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E-raamat: Stochastic Dynamics. Modeling Solute Transport in Porous Media

, (Centre for Advanced Computational Solutions (C-fACS), Applied Computing, Mathematics and Statistics Group, PO Box 84, Lincoln University, Canterbury, New Zealand)
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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.

Arvustused

"As the authors state in their preface, the book is intended to encourage students and researchers in science and engineering to study the mathematics discussed in it, a goal which is reasonable to believe it can achieve." --Steve Wright (1-OAKL-MS; Rochester, MI) Mathematical Reviews, 2005.

Preface vii
Modeling Solute Transport in Porous Media
1(26)
Introduction
1(3)
Solute Transport in Porous Media
4(3)
Models of Hydrodynamic Dispersion
7(2)
Modeling Macroscopic Behavior
9(7)
Representative Elementary Volume
9(1)
Review of a Continuum Transport Model
10(6)
Measurements of Dispersivity
16(4)
Flow in Aquifers
20(3)
Transport in Heterogeneous Natural Formations
20(3)
Computational Modeling of Transport in Porous Media
23(4)
A Brief Review of Mathematical Background
27(42)
Introduction
27(5)
Elementary Stochastic Calculus
32(1)
What is Stochastic Calculus?
33(1)
Variation of a Function
34(3)
Convergence of Stochastic Processes
37(1)
Riemann and Stieltjes Integrals
38(1)
Brownian Motion and Wiener Processes
39(4)
Relationship between White Noise and Brownian Motion
43(1)
Relationships Among Properties of Brownian Motion
44(2)
Further Characteristics of Brownian Motion Realizations
46(3)
Generalized Brownian motion
49(1)
Ito Integral
49(4)
Stochastic Chain Rule (Ito Formula)
53(14)
Differential notation
53(2)
Stochastic Chain Rule
55(3)
Ito processes
58(4)
Stochastic Product Rule
62(2)
Ito Formula for Functions of Two Variables
64(3)
Stochastic Population Dynamics
67(2)
Computer Simulation of Brownian Motion and Ito Processes
69(14)
Introduction
69(1)
A Standard Wiener Process Simulation
69(4)
Simulation of Ito Integral and Ito Processes
73(5)
Simulation of Stochastic Population Growth
78(5)
Solving Stochastic Differential Equations
83(10)
Introduction
83(1)
General Form of Stochastic Differential Equations
83(2)
A Useful Result
85(5)
Solution to the General Linear SDE
90(3)
Potential Theory Approach to SDEs
93(18)
Introduction
93(3)
Ito Diffusions
96(2)
The Generator of an ID
98(1)
The Dynkin Formula
99(1)
Applications of the Dynkin Formula
100(2)
Extracting Statistical Quantities from Dynkin's Formula
102(7)
What is the probability to reach a population value K?
103(1)
What is the expected time to reach a value K?
104(2)
What is the Expected Population at a Time t?
106(3)
The Probability Distribution of Population Realizations
109(2)
Stochastic Modeling of the Velocity
111(16)
Introduction
111(2)
Spectral Expansion of Wiener Processes in Time and in Space
113(4)
Solving the Covariance Eigenvalue Equation
117(3)
Extension to Multiple Dimensions
120(1)
Scalar Stochastic Processes in Multiple Dimensions
120(4)
Vector Stochastic Processes in Multiple Dimensions
124(1)
Simulation of Stochastic Flow in 1 and 2 Dimensions
125(2)
1-D case
125(1)
2-D Case
126(1)
Applying Potential Theory Modeling to Solute Dispersion
127(42)
Introduction
127(5)
Integral Formulation of Solute Mass Conservation
132(7)
Stochastic Transport in a Constant Flow Velocity
139(10)
Stochastic Transport in a Flow with a Velocity Gradient
149(4)
Standard Solution of the Generator Equation
153(3)
Alternate Solution of the Generator Equation
156(5)
Evolution of a Gaussian Concentration Profile
161(8)
A Stochastic Computational Model for Solute Transport in Porous Media
169(36)
Introduction
169(1)
Development of a Stochastic Model
170(6)
Covariance Kernel for Velocity
176(1)
Computational Solution
177(4)
Numerical Scheme
177(3)
The Behavior of the Model
180(1)
Computational Investigation
181(8)
Hypotheses Related to Variance and Correlation Length
189(3)
Scale Dependency
192(1)
Validation of One Dimensional SSTM
193(11)
Lincoln University Experimental Aquifers
194(1)
Methodology of Validation
195(1)
Results
196(8)
Concluding Remarks
204(1)
Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length
205(14)
Introduction
205(3)
Approximate Solutions
208(4)
Results
212(5)
Conclusions
217(2)
A Stochastic Inverse Method to Estimate Parameters in Groundwater Models
219(14)
Introduction
219(1)
System Dynamics with Noise
220(5)
An Example
222(3)
Applications in Groundwater Models
225(6)
Estimation Related to One Parameter Case
225(4)
Estimation Related to Two Parameter Case
229(1)
Investigation of the Methods
230(1)
Results
231(1)
Concluding Remarks
232(1)
References 233(4)
Index 237


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