Reactive flow and transport in porous media is topic of intense research since decades. Since dispersion is one of the key parameters in solute transport, its accurate modeling is essential to avoid wrong predictions of flow and transport behavior. In this research, we investigate novel effective dispersion models for reactive transport of electrically charged chemical species in a thin, potentially evolving strip taking into account Taylor–Aris and electroosmotic‐induced dispersion as well as their cross‐coupling effects. We prove positivity of the dispersion coefficient and the existence and uniqueness of strong solutions in the fixed geometry setting. Moreover, we numerically investigate scenarios for both the fixed and evolving geometry situation. The simulation results illustrate the possibility of separating charged species, such that the findings of this study can lead to a better understanding of mixing and separation processes of charged solutes and an improved prediction of breakthrough curves. Finally, we study the limits of vanishing channel width, precipitation layer thickness, and molecular diffusion. We show convergence of the solutions to the corresponding limit cases such as a hyperbolic model or the fixed geometry case. From these results, we can rate the impact of distinct dispersion mechanisms and evaluate the necessity of a detailed modeling for different parameter regimes.

Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.

The current article presents a degenerating diffusion-precipitation model including vanishing porosity and focuses primarily on uniqueness results. This is accomplished by assuming sufficient conditions under which the uniqueness of weak solutions can be established. Moreover, a proof of existence based on a compactness argument yields rather regular solutions, satisfying these unique conditions. The results show that every strong solution is unique, though a slightly different condition is additionally required in three dimensions. The analysis presents particular challenges due to the nonlinear structure of the underlying problem and the necessity to work with appropriate weights and manage possible degeneration.

We assess the complex coupling of biological, chemical and physical processes with the help of a mechanistic modeling approach extending [Rupp 2019] The aim is to study the interplay of relevant mechanisms in silico and consequently gain a model-based understanding of dynamics in soils.The hybrid discrete-continuum model used explicitly represents the pore structure and allows for dynamic structural organization of the medium at the pore scale. The movement of interacting entities - nutrients, bacteria and possibly charged chemicals - in the fluid is described by means of the diffusion and Nernst-Planck equations with Henry's law at the liquid/gas interfaces. Homogeneous chemical reactions are considered using for instance the mass action law whereas heterogeneous reactions on the solid surface are incorporated via a kinetic Langmuir isotherm. A biomass phase can develop from agglomerations of bacteria and stabilising sticky agents may grow or decay at the solid surfaces. Root cells and an explicit phase of exudate as well as attachment properties of root hairs can be included. In addition to solving the continuous partial differential equations, a discrete cellular automaton method [Ray et al. 2017, Rupp et al 2018, Tang and Valocchi 2013] is used, enabling structural changes in the solid and biomass/mucilage phases at each time step. The partial differential equations are discretised with a local discontinuous Galerkin method which is able to handle discontinuities induced by the evolving geometry. Upscaling techniques enable the incorporation of information from the pore scale into the macroscale.In this study, we illustrate the ability of the approach to advance the understanding of specific process mechanisms. Microaggregates are the fundamental building blocks of soils and are thus important for soil structure, properties, and functions. Although there has been much research investigating the dynamics, stability, and structure of microaggregates, there is still a substantial lack in quantifying the relationships between the major driving forces (soil fauna, microorganisms, roots, organic and inorganic matter, and physical processes). As an example, we study structure formation of microaggregates as a function of the size and shape of the solid building units taking into account the effect of attraction and repulsion by charges. Biomass development and root exudate can significantly alter the macroscopic soil hydraulic properties. Using the model in hand, this effect can be quantified for different amount and spatial distribution of root exudate with geometries from CT-scans. In the natural environment, microbial communities are highly diverse. We employ the model to investigate the way spatial distribution of organic matter can influence bacterial dynamics.

Key functions of soils, such as permeability or habitat for microorganisms, are determined by structures at the microaggregate scale. The evolution of elemental distributions and dynamic processes can often not be assessed experimentally. So mechanistic models operating at the pore scale are needed. We consider the complex coupling of biological, chemical, and physical processes in a hybrid discrete-continuum modeling approach. It integrates dynamic wetting (liquid) and non-wetting (gas) phases including biofilms, diffusive processes for solutes, mobile bacteria transforming into immobile biomass, and ions which are prescribed by means of partial differential equations. Furthermore the growth of biofilms as, e.g., mucilage exuded by roots, or the distribution of particulate organic matter in the system, is incorporated in a cellular automaton framework (CAM) presented in [1, 2]. It also allows for structural changes of the porous medium itself (see, e.g. [3]). As the evolving computational domain leads to discrete discontinuities, we apply the local discontinuous Galerkin (LDG) method for the transport part. Mathematical upscaling techniques incorporate the information from the pore to the macroscale [1,4]. The model is applied for two research questions: We model the incorporation and turnover of particulate OM influencing soil aggregation, including ‘gluing’ hotspots, and show scenarios varying of OM input, turnover, or particle size distribution. Second, we quantify the effective diffusivity on 3D geometries from CT scans of a loamy and a sandy soil. Conventional models cannot account for natural pore geometries and varying phase properties. Upscaling allows also to quantify how root exudates (mucilage) can significantly alter the macroscopic soil hydraulic properties. [1] Ray, Rupp, Prechtel (2017). AWR (107), 393-404. [2] Rupp, Totsche, Prechtel, Ray (2018). Front. Env. Sci. (6) 96. [3] Zech, Dultz, Guggenberger, Prechtel, Ray (2020). Appl. Clay Sci. 198, 105845. [4] Ray, Rupp, Schulz, Knabner (2018). TPM 124(3), 803-824.

A novel, comprehensive modeling approach extending (Ray et al., 2017, Rupp et al., 2018, Rupp et al., 2019) is used to study the interplay between biogeochemical processes in the rhizosphere. Understanding these local interactions is crucial for the habitat as they influence processes in the root-soil system such as the water and nutrient uptake by the roots. The mechanistic model explicitly represents the pore structure and allows for dynamic structural organization of the rhizosphere at the single root scale. At this microscale, the movement of interacting entities - nutrients, bacteria and possibly charged chemicals - in the fluid is described by means of the diffusion and Nernst-Planck equations with a Henry transmission condition at the liquid/gas interfaces. A biomass phase can develop from agglomerations of bacteria and stabilising sticky agents may grow or decay at the solid surfaces. To take into account specific properties of the rhizosphere, root cells and an explicit phase of exudated mucilage as well as root hairs are included. In addition to solving the continuous partial differential equations, a discrete cellular automaton method (Tang and Valocchi 2013, Ray et al. 2017, Rupp et al., 2019) is used, enabling structural changes in the solid and mucilage phases at each time step. The partial differential equations are discretised with a local discontinuous Galerkin method which is able to handle discontinuities induced by the evolving geometry. The microscale model is not amenable to large scale computations because of its high complexity. Upscaling techniques enable the incorporation of information from the rhizosphere scale to the macroscale. We apply these techniques to dynamically evolving microstructures taking the spatiotemporal evolution of the rhizosphere into account. Although the setting is periodic, the underlying geometries can be arbitrarily complex. The resulting hydraulic properties (e.g. diffusion coefficient, permeability) are an important input for existing root-water uptake models, involving e.g., the effect of mucilage. In this study, we use two- and three-dimensional CT scans of maize root, and show how mucilage concentration as well as its distribution in the pore space result in changes of macroscopic soil hydraulic properties. The access of nutrients in small pores for the root is assessed in simulation studies and its effect on effective diffusivity is evaluated. N. Ray, A. Rupp and A. Prechtel (2017): Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media. Advances in Water Resources 107, 393-404. A. Rupp and K. Totsche and A. Prechtel and N. Ray (2018): Discrete-continuum multiphase model for structure formation in soils including electrostatic effects. Frontiers in Environmental Science, 6, 96. A. Rupp, T. Guhra, A. Meier, A. Prechtel, T. Ritschel, N. Ray, K.U. Totsche (2019): Application of a cellular automaton method to model the structure formation in soils under saturated conditions: A mechanistic approach. Frontiers in Environmental Science 7, 170. Y. Tang and A.J. Valocchi (2013): An improved cellular automaton method to model multispecies biofilms. Water Research 47 (15), 5729-5742.

In this article, we consider fluid flow and transport in evolving porous media including vanishing porosity. We analyze the corresponding equations for a given porosity function, which describes the evolution of the underlying saturated porous medium, and are particularly interested in partially clogged media. Thereby, the hydrodynamic parameters (permeability, diffusivity) are assumed to depend on the porosity and degeneracies arise in case of clogging. Introducing appropriate weighted function spaces and including the degenerate parameters as weights of Muckenhoupt class, we are able to handle the degeneracy and obtain analytical results. We solve the underlying equations via saddle-point theory or an adjusted Rothe method by applying the useful properties of such weighted function spaces. Moreover, we obtain nonnegativity and boundedness for the weak solution to the transport equation. Finally, we are interested in the decay behavior of this solution with respect to the porosity.

Dieses Lehrbuch führt verständlich und anschaulich in die wissenschaftliche Denkweise im Mathestudium ein und erleichtert so den Übergang Schule-Hochschule. In Verbindung mit den Grundlagen der Mathematik bietet das Buch eine Einführung in die Programmiersprache Python mit vielen Beispielen.

Since dispersion is one of the key parameters in solute transport, its accurate modeling is essential to avoid wrong predictions of flow and transport behavior. In this research, we derive new effective dispersion models which are valid also in evolving geometries. To this end, we consider reactive ion transport under dominate flow conditions (i.e. for high Peclet number) in a thin, potentially evolving strip. Electric charges and the induced electric potential (the zeta potential) give rise to electro-osmotic flow in addition to pressure-driven flow. At the pore-scale a mathematical model in terms of coupled partial differential equations is introduced. If applicable, the free boundary, i.e. the interface between an attached layer of immobile chemical species and the fluid is taken into account via the thickness of the layer. To this model, a formal limiting procedure is applied and the resulting upscaled models are investigated for dispersive effects. In doing so, we emphasize the cross-coupling effects of hydrodynamic dispersion (Taylor–Aris dispersion) and dispersion created by electro-osmotic flow. Moreover, we study the limit of small and large Debye length. Our results improve the understanding of fundamentals of flow and transport processes, since we can now explicitly calculate the dispersion coefficient even in evolving geometries. Further research may certainly address the situation of clogging by means of numerical studies. Finally, improved predictions of breakthrough curves as well as facilitated modeling of mixing and separation processes are possible.

Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its hydrodynamic parameters, in particular the medium’s effective permeability. An accurate, quantitative description of the permeability is, however, essential for predictive flow and transport modeling. Well-established relations such as the Kozeny–Carman equation or power law approaches including fitting parameters relate the porous medium’s porosity to a scalar permeability coefficient. Opposed to this, upscaling methods directly enable calculating the full, potentially anisotropic, permeability tensor. As input, only the geometric information in terms of a representative elementary volume is needed. To compute the porosity–permeability relations, supplementary cell problems must be solved numerically on this volume and their solutions must be integrated. We apply this approach to provide easy-to-use quantitative porosity–permeability relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, and real geometries obtained from CT-data. As a discretization method, we use discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We compare the outcome with the well-established relations and investigate the ranges of the validity. From our investigations, we conclude whether Kozeny–Carman-type or power law-type porosity–permeability relations are more reasonable for various prototypic representative elementary volumes. Finally, we investigate the impact of a microporous solid matrix onto the permeability.

In this article, we derive an upscaled model for crystal precipitation and dissolution in a saturated porous medium with a perforated solid matrix. We model the solid matrix itself at the pore scale as a porous medium. Hence, we consider at the pore scale a Darcy-Stokes system, where the Beavers-Joseph boundary condition is proposed at the corresponding interface. By asymptotic expansions we derive an upscaled model describing the process via Darcy's law, a transport equation, and corresponding effective coefficients given by the evolution of the microstructure. Weak solvability of the upscaled model is also investigated.