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Über dieses Buch

The advection-dispersion equation that is 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. This book presents an approach, based on sound theories of stochastic calculus and differential equations, which removes this basic premise. This leads to a multiscale theory with scale independent coefficients. This book illustrates this outcome with available data at different scales, from experimental laboratory scales to regional scales.



Chapter 1. Non-fickian Solute Transport

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 previous book is not disadvantaged to follow the material presented.

Don Kulasiri

Chapter 2. Stochastic Differential Equations and Related Inverse Problems

As we have discussed in Chap. 1, the deterministic mathematical formulation of solute transport through a porous medium introduces the dispersivity, which is a measure of the distance a solute tracer would travel when the mean velocity is normalized to be one. One would expect such a measure to be a mechanical property of the porous medium under consideration, but the evidence are there to show that dispersivity is dependent on the scale of the experiment for a given porous medium. One of the challenges in modelling the phenomena is to discard the Fickian assumptions, through which dispersivity is defined, and develop a mathematical description containing the fluctuations associated with the mean velocity of a physical ensemble of solute particles. To this end, we require a sophisticated mathematical framework, and the theory of stochastic processes and differential equations is a natural mathematical setting. In this chapter we review some essential concepts in stochastic processes and stochastic differential equations in order to understand the stochastic calculus in a more applied context.

Don Kulasiri

Chapter 3. A Stochastic Model for Hydrodynamic Dispersion

We have seen in Chap. 1 that, in the derivation of advection–dispersion equation, also known as continuum transport model (Rashidi et al. 1999), the velocity fluctuations around the mean velocity enter into the calculation of solute flux at a given point through averaging theorems. The mean advective flux and the mean dispersive flux are then related to the concentration gradients through Fickian–type assumptions. These assumptions are instrumental in defining dispersivity as a measure of solute dispersion. Dispersivity is proven to be scale dependant.

Don Kulasiri

Chapter 4. A Generalized Mathematical Model in One-Dimension

In the previous chapter we derived a stochastic solute transport model (Eq. 3.14); we developed the methods to estimate its parameters, and investigated its behaviour numerically. We see some promise to characterise the solute dispersion at different flow lengths, and there are some indications that Eq. (3.14) produce the behaviours that would be interpreted as capturing the scale-dependency of dispersivity.

Don Kulasiri

Chapter 5. Theories of Fluctuations and Dissipation

In the previous chapters, we see that the hydrodynamic dispersion is in fact a result of solute particles moving along a decreasing pressure gradient and encountering the solid surfaces of a porous medium. The pressure gradient provides the driving force which translates into kinetic energy, and the porous medium acts as the dissipater of the kinetic energy; any such energy dissipation associated with small molecules generates fluctuations among molecules.

Don Kulasiri

Chapter 6. Multiscale, Generalised Stochastic Solute Transport Model in One Dimension

In Chaps. 3 and 4, we have developed a stochastic solute transport model in 1-D without rosorting to simplifying Fickian assumptions, but by using the idea that the fluctuations in velocity are influenced by the nature of porous medium. We model these fluctuations through the velocity covariance kernel. We have also estimated the dispersivity by taking the realisations of the solution of the SSTM and using them as the observations in the stochastic inverse method (SIM) based on the maximum likelihood estimation procedure for the stochastic partial differential equation obtained by adding a noise term to the advection-dispersion equation.

Don Kulasiri

Chapter 7. The Stochastic Solute Transport Model in 2-Dimensions


Chap. 6

, we developed the generalised Stochastic Solute Transport Model (SSTM) in 1-dimension and showed that it can model the hydrodynamic dispersion in porous media for the flow lengths ranging from 1 to 10000 m. For computational efficiency, we have employed one of the fastest converging kernels tested in Chapter 6 for illustrative purposes, but, in principle, the SSTM should provide scale independent behaviour for any other velocity covariance kernel.

Don Kulasiri

Chapter 8. Multiscale Dispersion in Two Dimensions

In Chapter 7, we have developed the 2 dimensional solute transport model and estimated the dispersion coefficients in both longitudinal and transverse directions using the stochastic inverse method (SIM), which is based on the maximum likelihood method. We have seen that transverse dispersion coefficient relative to longitudinal dispersion coefficient increases as



increases when the flow length is confined to 1.0. In this chapter, we extend the SSTM2d into a partially dimensional form as we did for 1 dimension, so that we can explore the larger scale behaviours of the model. However, the experimental data on transverse dispersion is scarce in laboratory and field scales limiting our ability to validate the multiscale dispersion model. In this chapter, we briefly outline the dimensionless form of SSTM2d and illustrate the numerical solution for a particular value of flow length. We also estimate the dispersion coefficients using the SIM for the same flow length.

Don Kulasiri


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