Agricultural Water Management 210 (2018) 208–216

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Agricultural Water Management
journal homepage: www.elsevier.com/locate/agwat

Theoretical bounds for the exponent in the empirical power-law advancetime curve for surface ﬂow

T⁎Behzad Ghanbariana, , Hamed Ebrahimianb, Allen G. Huntc,d, M. Th. van Genuchtene,faPorous Media Research Lab, Department of Geology, Kansas State University, Manhattan, KS, USA
Department of Irrigation and Reclamation Eng., University College of Agriculture and Natural Resources, University of Tehran, Karaj, Iranc
Department of Physics, Wright State University, Dayton, OH, USAd
Department of Earth and Environmental Science, Wright State University, Dayton, OH, USAe
Center for Environmental Studies, CEA, São Paulo State University, UNESP, Rio Claro, SP, BrazilfDepartment of Nuclear Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazilb

A R T I C LE I N FO

A B S T R A C T

Keywords:
Advance time
Backbone fractal dimension
Percolation theory
Surface ﬂow

A fundamental and widely applied concept used to study surface ﬂow processes is the advance-time curve, whichindicates the time at which water arrives at any given distance along a ﬁeld length. The advance-time curve istypically characterized by an empirical power law with an exponent r and a numerical prefactor p (i.e., x = pt r ).
In the literature, diﬀerent values of r have been reported for various situations and types of surface irrigation.
Invoking concepts from percolation theory, a theoretical approach from statistical physics for quantifying
transport properties in complex systems, we relate the exponent r to the backbone fractal dimension Db (r = 1/
Db). The backbone, through which transport occurs predominantly in a system, is formed by multiply connected
loops composed of several paths that bring ﬂuid from an initial point to a ﬁnal point. The backbone fractal
dimension Db, characterizing the complex structure of the backbone, depends on two factors: dimensionality ofthe system (e.g., two or three dimensions) and percolation class (e.g., random or invasion percolation with/without trapping). We show that the theoretical bounds of Db are well in agreement with experimental ranges ofr reported in the literature for two furrow and border irrigation systems. We further use the value of Db from theoptimal path class of percolation theory to estimate the advance-time curves of four furrows and seven irrigation
cycles (i.e., 28 experiments). The optimal path is the most energetically favorable path through a system.
Excellent agreement was obtained between the estimated and observed curves. We also discuss the indirect
eﬀects of initial water content, inﬂow rate, surface slope, and inﬁltration rate (soil texture) on system dimensionality, percolation class and, consequently, the backbone fractal dimension value. More speciﬁcally, we
postulate that for closely-spaced furrows with steep slopes, low inﬁltration rates, and relatively high inﬂow rates,
the wetting front advance will be mostly quasi one-dimensional. Since the backbone fractal dimension in one
dimension is 1, one should expect the exponent r to be near 1.

1. Introduction

x = pt r

Modeling water ﬂow and solute transport in surface irrigation systems requires knowledge of how quickly water moves over the soil-air
interface along a furrow or border. Analyses of this type lead to theadvance time curve, which describes the balance among water that
already exists in a soil (referred to as the initial water content), waterthat inﬁltrates into the soil, and water that traverses on the surface. Oneof the most widely used forms in the literature for the advance time
curve, which indicates the time at which water arrives at a given distance in a ﬁeld, is the following empirical power law

in which t is the time needed for the wetting front to reach distance x, p
is a numerical prefactor and r is an empirical exponent. The advancetime curve is a function of various factors, such as initial watercontent,
soil texture, inﬂow rate, and surface slope. The eﬀects of such factors
are lumped into the parameters p and r in Eq. (1). However, since Eq.
(1) is based mostly on empiricism, it is not clear how p and r vary from
one ﬁeld or irrigation cycle to another.
Diﬀerent values of r have been reported experimentally. For example, Elliott and Walker (1982) estimated the parameters p and r bydirectly ﬁtting Eq. (1) to two measured data points of the advance-time⁎Corresponding author.
E-mail address: ghanbarian@ksu.edu (B. Ghanbarian).

https://doi.org/10.1016/j.agwat.2018.08.010
Received 8 February 2018; Received in revised form 31 July 2018; Accepted 6 August 20180378-3774/ © 2018 Elsevier B.V. All rights reserved.

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curve. Such a procedure, known as the two-point method, was ﬁrst used
for furrow irrigation and later applied to basin and border irrigation
systems. In basin irrigation, probably the most common form of surface
irrigation, a ﬁeld is level in all directions and encompassed by a ditch to
avoid runoﬀ. Basins are commonly square or rectangular in shape.
However, they may exist in all sorts of irregular conﬁgurations. Borderirrigation is a more general type compared to basin irrigation. Borders
are typically long rectangular in shape, with free draining conditions at
the lower end. In furrow irrigation, the ﬁeld surface is channeled along
its primary direction using furrows. The main diﬀerence between
furrow and border/basin irrigation is that for the former ﬂow into eachfurrow is independently set and controlled, while for the latter ﬂow iscontrolled on a border by border or basin by basin basis.
Shepard et al. (1993) ﬁxed r at 0.5 and optimized p by using typically the last measured point on the advance-time curve. Their method
is known as the one-point method. Valiantzas et al. (2001) later presented another one-point method. Although they did not ﬁx the value ofr, their method requires a trial and error procedure to determine r and pfrom the last point of the advance-time curve.
In addition to one- and two-point methods, the values of p and r canbe determined by directly ﬁtting Eq. (1) to advance-time data if measurements are available. For instance, Serralheiro (1995) estimated r fora Mediterranean soil (southern Portugal) using furrows of diﬀerent
slopes, and found 0.52 ≤ r ≤ 0.94 (see their Table 1). In another study,
Alvarez (2003) reported 0.58 ≤ r ≤ 0.72 for diﬀerent types of soils(such as Gleysols, Vertisols, Acrisols, and Ferrasols) for furrows oflengths between 240 and 380 m, with Manning’s resistance coeﬃcient n
ranging from 0.02 to 0.04. The Manning equation is an empirical formula for estimating the cross-sectional average velocity (v) of waterﬂowing along an open channel from the hydraulic radius Rh, the surface
slope S, and Manning’s resistance coeﬃcient n (i.e., v = R h2/3S1/2/n).
For the estimation of Manning’s resistance coeﬃcient for ﬂow in a ﬁeld,
see Harun-Ur-Rashid (1990) and Li and Zhang (2001).
In Table 1, we summarize the range of r value reported in diﬀerent

studies under various conditions. Experimental evidence indicatesclearly that r is a function of several factors, such as soil type, initialwater content, soil surface slope, and type of surface irrigation. Thus,
one should expect r to vary from one soil and/or an irrigation cycle toanother.
The main objective of this study was to invoke concepts from percolation theory, a theoretical approach from statistical physics forquantifying transport properties in complex systems, to shed light on
the value of r, and relate this parameter to the backbone fractal dimension Db that characterizes the backbone structure for percolation. In
what follows, we ﬁrst introduce percolation theory and present its
fundamental concepts. We then propose a power law from percolation
theory and show the link between the backbone fractal dimension Db
and the advance-time curve exponent r in Eq. (1). We next compare ourtheoretical results with experiments for furrow and border irrigation
systems.
2. Percolation theory
Percolation theory, introduced in its present form by Broadbent andHammersley (1957), provides a promising theoretical framework fromstatistical physics to study interconnectivity and its eﬀects on transport
properties in heterogeneous systems and networks, such as soils.
Broadbent and Hammersley (1957) studied plant disease spreading inan orchard whose trees were located on the intersections of a squarelattice. As expected, when the distance between aligned trees increases,
the probability of spreading a disease decreases. Eventually the distance
between trees would reach a critical value above which the diseasecannot spread over the orchard (Feder, 1988).
Percolation theory exists in three main forms: bond, site, and continuum percolation. In bond percolation, bonds are represented byrandomly distributed segments that are open to ﬂow and transport (seeFig. 1a). They act as microscopic conducting elements. In site percolation, sites are the line intersections and bonds are the segments

Table 1
Values of the exponent r in Eq. (1) reported in the literature for various conditions.
Reference

Irrigation system

No. of samples

r

Ave.
r

Remarks

Alvarez (2003)
Abbasi et al. (2003)
Khatri and Smith (2006)

Furrow
Furrow
Furrow
Furrow
Furrow
Furrow
Furrow
Furrow
Border
CFIa
FFI
AFI

12
3
27
17
4
3
6
5
6
7
7
7

0.58–0.72
0.62–0.91
0.73–0.97
0.62–0.85
0.57–0.81
0.69–0.91
0.62–0.91
0.52-0.77
0.48–0.68
0.65–0.86
0.65–0.70
0.63–0.75

0.66
0.76
0.86
0.72
0.67
0.77
0.76
0.63
0.58
0.76
0.68
0.69

From various references, see their Table 1
Maricopa Agricultural Center, Phoenix AZ
Cotton ﬁeld T, Southern Queensland
Cotton ﬁeld C, Southern Queensland
Four furrows of various lengths and soil textures
Seed & Plant Improvement Res. Inst., Karaj, Iran
Research Station for Tobacco, Urmia, Iran
From various references, see their Table 1
From various references, see their Table 2
Maize ﬁeld, Karaj, Iran
Maize ﬁeld, Karaj, Iran
Maize ﬁeld, Karaj, Iran

Bautista et al. (2009)
Abbasi et al. (2009a)
Abbasi et al. (2009b)
Ebrahimian et al. (2010)
Ebrahimian (2014)a

CFI: conventional furrow irrigation, FFI: ﬁxed alternate furrow irrigation, AFI: variable alternate furrow irrigation.

Fig. 1. Schematic (a) bond, (b) site, and (c)continuum percolation forms on a two-dimensional lattice. In bond percolation, segments
are randomly distributed within the lattice,
while in site percolation sites are the line intersections, randomly selected, and bonds are
the segments connecting sites. In continuum
percolation, circular voids are randomly placed
within the lattice.

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B. Ghanbarian et al.

(Feder, 1988). When p > pc, the incipient inﬁnite cluster still exists
besides the ﬁnite clusters (Kirkpatrick, 1973).
At or below the percolation threshold, the typical size of the largest
ﬁnite clusters is on the order of the ﬁnite correlation length. However,
the inﬁnite cluster above pc is diﬀerent from the one at pc (BenAvraham and Havlin, 2000). When the system size L is less than thecorrelation length (i.e., L < χ), the system is a heterogeneous, statistically self-similar fractal (Gefen et al., 1983; Sahimi, 1993). For L > χ
the system is macroscopically homogeneous, and the geometry is Euclidean (Feder, 1988; Sahimi, 1993).
Fig. 3a shows the same square lattice as in Fig. 2c, with a bondoccupation probability of 0.6. As can be seen, dead ends exist that do
not contribute to ﬂow. If such dead ends are removed from the cluster,what remains is called the backbone as shown in Fig. 3b. The backbone
is the multiply connected part of the sample-spanning cluster throughwhich ﬂuid ﬂow occurs. Multiply connected refers to the presence ofmultiple loops within the ﬂow path bringing ﬂuid from the same initialpoint to the same ﬁnal point. The complex structure of the backbone ischaracterized by a fractal dimension, called the backbone fractal dimension, whose value is theoretically only a function of system dimensionality and percolation class, as is discussed in what follows.
Lee et al. (1999) studied traveling times for tracer particles andmodeling ﬂow driven by a pressure diﬀerence between two points separated by Euclidean distance x. They demonstrated that the mostprobable traveling time, t, of particles on a percolation cluster is proportional to x Db . Therefore,

connecting sites (see Fig. 1b). Two nearest-neighbor sites are called
connected if they are both occupied. In continuum percolation, circularvoids are randomly placed within a solid system (see Fig. 1c). Suchcircular pores are connected through their partial overlap, and medium
properties depend on their abundance.
After the initial study by Broadbent and Hammersley (1957), various types of percolation theory have been used to model diﬀerent
phenomena. These include (1) random or ordinary (Stauﬀer, 1985), (2)invasion (Wilkinson and Willemsen, 1983), (3) directed (Obukhov,1980), and (4) gradient (Rosso et al., 1986) approaches. In random (orordinary) percolation, sites or bonds are randomly occupied, while ininvasion percolation they are invaded based on a least-resistance-path
algorithm. In directed percolation, bonds or sites are also randomly
occupied. However, transport through occupied bonds can only occur
along a preferred direction. In gradient percolation, bond or site occupation varies along some ﬁxed direction and depends linearly onbond or site locations that produce a gradient pattern. In this study wefocus on random and invasion percolation types. For simplicity, we
introduce here only the random bond percolation model by providing
an example, and refer interested readers to Stauﬀer (1985), Feder(1988), Sahimi (1994, 2011) and Hunt et al. (2014b) for further details
and other percolation types and forms.
Let us consider a simple regular square lattice as shown in Fig. 2a, inwhich only 10% of the bonds (the thick black segments) are occupied
randomly and independently of the occupation status of their neighbors. The probability p of a bond being occupied by a black segment is
0.1 in Fig. 2a. A set of connected bonds surrounded by unoccupied
bonds is called a cluster. In Fig. 2a, several small clusters consisting of
one or two bonds exist. As the bond occupation probability p (i.e., thefraction of bonds that are occupied) increases, more individual clusters
become connected to each other and form a larger interconnectedcluster (see Fig. 2b). The critical probability pc for bond percolation on
the square network is 0.5 (Stauﬀer, 1985). In Fig. 2c, in which p = 0.6(i.e., above the critical probability pc), a large cluster connects the rightside of the square network to its left side, and the top to the bottom.
This means that percolation across the network has occurred.
A fundamental concept in percolation theory is the existence of apercolation threshold pc, below which a system (or network) loses its
connectivity (see Fig. 2a, b). For p < pc, all clusters are ﬁnite in size,with the largest clusters having a typical size of the order of the ﬁnite
correlation length (χ), the mean distance between any two bonds on thesame ﬁnite cluster, while no spanning cluster exists (Kirkpatrick, 1973;
Ben-Avraham and Havlin, 2000). At pc, an incipient inﬁnite cluster
occurs along with other ﬁnite clusters (Kirkpatrick, 1973; Ben-Avrahamand Havlin, 2000). In this case, the mass of the spanning cluster increases with the size, L, of the lattice as a power-law L Df in which Df isthe mass fractal dimension of the fractal cluster whose universal valueis 91/48 (or 1.896) and 2.53 in two and three dimensions, respectively1x ∝ t Db

(2)

Fig. 3. (a) Bond percolation on a square lattice with occupation probability
p = 0.6 and (b) its backbone. The value of the backbone fractal dimension Db
for random and bond trapping invasion percolation is, respectively, 1.643 and1.217 (Sheppard et al., 1999). The values of Db for diﬀerent percolation classes
are listed in Table 2.

Fig. 2. Bond percolation on a square lattice with occupation probability p = 0.1 (a), 0.3 (b) and 0.6 (c). The critical probability pc on the square lattice is 0.5 for bondpercolation in two dimensions.
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1

Table 2
Theoretical Db and corresponding r values for various percolation classes in two
and three dimensions (Sheppard et al., 1999).
Dimension

Percolation classDbr ( = 1/Db)

2
2
2
2
2
3
3
3
3
3

RPa
Optimal path
Site TIP
Bond TIP
Shortest path
RP
Optimal path
Site TIP
Bond TIP
Shortest path

1.643
1.210
1.217
1.217
1.130b
1.870
1.420
1.861
1.458
1.376b

0.609
0.826
0.822
0.822
0.885
0.535
0.704
0.537
0.686
0.727

a
b

1
t ⎞ Dbxx = x max ⎛= 1/max
t Db
Db
t
tmax
⎝ max ⎠⎜⎟

(3)

where xmax is the furrow or border length, and tmax is the advance timeto the end of the furrow or border, corresponding to the last measured
data point on the advance-time curve. Comparing Eq. (3) with Eq. (1)clearly indicates that the empirical parameters p and r can be interpreted theoretically within the percolation theory framework and havephysical meaning, as will be discussed next.
3. Materials and methods
In this section, we ﬁrst present experimental advance-time curves
and then compare them with estimates using Eqs. (1) and (3).

RP is random percolation and TIP is trapping invasion percolation.
1.130 and 1.376 represent shortest path (or minimum) fractal dimensions.

3.1. Experimental data

where Db is the fractal dimension of the backbone. Eq. (2) and its
subsequent derivations have been evaluated successfully in their ability
to describe the weathering rate of porous media (Hunt et al., 2014a;
Hunt, 2015; Yu and Hunt, 2017), geochemical reaction rates (Huntet al., 2015), soil depth and soil production (Hunt, 2016a; Hunt and
Ghanbarian, 2016; Yu et al., 2017), and vegetation growth and soil
formation (Hunt and Manzoni, 2015; Hunt, 2016b, 2017; Hunt et al.,2017).
Comparing Eq. (2) with Eq. (1) indicates that the empirical exponent r can be related to the physically meaningful backbone fractal
dimension (i.e., r = 1/Db). Diﬀerent values of Db for various percolation
classes (e.g., for random and invasion percolation) and system dimensions are summarized in Table 2. As can be seen, within the percolation
theoryframework
1.217 ≤ Db ≤ 1.87
and,correspondingly,
0.535 ≤ r ≤ 0.826, which agrees with the range of r values reported by
Alvarez (2003), among others (see Table 1). If one assumes that thetraveling time follows the shortest path concepts from percolation
theory, then the exponent in Eq. (2) should change to Dmin, the shortest
path (or minimum) fractal dimension, whose value is 1.13 or 1.374 in
two or three dimensions, respectively (Porto et al., 1997). The value of
1.13 for Db = Dmin provides a higher upper bound and thus a wider
range of r (i.e., 0.535 ≤ r ≤ 0.885). As can be seen from Table 2, thevalues of Db for optimal path, site trapping invasion percolation (siteTIP) and bond trapping invasion percolation (bond TIP) are not greatly
diﬀerent in two dimensions. All of the three percolation classes correspond to a value of about 0.82 for r. Similarly, bond TIP, shortest pathand optimal path correspond to an r value of about 0.7 in three dimensions. In addition, the values of Db for random percolation (RP) andsite TIP in three dimensions in three dimensions (1.87 and 1.861, respectively), corresponding to a value of r of about 0.53.
For furrow and border irrigation systems, one may normalize Eq. (2)to obtain

Experimental data used in this study are from Kamali (2015) who
investigated surface ﬂow in four open-ended furrows of the same lengthof 110 m, and having a longitudinal slope of 0.012 m m−1 with rows
75 cm apart. The study site of rectangular shape was located at the
University of Tehran, Karaj, Iran. Maize (single cross 704) was planted
in-furrow for furrows 1 and 2 and on-ridge for furrows 3 and 4, andirrigated once a week over a period of seven weeks. Soil texture at threedepths was determined using a combination of hydrometer and sieving
methods at the beginning, middle and end of the ﬁeld site. Gravimetricwater contents at ﬁeld capacity (−33 kPa) and permanent wilting(−1500 kPa) points were measured using the pressure plate method.
Salient soil properties are presented in Table 3.
Initial gravimetric water contents were measured before irrigation
cycles 1 and 5. The average value for all furrows was about 8%, close to
the gravimetric water content at the permanent wilting point (seeTable 3). For each irrigation cycle, the advance-time was measured ateleven equally-spaced stations (i.e., at 10 m intervals) along eachfurrow. The inﬂow rate, measured with a Washington State ﬂume
(WSC) type II, was about 0.29 l/s in furrows 1 and 3, and about 0.44 l/s
in furrows 2 and 4. More detailed information are presented in Table 4.
3.2. Estimation of the advance-time curve
To estimate the advance-time curve for surface ﬂows using Eq. (3),one requires three parameters: Db, xmax and tmax. As noted earlier, tmax
is the advance time at the end of the furrow (xmax), corresponding to thelast measured data point on the advance-time curve, which can bemeasured and hence is assumed to be available. Recall thatxmax = 110 m for each furrow. All four furrows had approximately thesame longitudinal slope. The inﬂow rate in furrows 2 and 4 (i.e.,
0.44 l s−1) was 50% larger than that in furrows 1 and 3 (i.e., 0.29 l s−1).
We therefore postulate that the wetting front advance should be quasi
three-dimensional in furrows 1 and 3, while quasi two-dimensional in

Table 3
Salient soil properties of the ﬁeld site studied by Kamali (2015).
Location

Depth (m)TextureClay (%)

Silt (%)

Sand (%)

BDa(g cm−3)

FC (%)PWP (%)

EC(dS m−1)

Beginning

0–0.2
0.2–0.4
0.4–0.6
0–0.2
0.2–0.4
0.4–0.6
0–0.2
0.2–0.4
0.4–0.6

Clay loam
Clay loam
Sandy loam
Loam
Sandy clay loam
Sandy clay loam
Clay loam
Loam
Sandy loam

28.5
28.5
16.0
26.0
23.5
21.0
31.0
26.8
20.2

35.0
33.8
17.5
30.0
25.0
22.5
31.7
30.4
24.6

36.5
37.8
66.5
44.0
51.5
56.5
37.3
42.8
55.3

1.50
1.45
1.47
1.50
1.45
1.52
1.51
1.48
1.49

18.2
17.5
14.2
18.1
17.2
15.5
18.1
17.7
15.0

8.7
8.1
6.0
8.5
8.0
6.9
8.4
8.1
6.6

2.55
1.77
1.88
2.10
2.08
2.00
2.98
2.05
2.47

MiddleEnda
BD is bulk density, FC is gravimetric water content at ﬁeld capacity (−33 kPa), PWP is gravimetric water content at permanent wilting point (−1500 kPa), EC iselectrical conductivity of saturated soil sample.

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B. Ghanbarian et al.

Table 4
Properties of the investigated irrigation scheme with seven cycles irrigating four furrows.
Furrow

Maize planted

1

in furrow2

in furrow3

on ridge4

on ridgeaAdvance time at the end of furrow.

Parameter

Irrigation cycle−1

inﬂow rate (l s )run-oﬀ volume (l)inﬁltration volume (l)cut-oﬀ time (min)tmaxa (min)inﬂow rate (l s−1)
run-oﬀ volume (l)inﬁltration volume (l)cut-oﬀ time (min)tmax (min)inﬂow rate (l s−1)
run-oﬀ volume (l)inﬁltration volume (l)cut-oﬀ time (min)tmax (min)inﬂow rate (l s−1)
run-oﬀ volume (l)inﬁltration volume (l)cut-oﬀ time (min)tmax (min)

1

2

3

4

5

6

7

0.32
1696
2670
229
51
0.44
2634
3252
229
45.5
0.34
1720
2440
204
45
0.44
2405
2875
204
37

0.31
1837
2450
245
59.5
0.44
3104
2577
220
44
0.31
1934
2833
265
95
0.44
3043
2641
265
42.6

0.29
1202
2688
230
73
0.44
2721
2521
205
44.6
0.29
1316
2584
230
76
0.44
2184
3115
230
53.5

0.28
1067
2179
200
61.5
0.46
1907
2084
150
39.5
0.28
1369
2296
225
77.5
0.46
2136
2570
225
50

0.28
1093
2548
225
67.5
0.44
2255
2413
180
39.5
0.29
1318
2079
200
60.6
0.44
2381
2800
200
49.9

0.29
1433
1980
220
71
0.44
2652
2561
200
40
0.29
1232
1907
180
58.7
0.44
2113
2567
180
41

0.29
878
1921
165
63
0.44
1719
2033
145
41
0.29
953
1837
165
61.5
0.44
1601
2122
165
43.5

Fig. 4. Measured advance-time curves, as well as the estimated curves using Eq. (1) with r = 0.5 and p = x max / tmax (Shepard et al., 1993), represented by blue
curves, and Eq. (3) with Db = 1.42 (optimal path in three dimensions; see Table 2), denoted by red curves, using the advance time (tmax) values measured at the endof furrow 1 for irrigation cycles 1–7 (a–g). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)furrows 2 and 4. Of course, water ﬂow behind the front is inherently
three-dimensional since water traverses on the surface and simultaneously inﬁltrates into the soil. In general, the higher the inﬂow rate,

the faster the wetting front should move, and thus the shorter the advance time. This can be conﬁrmed by comparing the advance time
(tmax) values presented in Table 4 for all four furrows.
212Agricultural Water Management 210 (2018) 208–216

B. Ghanbarian et al.

Fig. 5. Measured advance-time curves, as well as the estimated curves using Eq. (1) with r = 0.5 and p = x max / tmax (Shepard et al., 1993), represented by blue
curves, and Eq. (3) with Db = 1.21 (optimal path in two dimensions; see Table 2), denoted by red curves, using the advance time (tmax) values measured at the end of
furrow 2 for irrigation cycles 1–7 (a–g). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Surface ﬂow over the soil-air interface involves two main processes:(1) run-on, run-oﬀ and water movement on the surface, and (2) inﬁltration into the soil. Since water streamlines in the run-oﬀ process
have less resistance compared to water ﬂow paths within the soil, we
postulate that the optimal path class of percolation theory should accurately describe the advance-time curve. Note that the optimal path is
the most energetically favorable path through a system. Since thewetting front advance in furrows 1 and 3 is probably quasi three-dimensional, and quasi two-dimensional in furrows 2 and 4, we accordingly use Db = 1.42 (the optimal path fractal dimension in three dimensions; see Table 2) for furrows 1 and 3 and Db = 1.21 (the optimal
path fractal dimension in two dimensions; see Table 2) for furrows 2
and 4. However, some other percolation classes may well be relevant
for diﬀerent types of experiments depending upon such parameters as
initial water content, inﬁltration rate, inﬂow rate and slope. We will
discuss percolation classes and relevant Db values in the Results andDiscussion section.
We further compared our results for the advance-time curve foreach furrow and irrigation cycle with estimates obtained using the onepoint method of Shepard et al. (1993) who recommended ﬁxing r in Eq.
(1) at 0.5. Using r = 0.5 and the last measured data point, one has
p = x max / tmax . Although this approach has been criticized since thevalue of r commonly diﬀers from 0.5, some studies showed that the
Shepard et al. (1993) method still can provide reasonable appximations
for furrow and border irrigation systems (e.g., Ebrahimian et al., 2010).

4. Results and discussion
The measured advance-time curves of 28 experiments are shown in
Figs. 4–7 for four furrows and seven irrigation cycles. Using xmax andtmax from the experiments, we estimated the advance-time curves using
Eq. (1) with r = 0.5, and Eq. (3) with Db = 1.21 and 1.42 for two- andthree-dimensional ﬂow patterns, respectively. As can be observed from
Figs. 4–7, Eq. (3) with Db = 1.21 and 1.42 (shown in red) could accurately estimate the advance-time curves for all experiments, while Eq.
(1) with r = 0.5 and p = x max / tmax (shown in blue) mostly underestimates the curves.
We found that the optimal path class of percolation theory with
Db = 1.21 (for two dimensions) and 1.42 (for three dimensions), corresponding respectively to r = 0.826 and 0.704 in Eq. (1) (see Table 2),
accurately estimated the advance-time curves for 28 experiments carried out by Kamali (2015). Still, one should note that other percolation
classes might provide more accurate estimates for other experiments.
The value of Db in Eq. (3), in fact, indirectly depends on several factors,
such as slope, inﬂow rate, initial water content, and inﬁltration rate.
The steeper the slope, the greater the inﬂuence of gravity and run-oﬀ,
and thus the shorter the advance time. This means that for furrows withmild slopes a higher portion of water at the wetting front would inﬁltrate into the soil while advancing through the furrow, as comparedto furrows having steep slopes (e.g., Esfandiari and Maheshwari, 1997).
This in turn implies that water ﬂow at the wetting front should be quasi
three- and two-dimensional in mild and steep furrows, respectively.
The inﬂow rate has an eﬀect on the advance-time curve similar tothe slope. The higher the inﬂow rate, the less water will inﬁltrate at the213Agricultural Water Management 210 (2018) 208–216

B. Ghanbarian et al.

Fig. 6. Measured advance-time curves, as well as the estimated curves using Eq. (1) with r = 0.5 and p = x max / tmax (Shepard et al., 1993), represented by blue
curves, and Eq. (3) with Db = 1.42 (optimal path in three dimensions; see Table 2), denoted by red curves, using the advance time (tmax) values measured at the endof furrow 3 for irrigation cycles 1–7 (a–g). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)time curves for furrows in which maize was planted, did hold for furrows in which maize was planted on ridges.
A comparison of the range (0.48 ≤ r ≤ 0.68) and average value(0.58) of r for border irrigation with those for furrow irrigation in
Table 1 indicates that one should expect a smaller value of r for borders
than that for furrows. We accordingly postulate that the wetting front
advance should follow a quasi three-dimensional pattern in borders,
while being more quasi two-dimensional in furrows. This means thatthe two-dimensional values of Db, shown in Table 2, should be relevant
for furrows, and the three-dimensional values for borders.
We also postulate that for closely-spaced furrows with steep slopes
and relatively high inﬂow rates, the wetting front advance will bemostly quasi one-dimensional. Since the backbone fractal dimension in
one dimension is 1, one should hence expect the exponent r in Eq. (1) tobe close to 1 (r = 1/Db). This would then be in agreement with experimental observations by Khatri and Smith (2006) who reported r
values as large as 0.97 (see Table 1).
Although we showed that the optimal path percolation class could
estimate precisely the advance-time curves, one should not expect
percolation theory, as developed mostly to study subsurface ﬂow, toaccurately describe surface ﬂow for all types of experiments. In fact,
additional and more comprehensive sets of experiments are required toinvestigate the eﬀects of initial water content, inﬂow rate, Manning’s
roughness coeﬃcient, inﬁltration rate, soil texture and surface slope on
the exponent r, and to ﬁnd out which percolation class and Db value
would provide more accurate estimates under various ﬁeld conditions.

front, thus making the wetting front advance quasi two-dimensional
along the furrow, as we showed for furrows 2 and 4. This is the main
reason why Db = 1.21 (for two dimensions) could accurately describe
the advance-time curves for furrows 2 and 4 with inﬂow rates of about
0.44 l s−1 while Db = 1.42 (for three dimensions) did hold in furrows 1
and 3 having inﬂow rates of near 0.29 l s−1.
The initial water content of soil prior to an irrigation event also
aﬀects the wetting front advance since it controls inﬁltration and hence
also run-oﬀ. The higher the initial water content, the lower the inﬁltration rate, and thus the faster the water will move along the furrow(Rasoulzadeh and Sepaskhah, 2003). Less inﬁltration into the soil at thewetting front in turn will cause water to traverse more quasi two-dimensional.
The rate of inﬁltration from a furrow into the soil is very muchcontrolled by the hydraulic conductivity of the soil, and hence texture.
Fine-textured soils with their lower hydraulic conductivities hence willshow lower inﬁltration rates. For this reason, one should expect thewetting front advance to be more three-dimensional in coarse-textured
and more two-dimensional in ﬁne-textured soils. However, if cracks
exist along the furrow surface, particularly in ﬁne-textured soils, preferential ﬂow paths will be created (e.g., Novák et al., 2000) that will
likely cause the advancing wetting front in the furrow to become quasi
three-dimensional. This still has to be demonstrated through either
experiments or numerical simulations, including how such on-site ﬁeld
conditions may change the values of r and Db in the advance-time
curves. Interestingly, planting maize in furrows or on ridges did not
substantially aﬀect the exponent r in Eq. (1) or Db in Eq. (3). As weshowed in Figs. 4–7, the same Db values used to estimate the advance214

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B. Ghanbarian et al.

Fig. 7. Measured advance-time curves, as well as the estimated curves using Eq. (1) with r = 0.5 and p = x max / tmax (Shepard et al., 1993), represented by blue
curves, and Eq. (3) with Db = 1.21 (optimal path in two dimensions; see Table 2), denoted by red curves, using the advance time (tmax) values measured at the end of
furrow 4 for irrigation cycles 1–7 (a–g). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

5. Conclusions

References

In this study, we proposed theoretical bounds for the exponent r in
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framework (r = 1/Db). A comparison with r values reported in the literature showed that the theoretical range of the inverse of Db (i.e.,
0.535 ≤ 1/Db ≤ 0.885) is well in agreement with the experimental
range found for furrow and border irrigation systems. We also showedthat the optimal path class from percolation theory could accurately
estimate the advance-time curves for 28 furrow experiments, while theone-point method of Shepard et al. (1993) mostly underestimated thecurves. We found that large and small values of r should be relevant,respectively, to quasi two- and three-dimensional wetting front advance. We also addressed the indirect eﬀects of various factors, such as
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high inﬂow rates, the wetting front advance will be mostly quasi onedimensional. Accordingly, one should expect the exponent r to be close
to 1, since the backbone fractal dimension in one dimension is 1.

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