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Soil hydrological processes (SHP) support ecosystems, modulate the impact of climate change on terrestrial systems and control feedback mechanisms between water, energy and biogeochemical cycles. However, land-use changes and extreme events are increasingly impacting these processes. In this Review, we describe SHP across scales and examine their links with soil properties, ecosystem processes and climate. Soil structure influences SHP such as infiltration, soil water redistribution and root water uptake on small scales. On local scales, SHP are driven by root water uptake, vegetation and groundwater dynamics. Regionally, SHP are impacted by extreme events such as droughts, floods, heatwaves and land-use change; however, antecedent and current SHP partially determine the broader effects of extreme events. Emerging technologies such as wireless and automated sensing, soil moisture observation through novel synthetic aperture radars satellites, big data analysis and machine learning approaches offer unique opportunities to advance soil hydrology. These advances, in tandem with the inclusion of more key soil types and properties in models, will be pivotal in predicting the role of SHP during global change.

We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus - Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

The paper presents a 2D upward infiltration experiment performed on a model porous medium consisting of fine sand background with two inclusions made of coarser sands. The purpose of the experiment was to investigate the effects of structural air trapping, which occurs during infiltration as a result of heterogeneous material structure. The experiment shows that a significant amount of air becomes trapped in each of the inclusions. Numerical simulations were carried out using the two-phase water-air flow model and the Richards equation. The experimental results can be reproduced with good accuracy only using a two-phase flow model, which accounts for both structural and pore-scale trapping. On the other hand, the Richards equation was not able to represent the structural trapping caused by material heterogeneity.

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 fast- est 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 trans- port 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

Looking at kriging problems with huge numbers of estimation points and measurements, computational power and storage capacities often pose heavy limitations to the maximum manageable problem size. In the past, a list of FFT-based algorithms for matrix operations have been developed. They allow extremely fast convolution, superposition and inversion of covariance matrices under certain conditions. If adequately used in kriging problems, these algorithms lead to drastic speedup and reductions in storage requirements without changing the kriging estimator. However, they require second-order stationary covariance functions, estimation on regular grids, and the measurements must also form a regular grid. In this study, we show how to alleviate these rather heavy and many times unrealistic restrictions. Stationarity can be generalized to intrinsicity and beyond, if decomposing kriging problems into the sum of a stationary problem and a formally decoupled regression task. We use universal kriging, because it covers arbitrary forms of unknown drift and all cases of generalized covariance functions. Even more general, we use an extension to uncertain rather than unknown drift coefficients. The sampling locations may now be irregular, but must form a subset of the estimation grid. Finally, we present asymptotically exact but fast approximations to the estimation variance and point out application to conditional simulation, cokriging and sequential kriging. The drastic gain in computational and storage efficiency is demonstrated in test cases. Especially high-resolution and data-rich fields such as rainfall interpolation from radar measurements or seismic or other geophysical inversion can benefit from these improvements.

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.

We analyze the movement of a solute cloud in a saturated aquifer, resulting from a point-like instantaneous solute injection. Physical heterogeneities of the medium due to spatial variations of the hydraulic conductivity, as well as the chemical heterogeneities due to variations in the linear adsorption coefficient and the degradation rate, are modeled as spatial stochastic processes with exponential autocorrelation functions. Furthermore, cross-correlations between the chemical properties and the conductivity are taken into account. For large transport times, the movement of the solute cloud is characterized by its center-of-mass velocity, by the macroscopic dispersion constant, and the macroscopic degradation rate. These quantities are evaluated using perturbation theory and two different averaging procedures. The first procedure derives the large-scale properties from the central moments of the concentration distribution in a given aquifer realization, and averages over the ensemble afterwards. The second method which is mathematically less demanding obtains large scale transport coefficients from the central moments of the ensemble-averaged concentration distribution. Under the assumption that both prescriptions lead to the same macro-scale quantities, the second approach is usually preferred in literature. The present paper is an extension of the work of Metzger et al. (1996). We show that the two averaging procedures lead to different results in one-dimensional systems, whereas the difference vanishes for higher dimensions. Taking into account the influence of small scale dispersion, we give explicit results for the macroscopic parameters characterizing the solute plume. We analyze the various contributions to these parameters and show how the physical origin of these contributions can be traced back uniquely to fluctuations in the retardation factor, in the flow field, and in the degradation rate, and to the cross-correlations between these inhomogeneities, respectively.