Stochastic Dynamics. Modeling Solute Transport in Porous Media
Regular price
53.750 KD
inc. VAT
Couldn't load pickup availability
Table of contents
- Cover
- Contentsix
- Prefacevii
- Chapter 1. Modeling Solute Transport in Porous Media1
- 1.1 Introduction1
- 1.2 Solute Transport in Porous Media4
- 1.3 Models of Hydrodynamic Dispersion7
- 1.4 Modeling Macroscopic Behavior9
- 1.5 Measurements of Dispersivity16
- 1.6 Flow in Aquifers20
- 1.7 Computational Modeling of Transport in Porous Media23
- Chapter 2. A Brief Review of Mathematical Background27
- 2.1 Introduction27
- 2.2 Elementary Stochastic Calculus32
- 2.3 What is Stochastic Calculus?33
- 2.4 Variation of a Function34
- 2.5 Convergence of Stochastic Processes37
- 2.6 Riemann and Stieltjes Integrals38
- 2.7 Brownian Motion and Wiener Processes39
- 2.8 Relationship between White Noise and Brownian Motion43
- 2.9 Relationships Among Properties of Brownian Motion44
- 2.10 Further Characteristics of Brownian Motion Realizations46
- 2.11 Generalized Brownian motion49
- 2.12 Ito Integral49
- 2.13 Stochastic Chain Rule (Ito Formula)53
- 2.14 Stochastic Population Dynamics67
- Chapter 3. Computer Simulation of Brownian Motion and Ito Processes69
- 3.1 Introduction69
- 3.2 A Standard Wiener Process Simulation69
- 3.3 Simulation of Ito Integral and Ito Processes73
- 3.4 Simulation of Stochastic Population Growth78
- Chapter 4. Solving Stochastic Differential Equations83
- 4.1 Introduction83
- 4.2 General Form of Stochastic Differential Equations83
- 4.3 A Useful Result85
- 4.4 Solution to the General Linear SDE90
- Chapter 5. Potential Theory Approach to SDEs93
- 5.1 Introduction93
- 5.2 Ito Diffusions96
- 5.3 The Generator of an ID98
- 5.4 The Dynkin Formula99
- 5.5 Applications of the Dynkin Formula100
- 5.6 Extracting Statistical Quantifies from Dynkin's Formula102
- 5.7 The Probability Distribution of Population Realizations109
- Chapter 6. Stochastic Modeling of the Velocity111
- 6.1 Introduction111
- 6.2 Spectral Expansion of Wiener Processes in Time and in Space113
- 6.3 Solving the Covariance Eigenvalue Equation117
- 6.4 Extension to Multiple Dimensions120
- 6.5 Scalar Stochastic Processes in Multiple Dimensions120
- 6.6 Vector Stochastic Processes in Multiple Dimensions124
- 6.7 Simulation of Stochastic Flow in 1 and 2 Dimensions125
- Chapter 7. Applying Potential Theory Modeling to Solute Dispersion127
- 7.1 Introduction127
- 7.2 Integral Formulation of Solute Mass Conservation132
- 7.3 Stochastic Transport in a Constant Flow Velocity139
- 7.4 Stochastic Transport in a Flow with a Velocity Gradient149
- 7.5 Standard Solution of the Generator Equation153
- 7.6 Alternate Solution of the Generator Equation156
- 7.7 Evolution of a Gaussian Concentration Profile161
- Chapter 8. A Stochastic Computational Model for Solute Transport in Porous Media169
- 8.1 Introduction169
- 8.2 Development of a Stochastic Model170
- 8.3 Covariance Kernel for Velocity176
- 8.4 Computational Solution177
- 8.5 Computational Investigation181
- 8.6 Hypotheses Related to Variance and Correlation Length189
- 8.7 Scale Dependency192
- 8.8 Validation of One Dimensional SSTM193
- 8.7 Concluding Remarks204
- Chapter 9. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length205
- 9.1 Introduction205
- 9.2 Approximate Solutions208
- 9.3 Results212
- 9.4 Conclusions217
- Chapter 10. A Stochastic Inverse Method to Estimate Parameters in Groundwater Models219
- 10.1 Introduction219
- 10.2 System Dynamics with Noise220
- 10.3 Applications in Groundwater Models225
- 10.4 Results231
- 10.5 Concluding Remarks232
- References233
- Index237
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.
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.