A macroscopic relationship for preferential flow in the vadose zone: Theory and Validation Page: 3 of 7
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(b) I = L/2 V
From Figure 1, it is obvious that there exists a box
size /1 < L satisfying:
(c) I = L/4 (d) I = L/8
Figure 1 Demonstration of the "box" counting procedure for several box
A fractal pattern exhibits similarity at different scales.
When df < D, the corresponding pattern does not fill
the whole space, but only part of it (Figure 1).
To incorporate the effects of fractal flow patterns, we
need to develop a simple scheme to characterize these
patterns in terms of parameters relevant to water flow
processes. Consider Figure 1(a) to be a gridblock
containing an active flow region and the corresponding
flow pattern to be fractal. In this case, only a portion of
the medium within a gridblock contributes to water flow
(Figure 1). This is conceptually consistent with the
preferential flow process. Note that in Figure 1, a box is
shadowed if it covers the active flow region.
Combining Equations (1) and (2) yields
[N(Z)]1/ d f = [N * (Z)]1/ D (3)
The average active water saturation (Se*) for the
whole gridblock (Figure la) is determined to be
where V is the total water volume (excluding residual
water) in the active region for the gridblock (Figure la),
and # is the effective porosity (corresponding to satu-
rated water content excluding residual water content).
Similarly, the average active water saturation (Sb*) for
shadowed boxes with size of l is
Based on Equations (3)-(6), the average saturation for
shadowed boxes with size i, Sbl*, can be expressed by
Sb1*= (Se *) D (7)
Because a fractal is similar at different scales, the
procedure to derive Equation (7) from a gridblock with
size L can be applied to shadowed boxes with the
smaller size of ih. In this case, for a given box size
smaller than i, the number of shadowed boxes will be
counted as an average number for those within the (pre-
viously shadowed) boxes with a size of i. Again, we can
find a box size l2 < i to obtain a saturation relation:
df d 2
Sb2* (Sb1*) D = (Se*) (8)
The procedure to obtain Equation (8) can be contin-
ued until it reaches an iteration level, n *, at which all the
shadowed boxes with a size of in cover the active region
only. The resultant average saturation for these shad-
owed boxes is
bn )e D (9
By definition, Sbn should be equivalent to the effective
water saturation (Sa) within the active region. Using f to
denote the fraction of the active region within the grid-
block and based on Equation (9), we have
f _ Se * (S *)Y (10)
Parameter yis defined between zero and one. If flow
pattern is uniform and does not contain preferential flow,
will be equal to zero (corresponding to df= D orf= 1).
Otherwise, y will be larger than zero and result in an f
(a) 1 = L
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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/m1/3/: accessed April 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.