for the uncertainties/heterogeneity in the transport process. One- and two-dimensional solute transport models are developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian-Eulerian Fokker-Planck equations. The resulting Fokker-Planck equations are linear and deterministic differential equations, and these equations can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Moreover, the proposed governing equations can explicitly indicate the effect of the corresponding drifts on the uncertainty of the transport process. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport models, various steady and unsteady uncertain flow and solute loading conditions are applied. The Monte Carlo simulation with stochastic flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker-Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker-Planck equation approach indicates that the proposed models can adequately characterize the ensemble behavior of the solute transport process under uncertain flow and solute loading conditions via the evolutionary probability distribution in space and time of the transport process.
Author: Tongbi Tu Publisher: ISBN: 9780438930575 Category : Languages : en Pages :
Book Description
Solute transport in natural flows is a complex process, which is affected by various factors, such as channel roughness, river geometry, upstream inflow, lateral inflow, solute loadings. These factors are often spatiotemporally heterogeneous and uncertain. As a result, the solute transport process by rivers and stream flows in natural environment is full of uncertainties and has been approached as a stochastic process. Traditional transport governing equations are at point scale and cannot appropriately represent the stochastic dynamics of the transport process when applied in a reach scale or beyond. Upscaled governing equations of solute transport process in open channel flows are proposed in this dissertation to account for the uncertainties/heterogeneity in the transport process. One- and two-dimensional solute transport models are developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian-Eulerian Fokker-Planck equations. The resulting Fokker-Planck equations are linear and deterministic differential equations, and these equations can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Moreover, the proposed governing equations can explicitly indicate the effect of the corresponding drifts on the uncertainty of the transport process. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport models, various steady and unsteady uncertain flow and solute loading conditions are applied. The Monte Carlo simulation with stochastic flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker-Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker-Planck equation approach indicates that the proposed models can adequately characterize the ensemble behavior of the solute transport process under uncertain flow and solute loading conditions via the evolutionary probability distribution in space and time of the transport process.
Author: Don Kulasiri Publisher: Elsevier ISBN: 0080541801 Category : Mathematics Languages : en Pages : 253
Book Description
Most of the natural and biological phenomena such as solute transport in porous media exhibit variability which can not be modeled by using deterministic approaches. There is evidence in natural phenomena to suggest that some of the observations can not be explained by using the models which give deterministic solutions. Stochastic processes have a rich repository of objects which can be used to express the randomness inherent in the system and the evolution of the system over time. The attractiveness of the stochastic differential equations (SDE) and stochastic partial differential equations (SPDE) come from the fact that we can integrate the variability of the system along with the scientific knowledge pertaining to the system. One of the aims of this book is to explaim some useufl concepts in stochastic dynamics so that the scientists and engineers with a background in undergraduate differential calculus could appreciate the applicability and appropriateness of these developments in mathematics. The ideas are explained in an intuitive manner wherever possible with out compromising rigor. The solute transport problem in porous media saturated with water had been used as a natural setting to discuss the approaches based on stochastic dynamics. The work is also motivated by the need to have more sophisticated mathematical and computational frameworks to model the variability one encounters in natural and industrial systems. This book presents the ideas, models and computational solutions pertaining to a single problem: stochastic flow of contaminant transport in the saturated porous media such as that we find in underground aquifers. In attempting to solve this problem using stochastic concepts, different ideas and new concepts have been explored, and mathematical and computational frameworks have been developed in the process. Some of these concepts, arguments and mathematical and computational constructs are discussed in an intuititve manner in this book.
Author: Don Kulasiri Publisher: Springer ISBN: 9783642431142 Category : Science Languages : en Pages : 0
Book Description
The advection-dispersion equation that is used to model the solute transport in a porous medium is based on the premise that the fluctuating components of the flow velocity, hence the fluxes, due to a porous matrix can be assumed to obey a relationship similar to Fick’s law. This introduces phenomenological coefficients which are dependent on the scale of the experiments. This book presents an approach, based on sound theories of stochastic calculus and differential equations, which removes this basic premise. This leads to a multiscale theory with scale independent coefficients. This book illustrates this outcome with available data at different scales, from experimental laboratory scales to regional scales.
Author: Publisher: ISBN: Category : Languages : en Pages :
Book Description
Analysis of flow and solute transport problem in porous media are affected by uncertainty inbuilt both in boundary conditions and spatial variability in system parameters. The experimental investigation reveals that the parameters may vary in various scales by several orders. These affect the solute plume characteristics in field-scale problem and cause uncertainty in the prediction of concentration. The main focus of the present thesis is to analyze the probabilistic behavior of solute concentration in three dimensional(3-D) heterogeneous porous media. The framework for the probabilistic analysis has been developed using perturbation approach for both spectral based analytical and finite element based numerical method. The results of the probabilistic analysis are presented either in terms of solute plume characteristics or prediction uncertainty of the concentration. After providing a brief introduction on the role of stochastic analysis in subsurface hydrology in chapter 1, a detailed review of the literature is presented to establish the existing state-of-art in the research on the probabilistic analysis of flow and transport in simple and complex heterogeneous porous media in chapter 2. The literature review is mainly focused on the methods of solution of the stochastic differential equation. Perturbation based spectral method is often used for probabilistic analysis of flow and solute transport problem. Using this analytical method a nonlocal equation is solved to derive the expression of the spatial plume moments. The spatial plume moments represent the solute movement, spreading in an average sense. In chapter 3 of the present thesis, local dispersivity if also assumed to be random space function along with hydraulic conductivity. For various correlation coefficients of the random parameters, the results in terms of the field scale effective dispersivity are presented to demonstrate the effect of local dispersivity variation in space. The randomness of local.
Author: Tongbi Tu
Author: Tongbi Tu
Author: Lan Liang
Author: Don Kulasiri
Author: Xueqing Zhou
Author: Don Kulasiri
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Author: Alexander Yishan Sun
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