J. Hydrol. Hydromech., 69, 2021, 2, X–X
DOI: 10.2478/johh-2021-0004
ISSN 1338-4333©2021. This is an open access article distributed
under the Creative Commons Attribution
NonCommercial-NoDerivatives 4.0 License

Effect of soil textural characteristics on longitudinal dispersion in saturated
porous media
Mojtaba G. Mahmoodlu1, 2*, Amir Raoof 2, Martinus Th. van Genuchten2, 3
1Faculty of Agriculture and Natural Resources, Gonbad Kavous University, Gonbad Kavous, Iran.
Department of Earth Sciences, Utrecht University, Utrecht, Netherlands.
3
Center for Environmental Studies, CEA, São Paulo State University Rio Claro, Brazil.
*
Corresponding author. E-mails: m.g.mahmoodlu@gmail.com / mmahmoodlu@gonbad.ac.ir
2

Abstract: This study focuses on the effects of soil textural heterogeneity on longitudinal dispersion under saturation
conditions. A series of solute transport experiments were carried out using saturated soil columns packed with two filter
sands and two mixtures of these sands, having d50 values of 95, 324, 402, and 480 µm, subjected to four different steadyflow rates. Values of the dispersion coefficient (D) were estimated from observed in-situ distributions of calcium chloride, injected as a short nonreactive tracer pulse, at four different locations (11, 18, 25, 36 cm). Analyses of the observed
distributions in terms of the standard advection-dispersion equation (ADE) showed that D increased nonlinearly withtravel distance and higher Peclet numbers+. The dispersion coefficient of sand sample S1 with its largest average particle
size (d50) was more affected by the average pore-water velocity than sample S4 having the smallest d50. Results revealed
that for a constant velocity, D values of sample S1 were much higher than those of sample S4, which had the smallest
d50. A correlation matrix of parameters controlling the dispersion coefficient showed a relatively strong positive relationship between D and the Peclet number. In contrast, almost no correlation was evident between D and porosity as well as
grain size. The results obtained with the four sandy matrices were consistent and proved that the dispersion coefficientdepends mainly on the particle size.
Keywords: Solute dispersion; Soil textural heterogeneity; Saturated columns; Peclet number; Flow velocity; Travel
distance.
INTRODUCTION
Solute dispersion is a key part of solute transport by affecting
such processes as subsurface pollutant transport, leaching ofpesticides and nutrients from the soil root zone, remediation ofcontaminated sites, ion exchange chromatography, and manyother applications in the soil, agricultural, hydrogeologic, petroleum, and chemical engineering disciplines (e.g., Bear, 1988;
Delgado, 2006; Fetter, 2008; Godoy et al., 2019; Lewis andSjostrom, 2010; Nimbi and Powers, 2003). The problem ofsolute dispersion has attracted much interest from the early
1900s onward. However, it gathered much more attention andbecame a broader topic only after the 1950s, gathering manystudies on with hydrodynamic dispersion and miscible displacement. Many or most previous studies investigated dispersion in
fully saturated porous media (e.g., Bear, 1988; Cui et al., 2019;
Goldobin and Maryshev, 2017; Saffman, 1959; Scheidegger,1974). Their findings suggest that the dispersion coefficient (D)increases more or less linearly with the fluid phase velocity andthe distance travelled, while also being affected by the size andshape of particles making up the porous medium, media heterogeneity and anisotropy, and pore system tortuosity, as confirmed further by many recent studies (e.g., Bandai et al., 2017;
Bromly et al., 2007; Delgado, 2006; Hunt et al., 2011; Poulsen
et al., 2008; Pugliese et al., 2013; Sharma and Poulsen, 2010).
Despite the above and many other studies, consistent andcomprehensive data sets to establish links between solute dispersion and soil textural characteristics remain limited. The
main objective of this study was to investigate the role of soil
textural properties on solute dispersion under saturated conditions. For this purpose, we compared tracer breakthroughthrough four sandy matrices: two sandy matrices (coarse andfine) exhibiting narrow pore size distributions, and two mixtures of these sandy mixtures exhibiting intermediate particle
sizes and bimodal distributions. For these matrices, we determined solute breakthrough curves (BTCs) by deriving total
resident concentrations from the bulk soil electrical conductivity measured with four-electrode salinity probes at several locations along columns packed submitted to saturated flow. Valuesof the parameters in the governing advection-dispersion equation (ADE) were estimated by fitting analytical solutions of thetransport equation to the observed BTCs. Solute mixing in thesaturated sands will be discussed in terms of the value of D as afunction of the pore-water velocity, the transport distance, theparticle Peclet number, and soil texture (particle size distribution, i.e., grain size mean heterogeneity).
MATERIALS AND METHODS
Laboratory experiments
Sand properties
For this study we used two different filter sands (a relatively
coarse sand sample S1 and a relatively fine sand sample S4)
together with two mixtures (samples S2 and S3) of sands S1and S4. Sample S2 was a mixture (in weight) of 80% of S1 and
20% of S4, and sample S3 was a mixture of 60% S1 and 40%
S4. X-ray micro-computed tomography images of the four
samples are shown in Fig.1a. The images of sand mixtures S2and S3 exhibited a relatively significant degree of heterogeneity
in terms of particle.
The sands originated from a river bed in Papendrecht, Netherlands (Filtersand, Filcom, Netherlands) and were sieved todifferent retained sizes as given in Table 1. The average particle
size (d50) varied from 90 µm to 480 μm (from sands S4 to S1).

1

Mojtaba G. Mahmoodlu, Amir Raoof, Martinus Th. van Genuchten

and at the outlet, at 36 cm, were used to obtain solute concentrations utilizing the measured electrical conductivity (EC). A
picture of the experimental laboratory setup is shown in Fig. 3.
Sand columns S1, S2, S3, and S4 were packed to porosity
values of approximately 0.38, 0.34, 0.32, and 0.40, respectively.
After packing, the sand columns were flushed with CO2 for a
few hours to replace air from the soil pore spaces, thus ensuring
as saturated as possible conditions once filled with water. Next,
the columns were gradually saturated using deionized waterfrom the bottom using a peristaltic pump (Ismatec, Switzerland). Although peristaltic pumps deliver pulsated flow, therelatively large diameter of the columns together with theirFig. 1. MicroCT scans of the sand samples (a), and their particle
size distributions (b). Samples S1 (black dotted line) and S4 (green
dashed line) represented relatively coarse and fine sands, respectively. Sample S2 (blue dashed line) is a mixture of 80% of S1 and
20% of S4, and sand sample S4 (red dash-dotted line) is a mixtureof 60% S1 and 40% S4.

X-ray fluorescence analyses showed that more than 96% of thesand grains consisted of silica (SiO2).
Coefficients of uniformity (Cu = d60/d10), curvature (Cc =d302/d10×d60), and sorting (So = (d75/d25)1/2) were estimated forall sands (Table 1). Since Cu was less than 4.0 for sands S1 andS4, these two samples can be considered to be uniformly graded by containing essentially identical particle sizes. Sand S1
was the most uniform. We further included in Table 1 the coefficient of curvature (Cc) as another useful measure of the shapeof the particle size distribution curve. This coefficient is sometimes used to differentiate the sorting degree of a soil. Sands S2and S3 clearly showed lesser degrees of sorting as reflected bytheir bimodal grain size distributions (Fig. 1). In our study, weused sand S1 as the main sand in view of its low Cu, So, and d50
values. Mixing different fractions of main sand S4 with sand S1
created far more heterogeneous media (Fig. 1). We expect two
mixtures with intermediate particle size and bimodal pore size
distributions to exhibit different transfer properties.

Fig. 2. Schematic of the experimental setup and sand columns usedin this study.

Experimental columns
Fig. 2 shows a schematic of the experimental setup used inour study. Plexiglass cylinders of 36 cm length and 9.0 cm
internal diameter were used to construct the columns, whichcontained stainless steel lids with a port at its inlet and outlet
faces. Three measuring probes at depths of 11, 18 and 25 cm

Fig. 3. Picture of the experimental laboratory setup.

Table 1. Selected properties of the natural sands used in this study1.
Sample
S1
7
S2
8
S3
S4
1

d10
(µm)
400
60
60
60

d25
(µm)
440
380
70
70

d30
(µm)
445
390
90
80

d50
(µm)
480
402
324
95

d60
(µm)
500
460
440
100

d75
(µm)
525
480
500
115

2

Cu

1.25
7.67
7.33
1.67

3

Cc

0.99
5.51
0.31
1.07

4

So

1.09
1.12
2.67
1.28

5

ϕ

pH

0.38
0.34
0.32
0.40

7
7
7
7

6
TOC
(g m–3)< 5.0 × 10–20< 5.0 × 10–20< 5.0 × 10–20< 5.0 × 10–20

Technical data sheet of filter sands (Filcom, Papendrecht, Netherlands): 2Cu: Coefficient of uniformity (Cu = d60/d10); 3Cc: Coefficient ofcurvature (Cc = d302/d10×d60); 4So: Sorting coefficient (So = (d75/d25)1/2); 5 ϕ: Porosity; 6TOC: Total organic carbon of dry sample; 7S2 = S1(0.8) andS4(0.2); 8S3 = S1(0.6) and S4(0.4).

2Effect of soil textural characteristics on longitudinal dispersion in saturated porous media

relatively long lengths (36 cm) attenuated the pulsed input andprovided a relative continuous form of flow. We subsequently
established constant flow rates, ranging from 0.9 to 5.3 cm3
min–1. For the outlet, we implemented a fixed hydraulic head at
the top of the sand columns using a small water container. Afterreaching fully saturated steady-state flow conditions, we applied a short CaCl2 pulse input having a concentration of 0.05
M (Sigma-Aldrich).
Dynamic measurements of soil moisture, temperature, andEC were carried out over time using several 5TE sensors (Decagon). Three 5TE sensors were inserted horizontally in thecolumns (Bear, 1988; Bunsri et al., 2008a; Dong et al., 2014;
Inoue et al., 2000) via measuring holes. Each probe consisted ofthree 5.2-cm long, 0.5-cm wide, and 0.17-cm thick prongs. The
5TE sensors provided simultaneous measurements of the electrical conductivity of the bulk medium (ECb) surrounding thesensors, as well as of the bulk medium dielectric permittivity(εb) for the same volume. Hilhorst (2000) and Scudiero et al.,(2012), among others, showed that a linear relationship exists
between ECb and the electrical conductivity of pore water(ECp):

EC p =εpECb
ε b − εσ b = 0(1)

where εp denotes the real portion of the dielectric permittivity ofsoil pore water, which is a function of soil temperature, definedas εp = 80.3–0.37(Tsoil–20), and where εσb=0 is the real portion of
the dielectric permittivity of the soil when the bulk electricalconductivity is zero.
BTC data analysis
As noted earlier, the hydrodynamic dispersion coefficient(D) depends on a range of microscopic and macroscopic factors
such as the fluid flow velocity, the tortuosity of the solute travel
path in the medium, soil texture, possible anisotropic conditions, and ionic or molecular diffusion processes. At the macroscopic scale, the classical model for describing non-adsorbing
solute transport is the advection-dispersion equation (ADE),
considering no water fractionation in the columns (van Genuchten and Wierenga, 1976). The ADE equation reads as follows
for a uniformly packed one-dimensional column subjected toconstant flow (Bear, 1988):

∂C
∂ 2C
∂C
= D 2 −v
∂t
∂x
∂x

(2)

where C is the average solute concentration, x is the longitudinal distance, t is time, D is the hydrodynamic dispersion
coefficient, and v the average pore-water velocity (the Darcian
flow velocity divided by porosity).
Assuming the application of a short solute pulse of duration
to at the inlet (at x = 0) to an initially solute-free semi-infinite
profile, the analytical solution for the tracer distribution within
the column is given by (Lindstrom et al., 1967):
0 < t ≤ to
 Co A( x, t )C ( x, t ) = Co [ A( x, t ) − A( x, t − to ) ] t > to(3)where C0 is the input concentration and A(x, t) is a term defined
as follow:

A( x, t ) = ( x − vt )2 
1x − vt v 2terfc 
exp  −+
πD
2
4 Dt 
 4 Dt 

1  vx v 2t  vx 
 x + vt − 1 + + exp   erfc 
2
D D 
D
 4 Dt (4)

As shown by several authors (e.g., van Genuchten and Parker, 1984), Eq. (4) is an appropriate solution for the volumeaveraged (resident) concentration within semi-finite systems.
The CXTFIT computer program (Toride et al., 1999) within
the STANMOD software (Šimunek et al., 2000; van Genuchten
et al., 2012) was used to estimate the parameters v and D in Eq.
(4) from the observed concentration BTCs obtained for eachtransport experiment at each location in the columns. CXTFIT
uses a Marquardt-Levenberg type nonlinear least squares fitting
procedure to optimize the fit between the observed data and theadvection-diffusion equation.
For saturated porous media, hydrodynamic dispersion is
known to be a function of diffusion and mechanical dispersion,
the latter resulting from local velocity variations, leading to thefollowing general relationship for one-dimensional transport(Bear, 1988; Fetter, 2008; Freeze and Cherry, 1979):
D = De + λ v n(5)where De is the effective diffusion coefficient, λ is the (longitudinal) dispersivity, and n is a mostly empirical parameter. As
shown by Eq. (5), the hydrodynamic dispersion includes bothionic or molecular diffusion (De) and mechanical dispersion due
to solute advection processes (λvn). The effects of diffusion onhydrodynamic dispersion can be evaluated by using a particle
Peclet number (Pe) given by:

Pe =

vd
De

(6)

where d is the mean size of soil particles (Kutílek and Nielsen,
1994; Toride et al., 2003). At relatively low values of the Peclet
number (Pe < 3), solute dispersion is dominated mostly by molecular diffusion processes, while at high values (e.g., Pe > 10), thecontribution of molecular diffusion becomes negligible (Delgado,
2007; Toride et al., 2003). At high Peclet numbers, the exponent
n of the mechanical dispersion component is generally very close
to unity (Bear, 1988; Bolt, 1979; Scheidegger, 1974; among
others). This then reflects a linear increase in hydrodynamic
dispersion with the pore-water velocity, at least for ideal media
such as glass beads, packed beds, and non-aggregated soils.
RESULTS AND DISCUSSION
Breakthrough curves (BTCs)
The measured BTCs at three different depths (i.e., at 11, 18,
25 cm) and of the effluent at 36 cm are shown in Fig. 4 in termsof normalized concentrations (C/Co) of CaCl2. The BTCs show
a clear dominant Gaussian shape at all depths, characteristic ofequilibrium transport with Peclet numbers larger than 1.0. The
symmetric shape of the BTCs agrees with many studies using
tracer transport in homogenously packed soil columns undersaturated conditions (Lee et al., 2014; Toride et al., 2003).
Using CXTFIT, we fitted the ADE analytical solution to theobserved BTCs to determine values of the pore-water velocity,v, and the solute dispersion coefficient, D, from which thesolute dispersivity, λ, was calculated. Here we assume that thecontribution of molecular diffusion is negligible and that theexponent n in Eq. (5) is equal to one.

3

Mojtaba G. Mahmoodlu, Amir Raoof, Martinus Th. van Genuchten
Table 2. Measured and calculated values of the pore-water velocity (v) together with their differences (Δv), the dispersion coefficient (D)fitted to the observed BTCs, and dispersivity values (λ) for various imposed flow rates (q) and travel distances (L).
Sampleq(cm3 min–1)

2.43

S1

3.90

5.30

1.55

S2

3.07

4.87

0.90

S3

2.80

5.00

2.43

S4

3.90

5.20

L
(cm)
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5
11
18
26
36.5

Measured v(cm min–1)
0.091
0.091
0.091
0.091
0.145
0.145
0.145
0.145
0.197
0.197
0.197
0.197
0.064
0.064
0.064
0.064
0.129
0.129
0.129
0.129
0.204
0.204
0.204
0.204
0.040
0.040
0.040
0.040
0.125
0.125
0.125
0.125
0.223
0.223
0.223
0.223
0.091
0.091
0.091
0.091
0.123
0.123
0.123
0.123
0.129
0.129
0.129
0.129

Fig. 4 shows excellent agreement between the simulated
macroscale (continuum scale) BTC’s using Eq. (4), and theexperimental data. Results of the optimized parameters areprovided in Table 2. R2 values for regression of the observed
versus predicted data for all experiments were very high,approximately 0.99 or closer to unity.

4

Fitted v(cm min–1)
0.079
0.074
0.079
0.076
0.147
0.143
0.145
0.150
0.210
0.200
0.198
0.198
0.071
0.068
0.064
0.061
0.123
0.123
0.124
0.124
0.206
0.203
0.204
0.201
0.038
0.039
0.039
0.040
0.115
0.112
0.113
0.12
0.218
0.220
0.221
0.220
0.067
0.078
0.082
0.090
0.120
0.121
0.118
0.122
0.125
0.123
0.129
0.128

Δv

Fitted D(cm min–1)

λ
(cm)

0.012
0.017
0.012
0.015
–0.0020.002
0.00–0.005–0.013–0.003–0.001–0.001–0.007–0.004
0.00
0.003
0.006
0.006
0.005
0.005
–0.002
0.001
0.00
0.003
0.002
0.001
0.001
0.00
0.010
0.013
0.012
0.005
0.005
0.003
0.002
0.003
0.024
0.013
0.009
0.001
0.003
0.002
0.005
0.001
0.004
0.006
0.00
0.001

0.003
0.007
0.009
0.010
0.005
0.012
0.015
0.018
0.008
0.014
0.018
0.020
0.001
0.004
0.004
0.005
0.003
0.006
0.008
0.009
0.005
0.009
0.011
0.012
0.001
0.002
0.003
0.004
0.003
0.006
0.008
0.010
0.005
0.012
0.015
0.018
0.002
0.004
0.006
0.008
0.0005
0.002
0.003
0.004
0.0005
0.002
0.003
0.004

0.034
0.092
0.121
0.131
0.034
0.083
0.103
0.119
0.039
0.071
0.091
0.101
0.021
0.056
0.067
0.079
0.022
0.053
0.066
0.075
0.023
0.043
0.052
0.061
0.037
0.062
0.079
0.091
0.026
0.058
0.073
0.087
0.025
0.053
0.067
0.081
0.014
0.033
0.047
0.063
0.004
0.014
0.027
0.036
0.007
0.022
0.038
0.049

Fig. 5 further shows satisfactory agreement between the estimated pore-water velocities (v) at the column scale, calculated
from the measured Darcian flux (q) and porosity (ϕ) of thecolumns (i.e., using v = q/ϕ), versus velocities estimated byfitting the measured BTC’s using Eq. (3).

Effect of soil textural characteristics on longitudinal dispersion in saturated porous media

Fig. 4. Measured (open circles) and calculated (lines) breakthrough curves at three different depths (11, 18, and 25 cm) and the effluent at
36 cm of the saturated columns. Assuming one dimensional macroscopic flow, Eq. (4) was used for the continuum scale simulations of the
BTCs. The curves are for two experiments with each soil sample involving two different flow rates as listed in Table.

Velocity effects
Fig. 6 shows that the longitudinal dispersion coefficient (D)increases with the pore-water velocity (v) and that for a given
velocity, the dispersion coefficient increases with travel distance. Furthermore, results revealed that the dispersion
coefficient of sand sample S1, which had the larger mean particle size (d50 = 480 µm), was affected considerably by the porewater velocity. In contrast, the dispersion coefficient of sample
S4, having a much smaller mean particle size (d50 = 95 µm),
was far less affected by the pore-water velocity.

Another way of visualizing the data in Fig. 6 is to plot foreach given value of the pore-water velocity the dispersion coefficient as a function of the dispersivity (λ). This is done in Fig.
7, which shows linear relationships between D and λ, with theslope in all cases being very close to the measured pore-watervelocity, as expected. The linear plots in Fig. 7 confirm ourassumptions that the effective diffusion coefficient (De) in
Eq. (5) is very small and can be neglected for all practical purposes (Pe > 3 for all sands except S4), and that the exponent n isessentially equal to 1.0. The plots in Fig. 7 further indicate thatthe dispersivity increases with travel distance.

5

Mojtaba G. Mahmoodlu, Amir Raoof, Martinus Th. van Genuchten

Fig. 5. Pore-water velocities estimated from the measured BTCs
using CXTFIT, versus measured velocities (using a total of 52
data) calculated from the Darcian flux and the porosity.

value of 480 μm, was much higher than the dispersion coefficient of sample S4 having a d50 of 95 μm. These results are
consistent with many earlier experimental studies using repacked soils showing a dependency of the longitudinal dispersion coefficient on travel distance and the pore-water velocity(Bromly et al., 2007; Kanzari et al., 2015; Nützmann et al.,
2002; Toride et al., 2003; Vanderborght and Vereecken, 2007).
Remarkably, Fig. 9 shows that the dispersivity values for the
lower travel distances are close to the average grain size foreach sample. As the travel distance increased, dispersivity
values also increased and became larger than the average grain
size for each sand sample.
Despite the dispersion coefficient increasing with the porewater velocity and travel distance, Fig. 8 suggests that D isfurther affected by grain size and consequently the pore size
and its distribution in a porous media. Results show that at avelocity of 0.129 cm min–1, the dispersion coefficient of sandsample S4 with a d50 of 95 μm is much smaller than the D ofsand sample S1 having a d50 of 480 μm at a velocity of 0.197
cm min–1.
Effect of Peclet number

Effects of travel distanceConsistent with the results in Figs. 6 and 7, Figs. 8 and 9
show that the macroscale ADE model at different locations
provided values for dispersion coefficients and solute dispersivity that increase with travel distance, respectively. The values slowly approach asymptotic values after a travel distance ofaround 25 or 30 cm. Our results reveal that for a constant velocity, the dispersion coefficient of sand sample S1, having a d50

We next calculated the dimensionless Peclet number (Eq. 6)
to evaluate the effect of molecular diffusion on the hydrodynamic dispersion. For samples S1 and S2, very high Peclet
numbers (Pe > 10) were obtained (Fig. 9). This confirms thevalidity of our assumption that the contribution of molecular
diffusion is negligible, and that the empirical constant n is equal
to one (Delgado, 2007; Toride et al., 2003).

Fig. 6. Plots of the estimated longitudinal dispersion coefficient (D) as a function of the pore-water velocity (v) for the different types ofporous media. S1, S2, S3 and S4 have average mean particle sizes of 480, 402, 324, and 95 µm, respectively.

6

Effect of soil textural characteristics on longitudinal dispersion in saturated porous media

Fig. 7. Estimated longitudinal dispersion coefficient (D) as a function of the dispersivity (λ) at given pore velocities (v).

Fig. 8. Effect of travel distance (L) on the longitudinal solute dispersion coefficient (D) for different pore-water velocity (v) values.

7

Mojtaba G. Mahmoodlu, Amir Raoof, Martinus Th. van Genuchten

water velocity (v), the travel distance (L) and the average grain
diameter (d50). Of these, three parameters (v, λ, and Pe) aredirectly related to D as expected (e.g., through Eq. 6), while theparameters of L and Grain Size Variation (GSV) are independent variables. A strong-positive correlation was found between
the dispersion coefficient and the Peclet number, the dispersivity, and the pore-water velocity as reflected by values of
0.709, 0.674, and 0.653 for the Pearson correlation coefficient,
respectively (Evans, 1996). Pearson's correlation coefficientmeasures the statistical relationship, or association, between
two continuous variables. We used the guidelines by Evans(1996) to evaluate the degree of positive and negative correlation, i.e., very weak (0.00–0.019), weak (0.20–0.39), moderate(0.40–0.59), strong (0.60–0.79), or very strong (0.80–1.0).
A moderate-positive correlation was found between D andboth the travel distance and the average grain diameter, having
Pearson correlation coefficients of 0.546 and 0.454, respectively. However, we found almost no correlation between D andporosity (ϕ) and grain size variation (GSV). As shown by Eq.
(6), the Peclet number accounts for both the pore-water velocity(v) and the average grain diameter (d50). As such, it is not surprising that a strong-positive relationship was obtained between
the Peclet number and D, leading to relatively high and low
values of the Pearson correlation coefficient with Pe.

Fig. 9. Effect of travel distance (L) on the solute dispersivity (λ) for
different pore-water velocity (v) values, “Low v”, “Mid v” and“High v” refer to low, medium and high flow rates, respectively.

CONCLUSIONSIn this study we used two sand samples (S1 and S4 having
d50 values of about 480 and 95 μm, with narrow particle size
distribution, respectively) and their two mixtures (S2 and S3
having d50 values of 402 and 324 μm with dual particle size
distribution) to explore the effects of soil textural characteristics
on longitudinal dispersion in 36-cm long saturated sand columns. Breakthrough curves (BTCs) at 3 depths (11, 18, 25 cm)
and of the effluent at 36 cm were obtained by measuring theelectrical conductivity (EC) using 5TE sensors.
Analyses of the BTCs in terms of the macroscopic equilibrium advection-dispersion equation revealed that the dispersion
coefficient (D) of the sands increased with the pore-water velocity (v). The D value of sample S1 with its relatively large
mean grain size was more affected by the pore-water velocity
as indicated by much larger dispersivities (λ), Furthermore, theD values increased with travel distance to slowly approach
asymptotic values at distances of 25 to 36 cm. Our results revealed that for a constant velocity, the longitudinal dispersion
coefficient of sample S1 (the coarser sand) was much higher
than D of sample S4 which had a small d50. In addition to increasing with the pore-water velocity and travel distance, D canbe affected also by the grain size distribution. Our results
showed that D of sand sample S4 (the finer sand) was smaller
than of the sample S1 at a same velocity.

Fig. 10. Effect of the particle Peclet number (Pe) on the dispersion
coefficient (D) for the different sand samples.

In contrast with S1, S2 and S3, fine sand sample S4 showed
relatively low Peclet numbers for all velocities (Pe < 3). This
indicates that solute dispersion for that sample may have been
affected also by molecular diffusion, in part because of therelatively low pore-water velocities for that sample. Results
further showed that with an increase in the Peclet number
and/or pore-water velocity, the dispersion coefficient increased
nonlinearly, which is consistent with earlier studies (d’Angelo
et al., 2007; Kantzas et al., 2012).
Correlation matrix of parameters controlling the dispersion
coefficient
A correlation matrix of the various parameters controllingthe dispersion coefficient is provided in Table 3. The data indicate a significant correlation between D and five parameters,
namely the Peclet number (Pe), the dispersivity (λ), the pore-

Table 3. Correlation matrix of various transport and porous media parameters affecting the dispersion coefficient.
1

v
λ
L
d50
GSV
ϕ
8
Pe
1
8

D
0.653**
0.674**
0.546**
0.454**–0.007ns–0.062ns
0.709**

2v–0.036ns
0.005ns
0.24ns0.138ns–0.104ns
0.834**

3

λ0.72**
0.47**–0.046ns–0.096ns
0.212ns

4

L

0ns
0ns
0.026ns–0.006ns

5d50

0.418*–0.506**
0.716**

6GSV–0.91**
0.296*

7

ϕ–0.314*

Dispersion coefficient; 2Pore-water velocity; 3Dispersivity; 4Travel distance; 5Average diameter of grains; 6Grain size variation; 7Porosity;
Peclet Number; *P ≤ 0.05 (significant difference); ** P ≤ 0.01(highly significant difference); nsP > 0.05(no significant difference).

8Effect of soil textural characteristics on longitudinal dispersion in saturated porous media

We further found very large particle Peclet numbers for thecoarser sand samples with their larger mean diameters (notably
S1, but also S2, and S3). This confirms the validity of our assumption that the contribution of molecular diffusion was negligible. In contrast to S1, S2 and S3, the sample S4 (with its
finer sand) showed a relatively low Peclet number (Pe) for all
velocities (Pe < 2). This indicates that dispersion in sample S4
was affected more by molecular diffusion. Our results also
showed that with an increase in the Peclet number and/or porewater velocity, the dispersion coefficient increased nonlinearly,
with an exponential shape.
A correlation matrix of the many parameters potentially affecting dispersion processes showed a strong-positive correlation between D and three parameters: The Peclet number, thedispersivity, and the pore-water velocity, which is in line with
Equations (5) and (6). A moderate-positive correlation between
the dispersion coefficient and travel distance and the average
diameter of the grains was found. However, almost no correlation existed between D and the porosity as well as grain-size
heterogeneity. Consequently, it seems that that the dispersion
coefficient depends on particle size more than other soil texture
or structure features (e.g., porosity, particle size distribution
modality).
The results from this study suggest using scale-dependent relations for the solute dispersivity, particularly for applications
where the scale- dependency of solute dispersion is important,such as at early times and/or short travel distances. We further
note that our study was limited to saturated conditions. Several
experimental (e.g., Bunsri et al., 2008b; Toride et al., 2003) andpore-scale modeling (e.g., Raoof and Hassanizadeh, 2013)
studies suggest that the dispersivity may increase when soils
become unsaturated up until some maximum at intermediate
fluid saturation values, with lower values again in relatively dry
conditions. Several studies also reported water fractionation
into mobile and immobile water fractions (e.g., Gaudet et al.,1977). These and other effects such as heterogeneity andmacropore flow processes also need further investigations at a
range of scales.
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Received 16 December 2019
Accepted 4 January 2021