STOCHASTIC DYNAMICS. MODELING SOLUTE TRANSPORT IN POROUS MEDIA
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By Don Kulasiri, Centre for Advanced Computational Solutions (C-fACS), Applied Computing, Mathematics and Statistics Group, PO Box 84, Lincoln University, Canterbury, New Zealand Wynand Verwoerd
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.
Contents
Table of Content
1.Modeling solute transport in porous media
1.1 Introduction
1.2 Solute transport in porous media
1.3 Models of hydrodynamic
dispersion
1.4 Modeling macroscopic behavior
1.4.1 Representative elementary volume
1.4.2 Review of continuum transport model
1.5 Measurements
of dispersivity
1.6 Flow in aquifers
1.6.1 Transport in heterogeneous natural formations
1.7 Computational modeling of solute transport
in porous media
2. A brief review of mathematical background
2.1 Introduction
2.2 Elementary stochastic calculus
2.3 What is stochastic
calculus?
2.4 Variation of a function
2.5 Convergence of stochastic processes
2.6 Riemann and Stieltjes integrals
2.7 Brownian motion
and Wiener processes
2.8 Relationship between white noise and Brownian motion
2.9 Relationships among properties of Brownian motion
2.10
Further characteristics of Brownian motion realizations
2.11 Generalized Brownian motion
2.12 Ito integral
2.13 Stochastic chain rule
(Ito formula)
2.13.1 Differential notation
2.13.2 Stochastic chain rule
2.13.3 Ito processes
2.13.4 Stochastic product rule
2.13.5 Ito
formula for functions of two variables
2.14 Stochastic population dynamics
3. Computer simulation of Brownian motion and Ito processes
3.1 Introduction
3.2 A standard Wiener process simulation
3.3 Simulation of Ito integral and Ito processes
3.4 Simulation of stochastic
population growth
4. Solving stochastic differential equations
4.1 Introduction
4.2 General form of stochastic differential equations
4.3 A useful result
4.4 Solution to the general linear SDE
5. Potential theory approach to SDEs
5.1 Introduction
5.2 Ito diffusions
5.3 The generator of an ID
5.4 The Dynkin formula
5.5 Applications of the Dynkin formula
5.6 Extracting statistical quantities from Dynkin's
formula
5.6.1 What is the probability to reach a population Value K?
5.6.2 What is the expected time for the population to reach a value
K?
5.6.3 What is the expected population at a time t?
5.7 The probability distribution of population growth realizations
6. Stochastic
modeling of the velocity
6.1 Introduction
6.2 Spectral expansion of Wiener processes in time and in Space
6.3 Solving the covariance
eigenvalue equation
6.4 Extension to multiple dimensions
6.5 Scalar stochastic processes in multiple dimensions
6.6 Vector stochastic
processes in multiple dimensions
6.7 Simulation of stochastic flow in 1 and 2 dimensions
6.7.1 1-D case
6.7.2 2-D case
7. Applying potential
theory modeling to solute dispersion
7.1 Introduction
7.2 Integral formulation of solute mass conservation
7.3 Stochastic transport in
a constant flow velocity
7.4 Stochastic transport in a flow with a velocity gradient
7.5 Standard solution of the generator equation
7.6 Alternate solution of the generator equation
8. A stochastic computational model for solute transport in
porous media
8.1 Introduction
8.2 Development of a stochastic model
8.3 Covariance kernel for velocity
8.4 Computational solution
8.4.1 Numerical scheme
8.4.2 The
behavior of the model
8.5 Computational investigation
8.6 Hypotheses related to variance and correlation length
8.7 Scale dependency
8.8 Validation of one dimensional SSTM
8.8.1 Lincoln University experimental aquifer
8.8.2 Methodology of validation
8.8.3 Results
8.9 Concluding remarks
9. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length
9.1 Introduction
9.2
Approximate solutions
9.3 Results
9.4 Conclusions
10. A stochastic inverse method to estimate parameters in groundwater models
10.1
Introduction
10.2 System dynamics with noise
10.2.1 An example
10.3 Applications in groundwater models
10.3.1 Estimation related to one-parameter
case
10.3.2 Estimation related to two-parameter case
10.3.3 Investigation of the methods
10.4 Results
10.5 Concluding remarks
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