A macroscopic relationship for preferential flow in the vadose zone: Theory and Validation Page: 2 of 7
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guage for describing many different natural and social
phenomena1'61. While a vast literature exists on the va-
lidity of the fractal concept for a great number of fields,
fractals have been found to be useful for representing
many spatial distributions in subsurface hydrology, in-
cluding soil particle size distribution, roughness of frac-
ture surface, distribution of permeability in heterogene-
ous formations, and large-scale solute dispersion proc-
Recent studies have also suggested that complex pref-
erential flow patterns in unsaturated systems can be
characterized by fractals. Ref.  may be the first au-
thors to report in the vadose zone hydrology community
that the geometry of dyed flow patterns in
two-dimensional images of soil profiles could be char-
acterized by fractals. Refs. [7,25] indicated that solute
leaching patterns, observed from field plots, could be
well represented by a diffusion-limited aggregation
(DLA) modef351. It has been documented that DLA gen-
erates fractal patterns57"7. Refs. [21,22] also noticed
that the dye penetrations for two test sites (correspond-
ing to a clayey and a sandy soil) are characterized by
power-law mean power spectrum, a signature of fractal
pattern. Ref.  reported that a field observation of
dyed flow pattern in an unsaturated test site is charac-
terized by multi-fractals. This finding is consistent with
that in many cases spatial distributions of hydraulic
conductivity are multi-fractals 
Related to preferential flow in unsaturated soils, frac-
tal flow patterns have often been observed in other un-
saturated and multi-phase flow systems. Ref.  first
showed that unsaturated flow in a single vertical fracture
is characterized by gravity-driven fingers, and the re-
sulting flow patterns could be modeled by an inva-
sion-percolation approach[361. Again, percola-
tion-based models generate fractal clustering patterns .
Viscous fingering in porous media has been experimen-
tally shown to be fractal51. Ref.  reported that
DNAPL fingering in water saturated porous media, ob-
served from sandbox experiments, is fractal. Ref. 
demonstrated that a spatial distribution of fractures with
mineral coatings is also fractal, while fracture coating is
roughly a signature of water flow paths.
3 A macroscopic relationship
As discussed in Section 2, highly non-uniform (prefer-
ential) flow patterns in unsaturated soil (and other un-
saturated and multi-phase flow systems) are fractal.
Therefore, it is critical to incorporate fractal flow pat-
terns for modeling preferential flow behavior. To do so,
we have developed a macroscopic relationship. The
main idea is that flow domain can be divided into active
and inactive regions. Flow occurs preferentially in the
active region (characterized by fractals) and inactive
(immobile) region is simply bypassed. The macroscopic
relationship links the active region with related
large-scale flow parameters. This section presents the
derivation of the relationship based on the fractal flow
The key parameter for a fractal pattern is fractal di-
mension. Fractal dimension, df, is generally a noninteger
and less than the corresponding Euclidean (topological)
dimension of a space, D. Different kinds of definitions
for fractal dimension exist (e.g., similarity dimension,
Hausdorff dimension, and box dimension), although
they provide very close fractal dimension values for
practical applications 51. The most straightforward defi-
nition is the so-called box dimension, based on a simple
"box-counting" procedure. This dimension is determined
from Equation (1) (below) by counting the number (N)
of "boxes" (e.g., line segments, squares and cubes for
one-, two-, and three-dimensional problems, respec-
tively) needed to cover a spatial pattern, as a function of
box size (0)51:
where L refers to the size of the entire spatial domain
under consideration. Figure 1 shows a box-counting
procedure for a spatial pattern with df = 1.6, in a
two-dimensional domain with size L[371.
Obviously, if a spatial pattern is uniformly distributed
in space, the fractal dimension will be identical to the
corresponding Euclidean dimension. In this case, the
number of boxes that cover the pattern, N*, and the box
size l have the following relation
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Liu, H.H. & Zhang, R.D. A macroscopic relationship for preferential flow in the vadose zone: Theory and Validation, article, February 15, 2010; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1014748/m1/2/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.