[1] Discrete fracture network (DFN) and stochastic continuum (SC) are two common modeling approaches used for simulating fluid flow and solute transport in fractured media. Fracture continuum approaches combine the merits of each approach; details of the fracture network are preserved, and a computationally efficient grid is utilized for the solution of fluid flow by assigning a conductivity contrast between the grid cells representing the rock matrix and those representing fractures. In this paper, we propose a fracture continuum approach for mapping individual fractures onto a finite-difference grid as conductivity fields. We focus on several issues that are associated with this approach, such as enhanced connectivity between fractures that would otherwise not be in connection in a DFN simulation and the influence of grid cell size. To addresses these issues, both DFN and the proposed approach are used to solve for fluid flow through two-dimensional, randomly generated fracture networks in a steady-state, single-phase flow system. The DFN flow solution is used as a metric to evaluate the robustness of the method in translating discrete fractures onto grid cell conductivities on four different regularly spaced grids: 1 Â 1 m, 2 Â 2 m, 5 Â 5 m, and 10 Â 10 m. Two correction factors are introduced to ensure equivalence between the total flow of the grid and the original fracture network. The first is dependent on the fracture alignment with the grid and is set to account for the difference between the length of the flow path on the grid and that of the fracture. The other correction is applied for areas in the grid with high fracture density and accounts for the artificial degree of connectivity that exists on the grid but not in the DFN. Fifteen different cases are studied to evaluate the effect of fracture statistics on the results of the proposed approach and by taking average results of 100 realizations in each case in a stochastic Monte Carlo framework. The flow equation is solved for the DFN, and total flow is obtained. The flow is also solved separately for the four-grid resolution levels, and comparisons between the DFN and the grid total flows are made for the different cases and the different grid resolution levels. The approach performed relatively well in all cases for the fine-grid resolution, but an overestimation of grid flow is observed in the coarse-grid resolution, especially for cases wherein the network connectivity is controlled by small fractures. This overestimation shows minor variation from one realization to another within the same case. This allowed us to develop an approach that depends on solving limited number of DFN simulations to obtain this overestimation factor. Results indicate that the proposed approach provides improvements over existing approaches and has a potential to provide a link between DFN and SC models.

Statistical analysis and interpretation of heterogeneous sediment hydraulic properties is important to produce reliable forecasts of water and solute transport dynamics in the unsaturated zone. Most field characterizations to date have focused on the shallow 2‐m root zone. We characterized the geologic and hydraulic properties of a 16‐m‐deep, alluvial vadose zone consisting of unconsolidated sediments typical of the alluvial fans of the eastern San Joaquin Valley, California. The thickness of individual beds varies from <5 cm for some clayey and silty floodplain material to >2.5 m for large sandy deposits associated with buried stream channels. Eight major geologic units (lithofacies) have been identified at the site. Unsaturated hydraulic properties were obtained from multistep outflow experiments on nearly 100 sediment cores. Multivariate analysis of variance and post hoc testing show that lithofacies and other visual‐ and texture‐based sediment classifications explain a significant amount of the spatial variability of hydraulic properties within the unsaturated zone. Geostatistical analysis of hydraulic parameters show spatial continuity of within‐lithofacies variability in the horizontal direction in the range of 5 to 8 m, which is approximately an order of magnitude larger than spatial continuity in the vertical direction. Low nugget/sill ratios suggest that 1‐ to 10‐m sampling intervals are adequate for detection of horizontal spatial structure. The existence of thin clay or silt layers within lithofacies units results in only moderate spatial continuity in the vertical direction, however, suggesting inadequate sampling frequency for hydraulic parameter variogram development in that direction.

Heterogeneity in unsaturated soils and sediments is well known to exist at different scales, from microscopic scale to macroscopic scale. Characterization of different types of heterogeneity in deep vadose zones is challenging because of the usual lack of information at such sites. In this paper, we considered a site with detailed geological, chemical, and hydraulic properties measurements throughout an approximately 16‐m deep vadose zone consisting of unconsolidated, alluvial deposits typical for the alluvial fans of the eastern San Joaquin Valley, California. At the agricultural site, data were also available for a 7‐yr long field fertilization experiment that we used to independently estimate the amount of nitrate stored within the vadose zone at the end of the experiment. Simple mass balance calculations were performed and compared to six conceptually different two‐dimensional and three‐dimensional vadose zone numerical models that were implemented to represent varying degrees of hierarchical details of heterogeneity. Despite widely differing structure and heterogeneity of unsaturated flow, all models resulted in a narrow range of estimated nitrate storage in the deep vadose zone for near‐cyclically repeated water and nitrogen fertilizer applications. Simulated nitrate storage was found to be approximately six to eight times larger than the measured storage at the field. Simulated nitrate variability, while qualitatively similar in pattern, was considerably lower than measured, despite the large simulated hydraulic variability. This study underscores that physical heterogeneity of deep vadose zones may have limited effects on the transfer of conservative contaminants applied repeatedly to the land surface. It also raises questions about our understanding of the chemical fate of nitrate in the vadose zone; and suggests the presence of a significant immobile moisture domain within the deep vadose zone that is not explainable by heterogeneity of Richards equation parameters, yet needs to be considered for simulating nitrate transport under conditions of cyclical infiltration with gravity dominated convective flux.

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Stochastic theories have been developed that linkThe concept of block-effective dispersivity the kinematics of dispersion to the spatial covariance of represents the difference between the classic hydraulic conductivity and derive formulas to calculate macrodispersivity values and the dispersivity captured the macrodispersivity as a function of time and travel by the numerical grid.We use Monte Carlo distance [1][2][3][4]. In these theories, the natural logarithm simulations offlow and transport in two-dimensional of hydraulic conductivity (ln K) is modeled as a random conductivity, porosity, and distribution random field; and the resulting macrodispersivities coefficient fields to explore the influence of spatial become functions of the statistical nature of the variability on the block-effective dispersivity. Diferent hydraulic conductivity distribution. correlation structures between porosity, distribution This macrodispersion is arising from the coefficient, and hydraulic conductivity are assumed; heterogeneous local-scale material structure.A positive correlation, negative correlation, and nofundamental assumption associated with the concept of correlation using random fields with exponential macrodispersion is that the velocity field is represented covariance. Different grid sizes are also simulated only through its expected value and that the from very fine grids (grid-cell size is smaller than the macrodispersivity tensor accounts for all the effects of correlation length) to coarse grid (grid-cell size is unmodeled spatial variability of the velocity field. On larger than the correlation length). Results suggest the other extreme, a detailed description of the velocity that it is important to examine the role of distribution field can be modeled with only a local dispersivity coefficient andlor porosity variability, and the possible value (in the order of the one measured in the lab).correlation between them in calculating block-effectiveThe latter approach has been extensively used in many dispersivity.When porosity andlor distribution large scale numerical simulations to validate the coefficient is positively correlated with conductivity, applicability of the stochastic theories. In this block-effective longitudinal dispersivity is smaller than approach, numerical models with grid-cell size (A) the case of random conductivity with constant smaller than the correlation scale of hydraulic variables while the block-effective transverse conductivity (A) are modeled to capture all the spatial dispersivity is larger than the case of constant variability of the velocity field [5][6][7][8]. Modeling the variables. The negative correlation case leads to the velocity field through a detailed description of the opposite results. When porosity andlor distribution hydraulic conductivity field and a local dispersivity coefficient is spatially variable but uncorrelated to the may be computationally intensive and an upscaled hydraulic conductivity, block-effective numerical grid with cells larger than the correlation ...

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