In this study, geostatistical and stochastic methods are used to study groundwater flow and solute transport in a multiscale heterogeneous formation. The formation is composed of various materials, and distributions of conductivity and chemical sorption coefficient within each material are heterogeneous. The random distributions of materials in the formation are characterized by an indicator function. The conductivity and chemical sorption coefficient fields in each material are assumed to be statistically stationary. On the basis of these assumptions a general expression is derived for the covariance function of the composite field in terms of the covariance of the indicator variable and the properties of the composite materials. Darcy's law and perturbation method are applied to develop the covariance of the retarded velocity. The numerical method of moments [Zhang et al., 2000; Wu et al., 2003a, 2003b] is used to study the effects of various uncertain parameters on flow and transport predictions. Case studies have been conducted to investigate the influences of a medium's physical and chemical heterogeneity and nonstationarity on solute flux prediction. The study results indicate that the large-scale heterogeneity dominates the effcts on flow and solute transport processes, and the effect of small-scale heterogeneity is secondary. It is also shown from the case studies that the numerical method of moments is applicable to studying flow and solute transport in complex subsurface environments, especially for the uncertainty analysis. Monte Carlo simulation is also conducted, and the results are compared with those obtained through method of moment. The calculation results of the mean total solute flux by the two methods match very well, but the variance of total solute flux obtained by the method of moments is smaller than that by the Monte Carlo method, especially for the cases with large total variances of the conductivity and sorption coefficient. In comparison with the Monte Carlo simulation, the method of moments is much more efficient in calculation.

A Lagrangian perturbation method is applied to develop a method of moments for reactive solute flux through a three-dimensional, nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The reactive solute fluxis described as a space-time process where time refers to the solute flux breakthrough in a controlplane at some distance downstream of the solute source and space refers to the transverse displacementdistribution at the control plane. The analytically derived moments equations for solute transport in a nonstationary flow field are too complicated to solve analytically; therefore, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The approach provides a tool to apply stochastic theory to reactive solute transport in complex ubsurface environments. Several case studies have been conducted to investigate the influence of the physical and chemical heterogeneity of a medium on the reactive solute flux prediction in onstationary flow field. It is found that both physical and chemical heterogeneity significantly affect solute transport behavior in a nonstationary flow field. The developed method is also applied to an environmental project for predicting solute flux in the saturated zone below the Yucca Mountain Project area, demonstrating the applicability of the method in practical environmental projects.

In this study, we make use of a nonstationary stochastic theory in studying solute flux through spatially nonstationary flows in porous media. The nonstationarity of flow stems from various sources, such as multi-scale, nonstationary medium features and complex hydraulic boundary conditions. These flow nonstationarities are beyond the applicable range of the 'classical' stochastic theory for stationary flow fields, but widely exist in natural media. In this study, the stochastic frames for flow and transport are developed through an analytical analysis while the solutions are obtained with a numerical method. This approach combines the stochastic concept with the flexibility of the numerical method in handling medium nonstationarity and boundary/initial conditions. It provides a practical way for applying stochastic theory to solute transport in complex groundwater environments. This approach is demonstrated through some synthetic cases of solute transport in multi-scale media as well as some hypothetical scenarios of solute transport in the groundwater below the Yucca Mountain project area. It is shown that the spatial variations of mean log-conductivity and correlation function significantly affect the mean and variance of solute flux. Even for a stationary medium, complex hydraulic boundary conditions may result in a nonstationary flow field. Flow nonstationarity and/or nonuniform distribution of initial plume (geometry and/or density) may lead to nonGaussian behaviors (with multiple peaks) for mean and variance of the solute flux. The calculated standard deviation of solute flux is generally larger than its mean value, which implies that real solute fluxes may significantly deviate from the mean predictions.

A three-dimensional numerical method of moments has been developed for solute flux through nonstationary flows in porous media. The solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. Flow nonstationarity may stem from various sources, such as the medium's conductivity nonstationarity and complex hydraulic boundary conditions. The first two statistics of solute flux are derived using a Lagrangian framework and are expressed in terms of the probability density functions (PDFs). These PDFs are given in terms of one-and two-parcel moments of travel time and transverse locations, and these moments are related to the Eulerian velocity moments. The moment equations obtained analytically for flow and transport are so complex that numerical techniques are used to obtain solutions. In this study, we investigate the influence of various factors, such as the grid resolution relative to correlation length and the number of solute parcels comprising a source, on the accuracy of the calculation results. It has been found that for the computation of means and variances using the developed moment equations, hydraulic head requires at least one numerical grid element per correlation length scale. At least two grid elements are required for velocity, and 1-2 grid elements for the solute flux variance. Five parcels are required per correlation length scale to approximate the initial solute source distribution. The effects of boundary and hydraulic nductivity nonstationarity on flow and transport are also considered. Flow nonstationarity caused by either hydraulic boundary condition or conductivity nonstationarity significantly influences the transport process. The calculation results of numerical method of moments are compared with Monte Carlo simulations. The comparison indicates that the two methods are consistent with each other for head variance, velocity covariance in longitudinal direction, and mean and variance of total solute flux, but numerical method of moment underestimates the velocity variance in transverse direction. The method is applied to an environmental project for predicting the solute flux in the saturated zone below the Yucca Mountain project area, demonstrating the applicability of the method to complex subsurface environments.