# Stochastic Dynamics. Modeling Solute Transport in Porous Media

**By**

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

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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### Book information

- Published: November 2002
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-51102-7

### Reviews

"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.

### Table of Contents

Table of Content1.Modeling solute transport in porous media1.1 Introduction1.2 Solute transport in porous media1.3 Models of hydrodynamic dispersion1.4 Modeling macroscopic behavior1.4.1 Representative elementary volume1.4.2 Review of continuum transport model1.5 Measurements of dispersivity1.6 Flow in aquifers1.6.1 Transport in heterogeneous natural formations1.7 Computational modeling of solute transport in porous media2. A brief review of mathematical background2.1 Introduction2.2 Elementary stochastic calculus2.3 What is stochastic calculus?2.4 Variation of a function2.5 Convergence of stochastic processes2.6 Riemann and Stieltjes integrals2.7 Brownian motion and Wiener processes2.8 Relationship between white noise and Brownian motion2.9 Relationships among properties of Brownian motion2.10 Further characteristics of Brownian motion realizations2.11 Generalized Brownian motion2.12 Ito integral2.13 Stochastic chain rule (Ito formula)2.13.1 Differential notation2.13.2 Stochastic chain rule2.13.3 Ito processes2.13.4 Stochastic product rule2.13.5 Ito formula for functions of two variables2.14 Stochastic population dynamics3. Computer simulation of Brownian motion and Ito processes3.1 Introduction3.2 A standard Wiener process simulation3.3 Simulation of Ito integral and Ito processes3.4 Simulation of stochastic population growth4. Solving stochastic differential equations4.1 Introduction4.2 General form of stochastic differential equations4.3 A useful result4.4 Solution to the general linear SDE5. Potential theory approach to SDEs5.1 Introduction5.2 Ito diffusions5.3 The generator of an ID5.4 The Dynkin formula5.5 Applications of the Dynkin formula5.6 Extracting statistical quantities from Dynkin's formula5.6.1 What is the probability to reach a population Value K?5.6.2 What is the expected time for the population to reach a value K?5.6.3 What is the expected population at a time t?5.7 The probability distribution of population growth realizations6. Stochastic modeling of the velocity6.1 Introduction6.2 Spectral expansion of Wiener processes in time and in Space6.3 Solving the covariance eigenvalue equation6.4 Extension to multiple dimensions6.5 Scalar stochastic processes in multiple dimensions6.6 Vector stochastic processes in multiple dimensions6.7 Simulation of stochastic flow in 1 and 2 dimensions6.7.1 1-D case6.7.2 2-D case7. Applying potential theory modeling to solute dispersion7.1 Introduction7.2 Integral formulation of solute mass conservation7.3 Stochastic transport in a constant flow velocity7.4 Stochastic transport in a flow with a velocity gradient7.5 Standard solution of the generator equation7.6 Alternate solution of the generator equation8. A stochastic computational model for solute transport inporous media8.1 Introduction8.2 Development of a stochastic model8.3 Covariance kernel for velocity8.4 Computational solution8.4.1 Numerical scheme8.4.2 The behavior of the model8.5 Computational investigation8.6 Hypotheses related to variance and correlation length8.7 Scale dependency8.8 Validation of one dimensional SSTM8.8.1 Lincoln University experimental aquifer8.8.2 Methodology of validation8.8.3 Results8.9 Concluding remarks9. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length9.1 Introduction9.2 Approximate solutions9.3 Results9.4 Conclusions10. A stochastic inverse method to estimate parameters in groundwater models10.1 Introduction10.2 System dynamics with noise10.2.1 An example10.3 Applications in groundwater models10.3.1 Estimation related to one-parameter case10.3.2 Estimation related to two-parameter case10.3.3 Investigation of the methods10.4 Results10.5 Concluding remarks