Partial Differential Equations in Porous Media Research
by Hans van Duijn
Mathematical modelling of flow and transport through porous media
plays an important role in environmental studies as well as in reservoir
engineering. Applications include the spread of pollutants from a landfill
through the soil system and of oil spills in the subsurface, the intrusion
of salt seawater in coastal aquifers, and new methods for enhanced oil
recovery and underground gas storage. At CWI several of such problems are
studied, ranging from very applied to theoretical research.
Although already intensively studied in the past, transport phenomena
in porous media still yield new and challenging mathematical problems in
the field of nonlinear partial differential equations (PDEs), free boundary
problems, and homogenization procedures. Moreover, detailed numerical studies
are necessary to improve the prediction capacity of the related models.
CWI's research focuses on the mathematical modelling of transport processes
in the subsurface, and on the qualitative analysis and numerical study
of the governing PDEs.
In the field of PDE research particular attention was given to systems
consisting of a convection-diffusion equation coupled with an ordinary
differential equation. A special case, modelling salt uptake by mangroves,
involves non-local convection. In a collaboration with the University of
Bonn the interface between fresh and salt groundwater in the presence of
wells is studied. The interface appears as a free boundary in an elliptic
problem in which, depending on the pumping rate of the wells, a cusp singularity
develops. In the case of heterogeneous media the interface was studied
numerically, using a moving mesh Finite Element Method. Knowledge of the
behaviour of the interface is important, eg, to estimate in how far seawater
intrudes in aquifers in the Dutch coastal area when drinking water is pumped
out. This research was highlighted for example at a tele-conference 'Mathematics
and environ-ment: problems related to water', an initiative of the European
Mathematical Society, which was held last December simultaneously at three
locations: Amsterdam, Madrid, and Venice.
The study of density-driven flow in porous
media concentrates on brine transport problems related to high-level radio-active
waste disposal in salt domes. High salt concentrations give rise to non-linear
transport phenomena such as enhanced flow due to volume (compressibility)
effects and the reduction of hydrodynamical dispersion due to gravity forces.
A non-linear dispersion theory for this case, proposed by S.M. Hassanizadeh
(Delft University of Technology), was verified using experimental data
from the Technical University of Berlin.
The pressure in active gas reservoirs in the Netherlands becomes insufficient
to meet the needs during the winter period. Therefore Gas Unie/NAM (Nederlandse
Aardolie Maatschappij) intends to store gas in depleted gas reservoirs
which are no longer in production. These reservoirs act as a buffer to
meet peak demand. In an ongoing project sponsored by NAM CWI studies gas
injection into a reservoir, in order to understand and quantify the mixing
(diffusion/ dispersion) of injected gas with residual gas in old reservoirs,
and develops a numerical code to predict this mixing.
Another project concerns soil remediation techniques. Organic contaminants
may be removed from the soil either by pumping methods or by injecting
air (air sparging), which enhances biodegradation and volatilization. The
corresponding flow of groundwater, organic contaminant and air is described
using multi-phase flow models. For air injection into groundwater in a
horizontally layered medium accurate numerical solutions of the full transient
two-phase flow equations were found and an almost explicit solution for
the steady-state air flow just below a less permeable soil layer was derived.
To model pumping of a lens of light organic liquid from an aquifer, multi-phase
seepage face conditions were applied at the well boundary. For two different
geometries of the lens similarity solutions provided good approximations
of the removal rate and the location of the remaining contaminant as a
function of time.
Finally, in a recently started project PDE's with higher order mixed
derivatives are studied. Such equations arise in models for unsaturated
groundwater flow, taking into account dynamic capillary pressure.
See also : http://dbs.cwi.nl/cwwwi/owa/cwwwi.print_projects?ID=5
Please contact:
Hans van Duijn - CWI
Tel: +31 20 592 4208
E-mail: Hans.van.Duijn@cwi.nl
