[1] Large-scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone. Upscaled flow models need to capture the impact of small-scale heterogeneities, which are not resolved by the model, on large-scale flow. We perform upscaling of the Richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques. We restrict the analysis to flow regimes in which the capillary-equilibrium assumption holds on the small scale. In order to account for effects of capillary entry pressure we apply the Brooks-Corey model for the soil retention and relative permeability curves and consider Leverett scaling for the coupling of intrinsic permeability and entry pressure. For this model we derive and analyze the ensemble-averaged parameter functions for the macroscopic flow equations. The effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability. We compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties. These results illustrate that soils with wellconnected coarse materials differ in the relative permeability from those with wellconnected fine materials or those without particular connectedness.
[1] In recent years, statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second-order stationary velocity fields. The merit of accurately estimating the mean and variance of concentration, however, remains unclear without knowing the shape of the probability density function (pdf). In a setup where a conservative solute is continuously injected into a domain, the concentration is bounded between zero and the concentration value in the injected solution. At small travel distances close to the fringe of the plume, an observation point may fall into the plume or outside, so that the statistical concentration distribution clusters at the two limiting values. Obviously, this results in non-Gaussian pdf's of concentration. With increasing travel distance, the lateral plume boundaries are smoothed, resulting in increased probability of intermediate concentrations. Likewise, averaging the concentration in a larger sampling volume, as typically done in field measurements, leads to higher probabilities of intermediate concentrations. We present semianalytical results of concentration pdf's for measurements with point-like or larger support volumes based on stochastic theory applied to stationary media. To this end, we employ a reversed auxiliary transport problem, in which we use analytical expressions for first and second central spatial lateral moments with an assumed Gaussian pdf for the uncertainty of the first lateral moment and Gauss-like shapes in individual cross sections. The resulting concentration pdf can be reasonably fitted by beta distributions. The results are compared to Monte Carlo simulations of flow and steady state transport in 3-D heterogeneous domains. In both methods the shape of the concentration pdf changes with distance to the contaminant source: Near the source, the distribution is multimodal, whereas it becomes a unimodal beta distribution far away from the contaminant source. The semianalytical and empirical pdf's differ slightly, which we contribute to the numerical artifacts in the Monte Carlo simulations but also to hard assumptions made in the semianalytical approach. Our results imply that geostatistical techniques for interpolation and other statistical inferences based on Gaussian distributions, such as kriging and cokriging, may be feasible only far away from the contaminant source. For calculations near the source, the beta-like distribution of concentration should be accounted for.
Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure-(semi)-implicit transport scheme for the advection-diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2-4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a 123 290 S. K. Matthäi et al.highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.
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