The C-Ride Modulefor HYDRUS (2D/3D)
Simulating Two-Dimensional Colloid-Facilitated Solute
Transport in Variably-Saturated Porous Media
Version 1.0

Jirka Šimůnek1, Miroslav Šejna2, and Martinus Th. van Genuchten3
September 2012
1

Department of Environmental Sciences
University of California Riverside, Riverside, CA, 92521, USA
2

PC Progress, Ltd., Prague, Czech Republic
3

Department of Mechanical Engineering
Federal University of Rio de Janeiro, Rio de Janeiro, Brazil© 2012 J. Šimůnek and M. Šejna. All rights reserved.

2

Table of Contents
Table of Contents................................................................................................................................. 3
List of Figures ...................................................................................................................................... 4
List of Tables ........................................................................................................................................ 5
Abstract ................................................................................................................................................ 7
1. Introduction ..............................................................................................................................9
2. Conceptual Model....................................................................................................................... 13
3. Colloid Transport...................................................................................................................17
3.1.

Colloid Transport Equation .........................................................................................17

3.2.

Colloid Mass Transfer .................................................................................................18

4. Colloid-Facilitated Solute Transport ...................................................................................21
4.1.

Transport Equation for Colloid-Facilitated Solute Transport ....................................21

4.2.

Solute Interactions with the Solid Phase .....................................................................23

4.3.

Solute Interactions with Immobile Colloids.................................................................23

5. Numerical Implementation ...................................................................................................21
6. Preprocessing and Input Data ..............................................................................................27
6.1.

Main Processes ............................................................................................................27

6.2.

Solute Transport - General Information ......................................................................27

6.3.

Solute Transport Parameters .......................................................................................28

6.4.

Solute Reaction Parameters .........................................................................................30

6.5.

Initial Conditions .........................................................................................................33

7. Post-Processing.......................................................................................................................35
7.1.

Results – Graphical Display ........................................................................................35

7.2.

Results – Other Information.........................................................................................35

8. Verification and Test Examples ............................................................................................37
8.1.

Example Set 1 - Transfer Between Different Phases Without Flow ..............................38

8.2.

Example Set 2 - Colloid and Solute Transport During Steady-State Flow .....................39

8.3.

Other Examples - Colloid and Solute Transport During Transient Flow.......................40

References .....................................................................................................................................413

List of Figures
Figure 1.

Schematic of the colloid transport model. ...............................................................13

Figure 2.

Schematic of the colloid-facilitated solute transport model. ...................................14

Figure 3.

The Main Processes dialog window. .......................................................................28

Figure 4.

The Solute Transport dialog window. .....................................................................28

Figure 5.

The Solute Transport Parameters dialog window. ..................................................29

Figure 6.

Parts of the "Reaction Parameters for Solute - 1" (colloids) dialog window. .........31

Figure 7.

Parts of the "Reaction Parameters for Solute - 2" (solute on colloids) dialog
window. ...................................................................................................................32

Figure 8.

Parts of the "Reaction Parameters for Solute - 3" (solute) dialog window. ............33

Figure 9.

The "Initial Conditions" part of the Data Tab of the Navigator Bar of the C-Ride
module. ....................................................................................................................33

Figure 10.

The "Results - Graphical Display" part of the Data Tab of the Navigator Bar of the
C-Ride module. .......................................................................................................35

Figure 11.

The "Concentration Options" part of the View Tab of the Navigator Bar for the CRide module. ...........................................................................................................35

Figure 12.

Colloid (mobile and immobile) and solute (liquid, sorbed, and on colloids)
concentrations for examples Test02a (top left), Test02b (top right), and Test02e(bottom). ..................................................................................................................38

Figure 13.

Breakthrough curves for colloids (black line), solute sorbed to colloids (blue line),and solute (red line) for examples Test03a (left) and Test03b (right). ......................40

4

List of Tables
Table 1.

Species considered in the colloid and colloid-facilitated solute transport model (L isthe length unit, M is the mass unit, and n is the number of colloids).. ........................13

Table 2.

Kinetic reactions considered in the colloid and colloid-facilitated solute transportmodel (T is the time unit).............................................................................................15

Table 3.

Soil specific solute transport parameters... ..................................................................29

Table 4.

Solute specific solute transport parameters..................................................................29

Table 5.

Reaction parameters for the first component (colloids)...............................................30

Table 6.

Reaction parameters for the second component (solute on colloids)... .......................31

Table 7.

Reaction parameters for the third component (solute).................................................32

Table 8.

List of various test examples used to verify various new options of the C-Ride
module..........................................................................................................................37

Table 9.

Mass balance calculations for test examples Test02a, Test02b, and Test02e... ...........395

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AbstractŠimůnek, J., M. Šejna, and M. Th. van Genuchten, The C-Ride Module for HYDRUS (2D/3D)
Simulating Two-Dimensional Colloid-Facilitated Solute Transport in Variably-Saturated PorousMedia, Version 1.0, PC Progress, Prague, Czech Republic, 45 pp., 2012.
Strongly sorbing contaminants (e.g., heavy metals, radionuclides, pharmaceuticals, pesticides,and explosives) in porous media are associated predominantly with the solid phase, which iscommonly assumed to be stationary. However, recent field- and laboratory-scale observations
have shown that mobile colloidal particles (e.g., microbes, humic substances, suspended clay
particles and metal oxides) can act as pollutant carriers and thus provide a rapid transport
pathway for strongly sorbing contaminants. To address this problem, we have developed a twodimensional numerical module C-Ride for the HYDRUS (2D/3D) software package thatincorporates various processes associated with colloid and Colloid-Facilitated Solute Transport(CFSTr) in variably-saturated porous media. The model accounts for transient variably-saturated
water flow, and for both colloid and solute movement due to advection, diffusion, anddispersion, as well as for solute movement facilitated by colloid transport. The colloid transportmodule additionally considers the processes of attachment/detachment to/from the solid phaseand straining. Various blocking and depth-dependent functions can be used to modify theattachment and straining coefficients. The solute transport module uses the concept of two-site
sorption to describe nonequilibrium adsorption-desorption reactions to the solid phase. The
module further assumes that contaminants can be sorbed onto surfaces of both deposited andmobile colloids, thus fully accounting for the dynamics of colloid transfer between different
phases. Applications of the model are demonstrated using several test examples.
The purpose of this report is to documents the C-Ride module for the HYDRUS (2D/3D)
software package [Šimůnek et al., 2011; Šejna et al., 2011] simulating two-dimensional variablysaturated water flow, colloid transport, and colloid-facilitated solute transport in porous media.
The module, as well as the description of the module in this report, is largely based on thetheoretical development of Šimůnek et al. [2006]. The report serves as both a Technical Manualand a User Manual, as well as a reference document of the Graphical User Interface of C-Ride
related parts of the HYDRUS software package.

7

DISCLAIMER
The C-Ride module was developed as a supplemental module of the HYDRUS (2D/3D) software
package to simulate the transport and reactions of colloid and colloid-facilitated transport in soils
and groundwater. The software has been verified against selected test cases. However, no warranty
is given that the program is completely error-free. If you do encounter problems with the code, finderrors, or have suggestions for improvement, please contact the senior author at
Jirka Šimůnek
Tel/Fax: 1-951-827-7854
Email: jiri.simunek@ucr.edu

8

1. Introduction
Colloidal particles (e.g., humic substances, suspended clay particles, metal oxides, andmicroorganisms) are commonly found in subsurface environments [Kretzschmar et al., 1999].
Many contaminants can sorb onto colloids in suspension, thereby increasing their concentrations
in solution beyond thermodynamic solubilities [Kim et al., 1992]. Experimental evidence now
exists that many contaminants are transported not only in a dissolved state by water, but alsosorbed to moving colloids. This colloid-facilitated transport has been illustrated in the literaturefor numerous contaminants, including heavy metals [Grolimund et al., 1996; Šimůnek et al.,2006], radionuclides [Von Gunten et al., 1988; Noell et al., 1998], pesticides [Vinten et al., 1983;
Kan and Tomson, 1990; Lindqvist and Enfield, 1992], pharmaceuticals [Tolls, 2001; ThieleBruhn, 2003], hormones [Hanselman et al., 2003], and other contaminants [Magee et al., 1991;
Mansfeldt et al., 2004]. Since mobile colloids often move at rates similar or faster as non-sorbing
tracers, the potential of enhanced transport of colloid-associated contaminants can be verysignificant (e.g., McCarthy and Zachara, 1989). Failure to account for colloid-facilitated solutetransport can severely underestimate the transport potential and risk assessment for thesecontaminants. Models that can accurately describe the various mechanisms controlling colloid
and solute transport, and their mutual interactions and interactions with the solid phase, areessential for improving predictions of colloid-facilitated transport of solutes in variably saturated
porous media.
The transport behavior of dissolved contaminant species has been studied for many years. By
comparison, colloid transport and the many interactions among contaminants, colloids, andporous media are less well understood. While colloids are subject to similar subsurface fate andtransport processes as chemical compounds, they are also subject to their own uniquecomplexities [van Genuchten and Šimůnek, 2004; Flury and Qiu, 2008]. Since many colloids andmicrobes are negatively charged, they are electrostatically repelled by negatively-charged solid
surfaces. This will lead to an anion exclusion process that can cause slightly enhanced transport
relative to fluid flow. The advective transport of colloids may similarly be enhanced by size
exclusion, which limits their presence to the larger pores [Bradford et al., 2003, 2006]. In
addition to being subject to adsorption-desorption at solid surfaces, colloids are also affected bystraining [Bradford et al., 2003, 2006] and may accumulate at air-water interfaces [Wan andWilson, 1994; Thompson and Yates, 1999; Wan and Tokunaga, 2002; Crist et al., 2005;
Torkzaban et al., 2008; Lazouskaya et al., 2011]. All of these complexities require colloidtransport models to be more flexible than regular solute transport models.
Models that consider colloid-facilitated transport are based upon mass balance equations for all
colloid and contaminant species. The various colloid-facilitated transport models that haveappeared in the literature differ primarily in the manner which colloid transport and contaminantinteractions are implemented. For example, Mills et al. [1991] and Dunnivant et al. [1992]
assume that colloids are non-reactive with the solid phase, Corapcioglu and Jiang [1993] andJiang and Corapcioglu [1993] consider a first-order kinetic attachment of colloids, Saiers andHornberger [1996] assume irreversible nonlinear kinetic attachment of colloids, and van deWeerd and Leijnse [1997] describe colloid attachment kinetics using the Langmuir equation. All
colloid-facilitated transport models account for interactions between the contaminants andcolloids. Various equilibrium and kinetic models have been used for this purpose.
9

Although already relatively complex, no existing model for colloid-facilitated contaminanttransport to our knowledge includes all of the major processes contributing to colloid andcolloid-facilitated transport. For example, most models for colloid-facilitated transport consider
flow and transport only in fully saturated groundwater systems, usually for steady-state flow, andthus do not account for colloid interactions with the air-water interface. Also, no colloidfacilitated transport model has considered straining and size exclusion as mechanisms of colloid
retention and transport, respectively. These two processes, and especially straining, have recently
received much attention since classical colloid transport models are often unable to describe
simultaneously both breakthrough curves versus time and concentration profiles versus depth[Bradford et al., 2003, 2006; Li et al., 2004; Tufenkji and Elimelech, 2005].
Straining involves the entrapment of colloids at grain-grain contact points, surface roughness
locations, and in down-gradient pores that are too small to allow particle passage. The criticalpore size for straining will depend on the size of the colloid and the pore-size distribution of themedium [McDowell-Boyer et al., 1986; Bradford et al., 2002, 2003]. Straining may havesignificant implications for colloid-facilitated solute transport, as illustrated by Bradford et al.
[2006]. Considering average capillary pressure-saturation curves for the 12 major soil texturalgroups given by Carsel and Parrish [1988], they calculated that a 2 µm colloid, which is the sizeof a clay particle, will be excluded or strained in 10 to 86% of the soil pore space for various soil
textures [Bradford et al., 2006]. These percentages should significantly increase if a soil
becomes unsaturated.
Size exclusion, a process closely related to straining, affects the mobility of colloids byconstraining them to flow domains and pore networks that are physically accessible [Ryan andElimelech, 1996; Ginn, 2002]. Electrostatic forces also play an important role in the distribution(and mobility) of colloids. Anionic colloids will be excluded from locations adjacent tonegatively charged solid surfaces; similar to the much reported anion exclusion process foranionic solutes [Krupp et al., 1972; Gvirtzman and Gorelick, 1991; Ginn, 1995]. In case of sizeor anion exclusion, colloids will tend to reside in larger pores and in more conductive parts of theflow domain. As a result, colloids will be transported faster than a conservative solute tracer[Reimus, 1995; Cumbie and McKay, 1999; Harter et al., 2000; Bradford et al., 2004].
Differences in the dispersive flux for colloids and a conservative solute tracer are also anticipated
as a result of exclusion [Scheibe and Wood, 2003]. Bradford et al. [2002] observed that thedispersivity of 3.2 µm carboxyl latex colloids was up to 7 times greater than bromide in saturated
aquifer sand. Conversely, Sinton et al. [2000] found in a field. microbial transport experimentthat the apparent colloid dispersivity decreased with increasing particle size.
Colloid and colloid-facilitated contaminant transport in partially saturated porous media is evenmore complex than in water-saturated systems. In addition to all of the processes and difficulties
discussed above, colloid transport in partially saturated porous media is further complicated by
the presence of an air phase, thin water films, and the air-water-solid contact line, in addition to
the solid and water phases present in saturated media. Wan and Wilson [1994] observed thatcolloidal particles deposit preferentially on the air-water interface, and that particle transport was
tremendously retarded since the air-water interface acted as a strong sorption phase. Anotherphysical restriction on colloid transport in unsaturated systems is imposed by thin water films;

10this process is often referred to as film straining [Wan and Tokunaga, 1997; Saiers and Lenhart,2003]. Wan and Tokunaga [1997] proposed that colloid transport in unsaturated systems depends
on the ratio of colloid size to water film thickness. The air-water-solid contact line has also beenrecognized to significantly influence colloid retention and release [Crist et al., 2004 and 2005;
Chen and Flury, 2005]. Corapcioglu and Choi [1996] developed a mathematical model
describing colloid transport in unsaturated porous media, and also studied the effects of colloids
on volatile contaminant transport and air-water partitioning in unsaturated porous media.
Most current models for colloid-facilitated transport assume that the number of colloids withrespect to the contaminant is large and that kinetic reactions coefficients are not dependent on thenumber of colloids in the system. Although this may be true for some systems, the number ofcolloids (or concentrations) is often highly variable, with colloids being mobilized (orimmobilized) due to changing chemical or hydrological conditions. Thus the reaction
coefficients need to be adjusted to the number of colloids in the system in different phases (i.e.,
mobile, immobile, attached to the air-water interface). This adjustment needs to be carried out
also for numerical stability reasons. For example, if the number of colloids in the systemdecreases dramatically and the sorption constants for solute to colloids are assumed to beconstant, this may lead to large sorbed concentrations, and hence numerical instabilities.
The main purpose of this report is to document a two-dimensional numerical module C-Ride thatincorporates processes associated with colloid and colloid-facilitated solute transport in variablysaturated porous media. C-Ride (which refers to Colloids providing a ride for solutes) was
developed specifically for the HYDRUS (2D/3D) software package [Šimůnek et al., 2011; Šejna etal., 2011]. The general conceptual basis of the module is discussed in a paper by Šimůnek et al.
[2006], and restated here for multi-dimensional systems. The module accounts for both colloid
and solute transport due to advection, diffusion and dispersion in variably-saturated media, as
well as for solute movement facilitated by colloid transport. The colloid transport module
additionally considers the processes of attachment/detachment to/from the solid phase, straining,
and/or size exclusion. The module allows for different pore water velocities and dispersivities forthe colloids and the solute. However, the module does not describe all of the processes discussed
above, such as attachment/detachment to the air-water interface. These processes likely will be
added in future versions of the module.
The documentation in this report focuses mostly only on the colloid and colloid-facilitated transportfeatures of the C-Ride module. All processes related to variably-saturated water flow and heattransport are described in detail in the HYDRUS (2D/3D) documentation [Šimůnek et al., 2011],
and will not be repeated here.
The C-Ride module may be used to analyze water and solute movement in unsaturated, partiallysaturated, or fully saturated porous media. C-Ride can handle flow domains delineated by irregularboundaries. The flow region itself may be composed of nonuniform soils having an arbitrary degreeof local anisotropy. Flow and transport can occur in the vertical plane, the horizontal plane, or in athree-dimensional region exhibiting radial symmetry about a vertical axis. The water flow part ofthe model considers prescribed head and flux boundaries, as well as boundaries controlled byatmospheric conditions.

11

The governing flow and transport equations are solved numerically using standard Galerkin-typelinear finite element schemes. The C-Ride module is fully supported by the HYDRUS (2D/3D)
graphical user interface [Šejna et al., 2011]. Applications of the C-Ride module are demonstrated
later in this report on several examples.

122. Conceptual Model
The conceptual model of colloid and colloid facilitated solute transport (Fig. 1) is based on
several assumptions. We assume that the porous medium consists of three phases (i.e., solid
phase, air, and water). Following van der Weerd and Leijnse [1997], we assume that (a) only one
type of colloid exists in the system, and that these colloids can be described by their mean
behavior, (b) colloids are stable and that their mutual interactions in the liquid phase may beneglected, (c) immobile colloids do not affect the flow and transport properties of the porous
medium because of their clogging of small pores, and (d) the transport of colloids is not affectedby sorption of contaminants on colloids. Colloids are assumed to exist in three states: mobile(suspended in water), attached to the solid phase, and strained by the solid phase (Fig. 1, Table 1).
Colloid phase changes are described in terms of nonlinear first-order processes.

Figure 1. Schematic of the colloid transport model.
We further assume that contaminants can be dissolved in the liquid phase, as well as can be
sorbed instantaneously and kinetically to the solid phase, and to colloids in all of their states. We
thus distinguish six different states of contaminants (Fig. 2; Table 1): dissolved, instantaneously
or kinetically sorbed to the solid phase, and sorbed to mobile or immobile colloids (the latter
either strained or attached to the solid phase). The coefficients k in Figures 1 and 2 (Table 2)
represent various first-order attachment/detachment or sorption/desorption rates, the coefficients
Kd and ω represent instantaneous and kinetic solute sorption to the solid phase, respectively,
while the factors ψ account for nonlinearity of various processes.

Table 1. Species considered in the colloid and colloid-facilitated solute transport model (L is thelength unit, M is the mass unit, and n is the number of colloids).
13

Species description

Variable

Units

Cc

nL-3

Colloids attached to the solid phase

Scatt, Sc1

nM-1

Strained colloids

Scstr, Sc2

nM-1

Dissolved contaminant

C

ML-3

Contaminant instantaneously sorbed to the solid phase

Se

MM-1

Contaminant kinetically sorbed to the solid phase

Sk

MM-1

Contaminant sorbed to mobile colloids

Smc

Mn- 1

Contaminant sorbed to immobile colloids (attached)

Sic1

Mn- 1

Contaminant sorbed to immobile colloids (strained)

Sic2

Mn- 1

Colloid species
Mobile colloids

Solute (contaminant) species

Figure 2. Schematic of the colloid-facilitated solute transport model.

14

Table 2. Kinetic reactions considered in the colloid and colloid-facilitated solute transport model(T is the time unit).
Reaction description

Variable

Units

Colloids attachment coefficient

kac

T-1

Colloids detachment coefficient

kdc

T-1

Colloids straining coefficient

kstr

T-1

Solute adsorption rate to mobile colloids

kamc

T-1

Solute desorption rate from mobile colloids

kdmc

T-1

Solute adsorption rate to immobile colloids

kaic

T-1

Solute desorption rate from immobile colloids

kdic

T-1

Solute sorption rate to the solid phase (soil)

ω

T-1

Colloid species

Solute (contaminant) species

15

16

3. Colloid Transport
3.1. Colloid Transport Equation
Colloid fate and transport models are commonly based on the advection–dispersion equation, but
modified to account for colloid filtration [Harvey and Garabedian, 1991; Hornberger et al.,
1992; Corapcioglu and Choi, 1996; Bradford et al., 2003]. A very general equation for colloidtransport describing colloid/matrix mass partitioning in multi-dimensional form is given by:∂θ c Cc∂S∂S∂  c ∂Cc  ∂qic Cc+ ρ c1 + =− µcwθ Cc − µcs ρ Sc
ρ c2
 θ Dij−∂t∂t∂t∂xi 
∂x j 
∂xi
where:
Cc
Sc

θc
ρ
qic
Dijc
t
xi

µcw
µcs

(1)

colloid concentration in the liquid phase (mobile colloids) [nL-3]
colloid concentration in the solid phase (attached, immobilized, immobile colloids;
two sites are considered and there may have different interpretation; for example, Sc1
may be represented as colloids attached to the solid phase, and Sc2 may be represented
as strained colloids) [nM-1]
volumetric water content accessible to colloids [L3L-3] (due to ion or size exclusion,θc may be smaller than the total volumetric water content θ)bulk density [ML-3]
Darcy’s flux for colloids [LT-1]
dispersion coefficient for colloids [L2T-1]
time [T]spatial coordinate [L]colloid decay in the liquid phase [T-1]
colloid decay in the solid phase [T-1]

The second and third terms on the left side of Eq. (1) represent mass transfer of colloids between
the aqueous and solid phase [nL-3T-1]. Some researchers have attributed attachment/detachment
and straining mechanisms of colloid retention to sites 1 and 2, respectively (e.g., Bradford et al.,
2003; Gargiulo et al., 2007, 2008). However, additional microscopic information is frequently
needed to substantiate this assumption. Here we do not attempt to attribute specific colloid
retention mechanisms to a given site without other experimental evidence. Rather, Eq. (1) isviewed as a simple and flexible approach to describe colloid breakthrough curves and retention
profiles that are not exponential with depth. For example, the first kinetic site can represent
reversible retention, whereas the second kinetic site can represent irreversible, depth-dependent
retention.
The first two terms on the right side of Eq. (1) represent dispersive and advective colloid fluxes,
respectively. The value of θc is defined asθ c= θ − θim

17

(2)

where θ is the total water content [L3L-3] and θim is the water content that is not accessible tomobile colloids [-] due to size exclusion [Bradford et al., 2006].

3.2. Colloid Mass Transfer
Colloid mass transfer between the aqueous and solid phases is traditionally described using
attachment-detachment models of the form:ρ

where:
kac
kdc

ψc

∂Sc1
= θ cψ c kac Cc − ρ kdc Sc1 − µcs ρ Sc1∂t(3)

colloid deposition (attachment) coefficient [T-1]
colloid entrainment (detachment) coefficient [T-1]
dimensionless colloid retention function or blocking coefficient

The attachment coefficient is generally calculated using filtration theory [Logan et al., 1995], aquasi-empirical formulation in terms of the median grain diameter of the porous medium (often
termed the collector), the pore-water velocity, and collector and collision (or sticking)efficiencies accounting for colloid removal due to diffusion, interception and gravitational
sedimentation [Rajagopalan and Tien, 1976; Logan et al., 1995; Tufenkji and Elimelech, 2004].
The first-order detachment coefficient in Eq. (3) accounts for colloid mobilization. While inreality this coefficient is presumably affected by changes in pore-water chemistry (ionic strength,ionic composition, and pH) and physical perturbations in flow, including changes in the flow rate
and the water content, it is an input constant in the C-Ride module.
To simulate reductions in the attachment coefficient due to filling of favorable sorption sites, ψc
is sometimes assumed to decrease with increasing colloid mass retention. Random sequential
adsorption [Johnson and Elimelech, 1995] and Langmuirian dynamics [Adamczyk et al., 1994]
equations have been proposed for ψc to describe this blocking phenomenon, with the latter
equation given by:ψ c = 1−

Sc1
Scmax
1

(4)

in which Sc1max is the maximum solid-phase colloid concentration [nM-1]. Conversely, enhanced
colloid retention during porous-medium ripening can theoretically be described using afunctional form of ψc that increases with increasing mass of retained colloids [Tien, 1989; andTien and Chiang, 1985; Deshpande and Shonnard, 1999] We refer to several recent studies[Ginn, 2002; DeNovio et al., 2004; Rockhold et al., 2004] for more detailed discussions of theattachment and detachment coefficients in Eq. (3). The attachment coefficient in Eq. (3) has beenfound to depend strongly upon water content, with attachment significantly increasing as the18water content decreases. This has been attributed to a variety of processes, including interfacial
attachment and film straining. Bradford et al. [2002, 2003, 2006] hypothesized that the influenceof straining and attachment mechanisms on colloid retention should be separated into two
distinct components: attachment and detachment per se, and straining (being the entrapment of
colloids in pore throats that are too small to allow passage). They modeled the influence ofstraining using an irreversible first-order expression as follows:ρ

where:
kstr

ψstr

∂Sc 2
= θ cψ str kstr Cc∂t(5)colloid straining coefficient [T-1]
dimensionless straining blocking coefficient

Application of the first-order attachment-detachment model given by Eq. (3) typically leads toexponential colloid distributions versus depth. Bradford et al. [2003] showed that suchexponential distributions are often inconsistent with experimental data. They obtained muchbetter results using a depth-dependent straining coefficient ψsstr in Eq. (5) of the form
d +x
ψ str =  50

 d50 −β

(6)where d50 is the median grain size of the porous media [L], β is a fitting parameter [-], and x is
distance from the porous-medium inlet [L]. Data from Bradford et al. [2002, 2003] for different
colloid diameters (dp) and porous media median grain sizes showed an optimal value of 0.43 forβ, while kstr could be described using a unique increasing function of the ratio of dp/d50.
Subsequent studies with layered soils [Bradford et al., 2005] showed the importance of straining
at and close to textural interfaces when flow occurs in the direction of coarse-textured tomedium- and fine-textured media. Straining was found to be a significant mechanism for colloid
retention for values of dp/d50 greater than 0.005.

19

20

4. Colloid-Facilitated Solute Transport
4.1. Transport Equation for Colloid-Facilitated Solute Transport
The colloid-facilitated solute transport model requires knowledge of colloid transport, dissolved
contaminant transport, and colloid-facilitated transport, and of various interactions between
colloids, solute, and soil. Transport and/or mass-balance equations must therefore be formulated
for the total contaminant, for contaminant sorbed kinetically or instantaneously to the solid phase,and for contaminant sorbed to mobile colloids as well as to colloids attached to the soil solid
phase, i.e., for contaminant in all different phases or pools.
Our colloid-facilitated transport model closely follows the formulation of many other models(e.g., Corapcioglu and Jiang, 1993; Jiang and Corapcioglu, 1993; Saiers and Hornberger, 1996;
van de Weerd and Leijnse, 1997; and Bekhit and Hassan, 2005). New features of our model
include consideration of water contents, and water and solute fluxes, that may change in time andspace (i.e., transient variably-saturated water flow), and adjustment of all kinetic rates to thenumber of colloids present in the system.
Mass-balance equation for the total contaminant. The combined dissolved and colloid-facilitatedcontaminant transport equation is given by:∂S∂S ∂θ C S
∂Sc Sic∂θ C∂ ∂C  ∂qi C+ ρ e + ρ k + c c mc + ρ=+ θ Dij
−
∂t
∂t
∂t
∂t
∂t
∂xi 
∂xi  ∂xi ∂q C S∂ c ∂C S+ θ c Dij c mc  − ic c mc − Rs1∂xi ∂x j 
∂xi

where:
C
Se
Sk
Smc
Sic
Rs1

(7)

contaminant concentration of the liquid phase [ML-3]
contaminant concentration sorbed to equilibrium (type 1) sites [MM-1]
contaminant concentration sorbed to kinetic (type 2) sites [MM-1]
contaminant concentration sorbed to mobile colloids [MM-1]
contaminant concentration sorbed to immobile colloids [MM-1]. Note that there are
two types of immobile colloids: attached (Sic1) and strained (Sic2) colloids
reactions/degradation of contaminant in all phases combined (dissolved, sorbed tosolid phase and sorbed to colloids) [ML-3T-1]

For numerical convenience, equation (7) can be divided into two separate transport equations,
one for the dissolved solute and one for solute associated with the colloids being transport in theliquid phase. The dissolved solute transport equation is as follows

21

∂S
∂S k
∂θ C
∂ 
∂C  ∂qi C+ρ e +ρ =− θ kamcψ mC − θ kimcψ imC + θ Dij
−
∂t
∂t
∂t
∂xi 
∂x j  ∂xi+ θ c kdmc Cc S mc + ρ kdic Sc Sic − Rs 2

(8)while the transport equation for solute associated with the transport of colloids is given by ∂q C S∂θ c Cc S mc
∂S S∂ c ∂C S+=ρ c ic θ c Dij c mc  − ic c mc − θ c kdmc Cc S mc − ρ kdic Sc Sic +
∂t
∂t
∂xi 
∂x j 
∂xi+ θ kamcψ mC + θ kaicψ imC − Rs 3

(9)which is similar to Eq. (1), but with colloid concentrations multiplied by solute concentration,
augmented with the additional processes of solute adsorption/desorption from colloids (particles)
and possible degradation.
In equations (8) and (9):kdmc desorption rate coefficient from mobile colloids [T-1]
kdic desorption rate coefficient from immobile colloids [T-1]
kamc adsorption rate coefficient to mobile colloids [T-1]
kaic adsorption rate coefficient to immobile colloids [T-1]ψm parameter adjusting the sorption rate to the number of mobile colloids present. kamc is
the optimum rate for the maximum amount of mobile colloids sorbed. If less colloids
are present, then kamc is reduced by ψ.
Cψ m = refc
Cc
ref
Cc
reference concentration of mobile colloids for which sorption rate kamc is valid [ML-3]
ψiparameter adjusting the sorption rate to the number of immobile colloids present. kaic is
the optimum rate for the maximum amount of immobile colloids sorbed. If lesscolloids are present, then kaic is reduced by ψ.
Sc
ψ i = maxSc
refSc
reference concentration of immobile colloids for which sorption rate kaic is valid [-]Rs2 reactions/degradation of contaminant (dissolved and sorbed to solid phase) [ML-3T-1]
Rs3 reactions/degradation of contaminant (sorbed to colloids) [ML-3T-1]
Reactions/degradation terms Rs1, Rs2, and Rs3 from equations (7), (8), and (9) are defined asfollows, respectively:
Rs1 = µ wθ C + µ s ρ fSe + µ s ρ S k + µmθ c Cc S mc + µi ρ Sc Sic
Rs 2 =µ wθ C + µ s ρ fSe + µ s ρ S k=
Rs 3 µmθ c Cc S mc + µi ρ Sc Sic

22

(10)

where:

µw
µs
µm
µi

degradation rate for contaminant dissolved in the liquid phase [T-1]
degradation rate for contaminant sorbed to the solid phase [T-1]
degradation rate coefficient for contaminant sorbed onto mobile colloids [T-1]
degradation rate coefficient for contaminant sorbed onto immobile colloids [T-1]

4.2. Solute Interactions with the Solid Phase
Equations (7) and (8) invoke the concept of two-site sorption for modeling non-equilibrium
adsorption–desorption reactions (e.g., van Genuchten and Wagenet, 1989). The two-site sorption
concept assumes that total sorption, S, can be divided into two fractions:
S = Se + Sk

(11)

with sorption Se [MM-1] on one fraction of the sites (type-1 sites) assumed to be instantaneous,and sorption Sk [MM-1] on the remaining sites (type-2 sites) being time dependent according toρ∂S k= ω [ (1 − f )Ψ (C ) − S k ] - µ s S k
∂t

(12)

where ω is the first order rate constant [T-1], f is the fraction of exchange sites assumed to be in
equilibrium with the solution phase [-], Ψ(C) is the adsorption isotherm [MM-1] that can be
expressed using Freundlich, Langmuir, Freundlich-Langmuir, or linear adsorption models, andRsk represents various chemical and biological reactions of the kinetically sorbed contaminant[ML-3T- 1].

4.3. Solute Interactions with Immobile Colloids
The mass-balance equation for contaminant sorbed to immobile colloids can be written as
follows

ρ∂Sc Sic= θψ c kac Cc S mc − ρ kdc Sc Sic − ρ kdic Sc Sic + θ kaicψ imC − µi ρ Sc Sic∂t

All terms have been defined above.

23

(13)

24

5. Numerical Implementation
For numerical convenience, colloid transport and colloid-facilitated solute transport is simulated
by considering the transport of three different species:a) Colloids (transport equation (1) and mass transfer equation (3))b) Solute associated with colloids (transport equation (9) and mass transfer equation (13))c) Solute (transport equation (8) and mass transfer equation (12))
The colloid transport is a nonequilibrium transport (since it involves kinetic processes ofattachment and detachment). It can also be nonlinear when blocking functions (for attachment or
straining) are considered. The same is true for solute associated with colloids, which mirrors thetransport processes of colloids. Finally, solute transport can be nonequilibrium (i.e., one canconsider kinetic sorption using the two-site model, in addition to kinetic desorption fromcolloids). It can also be nonlinear, considering Freundlich or Langmuir isotherm.
Numerical schemes used to solve the transport equations (both linear and nonlinear) are the sameas in the standard HYDRUS module. These numerical schemes are described in detail in thetechnical manual [Šimůnek et al., 2011]. Additional details about the numerical implementation
can be found in Šimůnek et al. [2006].

25

26

6. Preprocessing and Input Data
6.1. Main Processes
The C-Ride module is activated in the Main Processes window (Fig. 3) by selecting the Solute
Transport check box and the Colloid-Facilitated Solute Transport radio button.

Figure 3. The Main Processes dialog window.

6.2. Solute Transport - General Information
When the C-Ride module is used, basic information needed for defining solute transport
problem is entered in the Solute Transport dialog window displayed in Figure 4. In this window
users specify again the Space and Time Weighting Schemes, and additional Solute
Information such as mass units. Note that only one type of mass units can be specified here,
while colloids and solute concentrations may clearly use different mass units. The text specified
in this edit box is used only in texts in graphs and is not involved in any calculations, and thus it
is inconsequential what is specified here. It is important that mass units used for colloids and forsolute are consistent.
The "Attachment/Detachment Concept (Virus, Bacteria Transport)" check box needs to bechecked for the C-Ride Module to be active. Users have to set the "Number of Solutes" to 3(colloids, solute attached to colloids, and solute).

27

Figure 4. The Solute Transport dialog window.
6.3. Solute Transport Parameters
The standard structure of HYDRUS input parameters is preserved also for the colloid-facilitatedsolute transport module. The input parameters for individual components (colloids, soluteassociated with colloids, and solute) are described below. Figure 5 shows the Solute TransportParameters dialog window, in which both soil-specific and solute-specific transport parameters
are specified.

28

Figure 5. The Solute Transport Parameters dialog window.
a) Soil Specific Parameters
These parameters are common for all three components, but different for different soil layers.
Table 3. Soil specific solute transport parameters.
Parameter Notation
Description
3
Bulk. D.
Bulk density [M/L ]ρDisp.L.
Longitudinal dispersivity [L]λL
Disp.T.
Transverse dispersivity [L]λT
Fract.
fFraction of sorption sites in equilibrium with the soil solution [-]. Thisparameters is used only for the third component (solute).
ThImob.
Immobile water content. In the C-Ride module, this parameter can beθimused to account for colloid exclusion (water content from whichcolloids are excluded). Immobile water content for the third component(i.e., solute) is in each case (even when colloids are considered to be
excluded from some fraction of water) considered to be zero.
b) Solute Specific Parameters
These parameters are common for different soil layers, but may be different for different
components.
Table 4. Solute specific solute transport parameters.
Parameter Notation
Description
Diffus. W.
Dw
Molecular diffusion in the liquid phase [LT-2]
D_soild50
Diameter of soil grains [L]. This parameter is used only in the Bradford
blocking straining function for colloids.

29

6.4. Solute Reaction Parameters
When the C-Ride module is used, the Solute Reaction Parameters are specified in the Reaction
Parameters dialog window displayed in Figures 6, 7, and 8, for the first, second, and third solutecomponent, respectively. The Reaction Parameters for the first, second, and third solutecomponent are listed in Tables 5, 5, and 7, respectively.
Table 5. Reaction parameters for the first component (colloids).
No Parameter Notation
Description
1
Kd
Kd
Should be equal to 0. This parameter is usually reserved for thesorption parameter Kd, which is not used with colloid transport[L3/M].
2
Nu
Should be equal to 0 (commonly used for Langmuir η [L3/M]).
η
3
Beta
Should be equal to 1 (commonly used for Freundlich exponent β).
β
4
Henry
KH
Should be equal to 0 (commonly used for the Henry's law constant
KH, which is not used with colloid transport).
5
SinkL1
Should be equal to 0 (commonly used for colloid immobilizationµcw
in the liquid phase µcw). This parameter should not be used when
the colloid-facilitated solute transport is considered.
6
SinkS1
Should be equal to 0 (commonly used for colloid immobilizationµcs
in the solid phase µcs).
7
iPsi2
Type of blocking on the second sorption sites (=0; no blocking;−=1: Langmuirian dynamics; =3: random sequential adsorption
model; =4: depth dependent blocking coefficient).
8
iPsi1
The same for the first sorption sites.
−
9
SMax2
Smax
Parameter in the blocking function for the second sorption sites.
10 AttachS2
kacFirst-order deposition (attachment) coefficient, kac (or kstr) [T-1],
for the second sorption sites.
11 DetachS2
kdc
First-order entrainment (detachment) coefficient, kdc [T-1], for thesecond sorption sites.
12
SMax1
Smax
Parameter in the blocking function for the first sorption sites.
13 AttachS1
kacFirst-order deposition (attachment) coefficient, kac (or kstr) [T-1],
for the first sorption sites.
14 DetachS1
kdc
First-order entrainment (detachment) coefficient, kdc [T-1], for the
first sorption sites.

30

Figure 6. Parts of the "Reaction Parameters for Solute -1" (colloids) dialog window.
Table 6. Reaction parameters for the second component (solute on colloids).
No
1

2
3
4
5
6
7
8
9
10
11
12
13
14

Parameter
Kd

Notation
Description
Kd
Should be equal to 0. This parameter is usually reserved for thesorption parameter Kd, which is not used with colloid transport[L3/M].
Nu
Should be equal to 0 (commonly used for Langmuir η [L3/M]).
η
Beta
Should be equal to 1 (commonly used for Freundlich exponent β).
β
Henry
KH
Should be equal to 0 (commonly used for the Henry's law
constant KH, which is not used with colloid transport).
SinkL1
Should be equal to 0 (commonly used for colloid immobilizationµcw
in the liquid phase µcw).
SinkS1
Should be equal to 0 (commonly used for colloid immobilizationµcs
in the solid phase µcs).
Not usedNot used.
Not usedNot used.
RefImColParameter adjusting the sorption rate to the number of immobile
ψicolloids present.
AttImCol
kaic
Solute adsorption rate to immobile colloids [T-1].
DetImCol
kdic
Solute desorption rate from immobile colloids [T-1].
RefMobCol
Parameter adjusting the sorption rate to the number of colloids
ψmpresent.
AttMobCol
kamc
Solute adsorption rate to mobile colloids [T-1]
DetMobCol
kdmc
Solute desorption rate from mobile colloids [T-1].

31

Figure 7. Parts of the "Reaction Parameters for Solute -2" (solute on colloids) dialog window.
Table 7. Reaction parameters for the third component (solute).
No Parameter Notation
Description
1
Kd
Kd
The sorption distribution parameter Kd [L3/M].
2
Nu
The sorption (Langmuir) parameter η [L3/M].
η
3Beta
The sorption (Freundlich exponent) parameter β.
β
4
Henry
KH
Henry's law constant KH.
5
SinkL1
Degradation rate for contaminant dissolved in the liquid phaseµw[T-1].
6
SinkS1
Degradation rate for contaminant sorbed to the solid phase [T-1].
µs
7
Not usedNot used8
Not usedNot used9
Not usedNot used10
Not usedNot used11
Not usedNot used12
Not usedNot used13
Not usedNot used
14
AlphaFirst-order rate coefficient for one-site or two-site nonequilibriumω
adsorption [T-1].

32

Figure 8. Parts of the "Reaction Parameters for Solute -3" (solute) dialog window.
When the depth-dependent straining coefficient ψsstr is defined using Eq. (6), cBnd6 and cBnd7
values in the Boundary Condition vector are interpreted as z-coordinates of the inlets to materials
1 and 2, respectively. This depth-depending function can thus be used only for up to twomaterials.

6.5. Initial Conditions
Since three solutes are considered when using the C-Ride module, the initial conditions have to
be given for all three solutes (i.e., colloids, solute on colloids, and solute). Initial conditions need
to be specified for (mobile) colloids suspended in the liquid phase (Concentration 1, Cc) and(immobile) colloids attached to the solid phase (Sorbed Noneq. Conc. 1, Scatt), for solute on
mobile colloids (Concentration 2, SmcCc), for solute on immobile colloids (Sorbed Noneq. Conc.
2, SicScatt), for solute in the liquid phase (Concentration 3, C), and for kinetically sorbed solute to
the solid phase (Sorbed Noneq. Conc. 3, Sk) (Figure 9).

Figure 9. The "Initial Conditions" part of Data Tab of the Navigator Bar for the C-Ride module.

33

34

7. Post-processing
7.1. Results – Graphical Display
The output for the C-Ride module is the same as the output for the standard HYDRUS module.
Multiple variables (all three main components) can be displayed in the View window (Figure 10).

Figure 10. The "Results - Graphical Display" part of the Data Tab of the Navigator Bar for the CRide module.
Other concentration modes (e.g., total concentration; concentration at sorption sites 2, totalsorbed concentration) can be selected at the View Tab of the Navigator bar (Figure 10).

Figure 11. The "Concentration Options" part of the View Tab of the Navigator Bar for the CRide module.

7.2. Results – Other Information
Similarly, solute fluxes across boundaries and mass transfers between different phases are displayed
for all three components. In addition to actual and cumulative boundary fluxes, of special interest is"Cumulative Non-Equil. Mass Transfer", which provides information about the mass transfer

35between the mobile and immobile colloids (for the first component - colloids), about the solute masstransfer associated with the mass transfer of mobile and immobile colloids (for the second
component - solute on colloids), and about kinetically sorbed solute (for the third component solute). For the second component (i.e., solute on colloids), "Cumulative Zero-Order Reaction"and "Cumulative First-Order Reaction" give solute mass adsorbed and desorbed to/from colloids,
respectively. Finally, for the third component (i.e., solute), "Cumulative Zero-Order Reaction"and "Cumulative First-Order Reaction" give solute mass desorbed and adsorbed from/to colloids,
respectively. "Cumulative First-Order Reaction" additionally also includes any first-orderdegradation, if considered.

36

8. Example Problems
The C-Ride module supplements the standard HYDRUS program, which includes variablysaturated water flow and solute transport model, and thus the water flow, solute transport andcolloids transport parts of the model have been tested earlier. Therefore, verification and test
examples presented in this section concentrate on the demonstration of the new features of themodel.
Table 8. List of various test examples used to verify various new options of the C-Ride module.
Name
Test01a
Test01b
Test02a
Test02b
Test02c

Flow+
No
No
No
No
No

Description
Only mobile colloids, desorption from mobile colloids.
Only mobile colloids, sorption/desorption from mobile colloids.
Mobile (initially) and immobile colloids, sorption/desorption from colloids.
Mobile and immobile (initially) colloids, sorption/desorption from colloids.
Mobile (initially) and immobile colloids, sorption/desorption from water tocolloids.
Test02d
No
Mobile (initially) and immobile (nonlinear) colloids, sorption/desorption fromcolloids.
Test02eNo
Mobile (initially) and immobile colloids, nonequilibrium sorption/desorption
from colloids.
Test02f
No
Mobile and immobile (initially) colloids, sorption/desorption from colloids, root
water uptake.
Test03a
SSF Colloids and solute enter the column independently.
Test03b SSF Solute enters the column attached to colloids.
Test03c
SSF Solute enters the column attached to colloids, Colloid exclusion.
Test04a
TF
Infiltration of mobile colloids, solute desorption from mobile colloids.
Test04b
TF
Infiltration of mobile colloids (attachment/detachment), solute desorption fromcolloids.
Test04c
TF
Infiltration of mobile colloids (attachment/detachment), solute
sorption/desorption from colloids.
Test04d
TF
Infiltration of mobile colloids (attachment/detachment - 2nd site), solute
sorption/desorption from colloids.
Test04e
TF
Infiltration of mobile colloids (attachment/detachment), solute
sorption/desorption from colloids, root water uptake.
Test04f
TF
Infiltration of mobile colloids (attachment/detachment), solute
sorption/desorption from colloids, nonequilibrium sorption to soil.
Test04g
TF
Infiltration of mobile colloids (attachment/detachment), solute
sorption/desorption from colloids, nonlinear sorption to soil.
Test05
TF
Transient water flow and CFSTr in a field soil profile under grass.
+
No - no flow, SSF - steady-state flow, TF - transient flow
The new module and most of its new features have been extensively tested on test problems ofincreasing complexity. The first set of examples (Test01 and Test02; see Table 8) described in
Section 8.1 was run to evaluate correct mass transfers between different phases, such as colloids

37between mobile and immobile phases, while solute was transferred between liquid and sorbed
phases, between mobile and immobile colloids, and between colloids and liquid phase. This set ofexamples did not consider water flow. The second set of examples (Test03a and Test03b; see Table8) described in Section 8.2 considered steady-state water flow conditions, and solute entering thesoil column either sorbed to mobile colloids (Test03a) or independently of colloids (Test03b).

8.1. Example Set 1 - Transfer Between Different Phases Without Flow
A set of test examples was run to evaluate correct mass transfer between different phases,
considering colloids being mobile and immobile (considering attachment, detachment, andstraining), solute being sorbed and desorbed to/from colloids from/into the liquid phase, and being
sorbed both instantaneously and kinetically to the solid phase. Selected results for selected test runs
(Test02a, Test02b, and Test02e) are provided below.
In the Test02a example, colloids were initially in the liquid phase (mobile) and underwent both
attachment (kac =0.01 min-1) and detachment (kdc =0.005 min-1). Solute was initially sorbed to themobile colloids, and was sorbing and desorbing from both mobile and immobile colloids at a rate of
kamc = kaic = 0.05 min-1 and kdmc = kdic = 0.05 min-1, respectively. Once in the liquid phase, solute
was instantaneously distributed between the liquid and solid phase with a KD equal to 1.

Figure 12. Colloid (mobile and immobile) and solute (liquid, sorbed, and on colloids)
concentrations for examples Test02a (top left), Test02b (top right), and Test02e (bottom).

38

The same rate and distribution coefficients were used in the Test02b example. The only difference
was that colloids were initially attached to the solid phase (immobile). In the Test02e example,
solute was considered to be sorbed kinetically (rather than instantaneously) to the solid phase with arate coefficient ω=0.001 min-1.
Figure 12 shows concentrations of mobile and immobile colloids, and solute in the liquid phase,kinetically sorbed to the solid phase, and sorbed to mobile and immobile colloids for the test
examples Test02a, Test02b, and Test02e. Table 9 provides mass balance calculations for thesetest examples at the beginning and end of the simulations. It provides concentrations for different
phases, solute mass in different phases, as well as the total mass in the entire system for both
colloids and solute.
Table 9. Mass balance calculations for test examples Test02a, Test02b, and Test02e.

Variable
Kd

Test02aInitial
Final

θ
ρ

f
Colloids
Cc
Mw= θ CcSc
Ms= ρ sc
Total= Mw + Ms
SoluteC
Mw= θ C
Se= Kd C
Mse= ρ Se
Sk
Msk= ρ Sk
Cc Smc
Mcw= θ Cc Smc
Sc Sic
Mcs= ρ Sc Sic
Total=Mw+Mse+Msk+Mcw+Mcs

1

Test02b
Initial
Final
1
0.5
1.5
1

Test02eInitial
Final

0

1

0.340700.988910.34070.5
00.17035
0.2198010.494450.67040.5
00.17035
0.2197

00.5
00.32970.50005
0.18461.5
1.5
01.00561.50005
0.3686

00.5
00.32955
0.49990.3075000.09230.1846000.18430.3686000.153750.3075000.27690000.552900000.0795301

00.12870000.7349010.119295
0.22210.5
00.064350.04428010.36745
0.26340.5
00.11105
0.0771300.5

0.066420.499971.5
1.5

0.39511.49975

00.5

0.115695
0.49979

8.2. Example Set 2 - Colloid and Solute Transport Under Steady-State Flow Conditions
A 10 cm long soil column, with a porosity of 0.5 and a saturated hydraulic conductivity of 0.1
cm/min was considered in this set of examples. The flow was considered to be saturated at a flowrate of 0.1 cm/d. The pore volume was thus equal to 50 (Lθs /q = 10*0.5/0.1 = 50). Colloid and39

solute were applied for 60 minutes. In the Test03a example, colloids and solute entered the soilcolumn independently. In the Test03b example, solute entered the soil column sorbed to the colloids.
The same rate and distribution coefficients as in examples Test02 were used (i.e., kac =0.01 min-1,
kdc =0.005 min-1, kamc = kaic = 0.05 min-1, kdmc = kdic = 0.05 min-1, and KD = 1). The retardation factor
for colloids is equal to 1 and for solute to 4 (R=1+ρKd/θ=1+1.5*1/0.5=4).
Breakthrough curves for colloids, solute sorbed to colloids, and dissolved solute for examples
Test03a and Test03b are shown in Figure 13. In the Test03a example, in which colloids and soluteentered the column independently, colloids reached the bottom of the column after about 50
minutes (one pore volume), solute concentrations at colloids (blue line) increased only gradually
due to sorption and showed long tailing (corresponding to tailing of colloids), and most solute mass
appeared mostly at later times, corresponding to the retardation factor of 4. There was only a small
increase in dissolved concentrations earlier because of the solute transported by colloids. In the
Test03b example, in which solute entered the column sorbed to colloids, solute concentration at
colloids mostly mirrored colloid concentrations. Solute concentrations at colloids were slightly
lower due to desorption. The solute mass appeared at the end of the column earlier than in theprevious example, mainly because solute was for the most part transported adsorbed to mobile
colloids.

Figure 13. Breakthrough curves for colloids (black line), solute sorbed to colloids (blue line), andsolute (red line) for examples Test03a (left) and Test03b (right).
8.3. Other Examples - Colloid and Solute Transport Under Transient Flow Conditions
Additional test examples were run for transient water flow conditions. Test04 examples involved
water, colloids and solute infiltrating into the soil profile. In all cases, solute was initially sorbed toinfiltrating colloids. Colloids (except for Test04a) underwent processes of attachment/detachment,
while solute was sorbing and desorbing to/from both mobile and immobile colloids. Additional
processes, such as root water uptake (Test04e), kinetic sorption (Test04f) and nonlinear sorption(test04g), were also evaluated.
Finally, the Test05 example considered complex, 6-month long, transient flow conditions with daily
values of precipitation and evapotranspiration and the presence of groundwater in the soil profile.

40References
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