Forschung

Im Projekt soll die Strukturbildung in der Rhizosphäre, welche durch geochemische, mikrobiologische und physikalische Einflüsse gesteuert wird, modellbasiert untersucht werden. Ziel ist die Entwickling eines mechanistischen Modellansatzes, welcher die dynamische strukturelle Reorganisation der Rhizosphäre auf der Skala einzelner Wurzeln (Mikroskala) ermöglicht (einschließlich expliziter Darstellung der Heterogenitäten des Porenraums). Dieses
sich zeitlich verändernde Mikroskalenmodell ist wechselseitig mit der Makroskala gekoppelt mittels mathematischer Homogenisierung (upscaling) und erlaubt so die Ableitung effektiver Bodenfunktionen. Dabei betrachten wir also keine statische Rhizosphäre, sondern
vielmehr eine dynamische, d.h. eine sich durch Bildung von Aggregaten und geochemische Strukturen verändernde. Insbesondere werden durch die Erkenntnisse aus dem
Zentralexperiment - CT-Bilder in verschiedenen Wachstumsphasen und Feuchteverhältnissen - die Porenstruktur sowohl mit als auch ohne Wurzelhärchen deutlich, und damit auf deren Einfluss zur Aggregation schließen lassen. Mit Hilfe der Kooperationspartner soll
auch eine explizite Wurzelsekretphase modelliert sowie die Anlagerungseigenschaften von Aggregaten an Wurzelhärchen aufgenommen.

Nonlinear (multiphase) flow and reactive multicomponent transport problems in highly heterogeneous porous media and their numerical simulation are of great interest for evaluating site remediation, energy exploitation or CO2 sequestration scenarios.   The resulting advection-diffusion-reaction-systems are coupled nonlinear parabolic partial differential equations, and we have parabolic or elliptic nonlinear flow equations, possibly degenerate. The development of convergent and efficient numerical schemes is very challenging and the mixed (hybrid) finite element method M(H)FEM and the multipoint flux approximation MPFA are powerful locally mass conservative choices. They offer also the advantage of continuous flux approximations over the element faces.  Analogies between the two techniques should help to prove order of convergence estimates and monotonicity for the multicomponent transport problems, but also for multiphase flow.  Furthermore numerical diffusion of the schemes should be quantified to assess the accuracy of the methods.  Simulation examples should include realistic scenarios on heterogeneous, log normally distributed random parameter fields.

The project was a cooperation of a group of applied mathematicians with the Russian company Aeroservice for the development and optimization of new photocatalytic filter systems for air cleaning of nanoparticles and organic substances with the help of mathematical simulation tools. For the simulation of aerosol transport in the filter made of polypropylene fibers, which is used in hospitals or airports, e.g., mathematical models and efficient solution algorithms had to be developed. These allow on the one hand to take stochastic components into account, as the heterogeneous conductivity distribution in the filter. On the other hand these methods were coupled with highly accurate computation schemes as mixed finite element methods, which guarantee local mass conservation for the transport processes. The design parameters of real experiments can be optimized with the help of such simulation tools and their sensitivity with respect to filter efficiency analysed. Among the used methods are particle filtration in porous media, based on the Darcy equation, and coupled Eulerian and Lagrangeian simulation of transport processes, including Monte Carlo approaches with given filter geometries.

Soil colloids may influence the interaction between solutes and the immobile solid phase. A coupling to the fluid transport is possible by processes of sedimentation, flocculation, precipitation, filtration and deposition. The objective of this research project is the qualitative and quantitative examination of the crucial aspects of colloidal-influenced solute- and fluid transport by means of systematic, prognostic simulation. In detail,

1. the attachment and detachment of colloids under consideration air-water interface of the soil,
2. the transformation of the pore space and the thus induced coupling to the fluid transport in soil, and
3. the transformation of the surface properties of the solid phase and the thus induced coupling to the solute transport

have to be analyzed. The main hypothesis of this project states that the couplings incorporated in the model conception affect the praxis-relevant situations not only qualitatively, but also quantitatively in a significant way. The deterministic description of the physicochemical mechanisms on basis of the conservation laws for mass, impulse and energy results in systems of time-dependent non-linear partial differential equations. In order to make the model operative with respect to the problem formulation, one has to approximate it via numerical methods and to implement those in a software tool. For each level of complexity which has to be achieved, a comparison with existing experimental data has to be accomplished. In particular, these datasets have is to be used to obtain a realistic parametrization of the model via inverse modelling.

The evaluation of the potential of contaminated sites concerning natural attenuation needs comprehensive process descriptions and accurate, reliable numerical algorithms. Numerical errors may lead to qualitatively completely wrong conclusions concerning the potential of the site for degradation. It has been developed a comprehensive and flexible simulation tool, that is outstanding concerning the variety of processes, the quality and efficiency of the calculations ensured by modern numerical methods as well as the usability. The existing software platform RICHY has been extended, which is already intensely and successfully used by universities, institutes and consultants for the simulation of reactive transport and parameter identification. Among previous modules for coupled sufactant transport, preferential, unsaturated flow or carrier facilitated transport the project could realize new model components that surpass most of all existing software packages. The extensions contain complete descriptions of microbially catalysed degradation with arbitrary reaction partners and inhibition, general multicomponent reactions including the effects of ionic strength, as well as mineral dissolution and precipitation. The efficient and highly accurate, newly developed mathematical solution algorithms for the resulting coupled systems of partial differential equations could show their quality in complex international benchmark studies. Locally mass conserving, mixed hybrid finite element discretisations of the flow problem have been combined with globally implicit, reactive multicomponent models. Novel reduction methods for the latter rely on the linear transformation of the equation systems and variables and lead to the consideration of conservation quantities which can be handled efficiently, as a part of the transport – reaction – equations decouples. Another approach that has been pursued simultaneously relies on a modified Newton method and results in efficiency enhancements by the neglection of coupling terms in the Jacobian matrix. This algorithm can be applied fully adaptively, in 1D as well as in 2D. Both approaches could be combined with adaptive techniques for the automatic, efficient choice of time steps and spatial grid sizes, which makes the calculation of these complex problems feasible on PCs.

The project included the mathematical modelling of natural attenuation processes in the subsurface and the extension of a software tool for complex reactive multicomponent processes in the framework of mixed hybrid and conforming finite elements. New  parameter identification methods allow the parametrization of unknown functions or a formfree optimization, and help to overcome the dilemma of missing data in complex models. Work included instationary 3D simulations and scenarios of  contaminated sites explored by project partners. The findings of the joint research project resulted in guidelines for authorities and consulting engineers dealing with natural attenuation at contaminated sites.

Mathematical simulation tools allow the quantitative integration of competing transport and transformation processes which are relevant for a seepage water risk prognosis. Therefore model simulations have to contain a comprehensive process description, while they can serve for parameter identification by inverse modelling of suitable column or batch experiments, and allow to quantify the dependence of a key variable on parameters through a simultaneous sensitivity analysis. The software platform RICHY1D has been extended and is already intensively and successfully used in universities, institutes and by consultants for the 1D simulation of complex reactive transport and for parameter identification. It stands out by the application of efficient and highly accurate mathematical solution strategies for the resulting systems of partial differential equations (e.g. locally mass conserving mixed hybrid finite element discretisations, modified Newton’s method). Besides the formerly existing modules for coupled surfactant-water transport, multiphase flow, saturated-unsaturated flow or carrier facilitated transport, the extensions contain in particular source terms (boundary conditions, distributed sources, arbitrarily time dependent, nonlinear and multiple (de-)sorption kinetics, mobilisation from a residual NAPL phase), preferential flow with solute transport, and heat transport in soils with coupling to reaction parameters of the contaminant transport like Monod degradation parameters, e.g.. The parameter identification is possible for the model extensions as well, which allows the identification of multiple complex parametrizations from suitable experiments (for example for source terms or microbially mediated degradation, sorption characteristics and hydraulic parameters). There is no need to impose a certain functional shape of these nonlinearities, the so-called form-free identification is also feasible, and furthermore a closed-flow experiment design can be accounted for. The sensitivity analysis is provided separately for the evaluation of the dependence of a key variable like the concentration of arbitrary model parameters, what represents a powerful tool in a transport simulation to identify controlling factors and evaluate uncertainties of the data.