Problem

Infiltration of immiscible fluids into an aquifer results in a multi-phase flow system. These fluids can be hydrocarbons such as benzene, tar, PAHs, or chlorinated compounds such as TCE. These types of organic substances tend to dissolve poorly in water, given their non-polar molecular structure.
Collectively, they fall under the category of NAPL (Non-Aqueous Phase Liquid), and are further classified by their density (compared to that of water) as either light (Light NAPL) or dense non-aqueous phase liquid (Dense NAPL). Due to the fact that their density differs from that of water, their propagation in the aquifer - usually as a coherent phase - cannot be physically described by ordinary numerical models of single-phase solute transport. In order to obtain accurate predictions of NAPL propagation in soil or of the effectiveness of remedial actions, it is necessary to simultaneously model multiple liquid phases.

In the case of an encounter of two liquid phases in a given pore space, interactions emerge at the phase interface. The dominating parameters for these interactions are capillarity and permeability, both of which depend on the ratio of the saturation of water and NAPL.

Physical Model

The difference between wetting and non-wetting fluids arises as a consequence of the ybetween the phase interface and the surface of the grain. The figure to the right shows the differing behaviour of water and NAPL with respect to the matrix. In this case, water is the wetting fluid, since its surface tension is lower than that of the NAPL.

The capillary pressure increases with a decreasing radius of curvature of the phase interface, and therefore the wetting fluid is forced into the small pore throats. On the other hand, an increasing radius of curvature of the phase interface causes a decrease in capillary pressure, and so the non-wetting fluid fills the pore space. The figure to the left shows the distribution of wetting and non-wetting fluids in a typical pore space. The effect of a decreasing radius of curvature of the phase interface is apparent.

The capillarity is an important parameter for describing the movement of two liquids in a pore space. Each of the fluids builds up a certain pressure on the interface. The difference between the two fluid tensions causes an interfacial tension, which acts upon the whole meniscus and is directly related to the capillary pressure. The figure to the right shows that the capillary pressure for a given degree of saturation in a permeable medium (I) is smaller than in a less permeable medium (II).
The two liquid phases compete for propagation. This phenomenon is taken into account by the use of a relative permeability, which ranges from 0 to 1 depending on the degree of saturation.

Mathematical Model

The flow pattern of mobile liquid phases in a porous aquifer can be described by the equations of motion in a continuum. These equations are based on Darcy's Law, and are applied to a representative elementary volume.
The pressure and the degree of saturation of the wetting fluid (i.e. water) are the unknown variables in this approach.

For the non-wetting (nw) phase:




For the wetting (w) phase:

These differential equations must be solved. They are interrelated by the non-linear dependencies of both the capillary pressure and the relative permeability on the degree of saturation by the wetting phase. The solutions are time-dependent and can be solved only by specifying a set of initial values and boundary conditions.

Numerical Model

Method of Finite Elements in Space
2D-isoparametric Bilinear Test Function


Galerkin's Method => Weighting Function = Test Function



Method of Finite Differences in Time:

Control of Time Step Length

Linearization using a Damped Version of Newton's Method
Solving the Equation by Super LU
Taking Heterogeneities into Account

Application

In a laboratory experiment (Kueper and Frind, Water Resources Research, June 1991), a 70x50x6cm container
was filled with sands of different hydraulic conductivities. Initially, the sand was completely saturated with water. Water could only escape through the side walls. On the top, there was a hole that allowed the infiltration of a DNAPL.
The experiment began with the release of TCE. Since this chlorinated solvent is denser than water, it gradually flowed downwards under the influence of gravity. It pooled on top of a lens of less permeable material (3), and since it could not build up the pressure necessary to displace the water within the lens, it spread laterally and passed down the sides of the lens.
Eventually, the capillary pressure above the lens increased, and the NAPL infiltrated the lens.
Water was then displaced and ran off to the sides.

Numerical results Part 1: Numerical Reproduction of Experimental Results

The two figures below show the results of the modelling (coloured) and the results of the laboratory tests (black hatched).

Numerical Results - Part 2: Animation

Example of Heterogeneities