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The vadose zone is subject to dynamic boundary conditions in the form of infiltration and evaporation. A better understanding of implications for flow and solute transport, arising from these dynamic boundary conditions in combination with heterogeneous structure, will help to improve the prediction of the fate of solutes. We present laboratory experiments and numerical simulations of heterogeneous porous media under unsaturated conditions where controlled, temporally varying precipitation and evaporation are applied to study the effect of dynamic boundary conditions on solute transport in the presence of material interfaces. Dye tracers Eosine Y and Brilliant Blue FCF are utilized to visualize solute transport and analyze redistribution processes in a flow cell. Water and solute fluxes in and out of the flow cell are quantified. While in dynamic experiments application of small infiltration rates (significantly below the saturated hydraulic conductivities of the materials) led to a reversal of transport paths between infiltration and succeeding evaporation, larger infiltration rates altered downward transport such that flow and transport paths differed from those observed during evaporation. Differences in transport paths ultimately led to a redistribution and trapping of solute in one material which manifested as pronounced tailing in breakthrough curves. Trapping was induced not by the formation of a stagnant zone as result of large parameter contrast but by an interplay of dynamic boundary conditions and material heterogeneity. This study thereby highlights the importance to consider dynamic boundary conditions in predictions of solute leaching. © 2019. The Authors.

Models for flow in environmental systems are subject to uncertainty. Models can thus be interpreted as hypotheses on the validity of the underlying model assumptions. One important source of uncertainty in models for flow and transport processes in the subsurface is the model concept. While uncertain model parameters or forcing terms can be captured as random processes and random fields, this type of uncertainty cannot be included into a model in a straightforward manner. This is particularly true if established model descriptions of a given process are not known or are still being debated. In this contribution, we outline several examples of subsurface flow and transport modeling where uncertainty of the model concept plays an important role. We discuss the need for the development of methods and standards to deal with this type of uncertainty in model hypothesis testing. This article is protected by copyright. All rights reserved.

Editors of several journals in the field of hydrology met during the Assembly of the International Association of Hydrological Sciences - IAHS (within the Assembly of the International Union of Geodesy and Geophysics - IUGG) in Prague in June 2015. This event was a follow-up of a similar meeting held in July 2013 in Gothenburg (as reported by Bloschl et al. [2014]). These meetings enable the group of editors to review the current status of the journals and the publication process, and share thoughts on future strategies. Journals were represented in the 2015 meeting through their editors, as shown in the list of authors. The main points on fostering innovation and improving impact assessment in journal publications in hydrology are communicated in this joint editorial published in the above journals.

[1] When predicting flow in the unsaturated zone, any method for modeling the flow will have to define how, and to what level, the subsurface structure is resolved. In this paper, we use the Ensemble Kalman Filter to assimilate local soil water content observations from both a synthetic layered lysimeter and a real field experiment in layered soil in an unsaturated water flow model. We investigate the use of colored noise bias corrections to account for unresolved subsurface layering in a homogeneous model and compare this approach with a fully resolved model. In both models, we use a simplified model parameterization in the Ensemble Kalman Filter. The results show that the use of bias corrections can increase the predictive capability of a simplified homogeneous flow model if the bias corrections are applied to the model states. If correct knowledge of the layering structure is available, the fully resolved model performs best. However, if no, or erroneous, layering is used in the model, the use of a homogeneous model with bias corrections can be the better choice for modeling the behavior of the system.

We study the transport behavior of a tracer in a radially diverging heterogeneous velocity field. Making use of homogenization theory, we derive effective transport equations. These effective transport equations are very similar to those defined on the local scale. However, the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients. For smoothly varying heterogeneous media, explicit results for effective radial dispersion coefficients are derived. Starting with the purely advective transport behavior (infinite Peclet numbers), we extend our calculations to transport with finite Peclet numbers. We find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion: Including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations. In contrast, effective dispersivities including diffusion are not constant but depend on the radial distance. We compare the results found by homogenization theory with those derived by Neuweiler et al. (this issue) by standard method of moments.

[1] Estimates of effective parameters for unsaturated flow models are typically based on observations taken on length scales smaller than the modeling scale. This complicates parameter estimation for heterogeneous soil structures. In this paper we attempt to account for soil structure not present in the flow model by using so-called external error models, which correct for bias in the likelihood function of a parameter estimation algorithm. The performance of external error models are investigated using data from three virtual reality experiments and one real world experiment. All experiments are multistep outflow and inflow experiments in columns packed with two sand types with different structures. First, effective parameters for equivalent homogeneous models for the different columns were estimated using soil moisture measurements taken at a few locations. This resulted in parameters that had a low predictive power for the averaged states of the soil moisture if the measurements did not adequately capture a representative elementary volume of the heterogeneous soil column. Second, parameter estimation was performed using error models that attempted to correct for bias introduced by soil structure not taken into account in the first estimation. Three different error models that required different amounts of prior knowledge about the heterogeneous structure were considered. The results showed that the introduction of an error model can help to obtain effective parameters with more predictive power with respect to the average soil water content in the system. This was especially true when the dynamic behavior of the flow process was analyzed.

[1] We consider flow in a porous medium containing coarse-textured inclusions with a low value of air entry pressure, embedded in a fine-textured background material having high entry pressure. During imbibition some air remains trapped in the inclusions, while during drainage the inclusions become drained only after the capillary entry pressure exceeds the pressure of the background material. These effects can only be reproduced by a two-phase flow model, and not by the Richards' equation. However, if an upscaled form of the Richards' equation with appropriately modified capillary and permeability functions is used, the results are in a reasonable agreement with the two-phase flow model.

[1] We analyze the combined effects of the spatial variability of soil hydraulic properties and the water uptake by plant roots on unsaturated water flow. For this analysis, we use a simplified macroscopic root water uptake model which is usually applied only for homogeneous or layered soil and therefore we also determine whether it is applicable for multidimensional heterogeneous media. Analytical solutions for mean and variance of pressure head (first-order second-moment approximations) in layered media and numerical solutions of two-dimensional (2-D) autocorrelated multi-Gaussian and non multi-Gaussian parameter fields are analyzed for steady state and transient flow conditions. For non-Gaussian topological features, that have little influence on the mean and the variance of the pressure field if root water uptake is ignored, we test whether the influence is significant if root water uptake is accounted for. The results reveal that, in field structures with large patches of coarse material, local regions with pressure head values at the wilting point develop; these are surrounded by wet material. Without a compensation mechanism for local stress, the global transpiration demand is not met if local wilting occurs. Various compensation mechanisms are tested that depend, respectively, on the saturation, the relative conductivity or a strategy where the deficit in the global uptake rate is equally distributed to unstressed locations. The strategies lead to a global actual transpiration rate at the potential value and attenuate the formation of locally wilted areas. Wilted regions can, however, still occur, and may be an artifact of the simplified model concept as root-soil interactions are neglected. Therefore simplified macroscopic models for root water uptake should be used with caution in heterogeneous media.

[1] In this work, we studied the influence of heterogeneities, fluid properties, and infiltration rates on front morphology during two-phase flow. In our experiments, a sand box, 40 cm × 60 cm × 1.2 cm, was packed with two different structures (either random or periodic) composed of 25% coarse material and 75% fine material. The infiltration process was characterized by the capillary number, Ca, and the viscosity ratio, M, between the fluids. The displacing and the displaced fluid had the same densities, such that gravity effects could be neglected. Similar to the pore scale, the stability of the front depends on the relation between M and Ca. However, on the scale under study, depending on the structure, zones of immobilized wetting fluid developed during drainage. The lifetime of these zones depended on the flow regime. Here we show that immobilized zones have an influence on the length of the transition zone, which could lead to a different time behavior than for that of the front width.

5 [1] We study the influence of buoyancy and spatial heterogeneity on the spreading of the 6 saturation front of a displacing fluid during injection into a porous medium saturated 7 with another, immiscible fluid. To do so we use a stochastic modeling framework. We 8 derive an effective large‐scale flow equation for the saturation of the displacing fluid that 9 is characterized by six nonlocal flux terms, four that resemble dispersive type terms and 10 two that have the appearance of advection terms. From the effective large‐scale flow 11 equation we derive measures for the spreading of the saturation front. A series of 12 full two‐phase numerical solutions are conducted to complement the analytical 13 developments. We find that the interplay between density and heterogeneity leads to an 14 enhancement of the front spreading on one hand and to a renormalization of the evolution 15 of the mean front position compared with an equivalent homogeneous medium. The 16 quantification of these phenomena plays an important role in several applications, 17 including, for example, carbon sequestration and enhanced oil recovery.

[1] Many environmental systems are driven by complex gas-water processes in fractured aquifers. One example is degassing processes occurring in fractures in the vicinity of radioactive waste disposal sites. These fractures can represent a potential fast track for radioactive substances to reach the surface: due to buoyancy effects, the created gas phase can move relatively quickly to the surface. The aim of this paper is to investigate and model the involved processes. First, a model for a single fracture is developed. Based on statistical properties of real fractures, an aperture distribution (raster element model) is created. A percolation-renormalization model yields effective properties, such as relative permeabilities and capillary pressure, and provides the size of a representative elementary volume. These effective relationships represent the basis for modeling the migration of the created gas phase through the fractured aquifer of the subsurface. An example modeling degassing and flow in a single fracture is shown and compared to experimental data. The presented model represents a conceptual method for degassing processes and the subsequent migration of the gas phase in fracture-matrix systems, improving predictions on the fate of radioactive substances.

[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.

[1] In this study, we assess probability density functions of hydraulic heads and specific discharges in three-dimensional bounded heterogeneous porous media by Monte Carlo (MC) simulation. We discuss their empirical shapes and demonstrate that the intuitive use of obvious information on boundedness leads to parametric distribution functions, which fit surprisingly well. On the basis of statistical moments of hydraulic heads and velocities up to fourth order, we discuss the spatial dependence of the empirical distributions and their dependence on the variance of log conductivity. Comparison of the first and second central moment to the results from classical numerical first-order second-moment (FOSM) analysis reveals that FOSM predicts these moments surprisingly close for hydraulic heads. On the basis of this fact, we demonstrate that fitting the chosen parametric distributions for hydraulic heads to FOSM moments is promising for the sake of estimating exceedance probabilities. Our MC scenarios vary in variance of log conductivity (0.125 to 5.0), in the type of multivariate dependence, in correlation scale and types of boundary conditions. Our study illustrates that in contrast to the common assumption, FOSM is a reasonable choice for evaluating multivariate and univariate moments for heads, if used in conjunction with additional information on distribution shapes. In the absence of utilizable additional information, we demonstrate that second-moment methods are mostly inadequate for assessing distributions accurately. Significant deviations from Gaussian distributions occurred for discharge components even at a variance of log conductivity as low as 0.125, and we found that the distributions of transverse discharge components are extremely fat-tailed. The observed non-Gaussianity questions the results of approximate approaches in solute flux and dispersion studies where velocity fields are assumed to be multi-Gaussian and then directly represented by or generated from their covariances. The main implication is to apply more accurate schemes such as exact non-local methods, extensive MC or higher-order stochastic Galerkin approaches, and to include higher-order moments, at least if no additional assumptions on the shape of distributions are available or justifiable.

[1] Large-scale models of transient flow processes in the unsaturated zone require, in general, upscaling of the flow problem in order to capture the impact of heterogeneities on a small scale, which cannot be resolved by the model. Effective parameters for the upscaled models are often derived from second-order stochastic properties of the parameter fields. Such properties are good quantifications for parameter fields, which are multi-Gaussian. However, the structure of soil does rarely resemble these kinds of fields. The non-multi-Gaussian field properties can lead to strong discrepancies between predictions of upscaled models and the averaged real flow process. In particular, the connected paths of parameter ranges of the medium are important features, which are usually not taken into account in stochastic approaches. They are determined here by the Euler number of one-cut indicator fields. Methods to predict effective parameters are needed that incorporate this type of information. We discuss different simple and fast approaches for estimating the effective parameter for upscaled models of slow transient flow processes in the unsaturated zone, where connected paths of the material may be taken into account. Upscaled models are derived with the assumption of capillary equilibrium. The effective parameters are calculated using effective media approaches. We also discuss the limits of the applicability of these methods.

[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 well-connected coarse materials differ in the relative permeability from those with well-connected fine materials or those without particular connectedness.