Abstract: 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 longterm time frame of
experiments make the latter infeasible.
Current challenges in environmental engineering comprise problems
like, for instance, the longterm 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 MUFTEUG
(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 MUFTEUG 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
coarsegrid corrections, which are calculated for the primary
variables on the coarsegrid 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
MUFTEUG 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 setup, the
relative permeability on the coarser scale generally shows
directiondependent properties and therefore has to be described by an
anisotropic tensor (figure 3).
Figure 3: Upscaling example. Top left: heterogeneous setup. 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 advectiondominated 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.
References
 Helmig R., Multiphase Flow and Transport Processes in the
subsurface  A contribution to the Modeling of Hydrosystems, Springer
Verlag, 1997.
 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).
 Pruess K., Bielinski A., EnnisKing 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.
 Braun C., Helmig R., Manthey S., Determination of constitutive
relationships for twophase flow processes in heterogeneous porous
media with emphasis on the relative
permeabilitysaturationrelationship, Accepted for Publication in
Journal of Contaminant Hydrology (2002).
 Eichel H., Helmig R., Neuweiler I., Cirpka O., Upscaling of
TwoPhase Flow Processes in Porous Media, Submitted to Transport in
Porous Media (2003).
 Aavatsmark I., An Introduction to Multipoint Flux Approximations
for Quadrilateral Grids, Computational Geosciences 6 (2002), 405432.
