**Meet the editor**

Professor Don Kulasiri obtained B.Sc. (Hons) in Mechanical Engineering (1980) at University of Peradeniya, Sri Lanka, M.S. (1988) and PhD (1990) at Department of Biological Systems Engineering, Virginia Tech, U.S.A. His main research interests are in developing mathematical and computational models of biological and environmental systems to explain the experimental data, and

to pursue these interests he has been developing research programmes at Lincoln University, Christchurch, New Zealand since 1991. He was appointed to a personal chair (professorship) in 1999 at Lincoln University, and he has been a visiting professor at the Computation and Mechanics division, Stanford University (6 months in 1998), Department of Mathematics, Princeton University (6 months in 2004) and Mathematical Institute, Oxford University (6 months in 2008 and a month in 2010). He has been a supervisor or co-supervisor of 30 PhD students so far, and co-authored over 100 journal publications and three books. He has been active in Computational Molecular Systems Biology for the last six years.

Chapter 1

**Preface VII**

**NonFickian Solute Transport 1**

**A Stochastic Model for Hydrodynamic Dispersion 65**

**A Generalized Mathematical Model in One-Dimension 117**

**The Stochastic Solute Transport Model in 2-Dimensions 195**

**References 221**

**Index 233**

**Theories of Fluctuations and Dissipation 161**

**Multiscale, Generalised Stochastic Solute Transport Model in One Dimension 177**

**Multiscale Dispersion in 2 Dimensions 215**

**Stochastic Differential Equations and Related Inverse Problems 21**

Contents

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

## Contents

### **Preface XI**


**Index 233**

In this research monograph, we explain the development of a mechanistic, stochastic theory of nonfickian solute dispersion in porous media. We have included sufficient amount of background material related to stochastic calculus and the scale dependency

The advection-dispersion equation that is being used to model the solute transport in a porous medium is based on the premise that the fluctuating components of the flow velocity, hence the fluxes, due to a porous matrix can be assumed to obey a relationship similar to Fick's law. This introduces phenomenological coefficients which are dependent on the scale of the experiments. Our approach, based on the theories of stochastic calculus and differential equations, removes this basic premise, which leads to a multiscale theory with scale independent coefficients. We try to illustrate this outcome with available data at different scales, from experimental laboratory scales to regional scales in this monograph. There is a large body of computational experiments we have not discussed here, but their results corroborate with the gist presented here.

In Chapter 1, we introduce the context of the research questions we are seeking answers in the rest of the monograph. We dedicate the first part of Chapter 2 as a primer for Ito stochastic calculus and related integrals. We develop a basic stochastic solute transport model in Chapter 3 and develop a generalised model in one dimension in Chapter 4. In Chapter 5, we attempt to explain the connectivity of the basic premises in our theory with the established theories in fluctuations and dissipation in physics. This is only to highlight the alignment, mostly intuitive, of our approach with the established physics. Then we develop the multiscale stochastic model in Chapter 6, and finally we extend the approach to two dimensions in Chapters 7 and 8. We may not have cited many authors who have published research related to nonfickian dispersion because our intention is to highlight the problem through the literature. We refer to recent books which summarise most of the works and apologise for omissions as this monograph is

There are many who helped me during the course of this research. I really appreciate Hong Ling's assistance during the last two and half years in writing and testing Mathematica programs. Without her dedication, this monograph would have taken many more months to complete. I am grateful to Amphun Chaiboonchoe for typing of the first six chapters in the first draft, and to Yao He for Matlab programming work for Chapter 6. I also acknowledge my former PhD students, Dr. Channa Rajanayake of Aqualinc Ltd, New Zealand, for the assistance in inverse method computations, and Dr. Zhi Xie of National Institute for Health (NIH), U.S.A., for the assistance in the neural

of diffusivity in this book so that it could be read independently.

not intented to be a comprehensive review.

networks computations.

Preface
