Published online 23 January 2008
Published in Vadose Zone J 7:131-135 (2008)
DOI: 10.2136/vzj2007.0045
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
TECHNICAL NOTES
Alternative Unsaturated Flow Analyses for the Falling-Head Ring Infiltrometer
W. D. Reynolds*
Agriculture and Agri-Food Canada, Greenhouse and Processing Crops Research Centre, 2585 County Rd. 20, Harrow, ON, Canada N0R 1G0
* Corresponding author (reynoldsd{at}agr.gc.ca).
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 2 March 2007.
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ABSTRACT
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The Green–Ampt expression for determining field-saturated hydraulic conductivity (Kfs) and matric flux potential (
m) from falling-head ring infiltrometer measurements produces a singularity when the standpipe cross-section divided by the ring cross-section (R) equals the change in porous medium water content (
). As R = is clearly possible in wet, low-permeability materials, the objective of this study was to develop an alternative falling-head ring infiltrometer analysis that applies for 0 < R < +
, including R = . This was accomplished by replacing the natural logarithm in the Green–Ampt analysis with a Taylor series expansion, which removes the (R – ) divisor from the analysis. The Taylor series form of the Green–Ampt expression collapses to a simple algebraic expression when R = . Combining this algebraic expression with a similar analysis developed for gravitationless infiltration provides an improved approximate relationship for estimating Kfs when standpipe drawdown is small and R < .
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INTRODUCTION
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The falling-head ring infiltrometer (Fig. 1
) is used primarily for in situ measurement of field-saturated hydraulic conductivity, Kfs [L T–1], in unsaturated, low-permeability porous materials (e.g., Reynolds, 2007; Guyonnet et al., 2000). With some care, however, it can also be used to estimate the so-called "unsaturated flow" parameters (Reynolds, 2007), including matric flux potential, m [L2 T–1] (Gardner, 1958), sorptive number, 
* [L–1] (White and Sully, 1987), the effective wetting front pressure head, 
f [L] (Mein and Farrell, 1974), sorptivity, S [L T–1/2] (Philip, 1957), flow-weighted mean pore diameter, PD [L] (Philip, 1987), and the number of flow-weighted mean pores per unit area, NP [L–2] (Philip, 1987).
A falling-head ring infiltrometer measurement is taken by inserting the ring to depth d [L] into the porous medium, filling the standpipe reservoir with water to a known initial height, H0 [L], and then monitoring the fall of the standpipe water level with time, Ht [L], as the water infiltrates the porous medium inside the ring (Fig. 1). Specifics on falling-head ring infiltrometer operation and equipment can be found in Reynolds (2007) and Guyonnet et al. (2000).
The falling-head ring infiltrometer analysis assumes Green–Ampt infiltration (Green and Ampt, 1911) and can be written as (Philip, 1992; Guyonnet et al., 2000; Elrick et al., 2002)
 | [1] |
where t [T] is the time since initiation of ponded infiltration, R = As/Ar is the cross-sectional area of the standpipe reservoir (As = 
rs2) [L2] divided by the cross-sectional area of the ring (Ar = rr2) [L2], = (fs – i) [L3 L–3] is the water-fillable porosity determined as the difference between the field-saturated volumetric water content (fs) and the antecedent volumetric water content (i) of the porous medium, and b is a dimensionless wetting-front shape parameter with a range of (White and Sully, 1987)
 | [2] |
with the lower limit corresponding to a step-function wetting front (Green–Ampt soil) and the upper limit corresponding to a diffusion-type wetting front (linear soil). Note that b = 0.55 can be used for ponded infiltration into most natural porous materials with an error of 
10% (White and Sully, 1987). The Kfs, m, *, f, and S parameters are interrelated according to (White and Sully, 1987)
 | [3a] |
where
 | [3b] |
The * parameter quantifies the porous medium's "capillarity" (ability to sorb water), which ranges from very high in fine-textured, compacted materials to negligible in very coarse textured materials (Table 1
).
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TABLE 1. Texture–structure and capillarity categories for selecting the sorptive number, *, when using the "specified-*" procedure (adapted from Elrick et al., 1989). The capillarity designations assume unsaturated material with the antecedent water content sufficiently below the saturated water content to produce near-maximum capillarity. Saturated materials have zero capillarity regardless of texture and structure (Reynolds et al., 1985).
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Equation [1] is both nonlinear and implicit in Kfs and m, and can be solved using numerical inversion or the "specified-*" procedure (R, , and H0 measured independently). Numerical inversion involves fitting Eq. [1] to a sequence of Ht vs. t measurements (using, e.g., the Levenberg–Marquardt algorithm) to obtain simultaneous determinations of Kfs and m. The specified-* approach, on the other hand, obtains an estimate of Kfs only; and it involves assuming an * value based on the most appropriate texture–structure category in Table 1, measuring one (Ht, t) data pair, and then solving Eq. [1] for Kfs after substituting in Eq. [3b]. Further detail on the solution of Eq. [1] can be found in Reynolds (2007), Elrick et al. (1995, 2002) and Bagarello et al. (2004). Discussions of the specified-* procedure can be found in Reynolds (2007) and Reynolds et al. (1992).
The falling-head analysis is quite versatile, as H0 and R can be readily adjusted (e.g., by using standpipes of various heights and diameters) to optimize measurement time and accuracy. The above analysis cannot be applied, however, when R = as Eq. [1] produces a singularity. In addition, the analysis may become inaccurate as R 
because the (R – ) term in Eq. [1] then becomes very small and may therefore generate rounding errors. At the same time, situations where the optimum R is 
may not be uncommon in moderate- to low-permeability materials in a near-saturated state (e.g., fine- to medium- textured soils, caps and liners of waste impoundments). Hence, the primary objective of this study was to develop an alternative expression for the falling-head ring infiltrometer that remains accurate and applicable for all physically plausible R values, including R = . A secondary objective was to develop an improved approximate analysis for estimating Kfs when standpipe drawdown is small.
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Analysis
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The singularity and potential rounding error in Eq. [1] is due to the (R – ) divisor, which approaches zero as R . The (R – ) divisor can be removed by applying a Taylor series expansion to the natural logarithm term in Eq. [1] (see Appendix for details), and the resulting series solution has the form
 | [4] |
where A = H0 – Ht, B = H0 + [m/(2bKfs)] = H0 + (2b*)–1, and C = R – . Equation [4] is mathematically valid for 0 < R < + and 0 < < + (see Appendix for details); however, the physically practical range of R is on the order of 0.0001 R 1, and the physically relevant range of is 0 < Dq fs. Note that Youngs et al. (1995) developed a falling-head ring infiltrometer analysis that applies for = 0 (a saturated or field-saturated porous medium).
For the special case of R = , Eq. [4] collapses to
 | [5] |
where P1 = R = , and P2 = m/b = Kfs/(b*). Note that Eq. [5] is not only much simpler than Eq. [1] and [4], but it also demonstrates the classical "square root time" behavior [i.e., (H0 – Ht) 
t1/2] of gravitationless infiltration (Philip, 1958; White and Sully, 1987), even though gravity may or may not be negligible.
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Discussion
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Behavior of Equations [1] and [4] as R
The behavior of Eq. [1] and [4] as R is perhaps best illustrated using squared relative "drawdown" in the standpipe, h*, and relative time, t*:
 | [6] |
 | [7] |
as they scale the response curves between 0 and 1, and they produce a straight-line relationship with unit slope when Eq. [5] applies.
For R 
, plots of h* vs. t* from Eq. [1] and [4] (obtained using Mathcad 2001i, MathSoft Engineering & Education, Cambridge, MA) produced identical results for /R ranging from 50 to 0.001 (Fig. 2
). This confirms the validity and accuracy of Eq. [4], although it should be noted that the summation limit in Eq. [4] (i.e., n) may need to be large to maintain accuracy if Eq. [A4] approaches –1 (see Appendix). Note also that Eq. [1] and [4] converge on the straight-line unit-slope relationship from above and below, depending on whether /R decreases or increases toward unity. This further confirms that the falling-head analysis is indeed represented by Eq. [5] in the special case of R = . Equation [4] can be solved for Kfs or m using the same procedures as for Eq. [1], i.e., numerical curve fitting to a sequence of Ht vs. t measurements for simultaneous determination of Kfs and m, or specification of * (using Table 1) and measurement of one (Ht, t) data pair to obtain Kfs.
Application of Equation [5]
An unfortunate consequence of R = is that the analysis (Eq. [5]) becomes nonunique with respect to m; as a result, numerical curve fitting to Ht vs. t measurements does not produce reliable m estimates. Numerical curve fitting can still produce reasonable estimates (within about a factor of 2) of Kfs, however, and of course Eq. [5] can still be solved for Kfs using the specified-* procedure.
It is noted that Eq. [5] is nearly identical to the approximate ring infiltrometer analysis of Fallow et al. (1994) and Elrick et al. (1995):
 | [8] |
which provides good estimates of Kfs (but only rough approximations of m and *) for early-time infiltration when the gravitational force is small relative to the hydrostatic pressure and capillarity forces [hence, (H0 – Ht) t1/2 behavior]. This suggests, as a consequence, that Eq. [5] may also provide good estimates of Kfs for R when gravity is small relative to hydrostatic pressure and capillarity. To test this, Eq. [5] and [8] were applied (using the specified-* procedure) to Ht vs. t data calculated from Eq. [4] when R . For R < , Eq. [5] underestimated Kfs, while Eq. [8] overestimated it; for relative standpipe drawdowns less than about 30% [i.e., (H0 – Ht)/H0 0.3, Ht 
0.7H0], the underestimates and overestimates were about equal and within 20% of the actual Kfs value (Fig. 3a
). Hence, it appears that Eq. [5] and [8] can be used to "bracket" the Kfs value for R < and small standpipe drawdown. For R > , on the other hand, both Eq. [5] and [8] overestimate Kfs, with the degree of overestimate increasing with increasing R (data not shown).
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FIG. 3. Estimation of field-saturated hydraulic conductivity using (a) Eq. [5] and [8] and (b) Eq. [9a]. Estimated hydraulic conductivity, Kest, refers to the results obtained using Eq. [5], [8] or [9a], and field-saturated hydraulic conductivity, Kfs, is the true value. The solid symbols denote negligible porous medium capillarity (sorptive number * = 100 m–1), and the open symbols denote very high capillarity (* = 1 m–1) (see Table 1). The Ht values were determined using Eq. [4] with the initial height of water in the standpipe reservoir H0 = 1 m, the standpipe cross-section divided by the ring cross-section R = 0.001, the change in porous medium water content = 0.02, Kfs = 10–9 m s–1, and the above-mentioned * values.
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Improved Approximate Analysis for Estimating Field-Saturated Hydraulic Conductivity
Given that Eq. [5] and [8] provide nearly equal underestimates and overestimates, respectively, for Kfs when the drawdown is small and R < , then the average of the two relationships,
 | [9a] |
or
 | [9b] |
should provide greatly improved estimates of Kfs using either the specified-* procedure (Eq. [9a]) or numerical curve fitting to Ht vs. t data (Eq. [9b]). This does indeed occur. Estimates of Kfs using the specified-* procedure and Eq. [9a] remained accurate within about ± 3% for relative standpipe drawdowns 50% (Ht 0.5H0, H0 = 1 m) (Fig. 3b); incorrect selection of * from Table 1 imparted less than ± 10% error in the Kfs estimates for porous media with capillarity ranging from negligible (* = 100 m–1) to high (* = 4 m–1). Similarly, numerical curve fitting of Eq. [9b] to Ht vs. t data for small standpipe drawdown (Ht 0.5H0, H0 = 2 m) produced Kfs estimates that fell within 3 to –18% of the true Kfs value for capillarity ranging from negligible (* = 100 m–1) to very high (* = 1 m–1). As expected, curve fitting Eq. [9b] did not produce reliable estimates of m.
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Conclusions
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Applying a Taylor series expansion to the natural logarithm in the Green–Ampt falling-head ring infiltrometer analysis, Eq. [1], resulted in an alternative series expression, Eq. [4], which avoids both the singularity when R = and potential rounding errors when R is nearly the same magnitude as . Equation [4] thus maintains the applicability and utility of the Green–Ampt analysis for determining Kfs and m for all physically plausible R values, including R = and R .
Setting R = in Eq. [4] produced a simple algebraic relationship, Eq. [5], that exhibited the classical square root time behavior of gravitationless infiltration, even though gravity may not be negligible. Equation [5] does not generally produce reliable estimates of m, but it can still produce good estimates of Kfs via numerical curve fitting to Ht vs. t measurements or by applying the specified-* procedure.
When R < , Eq. [5] and the expression of Fallow et al. (1994) (Eq. [8]) bracketed the true Kfs value for small standpipe drawdown when the gravitational force is small relative to the hydrostatic pressure and capillarity forces. Averaging the two expressions produced a new approximate relationship, Eq. [9], which predicted the true Kfs within about ± 20% for Ht 0.5H0, regardless of whether numerical curve fitting or the specified-* procedure was used. Equation [9] is thus a viable alternative analysis for estimating Kfs from falling-head ring infiltrometer data when standpipe drawdown is small and R < .
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Appendix
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The natural logarithm term in Eq. [1] is of the form ln(a + x), which can be represented by the Taylor series expansion (Dwight, 1965):
 | [A1] |
where a = 1 and x is given by
 | [A2] |
Substituting Eq. [A1] and [A2] into Eq. [1] and expanding produces
 | [A3] |
where A = H0 – Ht, B = H0 + m/2bKfs = H0 + (2b*)–1, and C = R – . Rewriting Eq. [A3] in summation notation yields Eq. [4].
As implied by Eq. [A1], the valid range of Eq. [A3] and [4] is
 | [A4] |
Note, however, that the physics of the falling-head ring infiltrometer method requires
 | [A5] |
and consequently
 | [A6] |
in Eq. [A4]. Hence, the valid mathematical ranges of R and in Eq. [A3] and [4] are 0 < R < + and 0 < < +.
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ACKNOWLEDGMENTS
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Funding for this work was provided by Agriculture and Agri-Food Canada. The assistance of J. Gignac in the preparation of Fig. 1 is greatly appreciated.
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ABSTRACT
INTRODUCTION
