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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

### Don Kulasiri

- 1st Edition, Volume 44 - November 22, 2002
- Authors: Don Kulasiri, Wynand Verwoerd
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 4 - 5 5 1 8 7 - 0
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 5 1 1 0 2 - 7
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 5 4 1 8 0 - 8

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… Read more

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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.

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.

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

- No. of pages: 252
- Language: English
- Edition: 1
- Volume: 44
- Published: November 22, 2002
- Imprint: North Holland
- Paperback ISBN: 9780444551870
- Hardback ISBN: 9780444511027
- eBook ISBN: 9780080541808

DK

Affiliations and expertise

Centre for Advanced Computational Solutions (C-fACS), Applied Computing, Mathematics and Statistics Group, PO Box 84, Lincoln University, Canterbury, New ZealandRead *Stochastic Dynamics. Modeling Solute Transport in Porous Media* on ScienceDirect