Stochastic Dynamics. Modeling Solute Transport in Porous Media (eBook)

Don Kulasiri, Wynand Verwoerd (Autoren)

eBook Download: PDF

2002 | 1. Auflage
252 Seiten
Elsevier Science (Verlag)
978-0-08-054180-8 (ISBN)

Infos

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.

Inhalt

Cover 1
Contents 10
Preface 8
Chapter 1. Modeling Solute Transport in Porous Media 14
1.1 Introduction 14
1.2 Solute Transport in Porous Media 17
1.3 Models of Hydrodynamic Dispersion 20
1.4 Modeling Macroscopic Behavior 22
1.5 Measurements of Dispersivity 29
1.6 Flow in Aquifers 33
1.7 Computational Modeling of Transport in Porous Media 36
Chapter 2. A Brief Review of Mathematical Background 40
2.1 Introduction 40
2.2 Elementary Stochastic Calculus 45
2.3 What is Stochastic Calculus? 46
2.4 Variation of a Function 47
2.5 Convergence of Stochastic Processes 50
2.6 Riemann and Stieltjes Integrals 51
2.7 Brownian Motion and Wiener Processes 52
2.8 Relationship between White Noise and Brownian Motion 56
2.9 Relationships Among Properties of Brownian Motion 57
2.10 Further Characteristics of Brownian Motion Realizations 59
2.11 Generalized Brownian motion 62
2.12 Ito Integral 62
2.13 Stochastic Chain Rule (Ito Formula) 66
2.14 Stochastic Population Dynamics 80
Chapter 3. Computer Simulation of Brownian Motion and Ito Processes 82
3.1 Introduction 82
3.2 A Standard Wiener Process Simulation 82
3.3 Simulation of Ito Integral and Ito Processes 86
3.4 Simulation of Stochastic Population Growth 91
Chapter 4. Solving Stochastic Differential Equations 96
4.1 Introduction 96
4.2 General Form of Stochastic Differential Equations 96
4.3 A Useful Result 98
4.4 Solution to the General Linear SDE 103
Chapter 5. Potential Theory Approach to SDEs 106
5.1 Introduction 106
5.2 Ito Diffusions 109
5.3 The Generator of an ID 111
5.4 The Dynkin Formula 112
5.5 Applications of the Dynkin Formula 113
5.6 Extracting Statistical Quantifies from Dynkin's Formula 115
5.7 The Probability Distribution of Population Realizations 122
Chapter 6. Stochastic Modeling of the Velocity 124
6.1 Introduction 124
6.2 Spectral Expansion of Wiener Processes in Time and in Space 126
6.3 Solving the Covariance Eigenvalue Equation 130
6.4 Extension to Multiple Dimensions 133
6.5 Scalar Stochastic Processes in Multiple Dimensions 133
6.6 Vector Stochastic Processes in Multiple Dimensions 137
6.7 Simulation of Stochastic Flow in 1 and 2 Dimensions 138
Chapter 7. Applying Potential Theory Modeling to Solute Dispersion 140
7.1 Introduction 140
7.2 Integral Formulation of Solute Mass Conservation 145
7.3 Stochastic Transport in a Constant Flow Velocity 152
7.4 Stochastic Transport in a Flow with a Velocity Gradient 162
7.5 Standard Solution of the Generator Equation 166
7.6 Alternate Solution of the Generator Equation 169
7.7 Evolution of a Gaussian Concentration Profile 174
Chapter 8. A Stochastic Computational Model for Solute Transport in Porous Media 182
8.1 Introduction 182
8.2 Development of a Stochastic Model 183
8.3 Covariance Kernel for Velocity 189
8.4 Computational Solution 190
8.5 Computational Investigation 194
8.6 Hypotheses Related to Variance and Correlation Length 202
8.7 Scale Dependency 205
8.8 Validation of One Dimensional SSTM 206
8.7 Concluding Remarks 217
Chapter 9. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length 218
9.1 Introduction 218
9.2 Approximate Solutions 221
9.3 Results 225
9.4 Conclusions 230
Chapter 10. A Stochastic Inverse Method to Estimate Parameters in Groundwater Models 232
10.1 Introduction 232
10.2 System Dynamics with Noise 233
10.3 Applications in Groundwater Models 238
10.4 Results 244
10.5 Concluding Remarks 245
References 246
Index 250

Textauszug

Chapter 1

Modeling Solute Transport in Porous Media

Don Kulasiri; Wynand Verwoerd    Centre for Advanced Computational Solutions (C-fACS), Lincoln University, Canterbury, New Zealand

1.1 Introduction

The study of solute transport in porous media is important for many environmental, industrial and biological problems. Contamination of groundwater, diffusion of tracer particles in cellular bodies, underground oil flow in the petroleum industry and blood flow through capillaries are a few relevant instances where a good understanding of transport in porous media is important. Most of natural and biological phenomena such as solute transport in porous media exhibit variability which can not be modeled by using deterministic approaches, therefore we need more sophisticated concepts and theories to capture the complexity of system behavior. We believe that the recent developments in stochastic calculus along with stochastic partial differential equations would provide a basis to model natural and biological systems in a comprehensive manner. Most of the systems contain variables that can be modeled by the laws of thermodynamics and mechanics, and relevant scientific knowledge can be used to develop inter-relationships among the variables. However, in many instances, the natural and biological systems modeled this way do not adequately represent the variability that is observed in the systems’ natural settings. The idea of describing the variability as an integral part of systems dynamics is not new, and the methods such as Monte Carlo simulations have been used for decades. However 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, i.e. for the given sets of inputs and parameters we only see a single set of output values. The complexity in nature can not be understood through such deterministic descriptions in its entirety even though one can obtain qualitative understanding of complex phenomena by using them. We believe that new approaches should be developed to incorporate both the scientific laws and interdependence of system components in a manner to include the “noise” within the system. The term “noise” needs further explaining.

We usually define “noise” of a system in relation to the observations of the variables within the system, and we assume that the noise of the variable considered is superimposed on a more cleaner signal, i.e. a smoother set of observations. This observed “noise” is an outcome of the errors in the observations, inherent variability of the system, and the scale of the system we try to model. If our model is a perfect one for the scale chosen, then the ”noise” reflects the measurement errors and the scale effects. In developing models for the engineering systems, such as an electrical circuit, we can consider “noise” to be measurement errors because we can design the circuit fairly accurately so that the equations governing the system behavior are very much a true representation of it. But this is not generally the case in biological and natural systems as well as in the engineering systems involving, for example, the components made of natural materials. We also observe that “noise” occurs randomly, i.e. we can not model them using the deterministic approaches. If we observe the system fairly accurately, and still we see randomness in spatial or temporal domains, then the “noise” is inherent and caused by system dynamics. In these instances, we refer to “noise” as randomness induced by the system.

There is a good example given by Øksendal et al. (1998) of an experiment where a liquid is injected into a porous body and the resulting scattered distribution of the liquid is not that one expects according to the deterministic diffusion model. It turns out that the permeability of the porous medium, a rock material in this case, varies within the material in an irregular manner. These kinds of situations are abound in natural and other systems, and stochastic calculus provides a logical and mathematical framework to model these situations. 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 stochastic models purely driven by the historical data, such as Markov’s chains, capture the system’s temporal dynamics through the information contained in the data that were used to develop the models. Because we use the probability distributions to describe appropriate sets of data, these models can predict extreme events and generate various different scenarios that have the potential of being realized in the real system. In a very general sense, we can say that the probabilistic structure based on the data is the engine that drives the model of the system to evolve in time. The deterministic models based on differential calculus contain differential equations to describe the mechanisms based on which the model is driven to evolve over time. If the differential equations developed are based on the conservation laws, then the model can be used to understand the behavior of the system even under the situations where we do not have the data. On the other hand, the models based purely on the probabilistic frameworks can not reliably be extended to the regimes of behavior where the data are not available.

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. In relation to the above-mentioned diffusion problem of the liquid within the rock material, the scientific knowledge is embodied in the formulation of the partial differential equation, and the variability of the permeability is modeled by using random processes making the solving of the problem with the appropriate boundary conditions is an exercise in stochastic dynamics. We use the term “stochastic dynamics” to refer to the temporal dynamics of random variables, which includes the body of knowledge consisting of stochastic processes, stochastic differential equations and the applications of such knowledge to real systems. Stochastic processes and differential equations are still a domain where mathematicians more than anybody else are comfortable in applying to natural and biological systems. One of the aims of this book is to explain some useful 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 recent developments in mathematics. We have attempted to explain the ideas in an intuitive manner wherever possible without compromising rigor.

We have used the solute transport problem in porous media saturated with water 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. The applications of stochastic calculus and differential equations in modeling natural systems are still in infancy; we do not have widely accepted mathematical and computational solutions to many partial differential equations which occur in these models. A lot of work remains to be done. Our intention is to develop 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, we have experimented with different ideas, learnt new concepts and developed mathematical and computational frameworks in the process. We discuss some of these concepts, arguments and mathematical and computational constructs in an intuitive manner in this book.

1.2 Solute Transport in Porous Media

Flow in porous media has been a subject of active research for the last four to five decades. Wiest et al. (1969) reviewed the mathematical developments used to characterize the flow within porous media prior to 1969. He and his co-authors concentrated on natural formations, such as ground water flow through the soil or in underground aquifers.

Study of fluid and heat flow within porous media is also of significant importance in many other fields of science and engineering, such as drying of biological materials and biomedical studies. But in these situations we can study the micro-structure of the material and understand the transfer processes in relation to the micro-structure even though modeling such transfer processes could be mathematically difficult. Simplified mathematical models can be used to understand and predict the behavior of transport phenomena in such situations and in many cases direct monitoring of the system variables such as pressure, temperature and fluid flow may be feasible. So the problem of prediction can be simplified with the assistance of the detailed knowledge of the system and real-time data.

However, the nature of porous formation in underground aquifers is normally unknown and monitoring the flow is prohibitively expensive. This forces scientists and engineers to rely heavily on mathematical and statistical methods in conjunction with computer experiments of models to understand and predict, for example, the behavior of contaminants in aquifers. In this...

Produkt-Details

• Erscheint lt. Verlag: 22.11.2002
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Mineralogie / Paläontologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-054180-1 / 0080541801
• ISBN-13: 978-0-08-054180-8 / 9780080541808