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Institute of Hydraulic Engineering


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SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik,
Teilprojekt A3, Mehrphasenprozesse in porösen Medien
Project manager:Prof. Dr.-Ing. Rainer Helmig
Deputy:Dr.-Ing. Holger Class
Research assistants:Dr.-Ing. Jennifer Niessner
Duration:1.1.2004 - 31.12.2006
Funding:externer Link German Research Foundation (DFG)
Comments:Für weitere Informationen zum SFB 404 bitte hier klicken !
Poster:Poster (GIF) - Poster (PS)
Publications: Link


Computer simulations of processes in the subsurface are based upon mathematical models for multiphase flow in porous media. They treat the concurrent movement of more than one fluid phase in a solid skeleton (e.g. flow of water, oil, air in soil) and are subject to permanent further development (e.g. Helmig R., 1997 [1]). In many cases, these fluids are miscible and the models therefore also have to consider the multiple components within the phases. Often, phase change by evaporation/condensation takes place which is additionally coupled with an exchange of energy. The special relevance of these complex simulations lies in the fact that, in different fields of technical applications, they are the only practible forecasting instrument because the high costs or the long-term time frame of experiments make the latter infeasible.

Current challenges in environmental engineering comprise problems like, for instance, the long-term storage of the greenhouse gas CO2 by injection into the subsurface to stop global warming or the remediation of contaminated sites. To tackle these problems, today's engineers need reliable, fast and robust simulation algorithms. The current work uses the program system MUFTE-UG (developed in cooperation with the former SFB 404 project C4) as its basis (Helmig R. et al., 1998 [2]). It is designed as a research code to combine the modelling of the relevant physical processes with the application of new discretisation techniques and solution methods including, for example, multigrid and parallelisation strategies.

Focal Point I: Multigrid Solver and Boundary Conditions

As an example of the increasingly complex physics described by the multiphase multicomponent models, the ongoing research of the working group into the sequestration of CO2 can be taken (figure 1, Pruess K. et al., 2002 [3]).

Figure 1: Simulation of injection of CO2 into a geological formation and its spread

A numerical algorithm was implemented in the simulator MUFTE-UG which takes into account the appearance and disappearance of phases, i.e. the number and combination of the locally existing fluid phases together with the mathematical equations describing them may change. The model yields large nonlinear systems of equations which need multigrid preconditioners for a fast solution. But then the problem arises that a change of primary variables is possible. So the coarse-grid corrections, which are calculated for the primary variables on the coarse-grid nodes and then interpolated on the fine grid, may disagree with the expected values for the primary variables on the finer grid, which sometimes leads to numerical problems.

A possible way out of this situation lies in the implantation of knowledge about the processes that are occurring into the transfer operators of the multigrid algorithm. This is in some way unconventional as that part of the mathematical procedure classically knows nothing about "what's going on in the physics". However, it will bring more robustness into the solution procedure and speed things up. As an effect, more practical problems will become "simulatable" without numerical difficulties.

Another approach to ease exact mathematical modelling and thereby facilitate more realistic simulations of practical questions is to allow boundary conditions to secondary variables. This procedure will further push back the frontier of problems that can be addressed by MUFTE-UG in a physically correct way.

Focal Point II: Upscaling and Heterogeneities

The term upscaling denotes the transition from a finer to a coarser scale where the underlying model is probably adjusted to the new scale. This is done by sieving important from unimportant information. To design better upscaling sieves, a better acquisition of the essential and characteristic properties of the physical processes in heterogeneous systems is necessary. Techniques for this are already present (figure 2, Braun C. et al., 2002 [4]) and will be further developed in the ongoing project period.

Figure 2: Upscaling Toolbox

The gain in knowledge of the characteristics of the upscaled system properties leads directly to more advanced models. One new insight, for example, is the fact that, for a heterogeneous set-up, the relative permeability on the coarser scale generally shows direction-dependent properties and therefore has to be described by an anisotropic tensor (figure 3).

Figure 3: Upscaling example. Top left: heterogeneous set-up. Bottom left: anisotropic behaviour of upscaled relative permeability. Top right: upscaled flow with isotropic properties. Bottom right: upscaled flow with anisotropic properties

Consequently, this new knowledge has to be incorporated into a more sophisticated model. First results simulated on a Cartesian grid for a relative permeability tensor with main directions parallel to the grid lines can be found in Eichel H., 2004 [5]. At the moment, the present approach is being extended to be capable of coping with general tensors on unstructured grids, taking into account the need for upwinding schemes in advection-dominated scenarios.

At the same time, the alternative and more powerful instrument of multipoint flux approximations (MPFA, Aavatsmark I., 2002 [6]) is investigated. This is a whole class of numerical fluxes that is applicable to general unstructured grids and promises a great improvement compared to the conventional discretised fluxes as it reduces the influence of grid effects. This will result in qualitatively better predictions of the numerical simulation, but one has to pay a higher price in terms of simulation time.


  1. Helmig R., Multiphase Flow and Transport Processes in the subsurface - A contribution to the Modeling of Hydrosystems, Springer Verlag, 1997.
  2. Helmig R., Class H., Huber R., Sheta H., Ewing J., Hinkelmann R., Jakobs H., Bastian P., Architecture of the Modular Program System MUFTE_UG for Simulating Multiphase Flow and Transport Processes in Heterogeneous Porous Media, Mathematische Geologie 2 (1998).
  3. Pruess K., Bielinski A., Ennis-King J., Fabriol R., Le Gallo Y., Garcia J., Jessen K., Kovscek T., Law D.H.-S., Oldenburg C., Pawar R., Rutqvist J., Steefel C., Travis B., Tsang C.-F., White S., Xu Tianfu, Code Comparison Builds Confidence in Numerical Models for Geologic Disposal of CO2. Sixth International Conference on Greenhouse Gas Control Technologies, Kioto, Japan, Oktober 2002.
  4. Braun C., Helmig R., Manthey S., Determination of constitutive relationships for two-phase flow processes in heterogeneous porous media with emphasis on the relative permeability-saturation-relationship, Accepted for Publication in Journal of Contaminant Hydrology (2002).
  5. Eichel H., Helmig R., Neuweiler I., Cirpka O., Upscaling of Two-Phase Flow Processes in Porous Media, Submitted to Transport in Porous Media (2003).
  6. Aavatsmark I., An Introduction to Multipoint Flux Approximations for Quadrilateral Grids, Computational Geosciences 6 (2002), 405-432.