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Published online 1 August 2007
Published in Vadose Zone J 6:446-457 (2007)
DOI: 10.2136/vzj2006.0061
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Modeling Water Movement in Heterogeneous Water-Repellent Soil: 2. A Conceptual Numerical Simulation

M. Deurera,* and J. Bachmannb

a Sustainable Land Use Team, HortResearch, Tennent Dr., P.O. Box 11030, Palmerston North, New Zealand
b Institute of Soil Science, Univ. of Hannover, Herrenhäuserstr. 2, 30419 Hannover, Germany
* Corresponding author (markus.deurer{at}hortresearch.co.nz).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.


Received 19 April 2006.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Materials and Methods
 Results and Analyses
 Discussion
 Conclusions
 REFERENCES
 
A new approach by Bachmann et al. describes the wetting behavior of soil with reduced wettability with a macroscopic dynamic contact angle concept. The contact angle (CA) is hysteretic and depends on both time and water content. A maximum CA is reached at the permanent wilting point and decreases during rewetting until achieving the minimum CA at a water content close to field capacity. To simulate water flow in a soil with dynamic wettability, we expand the van Genuchten parameterization of hydraulic properties to include the macroscopic CA. Both pore and pore-surface properties can vary in space. We tested the new parameterization approach by simulating one drying–rewetting cycle in a forest soil profile. Our drying–rewetting cycle is a simplified scenario for a seasonal climate where a prolonged dry period is followed by a wet period. We could qualitatively reproduce many observations of water dynamics typically attributed to water-repellent soils. The infiltration of water into hydrophobic soil created a distribution, fingering, and redistribution zone. With rewetting, the fingers slowly widened and finally dissipated. We further analyzed the sensitivity of the simulated water flow processes to the different wettability parameters. Fingers form where the average maximum CA is large (≥90°) but not in case of subcritical water repellency (average maximum CA <90°). The magnitude of the critical water content and the degree of pore heterogeneity also influence the initiation and stability of fingers. The infiltration into the root zone and the drainage out of the soil profile are highly sensitive to the degree and persistence of water repellency.

Abbreviations: CA, contact angle • WDPTT, water drop penetration time test


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Materials and Methods
 Results and Analyses
 Discussion
 Conclusions
 REFERENCES
 
Water repellency occurs in soils of different texture, land use, and a variety of climatic conditions (DeBano, 1981; Doerr et al., 2000; Woche et al., 2005), and it influences environmentally relevant hydrologic processes. Water repellency reduces the infiltration of water (King, 1981) and causes preferential flow (Ritsema et al., 1993). At the catchment scale, water-repellent soils can promote overland flow (Imeson et al., 1992) and rapid stormflow (Doerr et al., 2003).

Several simulation models incorporate specific aspects of a soil's wettability (van Dam et al., 1990; de Rooij, 1995; Nieber et al., 2000; Ritsema et al., 2005). Locations within a soil profile are modeled to be fully wettable or hydrophobic, without significant moisture exchange between these soil domains. Hydrophobicity can be considered as static and either does not change as a function of time or water content (van Dam et al., 1990; de Rooij, 1995; Nieber et al., 2000) or has locations that switch from fully wettable to hydrophobic as long as the water content is below a critical water content (Ritsema et al., 2005). The typical zonation of a partly hydrophobic soil during the infiltration of water into distribution, fingering, and redistribution zones and the details of the fingers such as their number and geometry have been prescribed (de Rooij, 1995; Ritsema et al., 2005). In brief, most conventional studies of water transport in soils have focused either on completely wettable soils or on hydrophobic soils with extreme water repellency. A general analysis of pore structure and continuously changing wetting properties, as suggested by many field studies, has not as yet been conducted for soil.

In this study we first presented a new concept to parameterize the combined macroscopic pore and pore surface properties (Bachmann et al., 2007). In this paper, we implement the concept into a numerical simulation model. The macroscopic pore surface properties describe a soil's wettability over the entire possible range from a subcritical, nonhydrophobic state (CA <90°) to extreme water repellency, that is, hydrophobicity (CA ≥90°). The resulting hydraulic properties are functions of the actual CA; they are time and system dependent and vary in space.

Here, we analyze the sensitivity of the water dynamics in a soil profile with respect to key parameters regarding the wetting properties of the soil. Consequently, our objective is to describe water flow in water-repellent soil by using simple parameterization techniques for describing the soil structure. These properties were additionally combined with parameters describing wetting properties, each of which may also exhibit spatial variability. Compared with conventional studies in heterogeneous soil, the main advantage of this approach is that only a few more parameters are needed to quantify wetting properties of the soil, without any restriction on the intensity or persistence of water repellency. We focus in this paper on the technical description of the extended hydraulic model and on the general behavior of the modeled processes to demonstrate its flexibility and capability for clear and simplified hydrological conditions by avoiding an overlap of too many different complex mechanisms. We simulate and discuss a drying–rewetting cycle in a forest soil profile, including root water uptake, to show the performance of the model. Our drying–rewetting cycle is a simplified scenario for a seasonal climate in which a prolonged dry period is followed by a wet period.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Materials and Methods
 Results and Analyses
 Discussion
 Conclusions
 REFERENCES
 
Parameterization of Macroscopic Pore Properties
A simple rigid pore structure is assumed. The hydraulic behavior of a pore ensemble at the continuum scale is represented by a nonhysteretic water characteristic and hydraulic conductivity function. It is parameterized with the van Genuchten–Mualem model (van Genuchten, 1980). The heterogeneity of hydraulic properties for the hydrophilic medium is represented with the concept of Miller similarity (Miller and Miller, 1956). Specifically, we use a macroscopically Miller-similar medium (Sposito and Jury, 1985). Essentially, the hydraulic properties in a Miller-similar soil are described by the reference hydraulic functions and a set of linear scaling factors. Each location x has a scaling factor that specifies how the local hydraulic properties relate to the reference hydraulic functions. More details on the concept of Miller similarity are given elsewhere (Roth, 1995; Sposito and Jury, 1990). The reference hydraulic functions and the set of scaling factors relate to a wettable soil, as characterized often with conventional drainage water retention curves. The reference water characteristic function h*({Theta}) is defined as

Formula 1[1]
where {alpha}vG* (cm–1) and nvG* (dimensionless) are fit parameters, and the asterisk denotes the reference values. The relative saturation {Theta} (dimensionless) is calculated by

Formula 2[2]
where {theta}r (cm cm–3) is the residual water content and {theta}s (cm cm–3) is the water content at saturation. Using Eq. [1], the water characteristic function at x is

Formula 3[3]
where {alpha}h(x) (dimensionless) is the location-specific scaling factor for the water characteristic function. The reference hydraulic conductivity function K*({Theta}) is defined as

Formula 4[4]
where Ks* (cm h–1) denotes the saturated hydraulic conductivity of the reference and Kr* (dimensionless) is the relative hydraulic conductivity of the reference. Using Eq. [4], the hydraulic conductivity function at x is

Formula 5[5]
In our macroscopically Miller-similar medium, only one scaling factor is needed to describe the heterogeneity of hydraulic properties. However, for the convenience of numerical modeling, we use a separate scaling factor for the water retention [{alpha}h(x)] and the hydraulic conductivity function [{alpha}K(x)] (Simunek et al., 1994). In our macroscopically Miller-similar medium those scaling factors, {alpha}K(x) and {alpha}h(x), are directly related to each other by {alpha}K(x) = [{alpha}h(x)]–2 (Sposito and Jury, 1990).

Combination of Pore-Surface and Pore Properties
The macroscopic hydraulic properties combine both the pore-surface and the pore properties. Therefore, the formulations of the water characteristic (Eq. [1, 3]) and the hydraulic conductivity function (Eq. [4, 5]) are expanded to integrate the impact of the CA as well. This procedure is briefly explained in the following paragraph. The derivation of the actual CA, {kappa}(S,x) (°), and the underlying physical assumptions are found in Bachmann et al. (2007). The actual contact angle depends on the state of the system, S, such as its local water content and matric head (see Eq. 5 of Bachmann et al. [2007]). For the water characteristic function, the parameters {alpha}vG and nvG (Eq. [1]) are assumed to depend on the CA {kappa}(S,x). The surface tension of the soil solution is assumed to be constant. The water characteristic function of each location x is represented by a set of CA-dependent van Genuchten parameters, {alpha}vG'({kappa},x) and nvG'({kappa},x) (Bachmann et al., 2007):

Formula 6[6]
and

Formula 7[7]
where p (dimensionless) is a site-specific calibration factor. The factor p in Eq. [6] signifies that nvG is probably less sensitive to a change in the CA {kappa}(S,x) than {alpha}vG. Therefore, in our model, the possible range of the factor nvG'({kappa},x) becomes a discrete number of values (e.g., 10), where each value represents a CA range (e.g., 10–20°). The CA dependency of the factor {alpha}vG'({kappa},x) can be conveniently included into the location specific scaling factor {alpha}h(x). Combining Eq. [1, 3, 6, 7] yields the CA-dependent water characteristic function h({Theta},{kappa},x):

Formula 8[8]
with

Formula 9[9]
where nvG' is a short-form notation for nvG'({kappa},x) and {alpha}h'({kappa},x) is an effective scaling factor. For {kappa} ≥90°, the local scaling factor is set to the limit value of 0.01. In this case, the matrix repels water when the capillary pressure is zero or negative. Only in the case of positive water pressure larger than a specific threshold value can water penetrate into the matrix. The conceptual background to this retention and penetration behavior is discussed in Bachmann et al. (2007).

The hydraulic conductivity in the form of Eq. [4] is assumed to be unaffected by a CA <90°. For CAs ≥90° the hydraulic conductivity is set to a very small value because the remaining liquid phase is considered as discontinuous. This is realized in the model by setting {alpha}K(x) to the very small value of the isothermal water vapor diffusion coefficient in air (Bachmann, 1998). The CA-dependent hydraulic conductivity K({Theta},{kappa},x) becomes

Formula 10[10]
In total, six variables are needed to parameterize water repellency in our model, of which five are physically based (Table 1).


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  		TABLE 1. Variables that are needed for the parameterization of water repellency. The variables are explained in detail elsewhere (Bachmann et al., 2007.

 

   Materials and Methods
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
Results and Analyses
 Discussion
 Conclusions
 REFERENCES
 
Simulation Model
The water flow in a two-dimensional rigid porous medium is numerically calculated as a solution of the Richards' equation subject to specified initial (Fig. 1F ) and boundary conditions, respectively. The hydraulic properties depend on the system-dependent CA and are updated after every constant and specified time step ti (e.g., 1 h) during the simulation period. For every time step, ti, a loop consisting of four steps is executed:
  1. The distribution of actual CAs, {kappa}ti(S,x), is calculated on the basis of the distribution of matric heads, hti(x), relative water saturations , and rewetting times tirw(x) in the matrix. The values of hti(x) and {Theta}ti(x) are outputs of the numerical calculations of the last time step or constitute the initial condition. The rewetting times, tirw(x), are calculated first (Eq. [4] in Bachmann et al. [2007]), and then the actual CAs, {kappa}ti(S,x), are calculated (Eq. [5–7]GoGo in Bachmann et al. [2007]).
  2. The hydraulic properties are updated using the newly calculated {kappa}ti(S,x) according to Eq. [7–10]GoGoGo.
  3. The initial condition for the next time step, ti+1, is the distribution of matric heads hti+1ini(x). It is calculated with the updated hydraulic properties as a function of the relative water saturations, {Theta}ti(x), that were numerically calculated in the previous time step.
  4. The water flow is numerically calculated for the next time step, ti+1, based on the updated hydraulic properties (step 2), the new initial condition hti+1ini(x) (step 3), and subject to specified boundary conditions. The standard finite-element model SWMS_2D (Simunek et al., 1994) is used to solve the Richards' equation iteratively.
The solution for water flow at ti+1 yields the distribution of hti+1(x) and {Theta}ti+1(x). A new loop starts with the calculation of the new rewetting times trwi+1(x) and actual CAs {kappa}ti+1(S,x) (step 1).


Figure 1
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  		FIG. 1. The (A, B, C, D) soil properties, (E) root density distribution, and (F) initial condition of the simulations. (A) Distribution of scaling factors for the water characteristic function. (B) Distribution of scaling factors for the hydraulic conductivity function. (C) Distribution of the minimum contact angle; the mean contact angle is 5 ± 1.5°. (D) Distribution of the maximum contact angle; the mean contact angle is 90 ± 20°. (E) Distribution of the root density factor b. (F) Distribution of matric heads as an initial condition for the simulation; the averaged matric pressure head of the soil profile is 60 hPa.

 
We wrote an interface for the program SWMS_2D in the language C. For every time step, SWMS_2D iteratively solves the water dynamics with the Richards' equation. The interface retrieves the respective data output, updates CAs and hydraulic properties (steps 1–3), and starts the new time step (step 4). In addition to the standard output of the model SWMS_2D (Simunek et al., 1994), the distribution of {kappa}(S,x), trw(x), {alpha}vG (x) and nvG(x) is generated for each user specified time step. The global mass balance errors of the simulations are <1%, with the exception of the first 2 h of the simulated rewetting phase, where they are <3.7%.

Simulation Scenarios
The water dynamics of one cycle of drying and rewetting in a two-dimensional soil profile with heterogeneous hydraulic properties and with root water extraction was simulated. The soil profile extends 200 cm in the horizontal and 125 cm in the vertical direction (Fig. 1). The grid resolution is 2.5 cm. An atmospheric boundary condition is prescribed at the surface, and free drainage is prescribed to the bottom of the profile. The initial distribution of matric heads results from simulating the drainage of a fully saturated soil profile to a mean matric head of 60 hPa (Fig. 1F) in a previous run. This initial condition represents a wettable soil profile.

The root water extraction function S(h) in SWMS_2D depends on the matric head (Feddes et al., 1978) according to:

Formula 11[11]
where the response function ar(h) is a prescribed dimensionless function with four characteristic matric heads: h1 (0 hPa), h2 (0.001 hPa), h3 (200 hPa), and h4 (10,000 hPa). There is no root water extraction for matric heads smaller than h1, where anaerobiosis is presumed to prevail. Above h2, roots start to extract water at the maximum possible rate up to h3. At matric heads larger than h3, root water extraction decreases and is zero at h4, where the permanent wilting point is reached. It is not clear how root water extraction reacts on water repellency. Therefore, root water extraction is set to zero in areas with reduced wettability ({kappa} > {kappa}min). Root water extraction takes place in a rectangle that extends 120 cm in the horizontal direction and 70 cm in the vertical direction (Fig. 1E). The geometry of the root density distribution is simplistic (Fig. 1E), mimicking the root system of an individual tree, and is represented by a dimensionless root density factor b(x) (Vogel, 1987):

Formula 12[12]
where Tp (cm h–1) is the potential transpiration rate and Lt (cm) is the width of the soil surface associated with the transpiration process. Integrating the function b(x) over the entire root zone {Omega}R yields unity:

Formula 13[13]
where {Omega} denotes the entire flow region.

In the simulated scenarios the soil dries out for 21 d. During this time there is no precipitation, and the potential evapotranspiration rate is 0.42 cm d–1. After Day 21, a rewetting period of 7 d occurs in response to a constant precipitation rate of 2 cm d–1, and a potential evapotranspiration rate of 0.42 cm d–1 prevails.

The heterogeneity of the hydraulic properties is implemented in the form of four random fields (Fig. 1A–1D). For the Miller-similar pore structure, one set of reference parameters ({alpha}vG = 0.047 cm–1, nvG = 1.8, {theta}r = 0.005 cm3 cm–3, {theta}s = 0.3 cm3 cm–3, Ks = 4 cm h–1) is used. This represents the average hydraulic properties of a wettable soil profile with sandy texture. We assume that the hydraulic conductivity and, therefore, {alpha}K(x) are lognormally distributed. Following a classical paper of water flow in a Miller-similar medium (Roth, 1995), the generation of scaling factors proceeds in two steps. First, a random normal deviate l with variance {sigma}l2 = 0.1, expectation <l> = 0, and an exponential autocovariance function with a correlation length of 12 cm is generated using spectral methods (Robin et al., 1993). Next, the random variable l is backtransformed to yield {alpha}h(x) and {alpha}K(x) with the following choice (Roth, 1995):

Formula 14[14]

Formula 15[15]

By the use of Eq. [14] and [15], only the expectation of the hydraulic conductivity function equals that of the reference state and is independent of {sigma}l2 (Jury et al., 1987), while the expectation of the water characteristic function varies with {sigma}l2 (Roth, 1995).

The random fields for {kappa}min(x) and {kappa}max(x) are assumed to be normally distributed and perfectly and positively correlated. The exponential autocovariance function has a correlation length of 2.5 cm (Fig. 1C and 1D). With a grid resolution of 2.5 cm, this is equivalent to implementing a pure nugget effect. The sensitivities of the water dynamics to a variation of the key parameters of the pore-surface properties are analyzed. The maximum CA, {kappa}max(x), assumes mean values of 5°, 70°, 80°, 90°, 100°, 110°, and 130°, with a standard deviation of 1.5° for a mean value of 5° and 20° for the other values. We use the characteristic rewetting time t0 of 10, 11, 12, 13, 14, and 20 d, the critical water content, {theta}crit (i.e., the water content where the CA switches to the maximum value during drying), with the maximum CA of 0.01, 0.015, 0.02, 0.025, 0.03, and 0.035 cm3 cm–3, and with the full wettability ({theta}FK) reached at matric heads of 10, 20, 40, 60, 80, and 100 hPa.


   Results and Analyses
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Analyses
Discussion
 Conclusions
 REFERENCES
 
Simulated Drying–Rewetting Event
After a dry period of 20 d with a high average potential evapotranspiration rate, the volumetric water contents within the root zone are low, and often below the critical water content of 0.015 (Fig. 2A and B ). The zone with the highest root water extraction dries out most. This zone is irregularly shaped, a result of an interaction of the rectangular and uniform root density distribution (Fig. 1E) and the heterogeneous Miller-similar pore structure (Fig. 1A and 1B). The actual CA increases to the maximum CA (on average, 90°) wherever the volumetric water content is below the critical water content (Fig. 2B). As a consequence, the local hydraulic properties in the form of the van Genuchten parameters {alpha}vG and nvG change (Eq. [6] and [7]), reflecting the degree of actual water repellency.


Figure 2
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  		FIG. 2. The distribution of volumetric water contents at three times during the simulated drying–rewetting cycle (B, D, E; right-hand side) with and (A, C, E; left-hand side) without considering water repellency. Locations where the actual contact angle is above the minimum contact angle (reduced wettability) are marked by dots. (A, B) End of the drying phase: after 20 d without precipitation and at a constant potential evapotranspiration rate of 0.42 cm d–1. (C, D) Start of the rewetting phase: after 2 d with a constant precipitation rate of 2 cm d–1. (E, F) End of the rewetting phase: after 7 d with a constant precipitation rate of 2 cm d–1.

 
The drying event is followed by a 7-d period of constant precipitation (2 cm d–1). The dry root zone impedes water infiltration at the start of the rewetting phase and triggers lateral flow (Fig. 2C and 2D). The reduced wettability, with a mean maximum CA of 90°, greatly enhances this effect (Fig. 2D). The water front moves downward as a sharp homogeneous front in a fully wettable root zone (Fig. 2C). A large fraction of the infiltrated water bypasses the water-repellent root zone and forms a distribution zone (Fig. 2D). At the top of the water-repellent root zone, the volumetric water contents locally rise to close to saturation. The actual CAs and the resistance to flow at these points rapidly decrease, and the water breaks through in the form of fingers (Fig. 2D). The fingers dissipate in the fully wettable subsoil (see Fig. 2D, at about 95-cm horizontal and 45-cm vertical distance). The water that is channeled through the fingers and the water that bypasses the water repellent root zone form a redistribution zone below the fingering zone.

The simulated increase of wettability in the root zone during the rewetting event depends also on the persistence of water repellency (Bachmann et al., 2007). Therefore, even after 140 mm of cumulative infiltration, small areas with reduced wettability and small volumetric water contents remain (Fig. 2F).

If no water repellency is considered, then a dry location (grid node in the model) will be rewetted by a rapid decrease of the matric head over several orders of magnitude (Fig. 3A ). This is accompanied by a corresponding gradual increase of the volumetric water content (Fig. 3B). The rewetting is completed once the water front has passed the grid node (Fig. 2C).


Figure 3
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  		FIG. 3. The impact of water repellency on the simulated behavior of two locations (grid node 1003 at 75-cm horizontal and 30-cm vertical distance, and grid node 1089 at 87.5-cm horizontal and 32.5-cm vertical distance) during the rewetting phase. The time 0 is the start of the rewetting phase. Water repellency is not considered for A–B and is considered for C–H. Simulation without considering water repellency: (A) Matric heads and actual contact angles (dashed line) at node 1003. (B) Volumetric water contents at node 1003. Simulation with considering water repellency: The pore surface properties are on average {kappa}max 90° ± 20° and {kappa}min 5° ± 1.5°; at node 1003 they are {kappa}max = 93.6° and {kappa}min = 5.27°; at node 1089 they are {kappa}max = 67.1° and {kappa}min = 3.28°. Node 1003 is outside of fingers and node 1089 is inside a finger. (C) Matric heads and actual contact angles (dashed line) at node 1003. (D) Volumetric water contents and actual contact angles (dashed line) at node 1003. (E) The values of the parameter {alpha}vG during the simulation and the actual contact angles (dashed line) at node 1003. (F) The values of the parameter nvG during the simulation and the actual contact angles (dashed line) at node 1003. (G) Matric heads and actual contact angles (dashed line) at node 1089. (H) Volumetric water contents and actual contact angles (dashed line) at node 1089.

 
Details of the Simulated Rewetting Process
The Gradual Decrease of the Contact Angles
In a dry and hydrophobic root zone, a grid node can be gradually or rapidly rewetted. A gradual rewetting extends from the area surrounding the water-repellent node, for example, from locations outside the root zone that are wetter than the critical water content. This mimics the transfer of water vapor and initializes a gradual decrease of the actual CA. As a consequence, the matric heads of the water-repellent node slowly increase, creating a hydraulic gradient that drives the rewetting process of this node (Fig. 3C). The increase of matric heads at the node during rewetting is spike-like (Fig. 3C). The persistence of water repellency is a function of the water content in the area surrounding the water repellent node and of the characteristic rewetting time, t0 (Bachmann et al., 2007).

The decrease of the actual CA at a grid node continuously modifies its hydraulic properties. The parameters {alpha}vG (Fig. 3E) and nvG (Fig. 3F) change and cause a spike-like increase of the matric heads. This is followed by an equilibration phase during which the matric head of the water-repellent node equilibrates with the matric heads of the surrounding less water repellent nodes (Fig. 3C). During the gradual rewetting process, the volumetric water contents slowly increase (Fig. 3 D). The water contents rise sharply when the rewetting water front captures the entire zone around the previously water repellent node (Fig. 3D after about 120 h of precipitation).

The Occurrence of Fingering
The rapid rewetting process is maintained through fingering. The fingers connect nodes along a path of least resistance to flow. Compared with the surrounding nodes, these locations tend to have smaller CAs and smaller values of the parameters {alpha}vG and nvG (Eq. [6], [7]) and, therefore, more negative water entry pressures. Laboratory experiments showed that for the wetting process, fingers start and propagate according to the spatial and temporal distribution of the water entry pressures (Wang et al., 1998). Individual grid nodes can be only subcritically water repellent even if the actual CA in the water-repellent root zone is on average 90°. When the finger meets a node, then its matric head sharply decreases (Fig. 3G) and its volumetric water content sharply increases (Fig. 3H). This is followed by a slow increase of the wettability of the nodes surrounding the finger and leads to a gradual widening of the finger. As a consequence, the matric heads in the core of the finger slightly increase (Fig. 3G) and the water contents slightly decrease (Fig. 3H). This behavior is typical for a region behind the tip of a finger in a hydrophobic soil (Bauters et al., 2000).

In most of our simulated scenarios, the water repellencies of the nodes where fingering originates are subcritical, and the water entry pressures are negative. Therefore, we simulated that fingers are mostly initiated under unsaturated conditions. In the field, the water contents in the distribution zone, and in the fingers, were also observed to be below saturation (Ritsema et al., 1993). Fingers are stabilized because of the hysteresis of the water characteristic function. Nodes outside a finger mostly have larger actual CAs than nodes within a finger. Consequently, nodes outside a finger also have mostly larger values of the parameters {alpha}vG and nvG. This creates the necessary hysteresis of the water characteristic function. If the degree of hysteresis is strong enough, fingers are predicted to be stable (Nieber, 1996; Nieber et al., 2000). However, with the breakdown of water repellency the fingers also vanish. The water repellency and the values of the parameters {alpha}vG and nvG in the core of the finger decrease (see Fig. 3E and 3F) while the capillary forces of nodes bordering the fingers increase with time during infiltration. As a consequence, capillary diffusion increases, and the fingers widen and slowly dissipate.

The degree of the hysteresis of the water characteristic function in a dry water-repellent soil, with water contents below {theta}crit, depends on, among other factors, the mean maximum CA. Fingers form in a hydrophobic root zone (mean {kappa}max ≥90°, Fig. 4B and 4C ) but not in a root zone with subcritical water repellency (mean {kappa}max = 70°, Fig. 4A). Here, the hysteresis of the water characteristic functions is too small. The local {kappa}max determines the magnitude of change of {alpha}vG and nvG due to water repellency.


Figure 4
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  		FIG. 4. The impact of a different parameterization of water repellency on the distribution of volumetric water contents at the start of the rewetting phase (after 3 d of precipitation). All simulations use the parameter set with {sigma}l2= 0.1, {kappa}min = 5° ± 1.5°, {kappa}max = 90° ± 20°, {theta}crit = 0.015, t0 = 10 d, and {theta}FK at 60 hPa. Only one of these parameters is modified for each of the simulations: (A) {kappa}max = 70° ± 20°, (B) {kappa}max = 90° ± 20°, (C) {kappa}max = 130° ± 20°, (D) {theta}crit = 0.03, {sigma}l2 (E) = 0.05, (F) t0 = 20 d.

 
We formulate both {alpha}vG and nvG as functions of the actual CA. Both the water entry pressure and the slope of the imbibing curve of the water characteristic function vary for soils with different degrees of water repellency (Bauters et al., 1998; Nguyen et al., 1999). Others have suggested using only a CA specific linear scaling factor in combination with the water characteristic function of the fully wettable soil (Bauters et al., 2000).

Sensitivity of Water Flow to Water Repellency–Related Parameters
The probability that the water flow encounters a node with only subcritical water repellency decreases with an increase in {kappa}max. Fingers originate under unsaturated conditions at nodes with only subcritical water repellency. Therefore, after 3 d of precipitation, five fingers have occurred when the mean of {kappa}max is 90° (Fig. 4B) but only one finger when the mean of {kappa}max is 130° (Fig. 4C). Whether the mean value of {kappa}max, its geostatistics, or other key pore-surface parameters such as t0 remain invariant in consecutive drying–rewetting cycles is not clear. The stability of the fingers depends on the persistence of water repellency. A larger characteristic rewetting time slows the widening of the fingers down. Also, water infiltrates more slowly into the hydrophobic root zone. For example, a t0 of 20 d (Fig. 4F) compared with 10 d (Fig. 4B) reduces the number of fingers. The hydrophobic area that remains after Day 24 is larger.

The occurrence of fingers also depends on the interaction of the critical water content with the heterogeneity of the pore structure. In a Miller-similar porous medium, there is no heterogeneity of volumetric water contents at {theta}r and {theta}s. The closer the critical water content is to the residual water content, the smaller the variation in local water contents will be. Consequently, once the water content somewhere within the root zone drops below the critical water content, a large continuous area will become hydrophobic within a short time interval. If the critical water content is considerably higher than the residual water content, then large variations in local water contents exist when the first patches within the root zone drop below the critical water content. Only small patches become water repellent during a small time interval, while bordering areas remain wettable. During infiltration water moves along the wettable patches and fingers do not occur (Fig. 4D).

A very small heterogeneity of the pore structure, here implemented by a small variance of the scaling factors {sigma}l2, can also reduce the variability of local water contents. This, for example, can lead to larger areas that become water repellent in a short time interval. Therefore, under otherwise identical conditions, the water-repellent area at Day 24 is larger for {sigma}l2 = 0.05 (Fig. 4E) compared with {sigma}l2 = 0.1 (Fig. 4B).

The water repellency and water dynamics in the root zone (Fig. 5 ) are sensitive to the different parameters describing the actual CA. Once a part of the root zone dries out to volumetric water contents below {theta}crit, the average actual CA of the root zone rises above the mean {kappa}min of 5° (Fig. 5C), and the water repellent area increases (Fig. 5D). For the parameter sets with {theta}crit = 0.015 (parameter sets 1, 2, 3, 4, 6) and with the selected initial and boundary conditions, this takes about 14 d, while it starts after 12 d if {theta}crit = 0.030 (parameter set 5). With higher critical water content (parameter set 5 in Fig. 5), the water-repellent area within the root zone is interspersed with many wettable patches. Consequently, the average CA (Fig. 5C) and the water-repellent area (Fig. 5D) are smaller than for scenarios with a smaller {theta}crit. The wettability characteristics for the root zone (Fig. 5C and 5D) of the scenario with a mean {kappa}max of 90° and {theta}crit of 0.03 are almost identical to the scenario with a mean {kappa}max of 70° and {theta}crit of 0.015. Hence, not only the potential hydrophobicity ({kappa}max) but also other parameters govern a soil's actual wettability.


Figure 5
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  		FIG. 5. The impact of a different parameterization of water repellency on (A–E) the dynamics of water flow and repellency in the root zone and (F) the drainage rates out of the soil profile. All simulations use the parameter set with {sigma}l2 = 0.1, {kappa}min = 5° ± 1.5°, {kappa}max = 90° ± 20°, {theta}crit = 0.015, t0 = 10 d, {theta}FK at 60 hPa. Only one of these parameters is modified for each simulation set: 0: {kappa}max = {kappa}min, 1: {kappa}max = 70° ± 20°, 2: {kappa}max = 90° ± 20°, 3: {kappa}max = 130° ± 20°, 4: t0 = 20 d, 5: {theta}crit = 0.03, 6: {theta}FK at 20 hPa. Entire drying–rewetting cycle (vertical line denotes the start of the rewetting phase); the averages presented here are the arithmetic average of all nodes within the root zone (Fig. 1E): (A) Average volumetric water contents of the root zone. (B) Average matric heads of the root zone. (C) Average actual contact angles of the root zone. (D) Relative areas of the entire root zone where the actual contact angle >{kappa}min. Rewetting phase only: (E) Average infiltration rate along a transect at the top of the root zone. The transect extends in 12.5-cm depth from 75- to 125-cm horizontal distance. The rate is in percentage of the precipitation rate; each symbol denotes 24 h. (F) Average drainage rate at the bottom of the soil profile in percentage of the precipitation rate; each symbol denotes 24 h.

 
The parameter t0 and the matric head used to derive {theta}FK influence how quickly a water-repellent soil becomes again fully wettable. After the start of the infiltration, both a scenario with a longer characteristic rewetting time t0 (parameter set 4) and one with a smaller matric head for {theta}FK (parameter set 6), show a higher actual CA in the root zone (Fig. 5C). A larger water-repellent area remains over time (Fig. 5D) than for the standard parameter set (parameter set 2). Even if a soil is potentially severely water repellent (e.g., parameter set 3 in Fig. 5), the average actual CA in the root zone is, after 3 wk of drying, only about 50°. The soil in the root zone would be classified as subcritically water repellent. The impact of different degrees of water repellency on the average volumetric water contents or matric heads in the root zone (Fig. 5A and 5B) is small. Therefore, a time series of matric heads and water contents without high spatial resolution gives little indication, for example, if the soil within the root zone is partly severely (parameter set 3 in Fig. 5) or only subcritically water repellent (parameter set 1 in Fig. 5).

Infiltration into the root zone (Fig. 5E) and the drainage out of the soil profile (Fig. 5F) are highly sensitive to the degree and persistence of water repellency. Severe water repellency (e.g., parameter set 3) leads to a small infiltration rate into the root zone (Fig. 5E) and a bypass of water around the root zone (Fig. 4C). This reduces the soil-profile scale transport volume. Consequently, the drainage rate rapidly increases (Fig. 5F). At the catchment scale it might be a mechanism that generates rapid stormflow.

The more persistent the water repellency, as prescribed by a higher value for t0 (parameter set 4) or a smaller matric head used to derive {theta}FK (parameter set 6), the smaller the increase of the infiltration rate into the root zone (Fig. 5E) and, likewise, the faster the increase in the drainage rate (Fig. 5F). Higher critical water contents (parameter set 5), or smaller maximum CAs, increase the infiltration rate. This leads to a slower increase in the drainage rate. However, even a subcritical water repellency (parameter set 1) with a mean {kappa}max of 70° has a pronounced influence on infiltration and drainage (Fig. 5E and 5F) compared with a wettable soil (parameter set 0). Up to a cumulative infiltration of about 80 mm (~90 h after the start of the infiltration), the infiltration into a soil with a mean {kappa}max of 70° is smaller than that for a wettable soil (Fig. 5E).


   Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Analyses
 Discussion
Conclusions
 REFERENCES
 
We based our parameterization of water repellency on different experiments observed in our own investigations or reported in the literature (Bachmann et al., 2007). Currently, no single soil data set is available from which both the pore structure– and the pore surface–related hydraulic properties can be derived simultaneously. Our simulations are, therefore, conceptual. The objective of the simulations was to analyze qualitatively the usefulness of our new concept to parameterize soil water repellency (Bachmann et al., 2007). As the next step, we suggest using the model to design an appropriate experiment to parameterize and validate the model. Properties and processes other than water repellency that influence water flow are represented simplistically in the simulations in order to distinguish the consequences of the parameterization of water repellency on water flow processes from other factors. Following other simulation studies (Birkholzer and Tsang, 1997; Roth, 1995; Russo and Dagan, 1991; Tseng and Jury, 1994) we use only a single scaling factor to include the pore structure–related heterogeneity of hydraulic properties. One (Rockhold et al., 1996) or several scaling factors (Deurer et al., 2001; Hammel et al., 1999; Jury et al., 1987; Warrick et al., 1977) were needed to describe heterogeneous hydraulic properties in the field. The magnitude of heterogeneity of the scaling factor was taken from a study in a sandy soil (Russo and Bresler, 1980) and was small compared with other field studies (Eching et al., 1994; Jury et al., 1987).

We used a forest soil profile with a sandy texture without pronounced horizons. The water repellency–related parameters also have no depth dependency. No systematic depth dependency of the maximum CA was observed in the root zone of some soils under forest (Woche et al., 2005). Our root density distribution is rectangular and uniform. We did not consider a linear or nonlinear variation of the root distribution with depth (Hao et al., 2005) or a multidimensional root water uptake function (Vrugt et al., 2001).

The boundary conditions represent a simplified scenario for a seasonal climate in which a prolonged dry period (e.g., summer) is followed by a wet period (e.g., autumn–winter). Many field studies have observed a seasonal pattern for water repellency with a maximum in the dry summer and a minimum in the wet winter (Burch et al., 1989; Leighton-Boyce et al., 2005; Doerr and Thomas, 2000; Imeson et al., 1992; Jungerius and de Jong, 1989).

Simulated Drying–Rewetting Behavior
We simulated that after 3 wk of intensive evapotranspiration, a soil at field capacity becomes water repellent. In soils of a sandy texture under Eucalyptus plantations in Portugal, a period of 3 to 9 wk of dry weather was required for water repellency to become reestablished after the previous wet period (Leighton-Boyce et al., 2005). However, a different study reported that a period as short as 6 to 9 d of hot dry weather was long enough to reestablish water repellency (Crockford et al., 1991). The behavior of the water-repellent soil profile during rewetting agrees with field observations. The simulated water flow in the water repellent soil profile creates a distribution, fingering, and redistribution zone similar to that described in field experiments (Ritsema et al., 1993). Other models prescribe the spatial organization of flow domains within a water-repellent soil profile a priori. For example, Ritsema et al. (2005) divided the soil profile into a distribution, finger, and redistribution zone. We simulated the occurrence of fingering through the water-repellent zone and bypass-flow around this zone as was reported from field experiments (van Dam et al., 1990; Hendrickx et al., 1993; Ritsema et al., 1993). The simulated fingering is a result of the interaction of the boundary conditions and the parameterization of hydraulic properties. In other models the spatial locations of fingers have to be specified and triggered with a prescribed initial condition (Nieber, 1996; Nieber et al., 2000).

We selected a rewetting scenario with a cumulative infiltration amount of 140 mm, the amount needed to rewet the entire root zone of the fully wettable soil (Fig. 2E). This will take longer in the case of the water-repellent soil. After a cumulative infiltration of 140 mm, small areas with reduced wettability and small volumetric water contents remain (Fig. 2F). In the field water-repellent areas were observed to remain dry during periods of heavy rainfall or even throughout the wet season (Dekker and Ritsema, 2000; Doerr et al., 2000). Within these dry islands, salts can accumulate, and moisture-dependent microbiological reactions such as the mineralization of organic compounds can be delayed.

Details of the Simulated Rewetting Process
In the field water repellency was observed to be a function of the water content (de Jonge et al., 1999), and an inverse relationship of soil moisture and water repellency has been identified (King, 1981; Witter et al., 1991). However, it is difficult to interpret field data that show a connection between water repellency and water content. For example, the persistence of hydrophobicity or the size of the entire hydrophobic area can also affect its breakdown. This may explain the great variation in times, ranging from <1 h to >30 d, needed for the breakdown of hydrophobicity in soils under Pinus and Eucalyptus species in Portugal (Doerr and Thomas, 2000). It is not clear if the water-content dependency of water repellency is a unique function of the specific quality of organic matter (Doerr et al., 2000). Our parameterization of the water-content dependency of water repellency (Bachmann et al., 2007) needs experimental confirmation. The same applies to the characteristic rewetting time t0. The water drop penetration time test (WDPTT; Letey, 1969) is often used to describe the time dependency of hydrophobicity. However, the WDPTT cannot be used to analyze soils with a subcritical water repellency (Bachmann et al., 2003). It does not only represent the persistence of water repellency, but it is also dependent on the actual CA at the beginning of the test. This led to the distinction between actual and potential water repellency (Dekker and Ritsema, 1994). Consequently, the t0 values cannot be derived from a set of WDPTT data alone. We have assumed, therefore, the simplest case of a unique persistence of water repellency for a site with a single value for t0. Future experiments may show that t0 is instead a distribution of spatially correlated values.

In a soil with a heterogeneous pore and pore-surface distribution, the water entry pressures spatially vary. Fingers are automatically initiated, and their pathways meander (Bauters et al., 2000; Ritsema et al., 1998). The fingers follow the path of least resistance to flow. In other water-flow simulation studies that used soils with heterogeneous hydraulic properties, the water was also channeled along meandering pathways (Birkholzer and Tsang, 1997; Roth, 1995). If a soil is modeled with homogeneous pore and pore-surface properties, it is necessary to trigger fingers in perturbation zones with specific initial and boundary conditions, and the fingers propagate vertically downward, perpendicular to the soil surface (Nieber, 1996; Ritsema et al., 1998). The width of the fingers during wetting depends on the correlation lengths of the water entry pressures. The spatial distribution of water entry pressures in the dry water repellent area of the modeled soil with small pore heterogeneity is determined by the geostatistics of the maximum CA. On the basis of the results of others (Hallet et al., 2004), we used a pure nugget effect for the spatial structure of {kappa}max. Consequently, the width of isolated fingers has the dimension of the grid resolution (2.5 cm). Once fingers merge, their width increases. In chamber experiments with water-repellent and non–water-repellent dry sandy soils, the finger width during wetting was about 3 cm (Bauters et al., 1998). For the first time, we managed to simulate the gradual dissipation of fingers in a water-repellent soil. This process has been observed during infiltration in a water-repellent sandy soil. The fingers formed and then broadened and converged with a breakdown of repellency (Carrillo et al., 2000). We also simulated that fingers form in a hydrophobic soil but not in a soil with only subcritical water repellency. The more severe the water repellency, the higher was the tendency for finger formation in a sandy soil during infiltration (Carrillo et al., 2000).

Sensitivity of Water Flow to Water Repellency–Related Parameters
In our model the potential hydrophobicity (maximum CA, {kappa}max) varies in space but not in time. This may be appropriate for our conceptual scenario of one drying–rewetting cycle. More experiments are needed to understand the behavior of {kappa}max over longer periods of time. It is not clear what happens with the organic compounds that caused water repellency after rewetting (Doerr et al., 2000). Some experiments suggest that a new input of hydrophobic substances is necessary to reestablish hydrophobicity (Doerr and Thomas, 2000), whereas other authors reported water repellency at the same level after repeated wetting–drying cycles (Bachmann et al., 2007). Following King (1981) we implemented a water-content dependency of water repellency between two water-content thresholds, {theta}crit and {theta}FK. Several studies have reported the influence of water content on water repellency over a specific range of water contents. Soils with loamy sand to sandy texture in Eucalyptus plantations in Portugal were water repellent when the soil moisture was <14% and fully wettable when the soil moisture was >27% (Leighton-Boyce et al., 2005). Such a range has been termed a "transition zone" (Dekker et al., 2001). The interaction of the weather, vegetation, and combined pore and pore-surface properties governs the water dynamics of a soil. In our simulated scenarios it is in the area of the root water extraction where the water contents vary most and where water repellency occurs. The vegetation plays a key role not only as a source of hydrophobic organic compounds but also as a seasonal sink for water. The spatial frequency of repellency followed a moisture-related seasonal cycle, where water repellency was contiguous in dry late-summer conditions. But it was entirely absent after wet winter conditions (Leighton-Boyce et al., 2005). Others have also suggested a seasonal cycle of water repellency (Doerr and Thomas, 2000; Keizer et al., 2005).

With our simulations we have explored the sensitivity of infiltration and bypass flow to water repellency related parameters. Our simulations reproduced the experimental finding that the infiltration capacity is lowest for dry conditions and increases with increasing moisture (Dekker and Ritsema, 1996). The opposite is true for wettable soils (Beven, 2001). At the soil-profile scale, we found that the infiltration rate is a function of the CA, as was already shown by King (1981). The impact of water repellency on large-scale infiltration-related processes such as rapid stormflow (Doerr et al., 2003) and erosion (Shakesby et al., 2000) is still poorly understood. Here, our model may be useful to design appropriate experiments. Many soils exhibit at least a subcritical water repellency (Hallet et al., 2001; Lamparter et al., 2006; Wallis et al., 1991). It should, therefore, be a standard procedure to incorporate time- and system-dependent wettability into large-scale hydrological models.


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Analyses
 Discussion
 Conclusions
REFERENCES
 
We have proposed a new approach to describe the wetting properties of soil with a dynamic CA concept, which allows the simulation of moisture transfer in soil. Our model to simulate water movement in water-repellent soil introduces only a few additional parameters needed to consider water repellency. There are four major differences to other two-dimensional numerical models that consider water repellency:

  1. Water repellency is represented and parameterized as an intrinsic hydraulic property using a macroscopic CA. Other models lump the effect of water repellency on the water retention together with the effect of pore structure–related properties (Nieber, 1996; Nieber et al., 2000; Ritsema et al., 2005).
  2. Hydraulic properties are formulated as functions of the actual water repellency. As such they are continuous functions of the soil system's state (water content, time since rewetting).
  3. Pore structure and key water-repellency parameters have specified autocovariance functions. The parameters of the hydraulic properties have no specified autocovariance functions in other models (Nieber, 1996; Nieber et al., 2000; Ritsema et al., 2005).
  4. The impact of water repellency on water and solute flow is not prescribed. Other models prescribe the spatial organization of flow domains within a water-repellent soil profile. Ritsema et al. (2005) divided the soil profile into a distribution, finger, and redistribution zone. The spatial locations of fingers during water flow through a water-repellent soil were prescribed and triggered with a prescribed initial condition in the model of Nieber (1996) and Nieber et al. (2000).
To test the impact of the time- and system-dependent parameterization of hydraulic properties, we simulated one drying–rewetting cycle in a forest soil profile. During 3 wk of drying, the interaction of the root density distribution and the heterogeneous pore structure generated a water-repellent area. Vegetation plays a key role both as a source of hydrophobic organic compounds and as a sink of water. The infiltration of water into the dry and partly hydrophobic soil profile shows the classical zonation that is expected for a water-repellent soil with distribution, fingering, and redistribution zones.

The rewetting of the water-repellent area is either by a gradual rewetting or by fingering. A gradual water content– and time-dependent decrease of the CAs is coupled with a gradual change of the parameters of the water characteristic function. During infiltration, fingers start at the top of the water-repellent part of the root zone and follow the locations where the actual CAs are smaller than in the surroundings. The fingers are stabilized by a hysteresis of the water characteristic function that is caused by the pore-surface properties. With time the CAs in locations bordering the fingers decrease and the fingers widen and slowly dissipate.

Fingers form in a hydrophobic root zone with a large hysteresis of the water characteristic function, but not in a root zone with subcritical water repellency where hysteresis is small. Also, the critical water content plays a key role in the initiation of fingers. If the critical water content is considerably higher than the residual water content, then large variations in local water contents exist when the first patches within the root zone drop below the critical water content. Therefore, only small patches become water repellent, while bordering areas remain wettable. During infiltration water moves along the wettable patches. Fingers do not occur. A small heterogeneity of the pore structure also reduces the variations in local water contents. It leads to larger continuous areas that become water repellent and can thus promote fingering. The characteristic rewetting time and the matric head where the soil is again wettable influence the breakdown of water repellency. Therefore, a soil's potential hydrophobicity governs the actual wettability along with other parameters. Overall, our model can qualitatively reproduce the water dynamics that were observed. The model's results also show the need for specific experiments to derive better parameters and processes, as, for example, the water-content dependency of water repellency.

Infiltration into the water-repellent domain and drainage out of the soil profile are both highly sensitive to the degree and persistence of water repellency. The most severe water repellency leads to the smallest infiltration rate into the root zone and a bypass of water around the root zone. Consequently, the drainage rate out of the profile quickly increases. At the catchment scale, this may be responsible for a rapid stormflow response. However, even a subcritical water repellency with a mean maximum contact angle of 70° still has a pronounced influence on infiltration and drainage. More persistent water repellency, for example, as prescribed by a higher value for the characteristic rewetting time, results in smaller increases of the infiltration rate into the root zone and faster increases in the soil profile drainage rate.


   ACKNOWLEDGMENTS
 
We thank Drs. B. Clothier and I. Vogeler for their review of an earlier draft of this paper.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Analyses
 Discussion
 Conclusions
 REFERENCES
 




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J. Bachmann, M. Deurer, and G. Arye
Modeling Water Movement in Heterogeneous Water-Repellent Soil: 1. Development of a Contact Angle Dependent Water-Retention Model
Vadose Zone J., August 1, 2007; 6(3): 436 - 445.
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