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Stochastic Dynamics. Modeling Solute Transport in Porous Media - 1st Edition - ISBN: 9780444511027, 9780080541808

Stochastic Dynamics. Modeling Solute Transport in Porous Media, Volume 44

1st Edition

Authors: Don Kulasiri Wynand Verwoerd
Hardcover ISBN: 9780444511027
eBook ISBN: 9780080541808
Imprint: North Holland
Published Date: 22nd November 2002
Page Count: 252
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Table of 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

  1. A brief review of mathematical background

  2. 1 Introduction

  3. 2 Elementary stochastic calculus

  4. 3 What is stochastic calculus?

  5. 4 Variation of a function

  6. 5 Convergence of stochastic processes

  7. 6 Riemann and Stieltjes integrals

  8. 7 Brownian motion and Wiener processes

  9. 8 Relationship between white noise and Brownian motion

  10. 9 Relationships among properties of Brownian motion

  11. 10 Further characteristics of Brownian motion realizations

  12. 11 Generalized Brownian motion

  13. 12 Ito integral

  14. 13 Stochastic chain rule (Ito formula)

  15. 13.1 Differential notation

  16. 13.2 Stochastic chain rule

  17. 13.3 Ito processes

  18. 13.4 Stochastic product rule

  19. 13.5 Ito formula for functions of two variables

  20. 14 Stochastic population dynamics

  21. Computer simulation of Brownian motion and Ito processes

  22. 1 Introduction

  23. 2 A standard Wiener process simulation

  24. 3 Simulation of Ito integral and Ito processes

  25. 4 Simulation of stochastic population growth

  26. Solving stochastic differential equations

  27. 1 Introduction

  28. 2 General form of stochastic differential equations

  29. 3 A useful result

  30. 4 Solution to the general linear SDE

  31. Potential theory approach to SDEs

  32. 1 Introduction

  33. 2 Ito diffusions

  34. 3 The generator of an ID

  35. 4 The Dynkin formula

  36. 5 Applications of the Dynkin formula

  37. 6 Extracting statistical quantities from Dynkin's formula

  38. 6.1 What is the probability to reach a population Value K?

  39. 6.2 What is the expected time for the population to reach a value K?

  40. 6.3 What is the expected population at a time t?

  41. 7 The probability distribution of population growth realizations

  42. Stochastic modeling of the velocity

  43. 1 Introduction

  44. 2 Spectral expansion of Wiener processes in time and in Space

  45. 3 Solving the covariance eigenvalue equation

  46. 4 Extension to multiple dimensions

  47. 5 Scalar stochastic processes in multiple dimensions

  48. 6 Vector stochastic processes in multiple dimensions

  49. 7 Simulation of stochastic flow in 1 and 2 dimensions

  50. 7.1 1-D case

  51. 7.2 2-D case

  52. Applying potential theory modeling to solute dispersion

  53. 1 Introduction

  54. 2 Integral formulation of solute mass conservation

  55. 3 Stochastic transport in a constant flow velocity

  56. 4 Stochastic transport in a flow with a velocity gradient

  57. 5 Standard solution of the generator equation

  58. 6 Alternate solution of the generator equation

  59. A stochastic computational model for solute transport in porous media

  60. 1 Introduction

  61. 2 Development of a stochastic model

  62. 3 Covariance kernel for velocity

  63. 4 Computational solution

  64. 4.1 Numerical scheme

  65. 4.2 The behavior of the model

  66. 5 Computational investigation

  67. 6 Hypotheses related to variance and correlation length

  68. 7 Scale dependency

  69. 8 Validation of one dimensional SSTM

  70. 8.1 Lincoln University experimental aquifer

  71. 8.2 Methodology of validation

  72. 8.3 Results

  73. 9 Concluding remarks

  74. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length

  75. 1 Introduction

  76. 2 Approximate solutions

  77. 3 Results

  78. 4 Conclusions

  79. A stochastic inverse method to estimate parameters in groundwater models

  80. 1 Introduction

  81. 2 System dynamics with noise

  82. 2.1 An example

  83. 3 Applications in groundwater models

  84. 3.1 Estimation related to one-parameter case

  85. 3.2 Estimation related to two-parameter case

  86. 3.3 Investigation of the methods

  87. 4 Results

  88. 5 Concluding remarks


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.


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© North Holland 2002
22nd November 2002
North Holland
Hardcover ISBN:
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"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.

Ratings and Reviews

About the Authors

Don Kulasiri

Affiliations and Expertise

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

Wynand Verwoerd