# Researches

Dynamic Capillary Fringes

Modelling of Transport in strongly heterogeneous soils

Parameter Estimation for heterogeneous porous media

Development of an efficient and flexible Software Framework for partial differential equations (DUNE)

The Iterative Solver Template Library

Unfitted Discontinuous Galerkin Methods

Two Phase Navier Stokes Flow in Complex Domains

Large-scale numerical simulation of processes during CO2 storage in geologic formations

Geostatistical Inversion of Coupled Processes in Heterogeneous Porous Media

## Adaptive Modelling of Coupled Hydrological Processes with Applications in Water Economy

Funded by: | BMBF |

Project Director: | Peter Bastian |

Executed by: | Christian Engwer |

Partners: | Freiburg University, FU Berlin, Münster University |

The water balance of the atmosphere, the biosphere, and the pedosphere are interconnected by the hydrological cycle. This results in coupled hydrological processes, which determine the availability of water and trigger hydrological hazards like flood waters and droughts. The impact of the different hydrological processes relevant for modelling of the water balance changes depending on the local scale of the considered catchment area and the temporal scale of the observation. A model for the regeneration of ground water in some catchment with a characteristic scale of 1-10 km the relevant time scale is at least some years. On the other hand, for a thorough flood water prediction with a spatial scale of 100-1000 km the required temporal resolution ranges from some days to some weeks. By adaptive modelling of the coupled hydrological processes depending on the data and the characteristic scales in space and time, an integrated simulation platform for applications in hydrology and water economy is to be developed.

This project is structured into four sub projects. This group contributes with the design and implementation of a flexible and efficient software infrastructure for the solution of coupled systems of differential equations. Within the context of a domain decomposition approach this includes the coupling of equations, discretizations and linear solvers. The development is based on the software framework DUNE.

The prediction of flood waters requires the consideration of large areas and very complex local geometries. Hence, the parallel computability of the resulting problems is of great importance for this project. The storage and visualization of the computed results inevitably requires corresponding capabilities for data I/O and parallel rendering.

## Dynamic Capillary Fringes

Funded by: | DFG |

Project Director: | Peter Bastian, Olaf Ippisch |

Executed by: | N.N. |

Partners: | K. Roth (Heidelberg), J. Winter (Karlsruhe), P. Gratewohl (Tübingen), F. Frimmel (Karlsruhe) |

The water balance of the atmosphere, the biosphere, and the pedosphere are interconnected by the hydrological cycle. This results in coupled hydrological processes, which determine the availability of water and trigger hydrological hazards like flood waters and droughts. The impact of the different hydrological processes relevant for modelling of the water balance changes depending on the local scale of the considered catchment area and the temporal scale of the observation. A model for the regeneration of ground water in some catchment with a characteristic scale of 1-10 km the relevant time scale is at least some years. On the other hand, for a thorough flood water prediction with a spatial scale of 100-1000 km the required temporal resolution ranges from some days to some weeks. By adaptive modelling of the coupled hydrological processes depending on the data and the characteristic scales in space and time, an integrated simulation platform for applications in hydrology and water economy is to be developed.

This project is structured into four sub projects. This group contributes with the design and implementation of a flexible and efficient software infrastructure for the solution of coupled systems of differential equations. Within the context of a domain decomposition approach this includes the coupling of equations, discretizations and linear solvers. The development is based on the software framework DUNE.

The prediction of flood waters requires the consideration of large areas and very complex local geometries. Hence, the parallel computability of the resulting problems is of great importance for this project. The storage and visualization of the computed results inevitably requires corresponding capabilities for data I/O and parallel rendering.

## Modelling of Transport in strongly heterogeneous soils

Executed by: | O. Ippisch |

## Parameter Estimation for heterogeneous porous media

Executed by: | O. Ippisch |

## Development of an efficient and flexible Software Framework for partial differential equations (DUNE)

Executed by: | P. Bastian, M. Blatt, C. Engwer, O. Ippisch, S. Lang |

Partners: | A. Dedner (Freiburg University), R. Klöfkorn (Freiburg University), M. Nolte (Freiburg University), M. Ohlberger (Münster University), O. Sander (FU Berlin) |

Most finite element, or finite volume software is built around a fixed mesh data structure. Therefore, each software package can only be used efficiently for a relatively narrow class of applications. For example, implementations supporting unstructured meshes allow the approximation of complex geometries but are in general much slower and require more memory than implementations using structured meshes.

By developing algorithms based on abstract interfaces, it is possible to avoid these restrictions, thus allowing multiple implementations of the same interface. Modern methods of generic programming (e.g. templates in C++) allow an efficient realization of this concept without restricting the flexibility of the resulting code. A new framework for partial differential equations - the “Distributed and Unified Numerics Environment” (DUNE) - is being developed according to this paradigm and in cooperation with many other universities. Currently, DUNE is applied in a vast variety applied sciences including the research on porous media, neuro transmitters, and particle accelerators.

## The Iterative Solver Template Library

Executed by: | Markus Blatt |

Sparse matrices obtained from finite element discretisations exhibit a lot of structure (e. g. discretisation of three-component system with linear finite elements and point-wise ordering) already known at compile time. As the knowledge is known already at compile time it can be exploited for efficiency using generic programming like in C++. The Iterative Solver Template Library (ISTL) provides a generic matrix/vector interface supporting a recursive block structure.

Solving large sparse linear systems is an ubiquitous task in the numerical solution of partial differential equations (PDEs). Increasing demands of computationally challenging applications both in problem size and algorithm complexity have lead to the development of parallel scalable solver libraries for these tasks. One of the most effective ways to achieve scalability is the use of multigrid or multilevel techniques. Algebraic Multigrid (AMG) is a very efficient algorithm for for solving large sparse problems on unstructured grids.

Our parallel AMG algorithm based on aggregation is designed to exploit the block structure of the matrices. Thus it can cope efficiently with scalar matrices as well as coupled block representations for systems of PDEs during the setup phase as well as during the solve phase. The algorithm proves to be a robust, efficient and scalable preconditioner within Krylow methods for the simulation of flow through heterogeneous media.

As a next step we want to tune the algorithm for solving linear systems stemming from Discontinuous Galerkin discretisations. Preliminary tests show that our algorithm is a potentially robust and efficient solver for these systems.

## Unfitted Discontinuous Galerkin Methods

Executed by: | Christian Engwer |

Simulation of physical, biological and chemical processes often involve complex shaped domains. Common problems are flow through root networks, solute transport on the pore scale of porous media or exchange processes through cell membranes.

Classical numerical methods require a grid resolving the complex geometry. Creating such grids is a very sophisticated process and therefore methods without this requirement are of great interest.

In this work a new approach to simulations on complex shaped domains was developed. The method is based on a Discontinuous Galerkin (DG) method with trial and test functions defined on a structured grid. Thus the number of degrees of freedom is proportional to the number of elements in the structured grid. The support of the trial and test functions is restricted according to the shape of the geometry. Essential Boundary conditions are imposed weakly via the Discontinuous Galerkin formulation. This method offers a discretization where the number of unknowns is independent of the complexity of the domain.

The method was successfully applied to stationary as well as time dependent problems.

## Two Phase Navier Stokes Flow in Complex Domains

Funded by: | DFG |

Project Director: | Peter Bastian, Olaf Ippisch |

Executed by: | Felix Heimann |

Partners: | K. Roth (Institute for Environmental Physics, Heidelberg University), H. J. Vogel (Helmholtz-Zentrum für Umweltforschung Halle), R. Hilfer (Institut für Computerphysik, Stuttgart University) |

In this project, a robust and efficient method for the simulation of the laminar flow of two incompressible and immiscible fluids within a complex geometry is developed. The method for the solution of the corresponding Navier Stokes equations is based on the Unfitted Discontinuous Galerkin method which has been specifically developed for applications in complex domains. This nonconforming method does not require a global grid generation and realizes local conservation of impulse and mass. Furthermore, the pressure jump across the two phase interface can be approximated by the discontinuous base functions without introducing any numerical error due to the regularization of the pressure field near the interface.

The target applications of this model are the determination of hydraulic parameters for porous media multi phase flow models on a continuous macro scale. Especially the most recent models which take into account the immobile parts of the fluids and their interfacial area could greatly benefit from reliable simulations on the micro scale. The analysis of phenomena like hysteresis and fingering, which still defy all attempts of reliable modelling on the macro scale, are also part of this project.

Geometries of interesting soil samples are provided by the cooperation partner by means of x-ray tomography.

## Large-scale numerical simulation of processes during CO2 storage in geologic formations

Funded by: | Landesstiftung Baden-Württemberg |

Project Director: | Peter Bastian, Olaf Ippisch |

Executed by: | Rebecca Neumann |

Partners: | Rainer Helmig (Stuttgart University) |

Climate change as a consequence of anthropogenic greenhouse gas emissions is now a fact. The european countries have agreed upon reducing emission of CO2 as the most important greenhouse gas by 20% until 2020. Carbon Capture and Storage (CCS) is a recently discussed new technology, aimed at allowing an ongoing use of fossil fuels while preventing the produced CO2 to be released to the atmosphere. This is of particular importance on an intermediate time scale, as long as the development and implementation of renewable energies has not yet reached the still growing demand of electrical energy. In this project, we propose to develop an efficient parallel numerical simulator in three space dimensions that is able to simulate the full range of hydrological and geochemical processes necessary to describe the injection of CO2 on a regional spatial scale (say 30 km times 30 km times 300m), and on time scales relevant for the preferable mineral trapping of CO2. So far, either a limited number of processes or other simplifying assumptions could be treated in the existing simulators. With the new simulator, we will be able to predict more accurately the CO2 storage capabilities of a reservoir and provide information about site management, e.g. about the placement of water production wells for an active pressure management.

## Geostatistical Inversion of Coupled Processes in Heterogeneous Porous Media

Funded by: | Landesstiftung Baden-Württemberg |

Project Director: | Peter Bastian, Olaf Ippisch |

Executed by: | Vacant |

Partners: | Olaf Cirpka (Tübingen University) |

The assessment of the three-dimensional distribution of hydraulic parameters in groundwater bodies is of uttermost importance for the management of groundwater resources used for water supply, for the evaluation of anthropogenic impacts on water bodies including the loading with contaminants, and for the design of remediation schemes at contaminated sites. The distribution of hydraulic conductivity determines capture zones, flow paths, and travel times. Spatial variability occurs on practically all scales so that traditional conceptual models of the subsurface, in which the domain is subdivided into a few layers and zones with uniform coefficients, may be put into question. In geostatistical characterization, by contrast, the hydraulic parameters are considered as correlated random space variables. This allows applying inversion schemes, in which the estimated parameter fields continuously vary in space, meet the measurements of dependent quantities, and show the required spatial correlation. State-of-the-art geostatistical inversion schemes on serial computers can be used to estimate up to one million parameters. This, however, is hardly sufficient for a full three-dimensional representation of the heterogeneous subsurface. Additional restrictions of serial computing concern the inversion of large sets of measurements and the generation of multiple realizations in conditional Monte-Carlo simulations. The latter restrictions hamper the full use of geophysical data in joint inversion of hydraulic and geophysical surveys, and the use of steady-state concentration measurements for the estimation of hydraulic-conductivity fields.

The objective of the proposed project is to develop a program environment for geostatistical inversion of data originating from coupled flow, transport and geophysical assessment processes in heterogeneous porous formations using high-performance-computing techniques. As platform for the discretization and solution of forward and adjoint partial differential equations for groundwater flow, solute transport, and geophysical monitoring, the software framework “Distributed and Unified Numerics Environment” (DUNE), developed by the parallel computing group at University of Heidelberg and others will be adopted and improved. The geostatistical inversion methods developed by the hydrogeology group at University of Tübingen will be integrated into this framework, allowing fully parallelized computations for inversion and conditional Monte-Carlo simulations on three levels: domain decomposition, parallel evaluation of sensitivities, and parallel generation of conditional realizations.

The software framework developed in the proposed research will push the capabilities of jointly analyzing groundwater data from hydraulic experiments (heads, concentrations, flowmeter and pumping tests) and geophysical surveying (mainly geoelectrical tomography) onto a new level, and thus helps making better decisions for groundwater management and remediation under uncertainty.