Stochastic Dynamics. Modeling Solute Transport in Porous Media

Table of Content

1.Modeling solute transport in porous media
1.1 Introduction
1.2 Solute transport in porous media
1.3 Models of hydrodynamic dispersion
1.4 Modeling macroscopic behavior
1.4.1 Representative elementary volume
1.4.2 Review of continuum transport model
1.5 Measurements of dispersivity
1.6 Flow in aquifers
1.6.1 Transport in heterogeneous natural formations
1.7 Computational modeling of solute transport in porous media

2. A brief review of mathematical background
2.1 Introduction
2.2 Elementary stochastic calculus
2.3 What is stochastic calculus?
2.4 Variation of a function
2.5 Convergence of stochastic processes
2.6 Riemann and Stieltjes integrals
2.7 Brownian motion and Wiener processes
2.8 Relationship between white noise and Brownian motion
2.9 Relationships among properties of Brownian motion
2.10 Further characteristics of Brownian motion realizations
2.11 Generalized Brownian motion
2.12 Ito integral
2.13 Stochastic chain rule (Ito formula)
2.13.1 Differential notation
2.13.2 Stochastic chain rule
2.13.3 Ito processes
2.13.4 Stochastic product rule
2.13.5 Ito formula for functions of two variables
2.14 Stochastic population dynamics

3. Computer simulation of Brownian motion and Ito processes
3.1 Introduction
3.2 A standard Wiener process simulation
3.3 Simulation of Ito integral and Ito processes
3.4 Simulation of stochastic population growth

4. Solving stochastic differential equations
4.1 Introduction
4.2 General form of stochastic differential equations
4.3 A useful result
4.4 Solution to the general linear SDE

5. Potential theory approach to SDEs
5.1 Introduction
5.2 Ito diffusions
5.3 The generator of an ID
5.4 The Dynkin formula
5.5 Applications of the Dynkin formula
5.6 Extracting statistical quantities from Dynkin's formula
5.6.1 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 realizations

6. Stochastic modeling of the velocity
6.1 Introduction
6.2 Spectral expansion of Wiener processes in time and in Space
6.3 Solving the covariance eigenvalue equation
6.4 Extension to multiple dimensions
6.5 Scalar stochastic processes in multiple dimensions
6.6 Vector stochastic processes in multiple dimensions
6.7 Simulation of stochastic flow in 1 and 2 dimensions
6.7.1 1-D case
6.7.2 2-D case

7. Applying potential theory modeling to solute dispersion
7.1 Introduction
7.2 Integral formulation of solute mass conservation
7.3 Stochastic transport in a constant flow velocity
7.4 Stochastic transport in a flow with a velocity gradient
7.5 Standard solution of the generator equation
7.6 Alternate solution of the generator equation

8. A stochastic computational model for solute transport in
porous media
8.1 Introduction
8.2 Development of a stochastic model
8.3 Covariance kernel for velocity
8.4 Computational solution
8.4.1 Numerical scheme
8.4.2 The behavior of the model
8.5 Computational investigation
8.6 Hypotheses related to variance and correlation length
8.7 Scale dependency
8.8 Validation of one dimensional SSTM
8.8.1 Lincoln University experimental aquifer
8.8.2 Methodology of validation
8.8.3 Results
8.9 Concluding remarks

9. Solving the Eigenvalue Problem for a Covariance Kernel with Variable Correlation Length
9.1 Introduction
9.2 Approximate solutions
9.3 Results
9.4 Conclusions

10. A stochastic inverse method to estimate parameters in groundwater models
10.1 Introduction
10.2 System dynamics with noise
10.2.1 An example
10.3 Applications in groundwater models
10.3.1 Estimation related to one-parameter case
10.3.2 Estimation related to two-parameter case
10.3.3 Investigation of the methods
10.4 Results
10.5 Concluding remarks