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Published online 9 April 2007
Published in Vadose Zone J 6:207-220 (2007)
DOI: 10.2136/vzj2006.0115
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Rivulet Approach to Rates of Preferential Infiltration

Peter Germann*, Andreas Helbling and Tomaso Vadilonga

Soil Science Section, Dep. of Geography, Univ. of Bern, Hallerstrasse 12, 3012 Bern, Switzerland
* Corresponding author (germann{at}giub.unibe.ch).

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 16 April 2006.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Materials and Methods
 Data
 Application
 Discussion
 Conclusions
 REFERENCES
 
Preferential infiltration is proposed to occur under atmospheric pressure but in unsaturated soil. Its domain is positioned between the domains of Richards' equation and Darcy's law. Rivulets in the form of tiny water streaks are considered to be the basic units of preferential infiltration. They move under atmospheric pressure and in pores that are filled with air before infiltration. Stokes flow relates variations in soil moisture with velocities of wetting and draining fronts, and with volume flux densities. We show that superposition of rivulets to rivulet ensembles leads to measurable soil moisture variations. The approach was applied to time series of soil moisture that result from sprinkler irrigation experiments and that were recorded with time domain reflectometry (TDR) equipment at depths of 0.1 and 0.2 m. A minimum water content, {theta}*, was identified that has to be exceeded at a particular depth before Stokes flow continues.

Abbreviations: TDR, time domain reflectometry.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Materials and Methods
 Data
 Application
 Discussion
 Conclusions
 REFERENCES
 
In situ research on infiltration at high degrees of saturation is still challenging, and numerous unexpected observations are ascribed to preferential flow. Thomas and Phillips (1991), for instance, used preferential flow to explain the broad spatial variation in capillary heads they observed in situ up to 2 d after heavy rains. Preferential flow in soils is fast. Kung et al. (2005), for example, reported wetting front velocities of preferential drainage of about 1 mm s–1. Not all fractions of soil water participate in preferential flow that takes place in macropores, such as animal burrows, root channels, and in cracks and fissures (Bouma and Dekker, 1978; Beven and Germann, 1982). Preferential flow can also occur in water repellent soils (Ritsema et al., 1993; Hendrickx et al., 1993), by funnel flow (Kung, 1990), and as finger flow across unstable wetting fronts (Glass et al., 1989). Except for the finger flow, preferential flow is frequently associated with a soil matrix having two distinct size ranges of pores (macropores and micropores). According to Gerke (2006), "Preferential flow severely limits the applicability of standard models for flow and transport that are mostly based on Richards' equation and the convection–dispersion equation," suggesting that preferential flow is not directly amenable to either Darcy's (1856) law or Richards' (1931) equation and hence that an alternate approach is needed.

Darcy's law applies to flow in completely water-saturated soils, thus allowing for separation of the hydraulic gradient from hydraulic conductivity, Ksat. Conductivity expresses momentum dissipation during flow. Richards' equation applies to capillarity-dominated flow, such as upward flow of water from a water table. What Richards (1931) called the "coefficient of proportionality" is since referred to as the hydraulic conductivity, K, being a function of the water content ({theta}) or the capillary head (h). Richards' equation is based on immediate equilibration of h in response to variations in {theta} (e.g., Sposito, 1986). Richards' equation expresses the diffusion of capillary potential during transient flow because capillarity affects simultaneously the hydraulic gradient and the hydraulic conductivity.

Germann and Beven (1981) proposed a soil moisture range for preferential flow between near-saturated conditions, where unsaturated flow does not obey Darcy's law, and a lower degree of saturation, where the capillary equilibrium requirement holds. Germann et al. (1997) postulated that D({theta}) /{eta} < 1 is a condition for the applicability of Richards' equation, where D is the soil water diffusivity (= K/C, where C = d{theta}/dh is the soil water capacity), and {eta} ({approx} 10–6 m2 s–1) is the kinematic viscosity. Conversely, D({theta}) > {eta} indicates a domain in which flow occurs mainly as laminar shear flow [i.e., where dissipation of momentum prevails in the moisture range from {theta}(D ≥ {eta}) to {theta}sat = {varepsilon}, where {varepsilon} is porosity]. Applying Campbell's (1974) K–h–{theta} relationships to the data of Clapp and Hornberger (1978), Germann et al. (1997) reported equivalent radii of capillary pores between 4 and 15 µm when D = {eta}. Conceptually, this range of pore widths marks the transition from diffusive to dissipative consumption of the flow-driving force. As with other dimensionless numbers in fluid mechanics, the proposed threshold at D({theta}) /{eta} = 1 is considered a conceptual limit. We focus on this limit in the present study.

Preferential flow within the near-saturated moisture range can be approached from two directions. One is to approach the domain of expected preferential flow from Richards' domain. Gerke and van Genuchten (1993), for example, applied Richards' equation to two hydraulically interacting pore domains, while Durner et al. (1999) adjusted the hydraulic property functions, K(h) and h({theta}), to accommodate preferential flow. Hassanizadeh et al. (2002), among others, investigated the dynamic potential to bridge the gap between capillary potentials at expected equilibrium and measured counterparts. The MACRO model of Jarvis (1994) approaches infiltration with Richards' equation but treats flow as a kinematic wave when h({theta}) exceeds a preset threshold.

The second direction of approaching preferential flow is from free-surface flow toward Richards' domain. Free-surface flow may be assumed to occur in shapes like films (e.g., De Quervain, 1973; Germann, 1985) or drops and bridges (e.g., Ghezzehei and Or, 2005) that move along corners and fissures (e.g., Tuller and Or, 2001) or in equivalent pipes (e.g., Germann and Hensel, 2006). Ghezzehei and Or (2005) used dimensionless bond numbers and capillary numbers to assess transitions between preferential flow and Richards-type flow. In contrast, Alaoui et al. (2003) explored the preferential flow domain from both sides. A similar procedure is used here.

We further waive the condition of immediate equilibration of capillarity in response to soil moisture variation. This allows water to move simultaneously in all pores of the preferential flow domain that are filled with air before infiltration. Ignoring gradients of capillarity leaves gravity as the only major driving force. Our conditions for preferential flow are met during flow in macropores as, for example, put forward by Beven and Germann (1982). However, pore diameters of about 7 mm are required of pores for not exerting capillarity under hydrostatic conditions (Germann and Hensel, 2006). Hence, we expect hydrodynamic principles to relate to more realistic pore sizes.

Significant preferential flow lasts about three to five times the duration of related infiltration (Germann, 1985). In contrast, Richards' flow continues until combined gradients of gravity and capillarity approach zero, which lasts about 10 to 100 times longer than substantial preferential flow. Wetting fronts move about 10 to 100 times faster during preferential flow than during Richards' flow, as simulations with HYDRUS (Simunek et al., 1999) have revealed (Germann and Hensel, 2006). The contrasting durations and velocities of wetting cause flow to occur in contrasting soil volumes. Thus, the differences in temporal and spatial scales justify dealing with preferential flow separately from other flow types.

We base our approach on Newton's shear flow. Germann and Di Pietro (1999) derived expressions of shear flow by simplifying the Navier-Stokes equations. Four canonical shear flow scenarios can be distinguished in fluid dynamics: (i) plane Couette flow, in which the fluid channel is defined by parallel walls that are moving relative to each other; (ii) plane Poiseuille flow, which is confined by stationary parallel walls and is typically driven by a pressure gradient along the channel (fluid next to the walls is stationary, which leads to a laminar parabolic velocity distribution across the channel); (iii) pipe flow in filled cylinders, also called Hagen-Poiseuille flow, which is similar to plane Poiseuille flow in that the walls are stationary and the flow is again typically driven by a pressure gradient (Fitzgerald, 2004); and (iv) Stokes flow, which occurs as film flow along a stationary wall with the other boundary being an air–water interface (White, 1991).

Beven and Germann (1981), among others, demonstrated that Stokes flow is amenable to kinematic wave theory according to Lighthill and Whitham (1955). Singh (1996) applied kinematic wave theory extensively to numerous one-dimensional flow phenomena at a wide variety of temporal and spatial scales. Colbeck (1972) similarly used kinematic wave theory to treat flow in layered snow, Sisson et al. (1980) applied it to infiltration, Charbeneau (1984) and Levy and Germann (1988) to solute transport, and Germann et al. (1987, 2002b) to microbial transport in soils. In still other studies, Smith (1983) combined kinematic wave theory with Richards' equation, Germann (1985) adapted it for drainage from a block of coarse sand, and Germann and Beven (1985) added a sink term to the wave equation to account for water transfer from preferential flow paths to the remaining soil.

Kinematic wave theory transforms the partial differential equations for flow to ordinary differential equations relating mobile soil moisture to the volumetric fluid flux density. Superposition of kinematic waves leads to an elegant procedure for process scaling. The theory provides the characteristics of flow properties like velocities of wetting and draining fronts, thus allowing for easy switching between Eulerian and Lagrangean schemes. The theory applies only to one-dimensional gravity-driven flow, which may be a limitation for some applications. Moreover, flow according to kinematic wave theory advances as a sharp-crested shock front, whereas observed drainage flow increases more gradually. Di Pietro et al. (2003) smoothed wetting shock fronts by superimposing a dispersive wave on the kinematic wave.

In this article we propose that preferential flow is composed of rivulets of shear flow. A rivulet consists of a water film that is limited in its horizontal extent, sustains a free surface to the soil atmosphere, and is continuous from the soil surface to the depth of investigation (typically to about 0.5 m). Germann (1987) described rivulets as one type of flow along the inner wall of an acrylic tube. Finger flow, according to Di Carlo (2004) and Nicholl and Glass (2005), is viewed as mechanism that most likely generates rivulets. Weisbrod et al. (2003) used films to explain observations at the wetting front. We suggest that the transient increase and decrease of soil moisture during and shortly after infiltration is related to various stages of a large number of superimposed rivulets. Furthermore, rivulets will form simultaneously in all air-filled pores that support preferential flow. The thinner a rivulet, the slower it advances. Observed monotonous increases in mobile soil moisture or drainage flow therefore represent the arrival of successively slower (i.e., thinner) rivulets. Our approach combines Stokes flow with elements of kinematic waves. We applied it to 16 soil moisture waves that resulted from 10 sprinkler infiltration experiments performed on a column of an undisturbed forest soil. Application of this conceptual model leads to frequency distributions of thicknesses and horizontal extents of rivulets, as well as to temporal variation in the volumetric flux density at two depths in the soil.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Materials and Methods
 Data
 Application
 Discussion
 Conclusions
 REFERENCES
 
A rivulet is considered the basic unit of preferential flow. Rivulets are very small and are not easily measured with equipment that is traditionally used in soil hydrological field investigations. Flow properties are derived from superimposing similar rivulets to form a rivulet ensemble. Rivulet ensembles are further superimposed to form a soil moisture wave, {theta}(Z,t), recorded at a particular depth, Z. Sprinkler irrigation during 0 ≤ t ≤ TS supplies water to the soil surface with a constant volumetric flux density, qS, where t and Ts refer to time and end of sprinkling, respectively.

Rivulet Flow
This section introduces a rivulet modeled as Stokes flow. Gravity is assumed to be the only force that drives flow, while viscous momentum dissipation impedes it. Linear momentum is transferred from the free surface toward the stationary plane that supports flow. Shear forces are generated in the planes parallel to the solid and act in a direction opposite to flow. Thus, viscous forces balance dynamically the weight, G, of a rivulet, as illustrated in Fig. 1 . In particular, the shear force in a water layer (i.e., a lamina) flowing at some distance f (0 ≤ f ≤ F) from a vertical plane at rest balances the weight of the remaining laminae from f (thickness variable) to F (thickness of the water film). The corresponding force balance at f is

Formula 1[1]
where {rho} (= 1000 kg m–3) is the density of water, v(f) is the velocity of a lamina at distance f from the surface, dv/df is the first derivative of the velocity with respect to f at distance f from the surface, dv/df is the rivulet's length of contact in the horizontal plane with the stationary parts, and H(t) is the progressive vertical length of a rivulet (i.e., position of its wetting front). We assume steady and gravity-driven flow in which the effects of capillarity on momentum dissipation are considered to be negligible. The left-hand side of Eq. [1] expresses momentum dissipation toward the area {ell} H(t), while the right-hand side represents the weight of the slab with thickness F – f.


Figure 1
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  		FIG. 1. Schematic representation of a rivulet during Stokes flow. F, thickness of the water film; f, thickness variable; df, thickness differential; l, length of contact between mobile soil moisture of the rivulet and the soil-water system at rest; H(t), vertical length of rivulet; G, weight of rivulet; v(f), velocity of the water at distance f from the parts at rest; {eta}, viscosity causing momentum dissipation toward the stationary parts.

 
Assuming no-slip at the solid-liquid boundary, v(0) = 0, integration of Eq. [1] leads to a semiparabolic velocity profile within a rivulet as follows:

Formula 2[2]
where v(f) is the velocity of a lamina at distance f from the surface. Equation [2] applies to the entire rivulet, 0 ≤ f ≤ F (i.e., boundary-layer flow is assumed), while momentum dissipation toward the air is assumed to be negligible because of low v(F).

At distance f the incremental volumetric flux is dQ = {ell}v(f)df. Integration of v(f) over 0 ≤ f ≤ F results in the rivulet's volumetric flux as

Formula 3[3]
in which F and {ell} are considered to adjust freely to the geometry of pores and to boundary and initial conditions.

The water content of a rivulet within a cross-sectional area A of soil is

Formula 4[4]
The contribution, qR, of a rivulet to volume flux density is

Formula 5[5]
where L = {ell}/A. Conductance of a rivulet is defined as

Formula 6[6]
From the fundamental volume balance requirement follows the wetting front velocity of a rivulet as

Formula 7[7]
Equation [7] expresses the velocity of a sharp wetting shock front of a viscous fluid. This equation can also be derived by considering the average velocity of water within a rivulet. A wetting front arrives at Z at time

Formula 8[8]
where Z is a point on the z-axis that points positively down.

The cessation of water input at time TS releases at once at the surface the back ends of all laminae of the rivulet. The fractions of the fluid flux density, dq, and the water volume, dw, are the two properties of a particular lamina that envelops an arbitrary water content, w (0≤ w ≤ wR). The velocity of the back end of a particular lamina follows from the volume balance requirement as

Formula 9[9]
where z is depth pointing vertically down. Rearranging the right part of Eq. [9] leads to the temporal position of an arbitrary water content as

Formula 10[10]
The outermost lamina at f = F that envelops w = wR moves the fastest as the leading edge of the rivulet. This edge is called the rivulet's draining front that moves with a velocity of

Formula 11[11]
The arrival time of the draining front at Z is

Formula 12[12]
The decrease of the moisture content w(Z,t) at z = Z (i.e., the trailing wave of the mobile volumetric moisture content at Z) during tD(Z) ≤ t ≤ {infty} is obtained by solving Eq. [12] for Z and inserting the result into Eq. [10] to give

Formula 13[13]
This conceptualization suggests that the draining front intercepts the wetting front. Combining Eq. [8] and Eq. [12] yields the time (TI) and depth of interception (ZI) as

Formula 14[14]
and

Formula 15[15]

The wetting front velocity reduces after interception, t > TI. By integrating the water content wave, Eq. [10], and considering that a rivulet stretching from the soil surface to the wetting front preserves its volume, qRTS, we obtain

Formula 16[16]
Equation [16] leads to the position of the wetting front after the draining front has caught up to the wetting front (i.e., TI ≤ t ≤ {infty}) as

Formula 17[17]
The water content of the wetting front is obtained by combining Eq. [17] with Eq. [10]:

Formula 18[18]
In accordance with Eq. [7], and using Eq. [18], the wetting front moves with a velocity of

Formula 19[19]

Figure 2 summarizes the properties of the model, where wR, TI, and ZI serve as scaling factors. Figure 2a shows the characteristics of the wetting front that was released at t = 0, and of some laminae that were released at t = TS. A characteristic is the trace of a flow property in a space-time diagram, and whose time derivative is the property's velocity. The draining front is determined by the fastest lamina released at TS. It starts at the surface and hits the wetting front at ZI and TI. Subsequent laminae move progressively slower and intercept the wetting front farther down at later times. Figure 2b gives the evolution of a rivulet's water content at four depths relative to ZI (i.e., two different depths above, one depth at, and one depth below ZI). Figure 2c presents the various stages of water content relative to wR of a rivulet before TS, between TS and TI, at TI, and some time after TI.


Figure 2
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  		FIG. 2. Properties of a rivulet in time and space (TI, ZI, and wR are scaling factors). (a) Characteristics of wetting and draining fronts and of four laminae. (b) w(Z,t) at four depths: z < ZI (twice), z = ZI, z > ZI. (c) w(Z,t) at four times: t < TS, TS < t < TI, t = TI, t > TI.

 
The duration, TS, and intensity, qS, of irrigation are the two parameters that usually can be controlled during experiments. Most wetting front velocities, vW, before front interception are between 0.1 to 1 mm s–1 (Germann and Hensel, 2006). Thus, TS in the range of 1 to 2 h is sufficiently long for ZI to be below the depth of most soil hydrological investigations, and the following expressions usually suffice for treating a rivulet that passes at z = Z < ZI:
0 ≤ t ≤ tW(Z):

Formula 20A[20a]

Formula 20B[20b]

tW(Z) < t < tD(Z):

Formula 21A[21a]

Formula 21B[21b]

t > tD(Z):

Formula 22A[22a]

Formula 22B[22b]

Equations [20a, 21a, 22a] hold for the water content, and Eq. [20b, 21b, 22b] apply to the volumetric fluid flux density of a rivulet.

Rearrangement of Eq. [8] leads to a rivulet's thickness as follows:

Formula 23[23]
Combining Eq. [4] with Eq. [23] results in a rivulet's contact length with stationary parts per unit cross-sectional area of soil as

Formula 24[24]

Superposition of Rivulets to Rivulet Ensembles
Superposition of Nr (1 ≤ j ≤ Nr) rivulets that pass at depth Z leads to macroscopic properties that are amenable to measurements. Superimposed trailing waves of rivulets result in a trailing wave that, using Eq. [22a], is given by

Formula 25[25]
where the index en refers to the ensemble of rivulets.

Stability of Rivulet Flow
Modeling a rivulet as a film that moves between pore spaces is a first-order simplification. One concern here is that vertical films are unstable and develop a wavy surface that can affect the flow properties. A flat film is considered to remain stable (i.e., nonwavy) when its Reynolds number is less than 3 (Lin and Wang, 1986). Using Eq. [23], the Reynolds number (Re) is given by

Formula 26[26]
where v is the average velocity of a fluid and µ = {eta}{rho} ({approx}10–3 Pa s) is the dynamic viscosity. The critical velocity and film thickness at Re = 3 are 31 mm s–1 and 97 {propto}m, respectively. The range of rivulet thickness and velocity for the data used in this study are below the critical values for instability, thus validating the applicability of the flat film flow approach.

Water Content Limits for Generation of Stokes Flow
The lower water content for generation of rivulets has been suggested to be at {theta}(D {approx} {eta}), and the uppermost possible limit is at full saturation when {theta}sat = {varepsilon}. These limits are explored in this section. Equation [5] at {theta}sat can be written as

Formula 27[27]
where {Delta}(h+ z)/{Delta}z is the hydraulic gradient in the vertical direction. The capillary potential vanishes at {theta}sat, and a unit hydraulic gradient prevails. The term (F3L)max is viewed as a measure of the intrinsic conductivity, k, and (g/{eta}) as fluidity. Thus, the hydraulic conductivity at saturation expresses continuous and complete momentum dissipation during flow that is exclusively driven by gravity. Ksat is considered a time-invariable soil property that expresses rivulet flow at saturation.

The rivulet approach assumes that gravity is the only force that drives flow and that no gradient of either pressure or capillary potential is involved. Thus, in analogy to Eq. [27],

Formula 28[28]
expresses a flux–water content relationship of rivulet flow in the range of {theta}(D {approx} {eta}) ≤ {theta} ≤ {theta}sat. Rousseau et al. (2004) derived similar relationships from the application of kinematic wave theory; however, they varied the exponent (e.g., 3 in Eq. [28]) to match the data.

Stokes Flow Compared with Hagen-Poiseuille Flow and Plane Poiseuille Flow
Hagen-Poseuille flow, Stokes flow, and plane Poiseuille delimit possible geometries of dissipative flow in soils (e.g., Germann and Di Pietro, 1996). Flow along corners (Tuller and Or, 2001) and other features fall between the three scenarios.

First, Stokes flow is compared with Hagen-Poiseuille flow in completely saturated cylindrical and vertical conduits. Germann and Hensel (2006) applied the general Hagen-Poiseuille relationship

Formula 29[29]
to the interpretation of wetting front velocities, where (–{Delta}p/{Delta}u) is the pressure gradient along the flow path in the general u direction (i.e., –{Delta}p/{Delta}u = {rho}g in this case), and r is the radius of the cylindrical pore. The volume balance requirement leads to the following expression for the velocity of a the wetting front in a vertical cylinder with radius rePp:

Formula 30[30]
where the subscript ePp refers to equivalent Poiseuille pores. The radius of equivalent Poiseuille pores of a rivulet ensemble that flows along similar cylindrical pores becomes

Formula 31[31]
and their number per cross-sectional area of soil is

Formula 32[32]
where wePp is the volumetric soil moisture content of the rivulet ensemble. The combination of Eq. [31] and Eq. [32] yields the contact length as

Formula 33[33]
The volumetric flux density of the rivulet ensemble becomes now

Formula 34[34]
From Eq. [24] and Eq. [33] follows the ratio of LepP and L as

Formula 35[35]
while from Eq. [23] and Eq. [28] we obtain similarly

Formula 36[36]
Equation [35] and Eq. [36] indicate a small difference in contact length and total film thickness of a rivulet ensemble when the parameters are derived for either Stokes flow or Hagen-Poiseuille flow. Moreover, the combination of Eq. [31, 33, 34] and Eq. [5, 23, 24], subject to wR = wePp, leads to

Formula 37[37]
Equation [37] demonstrates that the rivulet approach estimates volumetric flux densities rather independently from the presumed geometry of flow process.

Second, Stokes flow can be compared with plane Poiseuille flow. The water–air interface during Stokes flow allows for free adjustment of F and L to the actual flow conditions according to Eq. [23] and [24]. The velocities of laminae increase from the wall to the water–air interface, according to Eq. [2]. The resulting velocity profile follows a half-parabola with the maximum at the water–air interface. The relationship is limited to a distance B from the solid surface where velocity increase becomes less than parabolic and laminar flow may become unstable with Re > 3. Because Re << 3, we assume F << B such that conditions of laminar boundary-layer flow prevail. The maximum velocity is proportional to F2, and the volumetric flux density follows from Eq. [5].

Plane Poiseuille flow is confined between two solid walls that allow only L to adjust to the actual flow conditions, with F being equal to the distance between the walls. The velocity profile still follows Eq. [2]; however, it extends from both walls toward the plane that is parallel to and halfway between the walls. The maximum velocity is proportional to (F/2)2, and the volumetric flux density becomes now

Formula 38[38]
where the indexes pP and St refer to plane Poiseuille and Stokes flow, respectively. Thus, the volumetric flux density of the same film thickness may differ by a factor of 4, depending on whether flow is either unconfined Stokes flow or confined plane Poiseuille flow. Conversely, the same volumetric flux density during plane Poiseuille flow requires films that are 41/3 = 1.58 times thicker than during Stokes flow under the premise that L remains constant.


   Materials and Methods
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
Data
 Application
 Discussion
 Conclusions
 REFERENCES
 
Soil
The soil was located in a forest mixed of deciduous and coniferous trees along a slope of Mt. Bantiger near Bern (Switzerland) (47.00, 7.30). The soil is classified as a Mollic Cambisol according to FAO-Unesco (1994), with a sandy loam texture. Table 1 lists the pertinent properties.


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  		TABLE 1. Bulk density, porosity, texture, and hydraulic conductivity of soil.

 
Bulk density was determined from drying, at 100°C, three soil samples that were cut with a cylinder having a volume of 1000 cm3 and a height of 10 cm. Porosity of the mineral soil was calculated from the bulk density, assuming a particle density of 2650 kg m–3. The sand fraction was determined by sieving the mineral soil, the silt fraction was determined by the aerometer method, and the remainder was assigned to the clay fraction. The hydraulic conductivity at saturation was determined with a constant-head device using undisturbed samples having a volume of 100 cm3 and a height of 5 cm.

A column of undisturbed forest soil was prepared by driving a cylinder of stainless steel sheet metal 0.4 m deep into the ground. The cylinder had an inner diameter of 0.4 m and a length of 0.5 m, with the wall being 3 mm thick. In the laboratory the column was placed on a metal grid that permitted free drainage from the bottom. The edge of the soil surface was sealed against the cylinder wall using bentonite.

Instrumentation
Soil moisture variations, {theta}(Z,t), were recorded with a TDR100 device and a CR10X datalogger(Campbell Scientific, Logan, UT). The paired wave guides consisted of stainless steel rods having a diameter of 6 mm and a length of 150 mm, and were spaced 50 mm apart. The rods were electrically connected via a 50 {Omega} coax cable to a SDMX50 50W Coax Multiplexer, which was controlled by a CR10X Micrologger. A Campbell TDR100 device generated the electrical pulses and received the signals. Calibration was according to Roth et al. (1990). Precision of the {theta} measurements was assessed when flow had ceased (when the linear regression of {theta}(t) no longer showed a significant temporal trend). Standard errors sFormula 38 of various sets of 40 soil moisture readings never exceeded 0.001 m3 m–3. Instrument noise was thus set at 0.002 m3 m–3, and any variation in {theta} that exceeded ±0.002 m3 m–3 was considered significant. Two pairs of TDR wave guides were installed at depths of 0.1 and 0.2 m. Time intervals between readings were 12 s.

Capillary heads, h(Z,t), followed from tensiometer readings. Tensiometer cups were made of a ceramic pipe with an inner diameter of 6 mm and a length of 50 mm. Air entry pressure of the ceramic material exceeded 0.5 bar. The ceramic pipe was mounted on a perforated metal tube. The cup was hydraulically connected with a piezo-resistive pressure transducer (156 PC 15 GWL9 Microswitch, Honeywell International, Morristown, NJ; no longer available) that was electronically connected with a 21X Micrologger (Campbell Scientific, Logan, UT). The pressure range of the transducers was at least ±0.8 bars. Their precision was assessed when flow had ceased [i.e., when the linear regression of h(t) no longer showed a significant temporal trend]. The standard errors of various sets of 400 readings did not exceed sFormula 38 = 1 mm. The instrument noise was thus set at 2 mm. Two tensiometers were installed at depths 0.1 and 0.2 m through the wall of the cylinder about 0.1 m into the soil. Time intervals between tensiometer readings were 0.4 s.

The rain simulator consisted of a rotating disk of sheet aluminum with a diameter of 0.4 m that carried 60 nylon tubes with inner diameters of 1 mm. Controlled water supply was from a pump via a rotating manifold to the tubes. The tube outlets ended 0.15 m above the soil surface.

Regarding instrument sensitivity, the transducers required only a few cubic millimeters of water to cover the entire range of pressure reading. However, to produce a significant TDR signal, 2000 to 6000 mm3 of water had to enter the soil volume of about 1 to 3 L that is sensed by the TDR wave guides (Ferré et al., 1998). A ratio of about 1:5000 emerges for the water volumes necessary for a tensiometer and a TDR probe to produce significant signals. On the other hand, the projection on a horizontal plane of the sensitive areas of a tensiometer cup amounts to 3 to 5 cm2 and of a TDR wave guide to about 300 to 500 cm2, their ratio being about 1:100. Thus, the sensitive horizontal areas covered by the two instruments combined with the minimum soil moisture changes required to produce significant signals rendered the tensiometers about 50 to 100 times more sensitive than the TDR wave guides.

Experiments
We ran 10 sprinkler infiltration experiments on the column (Runs 1–10). The durations, TS, and intensities, qS, of sprinkling are shown in Table 2. Sprinkling rates represent annual hourly maxima for Switzerland with return periods between 10 and 100 yr. The rates never produced ponding on our soil column. Data recording began about 15 min before sprinkling and lasted about an hour longer than sprinkling. Only the instruments at the 0.1-m depth were in operation during Runs 1 through 4.


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  		TABLE 2. Dates (2004), durations, and rates of sprinkling.

 

   Data
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Data
Application
 Discussion
 Conclusions
 REFERENCES
 
Figure 3 presents all recorded data of the four waves [more than 600 points for each {theta}(Z,t) series, and about 20 000 points for each h(Z,t) series] for Run 10. The smoothness of the h(Z,t) curves is particularly worth noting, whereas the {theta}(Z,t) curves always showed substantial fluctuations.


Figure 3
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  		FIG. 3. Measured volumetric soil moisture, {theta}(Z,t), and capillary head, h(Z,t), at depths 0.1 m and 0.2 m due to infiltration at a rate of qS = 45.5 mm h–1 during TS = 1h for Run 10.

 
A {theta}(Z,t) series is referred to here as a moisture wave. Figure 4 illustrates key points of the time series of {theta}(Z,t) and h(Z,t). The time duration, TS, indicates the end of sprinkler irrigation that started at time t = 0. The {theta} data are bounded by {theta}in before increase and {theta}max after attaining the upper plateau. The times when soil moisture first significantly increased and when it started to level off at the plateau are given by tW and tP, respectively, while {theta}F and {theta}out are volumetric soil moisture contents at the end of measurements, tF, and when drainage has ceased, respectively. Divergence is defined as div{theta} = ({theta}out – {theta}in), and the wave's amplitude is wS = ({theta}max – {theta}out). Accordingly, hin and hmax are points of the h(Z,t) series before thW and after thP, respectively. Table 3 lists the most important data of our 16 {theta}(Z,t) and h(Z,t) series. In all cases {theta}max < {varepsilon}, which suggests that the soil never reached saturation.


Figure 4
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  		FIG. 4. Definitions of parameters and variables, shown for Run 7, depth 0.1 m. Times: thW, arrival of first pressure; thP, time of attaining pressure maximum; tW, arrival of first (measurable) moisture increase; tP, time of attaining steady soil moisture; TS, end of water input to the soil surface; tD, arrival of draining front; tF, end of measurements. Capillary heads: hinit, before infiltration; hmax, maximum. Volumetric soil moisture: {theta}in, initial; {theta}max, maximum; {theta}F, end of measurements; {theta}out, end of drainage or trailing wave; div{theta} (= {theta}out – {theta}in), divergence; wS (= {theta}max – {theta}out), amplitude of moisture wave.

 

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  		TABLE 3. Key points in the measured time series of {theta}(Z,t) and h(Z,t): Initial and maximum of soil moisture, {theta}ini and {theta}max, and of capillary heads, hinit and hmax; times of their arrival, tW, tP, thW, and thP, at the depths of measurement; and average velocities, vW, of wetting fronts.

 
Within short periods capillary heads at both depths repeatedly increased from initial pressure to close to atmospheric pressure, h {approx} 0 (Fig. 3 and 4). Capillary heads never extended into the positive range (i.e., h < 0 always), although the transducers were able to record positive pressures. Moreover, the capillary head always reacted at 0.1 m before reacting at 0.2 m, which indicates that wetting was from top to bottom during all 10 runs.

Velocities of wetting fronts shown in Table 3

Formula 39[39]
were between 0.05 and 0.27 mm s–1. (Front velocities are given in mm s–1 to distinguish them from volumetric flux densities that are given in m s–1). While the wetting front velocities we obtained represent preferential flow according to Kung et al. (2005), they are in the lower half of the 215 velocities reported by Germann and Hensel (2006). Application of Eq. [39] produces minimum velocities because wetting fronts of rivulets are here assumed to start moving at the onset of sprinkling at t = 0. However, it is feasible that a minimum saturation in the top soil layer has to be met before rivulets may form. Germann and Hensel (2006) assessed the initiation of wetting fronts from the first significant moisture increase at the uppermost TDR wave guides and not from beginning of sprinkling, which may explain their higher wetting front velocities. However, Fig. 2a suggests that we have to adhere here to the assumption of simultaneous release of wetting fronts at t = 0 because of a lack of information about the formation of rivulets close to the surface.

In view of the data in Table 3 and our instrument sensitivities, we consider the increases in capillary heads at both depths to have occurred at the same time as the moisture increases. The concurrence indicates that flow behind the wetting fronts happens at pressure close to atmospheric pressure, thus supporting the force balance of Eq. [1].


   Application
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Data
 Application
Discussion
 Conclusions
 REFERENCES
 
Individual rivulets are considered to be the units of flow. The rapid increase in h(t), the smooth and monotonous evolution of h(Z,t), and confinement to hmax ≤ 0 indicate that viscous momentum dissipation continuously and completely consumed the driving force. While some capillarity likely occurs at the edges of rivulets (Ghezzehei and Or, 2005), we ignore its gradient as driving force, also in view of the fact that h {approx} 0.

Rivulet ensembles according to Eq. [25] are highly amenable to TDR measurements. They are derived from partitioning a moisture wave into an arbitrary number of Ne rivulet ensembles. Each rivulet ensemble yields values for F, L, and q, which then serve as footholds for their respective continuous distribution across all rivulet ensembles of a moisture wave. Note that soil water retention curves, h({theta}), are similarly derived in that data pairs of h and {theta} are experimentally determined, usually at preset h levels, with smooth interpolations between the points.

The velocity of wetting fronts is related to the film thickness (Eq. [23]). Faster rivulets are thicker and therefore become more easily trapped. Accordingly, div{theta} > 0 is caused by the rivulets that arrive early, whereas the slower and thinner rivulets move on as drainage flow that is expressed in the trailing wave of {theta}(Z,t) during t > tD(Z). For this reason we define the amplitude and divergence of a moisture wave as wS = ({theta}max – {theta}out) and div{theta} = ({theta}out – {theta}in), respectively.

Partitioning of a Moisture Wave into Rivulet Ensembles and Their Superposition
Zero Divergence, div{theta} = 0
A moisture wave, {theta}(Z,t), is considered here to be the result of superimposing Ne (1 ≤ j ≤ Ne) ensembles of rivulets that arrive at Z between tW(Z) and tP(Z). The wetting front of the jth ensemble arrives at Z at time tW,j, and its moisture content is wR,j. The Ne arrival times, tWj, are deduced by partitioning the time period (tW ≤ t ≤ tP) into Ne sections. The soil moisture amplitude of the jth ensemble follows from

Formula 40[40]
where {theta}(tj) is soil moisture content that represents {theta}(Z,t) during the jth time interval.

Application of Eq. [12] to the tW,j array yields a series of arrival times of the draining fronts, tD,j(Z). Application of Eq. [22a] to wR,j for t > tD,j(Z) provides the trailing wave of the jth ensemble. Trailing waves of Ne ensembles are superimposed according to Eq. [25] to produce the trailing wave, wsup(Z,t) = [{theta}sup(Z,t) – {theta}out(Z)]. Figure 5 illustrates the procedure with Ne = 3.


Figure 5
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  		FIG. 5. Superposition of three rivulet ensembles applied to the data of Run 6. {theta}sup(Z,t) declines faster than {theta}(Z,t) because div{theta} > 0 was ignored in the analysis.

 
Some uncertainty remains when data during a short period are considered. As indicated in Fig. 5, {theta}sup(Z,t) declines before the water content of Ensemble 3 at tD,3. The effect is mitigated by increasing Ne. The only requirement that limits Ne is a significant soil moisture increase in the data from tj to tj+1. The modeled variations in Ne between 5 and 20 demonstrate that the computed trailing waves of wsup(t) did not significantly vary if Ne > 10. Figure 6 shows an example with Ne = 10.


Figure 6
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  		FIG. 6. Ten rivulet ensembles applied to the data of Run 3, depth 0.1 m. Calibration: Approach applied to increasing limb of {theta}(Z,t). Validation: Superimposed trailing waves compared with data.

 
The interval tW(Z) ≤ t ≤ tF(Z) was divided into a calibration period, tW(Z) ≤ t ≤ tP(Z), and a validation period, tD(Z) ≤ t ≤ tF(Z). The superimposed trailing wave, {theta}sup(Z,t), serves as an objective function. No statistically significant differences were found between measured and superimposed moisture contents of the trailing wave in Fig. 6. Thus, the rivulet approach is considered valid when div{theta} = 0, as Fig. 7 demonstrates.


Figure 7
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  		FIG. 7. Linear regressions of measured vs. modeled mobile soil moisture of trailing wave (upper plot) and of residuals of modeled data (lower plot) for Run 3. The respective differences are not significant.

 
Divergence, div{theta} > 0
Only {theta}(Z,t) series of Run 2 and Run 3 at depth 0.1 m showed div{theta} = 0. For all other cases we found div{theta} > 0, most likely due to sorption of some faster rivulets in the soil layer that was sensed by the TDR wave guides. According to Ferré et al. (1998), the sensed soil layer should be about 60 mm thick. Because all rivulet ensembles that become trapped in the layer can no longer contribute to the trailing wave, their inclusion in the approach would distort {theta}sup(Z,t), as Fig. 5 demonstrates.

Conversely, the standard deviations of {theta}F across the experiments of 0.399 ± 0.007 and 0.380 ± 0.003 at depths 0.1 and 0.2 m, respectively, are relatively small (Table 3). They suggest that trailing waves at one depth may repeatedly drain to the same final water content, {theta}out. The average for a moisture wave, Formula 40, can be estimated by analyzing the M data points of a trailing wave, {theta}[Z,t > tD(Z)] using a version of Eq. [22a]:

Formula 41[41]
and

Formula 42[42]

The average

Formula 42

(±SE) of all 10 {theta}(Z,t) series at the depth of 0.1 m was 0.386 (± 0.00215 m3 m–3); the average of the 6 time series at the depth of 0.2 m was 0.367 (± 0.00195 m3 m–3). The small standard errors suggest that

Formula 42

is a property of a particular soil layer. This average, herein referred to as {theta}*, is considered to separate the pore domain that carries preferential flow beyond Z from the remaining pore space. Also, div{theta} > 0 requires the separation of rivulet ensembles of a moisture wave into an arrival-only group with {theta}(Z,t) < {theta}* and a flow-through group with {theta}(Z,t) ≥ {theta}*. The boundary, {theta}*, between the two groups follows from Eq. [41] and [42].

Film Thickness
The rivulet approach was applied to the entire increasing limb of the moisture waves. Film thickness, Fj, of the jth ensemble of rivulets follows from Eq. [23]. Thus, Fj depends only on tW,j(Z). Figures 8a and 8b display the film thicknesses at both depths. The films are thinnest at 2.4 and 2.7 {propto}m for the 10th ensemble of Run 8, at both depths, and thickest at 11.5 and 8.1 {propto}m for the first ensemble of Run 2 at 0.1m, and for Run 10 at 0.2 m.


Figure 8
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  		FIG. 8. Film thicknesses of the rivulet ensembles vs. volumetric soil moisture for Runs 1–10. Input rates: {nabla} 15.3 mm h–1; {square} 28.8 mm h–1; {Delta} 45.5 mm h–1 at depth (a) 0.1 m and (b) 0.2 m. {theta}* ± 1 SE indicates boundary between arriving-only and flow-through rivulets. F(D = {eta}) represents the upper and lower limits of the film thickness where diffusivity equals kinematic viscosity (Germann et al., 1997).

 
Comparing Fig. 8a with 8b indicates relationships among film thicknesses at the two depths. Linear regressions between F of the 10 ensembles at 0.1m and those at 0.2 m during the same run result in coefficients of determination of (r2) larger than 0.97, suggesting that the patterns of film thickness were maintained over the depth range from 0.1 to 0.2 m.

Contact Length
The rivulet approach was again applied to the entire increasing limb of the moisture waves to estimate the contact length, Lej, of ensemble j per cross-sectional area. This parameter follows from the film thickness and the water content, according to Eq. [4] and [24]. Because Lej depends on Ne, only the sums of the contact lengths, SL, that is,

Formula 43[43]
need to be further explored. Figures 9a and 9b display SL of all 10 ensembles of all runs at both depths. The maxima of SL reached 42 000 m m–2 at a depth of 0.1 m (Run 1) and 21 200 m m–2 at a depth of 0.2 m during Run 8. The minima were 8500 m m–2 at a depth 0.1 m (Run 2) and 7900 m m–2 at a depth 0.2 m (Run 6). No dependency of contact lengths with depth was discernible.


Figure 9
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  		FIG. 9. Sum of contact lengths of ensembles vs. volumetric soil moisture for Runs 1–10. Input rates: {nabla} 15.3 mm h–1; {square} 28.8 mm h–1; {Delta} 45.5 mm h–1 at depth (a) 0.1 m and (b) 0.2 m. {theta}* ±1 SE indicates boundary between arriving-only and flow-through rivulets.

 
Volumetric Flux Density
Considering Eq. [6] and [25], and applying Eq. [20b, 21b, 22b] to each rivulet ensemble of the flow-through group yields the volumetric flux density of drainage, q(Z,t). The higher Ne the smoother q(Z,t) will be. However, modeling Ne between 5 and 20 did not lead to significant differences in the computed flux density as long as Ne > 10.

Integration over time of Eq. [21b] and [22b] for a rivulet ensemble yields the total volume of flow, while summation of the volumes of flow over all rivulet ensembles results in the total volume of preferential flow. Table 4 lists the 16 ratios of total volume of preferential flow per total volume irrigated. Some of the ratios increased with depth, which is most likely because preferential flow was considerably delayed with respect to the beginning of sprinkling. This would effectively result in higher velocities of the wetting fronts and, hence, thicker rivulets.


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  		TABLE 4. Volumes and fractions of preferential flow compared with the volumes of infiltration for Runs 1–10.

 
Figure 10 presents four cases of drainage. Figures 10a and 10b refer to Run 6, depths 0.1 and 0.2 m, respectively. Although the total volumes of drainage can be considered equal at both depths (Table 4), the maximum rate increased with depth presumably due to flow convergence. Figure 10c depicts results for Run 2, depth 0.1 m. We had to increase wsup(Z,t) + {theta}* by 0.013 m3 m–3 to match it with {theta}(0.1 m, t). Run 2 was performed on the same day as Run 1 (Table 2), and there was not enough time for the trailing wave to drain to {theta}*, thus, the fraction of drainage-to-applied volume of 0.996 and maximum relative volumetric flux density of 1.05 (Table 4). High {theta}in is also reflected in Fig. 8a and 9a. Figure 10d shows the results for Run 5, depth 0.2 m. This run was performed 10 d after the previous experiment. The soil had drained during that period to such an extent that {theta}max barely reached {theta}*, shown in Fig. 8b and 9b. The drainage flow rate in this case was about 30 times less than during Run 6.


Figure 10
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  		FIG. 10. Modeled and measured mobile soil moisture, w(Z,t) = [{theta}(Z,t) – {theta}out], and volumetric flux densities, q(Z,t). (a) Run 6, depth 0.1 m: typical evolution of soil moisture and flow; (b) Run 6, depth 0.2 m: typical evolution of soil moisture and flow; (c) Run 2, depth 0.1 m: qmax {approx} qS due to {theta}in > {theta}*; (d) Run 5, depth 0.2 m: q(Z,t) is much reduced compared with other runs due to {theta}max {approx} {theta}*.

 

   Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Data
 Application
 Discussion
Conclusions
 REFERENCES
 
The rivulet approach for preferential infiltration used in this study was based on Stokes flow. Film thickness and contact length are basic properties of a rivulet, while water content, velocities of the wetting and draining fronts, and volumetric flux density of drainage are related to the basic properties. Superposition of rivulets leads to ensembles of rivulets (bottom-up procedure) that are units in terms of partitioning a moisture wave (top-down procedure). Good agreement between modeled and observed trailing waves renders the approach valid.

The rivulets extended to a depth of at least 0.2 m as suggested by the predictable reactions of the trailing waves on cessation of infiltration. But rivulets at some depth may break up into intermittent flow according, for instance, to Ghezzehei and Or (2005). In such cases the trailing wave may no longer serve as the objective function. Application of Eq. [26] to the maximum vW = 0.27 mm s–1 (Run 10, depth 0.1 m, Table 3) leads to a maximum Reynolds number, Re, of 2.5 x 10–3 (<< 3). Thus, rivulet flows in this investigation are stable and strictly laminar.

The rivulet approach at saturation, S = {theta}/{varepsilon} = 1, can also be elucidated. The film thickness and contact length follow from plane Poiseuille flow (Eq. [38] and [4]), which is applied to the maximum mobile soil moisture content, wmax = {varepsilon} – {theta}* = 0.184, as

Formula 44[44]
and

Formula 45[45]
while recognizing that vmax = (Ksat/wmax) = 0.24 mm s–1. The results appear plausible, although outside the range of Fig. 8. Equations [44] and [45] link Ksat with gravity-driven viscous flow when all pores are filled with water. Because Ksat is considered time invariant, its value may not depend on the imposed boundary condition. Equation [27] demands (F 3L)max to be constant, which, according to the rivulet approach, is only feasible when {theta}* remains constant. Therefore, we do not expect that boundary conditions affect {theta}*. Germann et al. (1997) found 4 ≤ rePp ≤ 15 µm from {theta}(D = {eta}), which translates to 3.5 ≤ F ≤ 14.5 {propto}m when considering Eq. [36]. Figures 8a and 8b show the lines of F distributions crossing the range of {theta}* at F({theta}*). The average and variation of F({theta}*) are 6.3 ±1.3 {propto}m at 0.1 m and 7.5 ±2.0 {propto}m at 0.2 m, respectively. They are well within the limits of the minimum and maximum F(D = {eta}), although F(Ksat) exceeds the maximum by about 20%.

We further compared the contact lengths and film thicknesses of two hypothetical porous media made up of equal spheres with diameters, DS. The spheres were arranged in vertical prisms with overall horizontal cross-sections of either squares or triangles. Table 5 shows the geometries and the pertinent relationships. Porosities of the two hypothetical media compared well with those of soils. The narrowest paths from one to the next layer of spheres was lined by the circumferences of the spheres that result in either Lsq or Ltr. The subscripts sq and tr refer to the arrangement of spheres as squares and triangles, respectively. Application of relationship (3) in Table 5 to the range of 7900 ≤ SL ≤ 42 000 m m–2 resulted in a range of particle diameters of either 7.5 · 10–5 ≤ DSsq ≤ 4 x 10–4 m or 8.6 x 10–5 ≤ DStr ≤ 4.6 x 10–4 m, which are equivalent to the particle diameters of fine to coarse sands. The diameters of the narrowest opening between the spheres (i.e., the lumen) in the range of either 31 ≤ DLsq ≤ 166 µm or 13 ≤ DLtr ≤ 71 µm exceeded the thickness of the coarsest films in the range of 2.4 ≤ F1 ≤ 11.5 µm (Fig. 8) by factors from 1.6 to more than 100; that is, the ensembles of all rivulets would pass through the assembled spheres. Also, the rough estimates from Ksat of Fmax and Lmax (Eq. [44, 45]) are within the range of their equivalents of the two hypothetical porous media. Although the two hypothetical particle arrangements are not the most realistic for soils, they put the parameters F and L of the rivulet approach in feasible perspectives with respect to porosity and particle diameter. The hypothetical spheres may represent, for instance, aggregates that are composed of silt, clay, or organic matter, with the voids between them representing structural pores.


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  		TABLE 5. Parameters of two hypothetical porous media made of equal spheres that are arranged in prisms with either squared (index sq) or triangular (index tr) horizontal cross-sections.

 
The rivulet approach does not discriminate between Stokes and plane Poiseuille flow. Di Carlo et al. (2003) reported acoustical crackling noises during fluid invasion that may indicate discrete transitions from Stokes to increasingly plane Poiseuille flow. Hence, relationships between F and L may not be as simple as hitherto assumed. Confined plane Poiseuille flow could include local fluctuations in the hydraulic pressure; however, this was not immediately apparent from the smooth increases in capillary heads that were recorded at intervals of 0.4 s. In contrast, fluctuations in {theta}(Z,t), Fig. 3 and 4, indicate variations in either L or F during flow. Because F(z) seems to maintain some structure with depth (r2 > 0.97), L may well vary substantially over shorter distances, presumably in accord with variations in the local pore geometry.

The rivulet approach may be used as a diagnostic tool to quantify preferential flow. It is directly applicable to measured moisture waves. Sprinkler irrigation experiments may produce moisture waves in situ, with the wave measurable with TDR equipment (e.g., Germann et al., 2002a). The approach is robust in terms of estimating volumetric flux densities since assumptions about the geometry of flow cancel out as indicated by Eq. [37].

Finally, free drainage is a mandatory requirement for the applicability of the rivulet approach. However, restrained drainage is easily recognized by convex trailing waves that differ considerably from concave trailing waves (Fig. 3–6GoGoGo). See, for instance, Germann et al. (2002a), who observed concave and convex trailing waves.

Recommended Procedure
Data interpretation is according to the following 10-point protocol:

  1. Determine {theta}* (Eq. [41, 42]).
  2. Subtract {theta}* from {theta}(Z,t) that yields w(Z,t).
  3. Partition the increasing limb of w(Z,t) into Ne ≥ 10 rivulet ensembles that yields wR (Eq. [40]).
  4. Assign arrival times of the wetting fronts, tW, to each rivulet ensemble.
  5. Calculate arrival times of the draining fronts, tD, for all rivulet ensembles (Eq. [12]).
  6. Calculate the trailing waves for all rivulet ensembles (Eq. [22a]).
  7. Determine the volumetric flux density of each rivulet ensemble (Eq. [20b, 21b, 22b]).
  8. Superimpose the volumetric flux densities of all rivulet ensembles (Eq. [25]) to get q(Z,t).
  9. From arrival times of the wetting fronts, obtain the film thickness, F, of the rivulet ensembles (Eq. [23]).
  10. Obtain the contact length per area, L (Eq. [4, 24]).


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Data
 Application
 Discussion
 Conclusions
REFERENCES
 
The rivulet approach to flow relates transient volumetric flux densities with temporal variations in the volumetric soil moisture content, while only viscous forces and gravity are considered. Thickness and contact length of rivulet ensembles demonstrate that macropores are not essential for preferential flow to occur.

The approach continues logically and seamlessly out of Darcy's law into the domain of preferential flow when soil moisture reduces from saturation to the threshold of Richards' domain. Depending on the degree of saturation, S = {theta}/{varepsilon}, the following flow types are suggested:

S = 1: qS > Ksat, h > 0, Darcy's law

S = 1: qS = Ksat, h = 0, Darcy's law and plane Poiseuille flow

{theta}*/{varepsilon} ≤ S < 1: qS < Ksat, h {approx} 0, plane Poiseuille and Stokes flow

S = {theta}*/{varepsilon}: qS < Ksat, h {approx} 0, plane Poiseuille, Stokes, Richards flow

S < {theta}*/{varepsilon}: qS < Ksat, h < 0, Richards flow

where h = 0 indicates atmospheric pressure in the mobile soil water. However, according to the assumptions underlying the rivulet approach, preferential infiltration may occur simultaneously in all pores that are filled with air before infiltration.

We approached the proposed threshold between the domains of preferential flow and Richards' flow from the domain of preferential flow using {theta}* and from Richards' domain by postulating {theta}(D = {eta}). The parameter ranges overlap (e.g., Fig. 8), encouraging further research.

The rivulet approach to preferential infiltration serves currently as a diagnostic tool. With growing experience it may develop into a prognostic tool to eventually meet Gerke's (2006) and similar demands for theoretical concepts that are easily and satisfactorily applicable to soil hydrological field research and related modeling.


   ACKNOWLEDGMENTS
 
Acquisition and first analyses of the data was during the assistantships of Andreas Helbling and Tomaso Vadilonga that were paid with funds from the University of Bern. Lively discussions with Dani Or and Marco Carizzoni initiated many ideas. Numerous reviewers and engaged Associate Editors positively challenged earlier versions of the manuscript, while Rien van Genuchten polished the final version with great care.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Data
 Application
 Discussion
 Conclusions
 REFERENCES
 




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