Posts by Nicolae Suciu

1. Introduction
1.1 Motivation
1.2 Diffusion Equations
1.2.1 Stationary Diffusion and Flow in Porous Media
1.2.2 Advection–Diffusion Processes
1.2.3 Advection–Diffusion–Reaction Processes
1.3 Numerical Methods
1.3.1 Solutions of Deterministic PDEs
1.3.2 Solutions of Stochastic PEDs
References

2 Preliminaries
2.1 Random Variables, Processes, and Fields
2.1.1 Random Variables
2.1.2 Random Functions and Stochastic Processes
2.1.3 Densities of Finite Dimensional Probability
Distributions
2.2 Markov and Diffusion Processes
2.2.1 Markov Processes
2.2.2 Diffusion Processes and Fokker–Planck Equations
2.2.3 Itô Equation and Stochastic-Lagrangian Framework
References

3 Stochastic Simulations of Diffusion Processes
3.1 Random Sequences
3.1.1 Programming Tools
3.1.2 Convergence
3.2 Numerical Solutions of Itô Equations
3.2.1 Strong and Weak Solutions
3.2.2 Strong and Weak Euler Schemes
3.3 Global Random Walk
3.3.1 Weak Approximations by Global Random Walk
3.3.2 Unbiased GRW
3.3.3 Biased GRW
3.3.4 GRW Solutions for Flow and Reactive Transport
References

4 Diffusion in Random Velocity Fields
4.1 Classical Stochastic Theories Revisited
4.1.1 Taylor’s Theory of Diffusion by Continuous Movements
4.1.2 Advection–Dispersion Models
4.1.3 The Eulerian Statistics of the Travel-Time
4.1.4 The Local Dispersivity Tensor
4.1.5 First Order Approximations for Small Variance of the Log-Hydraulic Conductivity
4.2 Diffusion with Space Variable Drift
4.2.1 Fokker–Planck Equation with Variable Drift
4.2.2 Dispersion and Memory Terms
4.2.3 Memory Effects and Transition Probabilities
4.3 Diffusion in Random Fields Model of Passive Transport
4.3.1 Ensemble-, Effective-, and Center of Mass-Dispersion
4.3.2 Statistical Homogeneity Properties
4.3.3 Anomalous Diffusion, Ergodicity, and Self-averaging
4.4 First Order Approximations
4.4.1 Lagrangian and Eulerian Representations of Diffusion in Random Velocity Fields
4.4.2 Explicit First Order Results for Transport in Aquifers
4.4.3 First Order Results for Power-Law Correlated ln K Fields
References

5 Monte Carlo GRW Simulations of Passive Transport in Groundwater
5.1 Numerical Investigations on Memory Effects and Ergodicity
5.1.1 Ergodicity of the Center of Mass
5.1.2 Dependence on Initial Conditions
5.1.3 Non-ergodic Effective Dispersion at Finite Times
5.1.4 Loss of Memory and Asymptotic Ergodicity
5.2 Numerical Simulations of Transport in Aquifers with Evolving Scale Heterogeneity
5.2.1 Quantifying Anomalous Diffusion by Memory Terms
5.2.2 Ensemble and Memory Coefficients
5.2.3 Breakthrough Curves and Cross-Section Concentrations
5.2.4 Anomalous Diffusion Behavior
References

6 Probability and Filtered Density Function Approaches
6.1 PDF/FDF Evolution Equations
6.1.1 Background on PDF/FDF Methods
6.1.2 PDF/FDF Equations for Reactive Transport
6.1.3 The Fokker–Planck Approach
6.2 Spatial Coarse-Graining and FDF Simulations
6.2.1 PDF/FDF Problem for Passive Transport in Aquifers
6.2.2 Mixing Models
6.2.3 Upscaled Velocity Fields and Diffusion Coefficients
6.2.4 Coarse-Grained Simulations of Transport
6.3 GRW Solutions for PDF/FDF Equations
6.3.1 Convergence of the Mean Concentration
6.3.2 Convergence of FDF Solutions to PDF Solutions
6.4 Issues and Future Developments of PDF/FDF Approach
6.4.1 Looking for Appropriate Mixing Models
6.4.2 Perspectives of the FDF Approach
References

7 Model, Scale, and Measurement
7.1 Sampling Volume and Sampling Time
7.1.1 Sampling Volume Approach
7.1.2 Spatio-Temporal Upscaling
7.2 Coarse-Grained Spatio-Temporal Upscaling
7.2.1 Coarse-Grained Space–Time Averages
7.2.2 Continuous Fields
7.2.3 CGST Average Concentration
References

Appendices
A Numerical Simulation of Diffusion Processes
A.1 Diffusion Processes Constructed with i.i.d. Variables
A.2 Itô–Euler Schemes.
A.3 Global Random Walk
A.3.1 One-Dimensional Unbiased GRW Algorithms
A.3.2 Two-Dimensional Unbiased GRW Algorithm for Advection-Diffusion Processes
A.3.3 Biased GRW Algorithms

B GRW Solutions of Fokker–Planck Equations
B.1 GRW Approximations for Continuous Diffusion Processes
B.2 Strict Equivalence Between GRW and the Weak Euler Scheme for Constant Velocity
B.3 Biased GRW Approximations for Continuous Diffusion Processes

C Numerical Generation of Random Fields
C.1 Homogeneous Gaussian Random Fields
C.1.1 Randomization Method
C.1.2 Analytic Properties of the Samples
C.2 Filtered Kraichnan Fields
C.3 Kraichnan Field Generators
C.3.1 Hydraulic Conductivity Fields
C.3.2 Kraichnan Approximations of Velocity Fields

D Correlation Structure of the Itô Process
D.1 Mean Value and Covariance Components
D.2 Insights from Itô–Taylor Expansions
D.3 Weak Solutions by Successive Approximations

E Derivation of PDF Equations by δ-Function Method
E.1 The PDF Equation
E.2 The Fokker–Planck Equation

F Upscaled Dispersion Coefficients
F.1 Self-averaging Estimations of Diffusion Coefficients
F.2 Coarse-Grained Dispersion Coefficients

References

Index

Stochastic approaches for transport processes in heterogeneous media are motivated by the need to use stochastic parameterizations of the model equations. Essentially, this results in modeling diffusion processes in random fields. For instance, in stochastic subsurface hydrology, random hydraulic conductivity parameters generate random groundwater flow velocity fields and solute transport is modeled by diffusion equations with random drift coefficients. Technically, modeling approaches are based on equivalent Fokker–Planck and Itô representations of the diffusion in random fields, that is, through trajectories of molecules or computational particles and the corresponding continuous fields.

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In this chapter, basic concepts used in stochastic modeling of transport processes are revisited. Random fields and stochastic processes will be introduced as particular random functions. The hierarchy of finite dimensional distributions will be introduced and particularized for Markov and diffusion processes. Itô and Fokker–Planck descriptions of the diffusion process will be used to introduce the stochastic-Lagrangian framework.

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