Preface

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

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 not intented to be a comprehensive review.

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 networks computations.

### XII Preface

This work is funded by the Foundation for Science and Technology of New Zealand (FoRST) through Lincoln Ventures Ltd. (LVL), Lincoln University. I am grateful to the Chief Scientist of LVL, my colleague, Dr. Ian Woodhead for overseeing the contractual matters to facilitate the work with a sense of humour. I also acknowledge Dr. John Bright of Aqualinc Ltd. for managing the project for many years.

Finally I am grateful to my wife Professor Sandhya Samarasinghe for understanding the value of this work. Her advice on neural networks helped in the computational methods developed in this work. Sandhya's love and patience remained intact during this piece of work. To that love and patience, I dedicate this monograph.

> **Don Kulasiri** Professor Centre for Advanced Computational Solutions (C-fACS) Lincoln University, New Zealand

NonFickian Solute Transport 1

This research monograph presents the modelling of solute transport in the saturated porous media using novel stochastic and computational approaches. Our previous book published in the North-Holland series of Applied Mathematics and Mechanics (Kulasiri and Verwoerd, 2002) covers some of our research in an introductory manner; this book can be considered as a sequel to it, but we include most of the basic concepts succinctly here, suitably placed in the main body so that the reader who does not have the access to the

The motivation of this work has been to explain the dispersion in saturated porous media at different scales in underground aquifers (i.e., subsurface groundwater flow), based on the theories in stochastic calculus. Underground aquifers render unique challenges in determining the nature of solute dispersion within them. Often the structure of porous formations is unknown and they are sometimes notoriously heterogeneous without any recognizable patterns. This element of uncertainty is the over-arching factor which shapes the nature of solute transport in aquifers. Therefore, it is reasonable to review briefly the work already done in that area in the pertinent literature when and where it is necessary. These interludes of previous work should provide us with necessary continuity of thinking

There is monumental amount of research work done related to the groundwater flow since 1950s. During the last five to six decades major changes to the size and demographics of human populations occurred; as a result, an unprecedented use of the hydrogeological resources of the earth makes contamination of groundwater a scientific, socio-economic and, in many localities, a political issue. What is less obvious in terms of importance is the way a contaminant, a solute, disperses itself within the geological formations of the aquifers. Experimentation with real aquifers is expensive; hence the need for mathematical and computational models of solute transport. People have developed many types of models over the years to understand the dynamics of aquifers, such as physical scale models, analogy models and mathematical models (Wang and Anderson, 1982; Anderson and Woessner, 1992; Fetter, 2001; Batu, 2006). All these types of models serve different purposes. Physical scale models are helpful to understand the salient features of groundwater flow and measure the variables such as solute concentrations at different locations of an artificial aquifer. A good example of this type of model is the two artificial aquifers at Lincoln University, New Zealand, a brief description of which appears in the monograph by Kulasiri and Verwoerd (2002). Apart from understanding the physical and chemical processes that occur in the aquifers, the measured variables can be used to partially validate the mathematical models. Inadequacy of these physical models is that their flow lengths are

**1.1 Models in Solute Transport in Porous Media**

in this work.

previous book is not disadvantaged to follow the material presented.

**NonFickian Solute Transport** 

**1** 
