Terrestrial Systems > Soil Physics > Research > Porous Media and Soils
Experimental Study of Fingering Flow in Porous Hele-Shaw Cells
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Fereidoun Rezanezhad, Hans-Jörg Vogel, Kurt Roth
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Observations:
Laboratory experiments have shown that the redistribution process
following the infiltration in homogeneous coarse sand is unstable,
producing a series of the fingers. A capillary barrier effect
exists between these two-layer structure where an inclined coarse
layer is overlaid by a fine layer which the coarse layer tend to
impede the flow from fine layer. See the video clip of the
fingering flow experiment during water and dye tracer infiltration
into a multi-layered medium (Movie-mpg,
2.7MB).
In
all experiments, the onset of a stable wetting front in the upper
fine-textured layer characterized by a relatively low saturated
hydraulic conductivity was observed which supplies uniformly water to
the underlying coarse-textured layer characterized by a high value of
saturated hydraulic conductivity. At the textural interface between
the fine and the coarse layer the water moves in preferred paths or
fingers at many discrete points induced by infiltrating flow. After
visualization of the stable infiltration fronts and subsequent
transition to instability in homogeneous coarse layer, the flow
fingers disturbed within the heterogeneous middle-layer and reappear
in the uniform layer below. Figure 1(a) shows a qualitative record of
the advance of the downward growing fingers through an initially dry
porous Hele-Shaw cell by time-lapse images during an experiment with
constant flux of 1.2 mm/min.
After water fingers had
fully developed, a dye tracer (Brilliant Blue) was added to visualize
local flow velocities. Figure 1 (a) shows photographs of the dye
wetting front patterns taken at different times during the second
cycle of infiltration into previous water fingers with the same flow
rate infiltration. It highlights the separation of the water phase
into a mobile (finger core) and an immobile one (finger
fringe) which has implications for solute transport. This shows
the high velocity in the center and stagnant water at the periphery
of the fingers. Here it could be hypothesized as separation of a
portion inside the finger core with convective gravity driven flow
and a portion at the boundaries of the finger fringe with slow and
diffusive flow (Figure 2).
As a finger grows downwards from
the textural interface, there is a narrow zone at the finger tip
where the water saturation increases rapidly. These maximum values
are required to advance the water front into the dry porous medium.
This phenomenon is called saturation overshoot.This pattern
consists of a region directly behind the wetting front with a high
water saturation called finger tip, followed by another region with a
lower water saturation called finger tail. An example of the downward
growth of the fingers in an initially dry porous medium and typical
finger moisture content structure during the water infiltration after
20 min infiltration is shown in Figure 1 (c,d). The bluest
color at the tip indicates the highest water saturation.
Figure
1: a) Fingering patterns observed by Light Transmission
Method (LTM) during water and dye infiltration into the multi-layered
medium (Movie-mpg,
2.7MB).
b) Original observed image. c) Normalized image to visualize
the water saturation. d) Deconvoluted image using the Point
Spread Function (PSF)
Figure
2: The separation of the water phase into a mobile component
(core) and an immobile one (fringe) observed using dye tracer
infiltration.
We analyzed the dynamics of water saturation within the finger tip, along the finger core behind the tip, and within the fringe of the fingers during radial growth. Our results confirm previous findings of saturation overshoot in the finger tips and revealed a saturation minimum behind the tip as a new feature. The finger development was characterized by a gradual increase in water content within the core of the finger behind this minimum and a gradual widening of the fingers to a quasi-stable state which evolves at time scales that are orders of magnitude longer than those of fingers' evolution. In this state, a sharp separation into a core with fast convective flow and a fringe with exceedingly slow flow was detected. All observed phenomena, with the exception of saturation overshoot, could be consistently explained based on the hysteretic behavior of the soil-water characteristic. The transverse saturation profile shows a maximum in the center of the fingers and slow lateral movement of the moisture from finger core regions creates less saturated surrounding finger. Hence, the fully developed fingers consist of a high saturated tip, a core with mobile water and a hull with immobile water fringe and once a finger is developed, the saturation profile along its core is invariant as the tip progresses.
The simultaneous measurements of pressure and saturation definitively show the nonuniform moisture profiles in flow paths created by fingered flow. In the evaluation of hydraulic states to describe the dynamics and stabilization of fingers, we demonstrated that the water content and matric potential at the wetting front of unstable fingered flow are on the wetting curve and behind the front are on the drying curve of the soil-water characteristic curve.
Based on the experimental results, we propose two stages of lateral expansion for fingers: a fast expansion in a short time to core stabilization and a slow and steadily expansion for long time to fringe lateral growth. The finger core stabilization well expressed by the interface between the finger core and the fringe zone approaching to the same potential on different hysteresis loop of soil-water characteristic curve, even through the water contents are quite different. A large fringe lateral expansion, however, is expressed by a hydraulic conductivity and water pressure gradient between the outer limit of the core and the fringe region of finger. This is severely hindered by the low conductivity in this dryer range and by the limited supply of water and as long as the matric potential in finger core is smaller than the surrounding zone, finger grows. Hence, within the core of the fingers, the fast convective flow is driven by gravity while at the boundaries (fringes), flow is slow and diffusive.
Stage I: Core Stabilization: A fast expansion of the finger width to stabilization of the finger core that conducts most of the flow by effects of infiltrating forces on a short time scale. In this stage, water infiltrates radially driven by the large hydraulic gradient between the center of the finger core and the outer limit of the core. The radial gradient decreases over time and with it the corresponding flux, approaching the center and the outer limit of the finger core to the same pressure. This stage of the development can be described by hysteretic behavior on soil-water characteristic curve where the tip is on a wetting branch and the core behind the tip on a drying branch. The radial water flux ceases when core and the outer limit of the core approach the same potential on different hysteresis loops, and hence, the finger core becomes stabilized.
Stage
II: Fringe Expansion: A slow expansion associated with growth
by effects of capillary forces and hydraulic gradient between the
outer limit of the core and fringe zone on a time scale much longer
than core stabilization. This low hydraulic gradient decreases the
hydraulic conductivity very rapidly and with it the radial flux. In
this stage, water flows vertically in the stabilized core and
laterally in the fringe zone which means that wetting fronts leave
the fingers and move laterally into the dry sand on either side of
the finger core areas. Over time, slow lateral movement of moisture
from finger core regions creates a less saturated surrounding fringe
region and the finger continues to expand.
Figure 3 shows the development of the finger core and fringe areas in time, for an experiment with the duration of the 10 days of continuous infiltration with a constant flux of 1.2 mm/min. These intensity measurements show that the finger core stabilized 16 minutes after the tip wetting front passed (stage I) and the lateral movement in the fringe zones continues to expand for a long time after stabilization of the core (stage II). As long as there is a radial hydraulic gradient between the outer limit of core and fringe zones, it continues to expand. This is an example showing that the growth of the fingers is a very slow process. Hence, the wetting and drying to equilibrium inside the finger core took place in less than an hour, while the wetting to equilibrium outside the fingers took place over several days.
Figure 3: Sixteen different times series visualizing the finger core and fringe areas development and the lateral cross-section intensity profiles of a finger at a fixed location (indicated by black dash line in images top left) during the passage for the duration of 10 days continuous infiltration. Over time, slow lateral movement of moisture from finger core regions creates a less saturated surrounding fringe region and finger continues to expand. The width of the fingers increases from 11 mm after the passage of tip to 60 mm after 10 days continuous infiltration.
