University of California


Unconditional and Conditional Simulation of Flow and Transport in Heterogeneous, Variably Saturated Porous Media

Thomas Harter


The University of Arizona, 1994


            Spatial heterogeneity of geologic media leads to uncertainty in predicting both flow and transport in the vadose zone. In this work an efficient and flexible, combined analytical-numerical Monte Carlo approach is developed for the analysis of steady-state flow and transient transport processes in highly heterogeneous, variably saturated porous media. The approach is also used for the investigation of the validity of linear, first order analytical stochastic models. With the Monte Carlo analysis accurate estimates of the ensemble conductivity, head, velocity, and concentration mean and covariance are obtained; the statistical moments describing displacement of solute plumes, solute breakthrough at a compliance surface, and time of first exceedance of a given solute flux level are analyzed; and the cumulative probability density functions for solute flux across a compliance surface are investigated. The results of the Monte Carlo analysis show that for very heterogeneous flow fields, and particularly in anisotropic soils, the linearized, analytical predictions of soil water tension and soil moisture flux become erroneous. Analytical, linearized Lagrangian transport models also overestimate both the longitudinal and the transverse spreading of the mean solute plume in very heterogeneous soils and in dry soils.

            A combined analytical-numerical conditional simulation algorithm is developed to estimate the impact of in-situ soil hydraulic measurements on reducing the uncertainty of concentration and solute flux predictions. In soils with large spatial variability and in dry soils, soil water tension measurements significantly reduce the uncertainty in the predicted solute concentration. Saturated hydraulic conductivity data are valuable in relatively wet soils. A combination of tension and saturated hydraulic conductivity data gives the best results, especially if some data are available on the unsaturated hydraulic conductivity function. It is also found that if soil heterogeneity is large, the conditional spatial moments of inertia of the mean concentration plume and the conditional mean breakthrough curves are poor means of depicting the actual solute plume distribution and the actual solute flux. Nevertheless, conditional simulation is one of the most rational approaches for modeling unsaturated flow and transport, if in-situ data are available.

Table of Contents

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1.         Introduction

PART II: Theoretical And Numerical Framework

2.         Heterogeneity, Probability, and Random Fields
            2.1.      Introduction: Heterogeneity and Stochastic Analysis
            2.2.      Principles of Probability Theory
            2.3.      Independence and Conditional Probability
            2.4.      Random Variables and Random Vectors
                      2.4.1.   Random Variables
                        2.4.2.   Random Vectors
            2.5.      Random Processes and Random Fields
                        2.5.1.   Definition
                        2.5.2.   Stationarity and Ergodicity of Random Processes
                        2.5.2.   Conditional Random Fields and Kriging
            2.6.      Spectral Representation of Random Variables

3.         Random Field Generators
            3.1.      Introduction
            3.2.      Unconditional Two-Dimensional Random Field Generation by the Spectral 
            3.3.      Conditional Two-Dimensional Random Field Generation by teh Spectral Method
            3.4.      Alternative Methods of Random Field Generation
                        3.4.1.   Turning Bands Method
                        3.4.2.   Matrix Decomposition
                        3.4.3.   Sequential Simulation
            3.5.      Performance Analysis of Random Field Generators

4.         Stochastic Theory of Unsaturated Flow in Two Dimensions
            4.1.      General Problem Formulation
            4.2.      The Governing Stochastic Equation: Formulation and Solution
            4.3.      Variance-Covariance Analysis
                       4.3.1  Head
                        4.3.2.   Unsaturated Hydraulic Conductivity
                        4.3.3.   Velocity
                        4.3.4.   Obtaining 2-D (Cross-)Covariance Functions from 2-D (Cross-)Spectral
                                    Density Functions by Inverse Fast Fourier Transform

5.         MMOC2 - a Numerical Model for Water Flow and Transport in Variably Saturated Porous Media

6.         Grid Design and Accuracy in Numerical Simulations of Variably Saturated Flow in Random Media: Review and Numerical Analysis
            6.1.      General Problem Statement
            6.2.      Review of Theoretical Considerations Regarding Numerical Accuracy
                        6.2.1.   Grid Size
                        6.2.2.   Block Subdiscretization
                        6.2.3.   Correlation Length
            6.3.      Numerical Simulation
                        6.3.1.   Model Parmaters, Initial and Boundary Conditions
                        6.3.2.   Model Verification
                        6.3.3.   Grid Design Sensitivity Analysis
            6.4.      Results and Discussion
                        6.4.1.   Random Field Generator
                        6.4.2.   Comparison to Analytical Solutions (Model Verification)
                        6.4.3.   Grid Design Sensitivity Analysis
                                      Variance and Covariance
                                      Rectangular vs. Square Elements
            6.5.      Evaluation of the Simulation Results
            6.6.      Conclusion and Summary

7.         ASIGNing: An Efficient Method for the Solution of Steady State Unsaturated Flow in Heterogeneous Porous Media
            7.1.      Introduction
            7.2.      Formulation of the Initial Guess Solution
            7.3.      Example Problems
            7.4.      Results and Discussion:
                        7.4.1.   The Quasi-Analytical, the ASIGNed, and the Transient Solution in Comparison
                        7.4.2.   Efficiency of the ASIGNed Solutions
                        7.4.3.   Limitations of ASIGNing
                        7.4.4.   Extensions of ASIGNing
            7.5.      Conclusion

PART III:  Implementation And Evaluation Of Stochastic Simulation

8.         Stochastic Analysis of Steady-State Flow in Heterogeneous Unsaturated Soils via Intensive Monte Carlo Simulation
            8.1.      Introduction
            8.2.      Monte Carlo Simulation and Sampling Accuracy
                        8.2.1.   Definition of Monte Carlo and it Sampling Accuracy
                        8.2.2.   General Computational Procedures
            8.3.      Parameters and Implementation
            8.4.      Input Random Fields of the Saturated Hydraulic Conductivity and the Pore Size
                        Distribution Parameter - General Assessment
            8.5.      Stochastic Analysis of the Unsaturated Hydraulic Conductivity
                        8.5.1.   General Observations
                        8.5.2.   Stochastic Analysis
            8.6.      Stochastic Analysis of the Soil Water Tension
                        8.6.1.   General Observations
                        8.6.2.   Stochastic Analysis
            8.7.      Stochastic Analysis of the Velocity
                        8.7.1.   General Observations
                        8.7.2.   Stochastic Analysis
            8.8.      Stochastic Analysis of the Cross-Covariance Functions
            8.9.      Summary and Conclusions

9.         Transport in Heterogeneous Unsaturated Soils
            9.1.      Introduction
            9.2.      Implementation of the Monte Carlo Simulation and Statistical Methods
                        9.2.1.   Transport Model
                        9.2.2.   Moment Analysis
                        9.2.3.   Parameters and Model Design
            9.3.      Spatial Analysis of Solute Transport under Uncertainty
                        9.3.1.   General Characteristics of Solute Movement and of its Statistical
                        9.3.2.   The Minimum CVc - an Empirical Functional Analysis
                        9.3.3.   Spatial Spreading of the Mean Plume, Mean Spatial Spreading of Plumes, and
                                     Variability of the Plume Center of Mass
            9.4.      Temporal Analysis of Solute Transport under Uncertainty
                        9.4.1.   Integrated Breakthrough at a Compliance Surface
                        9.4.2.   Local Compliance with Maximum Contamination Levels
                        9.4.3.   Effects of Mean Head and the Correlation of f and a
                        9.4.4.   Effects of Anisotropy
            9.5.      Summary and Conclusion

10.       Conditional simulation of unsaturated non-reactive transport
            10.1     Introduction
            10.2     Theory of Conditional Simulation by Cokriging
            10.3     Implementation of a Conditional Simulator
            10.4     "Field Test Sites" and Sampling Strategies: Methodology
            10.5     Unconditional vs. Conditional Moments of y, h, and v: selected examples
            10.6     Sampling Network Design Impacts on Concentration Prediction
                        10.6.1  The field-site plume
                        10.6.2  Concentration moments as functions of sampling networks, soil site
                        10.6.3  Comparison to a dry, anisotropic field site of equivalent variability
            10.7     Hydraulic conductivity variance and the effect of conditioning data
            10.8     Anisotropy ratio and the effect of conditioning data
            10.9     Conditional simulation under parameter uncertainty
            10.10   Mean displacement variance, moment of inertia, and the dilution index as a
                        measure for
                         the conditioning effect
            10.11   Local solute travel time predictions under conditional uncertainty
            10.12   Conditional integrated solute breakthrough at the compliance surface
            10.13.  Conditional modeling - what for?: A deterministic inverse approach in
            10.14   Discussion, summary, and conclusion

11.       References

A special Thank You to Diana Nix for setting the manuscript into PDF format!

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