A Solute Flux Approach to Transport through Bounded, Unsaturated Heterogeneous Porous Media

framework. In Eulerian approaches, measurements of a quantity, say the solute concentration, are taken at In this paper, we present a solute flux approach for analyzing solute fixed locations by preinstalled samplers. At any time transport statistics in statistically nonstationary, unsaturated flow. Flow nonstationarity in the vadose zone may arise from a number of the concentration field is characterized through meafactors. It is useful to develop a systematic approach that incorporates surements obtained at various locations. For transport these factors into an uncertainty analysis. We first derive the general in heterogeneous porous media, the concentration field forms for solute flux moments. The solute flux moments are associated is generally spatially and temporally nonstationary due with oneand two-particle joint probability distribution functions to the evolving nature of mass transport. In Lagrangian (JPDF). We illustrate our results for certain forms of one-particle approaches, the contaminant plume is envisioned as beand two-particle JPDFs, in which the particle travel time is assumed ing composed of a large number of particles. The modto be lognormally distributed and the particle transverse displacement eler tracks the movement of all particles. Thus, the Lanormally distributed. In the numerical examples, the Eulerian velocity grangian approach offers a flexible and conceptually moments is obtained by solving the head moment equations numeriintuitive way for modeling contaminant transport. The cally using a finite-difference method. Our results show that flow nonstationarity has a significant impact on the statistics of solute fluxes derivation of a Lagrangian framework is independent and solute breakthrough curves. of any specific velocity field. Consequently, once a Lagrangian framework is formulated, it can be used to analyze both saturated and unsaturated flow. One of the most difficult steps in Lagrangian analysis, however, M of solute transport in the unsaturated is relating variables in a Lagrangian framework to those zone has received broad attention during the last in an Eulerian framework, since by convention the vetwo decades. Such analyses can provide valuable inforlocity fields are given in the Eulerian form. Several mation to environmental risk assessment projects. For approximations exist for such purposes (see a review example, a travel time analysis is required in feasibility by Rubin, 1997). In the past, geohydrologists used both studies for potential nuclear waste repository sites. In Eulerian and Lagrangian approaches to derive macrocase of leakage, the contaminant-laden solute flux may dispersion coefficients. In recent years, as Rubin (1997) eventually reach the underlying groundwater aquifer pointed out, more recognition has been given to the and subsequently pose a direct threat to humans. It is true nature of the two approaches, with the Eulerian thus important to assess beforehand the risks associated approach being used mainly in static type of analyses with such events. Both the U.S. Nuclear Regulatory and Lagrangian approach in dynamic type of analyses. Commission and USDOE have specified site-selection Examples of the latter are particle travel time and solute criteria. For example, a disqualifying condition specified flux studies. Particle travel time problems are not new by USDOE is that “a site shall be disqualified if the and have been an ongoing topic of research in other pre-waste-emplacement groundwater travel time from disciplines, such as in chemical engineering and quanthe disturbed zone to the accessible environment is extum physics (van Kampen, 1992). pected to be less than 1,000 yr along any pathway of Shapiro and Cvetkovic (1988) derived moments of likely and significant radionuclide travel” (Altman et solute arrival times in terms of statistics of the hydraulic al., 1996). A numerical simulation performed by Altman conductivity field. While the flow field is three-dimenet al. (1996) revealed that particle travel times from the sional, the authors considered solute movement in the proposed Yucca Mountain repository horizon to the longitudinal direction only. They concluded that in the water table vary from approximately 50 yr to more than near field, the variance of arrival time, 2 , is a quadratic 1 million years. This large variability in travel time was function of the longitudinal travel distance. At large attributed to nonuniform infiltration rates at the ground distances from the injection point (i.e., in the far field), surface and also to spatially heterogeneity of geohydro2 varies linearly with distance, an indication of Fickian logic parameters. transport behavior. Current approaches for modeling solute transport in a To increase the dimensionality of earlier solute flux porous medium adopt either an Eulerian or Lagrangian analyses (e.g., Jury 1982; Simmons, 1982), Dagan et al. A.Y. Sun, Center for Nuclear Waste and Regulatory Analyses, South(1992) formulated a general Lagrangian framework for west Research Institute, 6220 Culebra Road, San Antonio, TX 78238; modeling conservative solute transport in three-dimenD. Zhang, EES-6, Los Alamos National Laboratory, Los Alamos, sional, steady-state flow. The starting point of their analNM 87545. Received 16 Apr. 2003. Special Section: Uncertainty in ysis was to replace the particle trajectory vector by an Vadose Zone Flow and Transport Properties. *Corresponding author ([email protected]). alternative representation, which consists of the particle Published in Vadose Zone Journal 3:513–526 (2004).  Soil Science Society of America Abbreviations: CP, control plane; JPDF, joint probability distribution function; PDF, probability distribution function. 677 S. Segoe Rd., Madison, WI 53711 USA