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Published online 26 July 2006
Published in Vadose Zone J 5:867-876 (2006)
DOI: 10.2136/vzj2005.0080
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Poiseuille Flow Geometry Inferred from Velocities of Wetting Fronts in Soils

P. F. Germann* and D. Hensel

Soil Science Section, Department of Geography, University of Bern, 3012 Bern, Switzerland
* Corresponding author (germann{at}giub.unibe.ch)

Received 30 June 2005.



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soil structure can significantly enhance or modify infiltration rates and flow pathways in undisturbed field soils. Relations between features of soil structure and features of infiltration were elucidated from the velocities of wetting fronts. To study flow geometries during wetting, 100 sprinkler infiltration experiments were performed in situ at 25 different sites. Applied sprinkler irrigation rates ranged from 20 to 100 mm h–1 and lasted 1 h. Time domain reflectometry (TDR)-based observations of soil moisture time series at various depths yielded 215 velocities of wetting fronts between 0.2 and 5.5 mm s–1. Application of Poiseuille's Law to the velocities resulted in radii and densities of equivalent Poiseuille pores in the ranges of 5 to 30 µm and of 2 x 106 to 2 x 108 m–2, respectively. From the equivalent Poiseuille pores capillary heads were estimated to be in the range of –2.0 to –0.2 m. They served as initial conditions for modeling infiltration with the HYDRUS-2D code using hydraulic properties representative of the sand, silt, loam, and clay soil textural classes. The observed in situ wetting velocities indicated preferential flow; however, the calculated equivalent pore radii suggested that no macropores were required to initiate preferential flow. HYDRUS-2D reproduced the bulk of observed wetting velocities when hydraulic properties typical of sandy soils were used.


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
INFILTRATION in structured field soils can lead to very irregular, nonuniform flow path geometries. Discussions about the geometry of flow paths in soils can be traced back to the middle of the 19th century when Schumacher (1864), a German soil physicist, claimed that flow in soils is mainly through large pores. He possibly opposed the concepts of Darcy's (1856) one-dimensional and Dupuit's (1863) two-dimensional approaches to flow in saturated porous media that did not discriminate among pore sizes.

Many studies at various spatial scales and with different levels of sophistication have followed since in attempts to relate the morphology of pores with flow. Bouma et al. (1977) were among the first to introduce the concept of macropores and by-pass flow. In some of these studies (e.g., Heijs et al., 1995), preferred flow paths were analyzed using nuclear magnetic resonance–supported computer tomography. Di Pietro and Germann (2001) more recently reported flow simulations with lattice gas automatons in virtual pore structures, as well as the ability of this approach to deal with flow through more realistic pores.

The approaches mentioned above assume continuous rivulets of water along the flow paths at least during times when the same flow conditions are maintained, as experiments on finger flow may demonstrate (e.g., Di Carlo, 2004). However, concepts of discontinuous water advancement in fractures and fissures are now also evolving. Ho (2004), for instance, considered asperities to induce episodic percolation, while Ghezzehei and Or (2005) presented liquid bridges moving along fissures. Physically based models, such as those by Jarvis (1994) and Köhne and Mohanty (2005), assume a priori a certain pore geometry that combines the effects of water sorbing into micropores with the flow of water into macropores, with relatively little resistance. Jarvis (2004) demonstrated the difficulties one may encounter when applying such models to flow and transport experiments without extensive calibration.

Other studies tried to infer the geometry of flow from measurable flow and transport parameters using theoretical considerations. Some of these studies assumed continuous equilibrium between the capillary potential and soil moisture (e.g., Sposito, 1986), thus focusing on the hydrostatic properties of soil–water–air systems. The various functions and parameters needed for these purposes are generally determined assuming steady or quasi-steady flow conditions. Other approaches have been based on dynamic potentials. The studies by Vachaud et al. (1972) and, more recently, by Hassanizadeh et al. (2002) attempted to bridge the gap between the more dynamic and the more static hydromechanical concepts of flow in soils. Flow geometries can also be quantified using principles of momentum dissipation (e.g., Germann and Di Pietro, 1999), as they are expressed in Poiseuille's Law. Watson and Luxmoore (1986) combined Poiseuille's Law with the capillary equation to interpret steady-infiltration data from tension infiltrometers. Relevant parameters may also be determined under transient conditions of either water flow or solute transport. In one study of this type, Kung et al. (2005) deduced radii of flow paths in a field soil by applying Poiseuille's Law to steady water flow but transient solute transport data.

We report the application of Poiseuille's Law to transient water flow as it shows in the rapid temporal variation of volumetric soil moisture, {theta}(Z,t). Thus we analyzed preferential flow at 25 different sprinkler-irrigated field sites in terms of velocities of wetting fronts that were calculated from {theta}(Z,t) series. The time series were recorded with TDR equipment at various depths, leading to 215 estimates of wetting front velocities, vmeas. From vmeas we estimated the radii of an equivalent Poiseuille pore, rePp. By dividing the measured soil moisture increases, {Delta}{theta}, by the volume of an equivalent Poiseuille pore, we obtained estimates of the number of pores per unit cross-sectional area of soil, NePp. From the vmeas data we further estimated the capillary heads, hePp, and used them as initial conditions for modeling infiltration with the HYDRUS-2D code of Simunek et al.(1999).


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soils
The data used in this study had been collected since 1995 as part of several soil hydrological investigations. Here we analyze 25 soil profiles. The soils belong to seven suborders: Udalfs, Ochrepts, Umbrepts, Aquods, Ferrods, Humods, and Udults (Soil Survey Staff, 1975).The depths of investigation ranged from 0.4 to 1.7 m. The soil profiles were located in the northern Pre-Alps and on the Plateau of Switzerland. Specific locations and further details of the sites are available from the authors on request.

Data Collection
In all cases water was sprinkled on the soil surface during preset durations from 30 min to 2 h at rates between 20 and 100 mm h–1 (i.e., 5.6 x 10–6 to 2.8 x 10–5 m s–1). The rates represent heavy storms that correspond to hourly annual maximum intensities with return periods between 10 and 100 yr for the northern part of Switzerland.

Water was supplied to the soil surface using a sprinkler system that consisted of 100 metal tubes with inner diameters of 2 mm. The tubes were mounted in a 0.1 by 0.1 m square pattern through a square sheet metal of 1 by 1 m. A gear moved the horizontally suspended sheet metal 50 mm forward and backward in both horizontal directions such that it took about 30 min for a drop to fall on the same spot. To prevent wind drifts, the tube outlets were mounted as close to the soil surface as possible (i.e., between 0.05 and 0.25 m above ground, depending on the slope and roughness of the soil surface). A battery-driven pump supplied water at preset rates from a tank through a manifold to the tubes. Germann and Zimmermann (2005) provide further details of the experimental setup.

We measured variations in the volumetric soil moisture content, {theta}(Z,t), as a function of time, t, at various depths, Z, in the soil profiles. Values of {theta} were recorded automatically with TDR equipment at intervals of 300 s. The paired wave guides were horizontally installed from a trench into the soil at desired depths. Each pair of wave guides consisted of two parallel stainless-steel rods, 50 mm apart, 6 mm in diameter, and 0.3 m long. The rods were electrically connected via a 50 {Omega} coax cable with a SDMX50 50W Coax Multiplexer which was controlled by a CR 10X Campbell Micrologger (Campbell Scientific, Logan, UT). A Campbell TDR 100 device generated the electrical pulses and received the signals. In earlier experiments we had used a Tektronix Cabletester 1502B (Tektronic, Inc., Beaverton, OR).

The dielectric numbers produced with the TDR equipment were transformed to volumetric soil moisture contents using the calibration approach of Roth et al. (1990), who separated the impact of the wave-guide geometry on the dielectric number from those of such soil properties as bulk density and the content of clay and organic matter. Before installation, each wave-guide pair was completely submerged in water, and the corresponding dielectric number was calibrated to an apparent water content of 1.0 m3 m–3.

The precision of the {theta} measurements was assessed when flow had ceased [i.e., when a linear regression of {theta}(t) did not longer show a significant temporal trend]. The standard errors s{theta} of various sets of 40 soil moisture readings never exceeded 0.001 m3 m–3. The instrument noise was thus set at d{theta} = 0.002 m3 m–3, and any variation in the water content measured with the wave guides that exceeded ±d{theta} was considered significant.

Data Selection
During the last 10 yr we collected more than 1000 {theta}(Z,t) series from sprinkler infiltration experiments. For the current study we selected those time series that showed distinct soil moisture increases that considerably exceeded d{theta} in response to sprinkling. The increase also had to be followed by a concave decrease after the drainage front had arrived at Z. The distinct increase in soil moisture shortly after the onset of sprinkling, and its concave decline some time after sprinkling has ceased, indicated that the soils were not completely saturated before sprinkling, and that they had drained to a final moisture content some time after sprinkling. Although we observed several other shapes of the {theta}(Z,t) series (Germann et al.,2002), they were not considered in the analyses below.

Estimation of Arrival Times
Figure 1 shows how we estimated the arrival time, {tau}, of a wetting front at Z. The instrument noise, d{theta}, was added to the linear regression that represents {theta}(Z,t) distinctly before {tau}. The resulting horizontal line was intersected with a linear regression line applied to {theta}(Z,t) data after {tau}, thus producing the estimated arrival time of the wetting front. Soil moisture increases, {Delta}{theta}, are further referred to as the amplitudes of the moisture waves, ranging from 0.005 to 0.10 m3 m–3. For further interpretation we selected 315 {theta}(Z,t) series from 25 sites that represented the seven soil suborders.


Figure 1
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  		Fig. 1. Example of determining the arrival time of the wetting front. Solid line and open triangles: measurements of the volumetric soil moisture content vs. time, {theta}(Z,t), since the beginning of infiltration at t = 0. Dashed-dotted line: linear regression fitted to the data before the arrival of the wetting front. Horizontal dashed line: instrument noise, d{theta}, added to the linear regression before the arrival of the wetting front. Slanted dashed line: linear regression fitted to the data of increasing soil moisture content. The amplitude, {Delta}{theta}, of the moisture wave is the difference between the maximum of {theta}(Z,t) and the soil moisture content before infiltration.

 
The average velocity, vmeas, of the wetting front was calculated from the time lapsed, {Delta}{tau} = {tau}j>1 – {tau}1, (1 ≤ j ≤ n, where n is the number of depths where TDR probes were installed in a particular soil) between the arrival of wetting at the uppermost pair of wave guides at depth Zj=1 and the significant moisture increase at the depth of a particular pair of wave guides at Zj>1. Hence,

Formula 1[1]
in which, for the purpose of this study, vmeas was expressed in millimeters per second, Z is in meters (positive down), and {tau} is in seconds. Thus, each vmeas represents an average velocity of the wetting front between the uppermost and the respective depth of measurement, and each velocity is subject to the same uncertainty of timing as shown in Fig. 1.

Discussion of Experimental Approach
We are aware of the somewhat arbitrary way we selected the various data sets for our analysis. We would have to discuss all of the cases separately, including those rejected from the analysis, if the study were to systematically characterize distinct spatial units, such as fields or hydrological catchments. However, we selected {theta}(Z,t) series that were consistent with the flow behavior illustrated in Fig. 1. If anything, we believe that the 25 sites and seven soil suborders thus selected reflect flow patterns that are mostly typical of situations with relatively high infiltration rates.

We are also aware that the determination of {tau} depends heavily on the sensitivity of the TDR equipment to detect soil moisture increases (i.e., on d{theta}). While faster wetting cannot be excluded, it is not amenable to our particular equipment. The resulting velocities of wetting fronts are thus considered minima in that somewhat higher vmeas would be expected with increasing spatial and temporal resolutions of {theta} measurements.

The estimation of vmeas, using Eq. [1], is based on the hypothesis that water has flowed as a result of sprinkler irrigation from the soil surface to depth Z. An alternative hypothesis states that water was pushed down in parcels, with each parcel of water moving along a very short distance. We tested these two mutually exclusive hypotheses—flow from the surface to Z versus flow over considerably shorter distances—with a salt tracer experiment. Bretscher (2002) provided the data for the following section.

Ions dissolved in soil water tend to increase the response of TDR wave guides. For instance, Muñoz-Carpena et al. (2005) and Ritter et al. (2005) found strong correlations (coefficients of determination R2 > 0.98) between the bulk soil electrical conductivity and the concentration of a KBr solution added to a column of undisturbed soil. This fact was used here to qualitatively track a pulse of a salt solution during the last sprinkler infiltration experiment at one of the sites. The dynamics of infiltrating salt solution was compared with five {theta}(Z,t) series that had been obtained before the tracer experiment for comparable initial and boundary conditions. For these two experiments, with and without the tracer, we applied water during 1 h at a rate of 60 mm h–1. Five previous infiltration experiments at the site showed that the repeated {theta}(Z,t) patterns were similar enough to unmistakably identify arrival of the tracer solution at the respective locations of the TDR wave guides during the last experiment.

Figure 2 shows three pairs of {theta}(Z,t) time series that were measured at depths Z = 0.05, 0.30, and 0.55 m. Three of them are denoted by {theta}(Z,t)ns (solid symbols; "ns" indicates no salt in the infiltration water) and the other three by {theta}(Z,t)ws (open symbols; "ws" indicates infiltration with salt, i.e., a 0.5 mol NaCl solution). The differences between the amplitudes of {theta}(Z,t)ns and {theta}(Z,t)ws are interpreted as the arrival of the salt tracer.


Figure 2
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  		Fig. 2. Results of the salt tracer experiment. Dashed lines, closed symbols, ns: infiltration of tap water. Solid lines, open symbols, ws: infiltration of a salt solution. Note that only the open circles, {theta} (0.05,t), indicate all of the measurements.

 
At a depth of 0.05 m (circles), the sudden increase of soil moisture during the ns experiment occurred only 300 s (i.e., one measurement interval) later than in the ws experiment. This difference between the arrival times of the wetting fronts is not considered significant. Initial and final soil moisture content of the ns curve were 0.335 and 0.378 m3 m–3, and the corresponding figures for the ws curve were 0.375 and about 0.50 m3 m–3, respectively. A small amount of water ({approx}0.003 m3 m–3) drained between the end of the ns curve and the beginning of the ws curve. The subsequent arrival of the salt tracer front is marked by the comparatively large apparent increase in soil moisture to about 0.65 m3 m–3.

A similar pattern is discernible at the 0.30-m depth (squares), except that the increase in soil moisture content is now much smaller. At the 0.55-m depth (triangles), a minor, yet still noticeable increase in soil moisture content with tap water occurred, while the soil moisture content remained constant throughout the tracer experiment. This shows that {theta}(0.55,t)ws was not affected by either water or the salt tracer. Therefore, we conclude that no water had moved to this depth during the very last infiltration experiment. These results from the paired infiltration experiments using water and a salt tracer demonstrate that soil moisture increases at depths of 0.05 and 0.3 m were due to both water and salt transport from the surface to the respective depths, but that neither water nor salt had moved to a depth of 0.55 m during the last infiltration experiment.


   RESULTS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Velocities of Wetting Fronts
Figure 3 shows a frequency distribution of the 215 vmeas values, their majority being between 0.2 and 0.4 mm s–1. The distribution is compressed to the left with a long tail to the right, typical of a lognormal distribution.


Figure 3
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  		Fig. 3. Histogram of wetting front velocities.

 
Impacts of Sprinkling Rates and Soil Texture
We tested the impacts of sprinkling rate, qS, and soil texture class, TC, on vmeas, using linear regressions. Each of the 25 profiles was assigned to one of four broad texture classes: sand, silt, loam, and clay.

The 215 vmeas data were further separated into two groups, one below and one above a certain threshold velocity. Coefficients of determination, R2, were determined for each group separately. The procedure was repeated for vmeas thresholds of 0.3 mm s–1 (approximate peak of the frequency distribution in Fig. 3), 0.5 mm s–1 (median of the vmeas-data), and 1.0 mm s–1, and for the vmeas data. Results are listed in Table 1. Because all R2 values were <0.02, we conclude that no statistically significant relationships existed between the velocities of the wetting fronts and both the sprinkling rate and soil texture class.


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  		Table 1. Coefficients of determination, R2, of wetting front velocities, vmeas, vs. infiltration rates, qS, and vs. texture classes, TC. The threshold of vmeas separates the data into slower and faster subgroups that indicate the weak impacts of qS and TC on vmeas.

 
Comparison with Other Studies
Flynn and Sinreich (2005) reported a tracer front that within 30 min had moved a vertical distance of 12 m in a fissured limestone (i.e., karst) formation in Switzerland. This corresponds to a velocity between 6 and 7 mm s–1. Their wetting velocity was similar to the highest values we observed, and about 20 times faster than most of our data. Kung et al. (2005) similarly reported relatively high velocities of wetting and tracer fronts in soils of about 1 m in 15 min ({approx}1 mm s–1). Rasmussen et al. (2000) found wetting front velocities in saprolites within the range of 0.2 to 0.4 mm s–1.

Hydromechanical Considerations
In this section we elucidate the geometry of flow by portraying the process in terms of Poiseuille's Law. Thus, the properties of hypothetical vertical equivalent Poiseuille pores, ePp, will be used to characterize flow. We note here that portraying the geometry of flow with idealized structures is inherently different from analyzing the geometry of pores that are occupied by the moving water; the latter is not the subject of this manuscript.

Our analysis involves first estimating the radius, rePp, of an equivalent Poiseuille pore, ePp, from vmeas, using the assumption that ePp is empty ahead of wetting and completely filled behind wetting. Next, the capillary equation is applied to vmeas via rePp to produce a capillary head, hePp. Finally, infiltration will be simulated numerically with HYDRUS-2D using hePp values that represent a range of rePp and vmeas data.

Radii and Number of Equivalent Poiseuille Pores
Poiseuille's Law states that

Formula 2[2]
where QePp is the volume flux, µ is dynamic viscosity of water, and {Delta}p/{Delta}{ell} is the hydraulic gradient along the macroscopic direction of the flow path, {Delta}{ell}. In Eq. [2] µ represents the properties of water in films that are thicker than about 10 nm (personal communication of Dr. M. Heuberger, Laboratory of Surface Sciences and Technology, ETH, Zürich; 10 May 2005).

The average vertical velocity, vePp, in the Z direction of a wetting front in one ePp follows from the volume balance as

Formula 3[3]
The hydraulic gradient, –{Delta}p/{Delta}{ell} in Eq. [3], expresses energy per unit volume of water per unit length in the direction opposite of flow or, equivalently, the driving force for flow acting on a unit volume of water. In the case of a vertical cylindrical pore, the driving force is the sum of the specific weight of water, {rho}g (where g is acceleration due to gravity), and the gradient of the capillary potential, ({rho}g)dh/dz. Moreover, dh/dz < 0 during infiltration from a soil surface when water pressure is close to atmospheric pressure, h {approx} 0 (i.e., the capillary gradient always acts in the same direction as gravity). From this it follows that

Formula 4[4]

Combining Eq. [1], [3], and [4] yields

Formula 5[5]

The number of vertical ePp per unit cross-sectional area, A, of soil that enable the wetting front to move with vmeas (i.e., the pore density) is given by

Formula 6[6]
where {Delta}{theta} is the mobile soil moisture content according to Fig. 1 (i.e., soil water that participates in the flow process). Our data set yielded a range of 0.005 to 0.10 m3m–3 for {Delta}{theta}. Accordingly, the contact length, LePp, of {Delta}{theta} per unit cross-sectional area is

Formula 7[7]
The effect of gravity {rho}g is ubiquitous and is assumed to be constant in time and space. The force due to capillary potential, ({rho}g)dh/dz, however, varies in space and time. Equations [5]Go through [7] indicate that any dh/dz < 0 reduces rePp and increases NePp and LePp.

Di Carlo (2004) observed complete saturation behind the wetting front of fingers. Adopting similar flow conditions we assume that dh/dz = 0 between the wetting front and the soil surface, with ponding being the corresponding boundary condition. This leads to maximum values for rePp and to minimum values for NePp and LePp. The bulk of the radii are then in the range 10 ≤ rePp ≤ 20 µm. Application of Eq. [5]Go to [7] to the vmeas data leads now to results compiled in Table 2 for the range 0.1 ≤ v(r) ≤ 1 mm s–1, resulting in 350 ≤ LePp ≤ 10100 m–1.


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  		Table 2. Characteristics of equivalent Poiseuille pores, ePp, for selected wetting front velocities, v, and soil moisture amplitudes, {Delta}{theta}.

 
Capillary Heads
The capillary equation expresses the capillary head, h, as function of the radius, r, of a cylindrical pore; that is,

Formula 8[8]
where {sigma} (= 0.073 N m–1) is the surface tension of water against air, and {alpha} (°) is the contact angle between the water–air interface and the solid. Combining Eq. [5] and [8] and neglecting the capillary gradient yields the capillary head as a function of the wetting velocity:

Formula 9[9]
Application of Eq. [9] to the range 0.2 ≤ vmeas ≤ 2.0 mm s–1 and to the peak of the distribution in Fig. 3 at 0.3 mm s–1 yields capillary heads in the range of –1.16 ≤ h(vmeas) ≤ –0.37 m, while h(0.3 mm s–1) = –0.95 m.

Beven and Germann (1982), among others, postulated that macropores carrying preferential flow will not exert much capillarity onto the moving water. Equation [8] assumes an ideal semispherical meniscus hanging in a vertical cylindrical capillary tube. The smallest capillary rise, |hmin|, occurs when |h| = r, which occurs when the bottom of the meniscus just touches the level of the water that surrounds the vertical capillary tube. Any r > |h| does not lift water above |hmin|. At the limit of |h| = r follows from Eq. [8] that

Formula 10[10]

Equation [10] estimates the minimum radius of a macropore according to Eq. [8], consistent with the assumption of Beven and Germann (1982). Application of Eq. [5] to the rePp values in Table 2 and using the rmax value from Eq. [10] shows that the ratio rmax/rePp is roughly in the range between 200 and 400. This suggests that macropores are not mandatory features for the transmission of soil water with the wetting velocities presented in Fig. 3.

Simulations with HYDRUS-2D
Richards (1931) introduced a widely applied approach to flow in partially saturated porous media that is based on equilibrium between soil moisture and its capillary head. The HYDRUS-2D model (Simunek et al., 1999) is a computer code that solves the Richards equation using, among others, van Genuchten's (1980) relationships for the unsaturated soil hydraulic properties.

We used this code to simulate wetting front velocities, vmod, so as to place vmeas within a broader context of flow in the four soil texture classes of our database (i.e., sand, silt, loam and clay). The textural classes covered the range of 25 soil profiles presented here. For the soil hydraulic parameters we used the pedotransfer functions of Carsel and Parrish (1988) that are provided with HYDRUS-2D and listed here in Table 3. The intent of using HYDRUS-2D was to cover the types of soil used in our study. The simulations hence were not intended to reproduce any vmeas values.


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  		Table 3. The van Genuchten (1980) soil hydraulic parameters used in HYDRUS-2D.

 
HYDRUS-2D, together with its structured mesh generator, was used to simulate vertical flow in the four soil types, all of which were considered to be homogeneous. We assumed ponded infiltration with h = 0 at the soil surface and implemented a seepage face at the bottom boundary at 1.0 m. The latter boundary condition assumes a zero flux during unsaturated conditions (h < 0) and zero tension during saturated conditions (h = 0).

Values of the simulated wetting front velocities, vmod, were calculated from the arrival times of wetting fronts at depths of 0.4 and 1.0 m. Simulations of wetting in the silt reached depths only of 0.42, 0.6, and 0.98 m after 1 h for the invoked initial conditions of –2.0, –1.0, and –0.5 m. For these cases vmod was calculated from the times when wetting reached the final depth. Results of the 12 modeling runs are summarized in Table 4. Figure 4 presents eight examples of modeled {theta}(Z,t) series for the sand and loam soils. Figures 1 and 2 may be consulted for comparison with the measured {theta}(Z,t) series. Figure 5 compares various vmeas and vmod values from the literature as well as results of this study. For the sand vmod was within the range of most vmeas, whereas vmod values for the other three textural classes were at the very low end of the vmeas range.


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  		Table 4. Wetting front velocities, vmod, from HYDRUS-2D simulations of infiltration.

 

Figure 4
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  		Fig. 4. Examples of HYDRUS-model outputs for (a)–(d) sand and (e)–(h) loam. (a) Sand: hinit: –0.5 m, Z: 0.4 m; (b) sand: hinit: –0.5 m, Z: 1.0 m; (c) sand: hinit: –1.0 m, Z: 0.4 m; (d) sand: hinit: –1.0 m, Z: 1.0 m. (e) Loam: hinit: –0.5 m, Z: 0.4 m; (f) loam: hinit: –0.5 m, Z: 1.0 m; (g) loam: hinit: –1.0 m, Z: 0.4 m; (h) loam: hinit: –1.0 m, Z: 1.0 m. See Table 4 and Fig. 1 and 2 for corresponding velocities of wetting fronts, vmod and shapes of {theta}(Z,t).

 

Figure 5
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  		Fig. 5. Summary of wetting front velocities as estimated by (1) Flynn and Sinreich (2005), (2) our investigations, (3) Kung et al. (2005), (4) Rasmussen et al. (2000), and from HYDRUS-2D simulations for (5) sand, (6) clay, (7) loam, and (8) silt.

 

   DISCUSSION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
CONCLUSIONS
 REFERENCES
 
From more than 1000 time series of {theta}(Z,t) we selected 315 data sets that showed patterns similar to the example presented in Fig. 1. Our results indicate that during heavy rains vmeas values may exceed 0.1 mm s–1 for 25% of the soils of our entire database. The depths and velocities of penetration sufficed to adsorb most of the heavy rains. Our sprinkling rates were between 20 and 100 mm h–1, which represent 1-h rain storms with return periods between 10 and 100 yr. These rates will potentially initiate preferential flow, according to Kung et al. (2005), who considered that at least their highest rate of 4.4 mm h–1 produced preferential flow. Their rates were factors of about 4 to 23 times lower than ours. Therefore, all our input rates should have produced preferential flow.

Kung et al. (2005) calculated wetting front velocities and Poiseuille pore radii from measured tracer breakthrough curves. Their radii were between 0.5 and 30 µm, with a frequency maximum near 1 µm. They also found a substantial increase in the pore radii when infiltration rates increased from 1.2 to 2.4 and to 4.4 mm h–1. Along the same lines, Germann and Zimmermann (2005) found for one soil profile that the soil moisture amplitude, {Delta}{theta} (Fig. 1), was positively correlated with the input rate in the range of 61 and 90 mm h–1 (i.e., 1.7 x 10–5 to 2.5 x 10–5 m s–1). In addition, they found positive correlations between the volume flux density, q, in the soil on one side, and both {Delta}{theta} and v. All three relationships were significant at the 1% error limit. Thus, the higher the input rate the higher {Delta}{theta} and v, leading to larger radii of the equivalent Poiseuille pores.

The densities of equivalent Poiseuille pores vary greatly among different investigations. Kung et al. (2005) found 1011 m–2, we found a range from 106 to 107 m–2, while Watson and Luxmoore (1986) reported 9 x 104 m–2. Differences among the various experimental and invoked theoretical approaches may explain this broad range in pore densities. Kung et al. (2005) included the entire width of the breakthrough curve in their analysis. They included also the finer pores with a much higher frequency. On the other hand, Watson and Luxmoore (1986) limited the capillary heads of their tension infiltrometer to h > –0.15 m, whereas –1.2 < h < –0.3 m was the range before infiltration in our investigations. Hence, it seems worthwhile to investigate more rigorously the relationships between Poiseuille-type laws and other properties of flow, in particular the velocities of tracers and wetting.

Straight cylindrical pores parallel to the flow direction, such as the equivalent Poiseuille pores assumed here, cause the least momentum dissipation (e.g., Germann and Di Pietro, 1996), while any other shape of the flow paths will increase momentum consumption. Conversely, apertures of flow-active pores have to increase for a given overall potential gradient when pore shapes deviate increasingly from ideal straight cylindrical pores. Thus, assuming the presence of equivalent Poiseuille pores should lead to the smallest possible pore diameters and the highest pore densities. As a consequence, the dimensions of the contact lengths, LePp, represent maxima. Wall roughness, tortuosity, rugosity, asperity, and vorticity may reduce areas of contact between the moving (mobile) water and the remaining parts of the liquid phase as a result of enhanced momentum dissipation, thus possibly leading to thicker films that require wider flow channels.

The contact lengths we found compare well with the range of 1300 to 26700 m–1 reported by Germann and Di Pietro (1999), who interpreted {theta}(Z,t) series from in situ experiments with kinematic wave theory. They also postulated film thicknesses in the range between 3.9 and 47 µm, which again are comparable with our rePp values compiled in Table 2.

The magnitudes of rePp and NePp indicate that, more realistically, flow occurs in tiny structures, possibly in the shapes of films and rivulets, or along angles, edges, and corners. Moreover, macropores are not mandatory for advancement of wetting fronts in the velocity range of 0.1 ≤ vmeas ≤ 1.0 mm s–1.


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
REFERENCES
 
At least 315 {theta}(Z,t) series of a total of about 1000 data sets followed patterns similar to the example shown in Fig. 1. The 315 time series are the resulted from 100 infiltration experiments performed on 25 soil profiles that represented seven soil suborders. From these data we could determine 215 wetting front velocities. The resulting frequency distribution (Fig. 3) shows a maximum at 0.3 mm s–1. Neither the rate of sprinkling, qS, nor the soil texture classes, TC, of sand, silt, loam, and clay were correlated with the velocities of wetting (Table 1).

Poiseuille's Law was applied to the velocities, from which equivalent pore radii were estimated. Application of the capillary equation to the equivalent pore radii yielded corresponding capillary heads, which were subsequently used as initial conditions in HYDRUS-2D to simulate the wetting front velocities in the sand, silt, loam and clay soils. HYDRUS-2D was able to produce wetting front velocities only for the sandy soils. The wetting front velocities were much smaller for the silt, loam, and clay soils, and in the silt and loam did not reach the bottom of the assumed soil profile within several hours of simulated infiltration and redistribution.

The observed wetting patterns shown in Fig. 1 are indicative of preferential flow. The observed velocities of the wetting fronts are similar to the velocities of the tracer fronts according to Kung et al. (2005). Infiltration in at least 25% of our {theta}(Z,t) series produced preferential flow for the relatively heavy rain fall rates (between 20 and 100 mm h–1) used for our experiments.

The concept of equivalent Poiseuille pores, including the velocities of wetting front, may serve as a set of measures at the extreme end of preferential flow. The ratio rmax/rePp was found to be between about 200 and 400, which intuitively seems quite high to us, although no direct evidence to the contrary could be provided. This underscores the need for assessing more realistic dimensions of the geometry of flow. As such it would be worthwhile to test the equivalent Poiseuille pore concept, or similar approaches, against an even larger database.

Finally, we note that the radii of most of the equivalent Poiseuille pores do not indicate typical macropores, but more likely belong to the class of mesopores according to Watson and Luxmoore (1986). Moreover, the distinction between Richards flow and Poiseuille flow, within a wide range of pore radii, may not depend on the actual geometry of pores of the matrix.


   ACKNOWLEDGMENTS
 
We are indebted to all the graduate students who laboriously conducted the many infiltration experiments and collected the numerous time series of water contents. John Nieber, Rien van Genuchten, and anonymous reviewers substantially shaped the manuscript.


   REFERENCES
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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P. Germann, A. Helbling, and T. Vadilonga
Rivulet Approach to Rates of Preferential Infiltration
Vadose Zone J., April 9, 2007; 6(2): 207 - 220.
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