Environmental Hydrogeology
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Density effects in heat and mass transfer in porous media
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Researchers:
ms. ir. A. J. Landman, dr. R.J. Schotting, prof.dr.ir S. M. Hassanizadeh
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Project description
In many practical applications of flow and transport in porous media the density of the fluid varies with temperature
or solute concentration. Examples are: groundwater pollution problems, storage of thermal energy in aquifers and seawater
intrusion. To simplify the modelling, density effects are often (partly) neglected. It is said that this is justified
when density gradients are small. A commonly used assumption is the Oberbeck-Boussinesq approximation.
When this approximation is used in studies of density dependent flows the density is treated as a constant
in the transport equations, whereas it remains variable (a function of concentration and/or temperature) in Darcy’s law.
To clarify why density variations can be neglected in one equation but not in the other, a study after the validity and
applicability of this assumption was performed as a first part of this Ph.D. research. For transport of heat
and solute, limits were derived that yield the equations according to the Boussinesq approximation. Furthermore,
self-similar solutions were obtained for a simultaneous heat and brine transport problem.
The second part of this Ph.D. research focuses on hydrodynamic dispersion under high-concentration-gradient conditions.
Large concentration gradients can occur for instance in flow of concentrated brines near salt formations. This example
is of interest in relation to possible storage of radioactive waste in deep salt formations. There is a growing awareness
that density gradients affect the spreading of a solute but it is still unclear how to model this in a correct way.
In the eighties, a series of displacement experiments were carried out to study the effect of density gradients on
dispersive mixing of a solute. In a gravitationally stable configuration a decrease in dispersivity with increasing
concentration difference is found. It is generally believed that gravity forces at the micro scale, induced by density
gradients, are the physical cause of this suppression of dispersive mixing. However, it is not fully clear yet how to
model this correctly. A possible approach is a modification of Fick’s law. With the commonly used linear law the
dispersive flux is overestimated in the case of large concentration gradients. A non-linear equation for the dispersive
flux, introduced by Hassanizadeh and Leijnse in 1995, has proven itself in the past few years by being able to model
displacement experiments for a wide range of density differences.
An important objective of this Ph.D. research is to test and strengthen this new dispersion theory. To this end,
displacement of brine in a heterogeneous column is simulated numerically. The basic assumption is that at the local
scale (the scale of heterogeneities) the standard porous media transport equations are valid, including Fick’s Law.
Gravity effects at this local scale influence the average spreading of the brine front. To simulate the small-scale
effects accurately, a well-tested and highly advanced computer code is used, running on a cluster of 80 parallel
processors. The numerically determined breakthrough curves are analysed and fitted with the non-linear dispersion theory.
In this way, the behaviour of the new dispersion parameter in relation to flow velocity, density difference and medium
properties is investigated. So far, only longitudinal dispersion has been studied but the research will be extended
to transversal dispersion. Another aim is to help in designing a new set of experiments to support the further
development of the non-linear dispersion theory. In addition, the problem is approached in a purely mathematical
way using homogenisation theory. The homogenisation procedure starts with the same presumptions at the local scale.
The goal is to arrive at a macro scale equation for the dispersive flux. An attempt is made to match the
homogenisation results with those obtained numerically. In this way a profound theory of high-concentration-gradient
dispersion can be established.
