788 SOIL SCI. SOC. AM. J., VOL. 63, JULY���AUGUST 1999 for estimating fractal dimensions of soil aggregates. Soil Sci. Soc. Tyler, S.W., and S.W. Wheatcraft. 1989. Application of fractal mathe- matics to soil water retention estimation. Soil Sci. Soc. Am. J. Am. J. 57:891���895. Shiozawa, S., and G.S. Campbell. 1991. On the calculation of mean 53:987���996. Tyler, S.W., and S.W. Wheatcraft. 1992. Fractal scaling of soil particle particle diameter and standard deviation from sand, silt, and clay fractions. Soil Sci. 152:427���431. size distributions: analysis and limitations. Soil Sci. Soc. Am. J. 56:362���369. Sokal, R.R., and F.J. Rohlfs. 1995. Biometry, 3rd ed. Freeman and Co., New York. Wu, Q., M. Borkovec, and H. Sticher. 1993. On particle-size distribu- tions in soils. Soil Sci. Soc. Am. J. 57:883���890. Turcotte, D.L. 1986. Fractals and fragmentation. J. Geophys. Res. 91:1921���1926. Young, I.M., J.W. Crawford, A. Anderson, and A. McBratney. 1997. Comment on number-size distributions, soil structure, and fractals. Turcotte, D.L. 1992. Fractals and chaos in geology and geophysics. Cambridge Univ. Press, Cambridge, UK. Soil Sci. Soc. Am. J. 61:1799���1800. Measuring Saturated Hydraulic Conductivity using a Generalized Solution for Single-Ring Infiltrometers L. Wu,* L. Pan, J. Mitchell, and B. Sanden ABSTRACT meter is a combination of both vertical and horizontal flow (Tricker, 1978). A method to calculate the Ks from Saturated hydraulic conductivity is a measure of the ability of a data obtained from a pressure or ring infiltrometer for soil to transmit water and is one of the most important soil parameters. New single-ring infiltrometer methods that use a generalized solution both early-time and steady-state infiltration was devel- to measure the field saturated hydraulic conductivity (Ks) were devel- oped by Reynolds and Elrick (1990), Elrick and Rey- oped and tested in this study. The Ks values can be calculated either nolds (1992), and Elrick et al. (1995). Their steady- from the whole cumulative infiltration curve (Method 1) or from state method uses a shape factor that was numerically the steady-state part of the cumulative infiltration curve by using a calculated based on Gardner���s (1958) relationship be- correction factor (Method 2). Numerical evaluation showed that the tween hydraulic conductivity and matric pressure head. Ks values calculated from the simulated infiltration curves of represen- Groenevelt et al. (1996) further extended this concept tative soil textural types were in the range of 87 to 130% of the real by developing a method to define the critical time that Ks values. Field infiltration tests were conducted on an Arlington fine separates early-time and steady-state infiltration. sandy loam (coarse-loamy, mixed, thermic, Haplic Durixeralfs). The By applying scaling theory, Wu and Pan (1997) devel- geometric means of the Ks values calculated from the field-measured infiltration curves by Method 1 and Method 2 were not significantly oped a generalized solution for single-ring infiltromet- different. The geometric mean of the Ks calculated from the detached ers. Wu et al. (1997) showed further that the infiltration core samples, however, was about twice that of the Ks calculated from rate of a single-ring infiltrometer was approximately f the infiltration curves, which was consistent with earlier findings. times greater than the one-dimensional (1-D) infiltra- Unlike the earlier approaches, Method 1 calculates Ks values from tion rate for the same soil, where f is a correction factor the whole infiltration curve without assuming a fixed relationship that depends on soil initial and boundary conditions and (a 5 Ks /fm) between saturated hydraulic conductivity and matric ring geometry. For a relatively small ponded head, the flux potential fm. 1-D final infiltration rate of a field soil is approximately equal to the field saturated hydraulic conductivity (Ks), which is valuable information for computer modeling, Saturated hydraulic conductivity is an important as well as for irrigation management. The objectives of soil parameter that measures the ability of a soil to this research were (i) to develop alternative methods transmit water. Measurement of field saturated hydrau- to calculate Ks by best fit of a generalized solution to lic conductivity (Ks) is often done by borehole permea- the infiltration curves that are measured by single-ring meters (Amoozegar and Warrick, 1986 Elrick and infiltrometers, and (ii) to compare and evaluate Ks val- Reynolds, 1992). In many cases, however, measurement ues calculated from infiltration curves of single-ring in- of the soil surface Ks is essential, especially in infiltra- filtrometers with those measured by the single head tion-related applications, such as irrigation manage- (SH) method (Elrick and Reynolds, 1992) and detached ment. soil core samples (Klute and Dirksen, 1986). Ring infiltrometers are often used for measuring the water intake rate at the soil surface. Water flow from THEORY a single-ring infiltrometer into soil is a three-dimen- A generalized infiltration equation developed by Wu and sional (3-D) problem (Reynolds and Elrick, 1990). The Pan (1997) has essentially the same form as the truncated total flow rate into the soil from a single-ring infiltro- Philip (1957) model of vertical infiltration. We propose here to measure infiltration curves in the field and then utilize the generalized equation to fit to the data in order to obtain L. Wu, Dep. of Environmental Sciences, Univ. of California, River- side, CA 92521 L. Pan, Earth Sci. Div., Lawrence Berkeley National the relevant parameters for estimating Ks. The generalized Lab., Univ. of California, Berkeley, CA 95720 J. Mitchell, Kearney equation (Wu and Pan, 1997) is Agri. Center, Univ. of California, Parlier, CA 93648 and B. Sanden, i/ic 5 a 1 b(t/Tc)20.5 [1] Univ. of California Coop. Ext., Kern County, Bakersfield, CA 93307. Received 8 June 1998. *Corresponding author (laowu@mail.ucr.edu). Abbreviations: SH, single head. Published in Soil Sci. Soc. Am. J. 63:788���792 (1999).

WU ET AL.: SINGLE-RING INFILTROMETER AND CONDUCTIVITY 789 where of the infiltration event has reached steady state. With this assumption, we can fit the linear equation ic 5 f Ks [2] I 5 At 1 c 5 a f Ks t 1 c [14] f 5 H 1 f9 m /Ks G* 1 1 [3] to infiltration data, and Ks can be calculated from Ks 5 A/(a f) [15] G* 5 d 1 r/2 [4] where A is the slope and c is the intercept from the linear f9 m 5 #0 hi K9(h)dh [5] regression, and f, which is defined by Eq. [3], can be estimated by Tc 5 Duls Ks 2 Ki ��� Du K 2 s f9 m [6] f ��� H 1 1/a G* 1 1 [16] where Du 5 u0 2 ui. The approximation in Eq. [6] follows since a 5 Ks/fm ��� Ks/f9 m, as shown later in this paper (Table from the definition 2) fm is the matric flux potential when Gardner���s (1958) hydraulic conductivity function is used. Method 2 is similar lS 5 1 Ks 2 Ki #0 hi K9 (h)dh [7] to the SH method of Elrick and Reynolds (1992), since a in Eq. [15] is very close to 1. and the fact that Ki ,, Ks under most field soil moisture conditions (Elrick and Reynolds, 1992). In Eq. [1] through MATERIALS AND METHODS [7], a and b are dimensionless constants (a 5 0.9084, b 5 0.1682) from the generalized equation, H is the ponded depth To test the reliability of the Ks measured and calculated in the ring, d is the ring insertion depth, r is the radius of the from the various methods, numerical tests were conducted ring infiltrometer, Ks and Ki are the hydraulic conductivity at to simulate infiltration curves for three representative sandy, saturated water content (u0) and at initial water content (ui), loamy, and clayey soils (Table 1). The axisymmetric form of h and hi are matric and initial matric pressure heads, and the Richards equation was solved numerically using the finite K9(h ) is the modified van Genuchten hydraulic conductivity��� volume (control volume) method (Pan and Wierenga, 1997). pressure head function (Wu and Pan, 1997). The model used a nonlinear, transformed pressure as the de- There are two ways to calculate Ks by applying the general- pendent variable with a modified Picard method. The hydrau- ized infiltration equation to the measured infiltration curves lic functions in the Richards equation were van Genuchten���s from a single-ring infiltrometer. Method 1 is based on the (1980) u(h ) and K(h ) relationships with m 5 1 2 1/n. The cumulative infiltration equation. By integrating Eq. [1] from numerical simulation domain was 100-cm radial by 100-cm t 5 0 to t 5 t, we have depth and was assumed to be homogeneous and isotropic. The initial soil water pressure head (hi) was assumed to be I 5 a f Ks t 1 2b f Ks(t Tc)0.5 [8] 21000 cm. Due to the differences in Ks and other hydraulic parameters, the time required to reach final infiltration rates or for the three different textural soils are different. The simu- I 5 At 1 B t 0.5 [9] lated infiltration duration here was 1 h, 1 d, and 10 d, respec- tively, for the representative sand (Berino fine sand: fine- Substituting Eq. [3] and [6] into [9], we can solve for Ks: loamy, mixed, thermic Typic Haplargids), sandy clay loam, and clay soil (Yolo clay: fine-silty, mixed, nonacid, thermic Ks 5 Du ls Tc [10] Typic Xerorthents). Parameters of the van Genuchten u(h ) and K(h ) relationships for the three representative soils used where the values of Wu and Pan (1997). The simulated infiltration curves were used to test Methods 1 and 2, and for comparison ls 5 1 2 [���(H 1 G*)2 1 4G*C 2 (H 1 G*)] [11] with the SH method of Reynolds and Elrick (1990). For Method 1, Eq. [9] was fit to the entire cumulative infiltration curve of each soil to obtain A and B (Fig. 1). For Method 2, Tc 5 1 4 1bA22 Ba [12] the last 20 min. of infiltration data from each soil were used to determine parameter A in Eq. [14] (Fig. 2). For the SH method, the simulated final infiltration rates of the soils were C 5 1 4Du 1B22 b a A [13] used to calculate Ks. The a values for the sand, loam, and clay soils were assumed to be 0.36, 0.12, and 0.04 cm21, respectively, for the SH method (Elrick and Reynolds, 1992). It is relatively easy to measure the initial soil water content, ui, at the time of an infiltration test, and u0 can be measured The field experiment was conducted at the University of California, Riverside Agricultural Field Experimental Station, or estimated from the bulk density and particle density. Method 2 is based on the assumption that the last part Moreno Valley, CA. The soil is classified as an Arlington fine Table 1. Comparison among the Ks values (cm min.21) calculated by Method 1 (M1), Method 2 (M2), and the SH method (SH) from the numerically simulated infiltration curves and the real Ks for the three test soils. Ks Soil M1 M2 SH Real Berino fine sand 4.87 3 1021 (130)�� 4.44 3 1021 (118) 4.61 3 1021 (123) 3.76 3 1021 Sandy clay loam 1.90 3 1022 (87) 2.45 3 1022 (112) 1.91 3 1022 (87) 2.18 3 1022 Yolo clay 7.72 3 1024 (105) 8.95 3 1024 (121) 7.04 3 1024 (95) 7.38 3 1024 �� Numbers in parentheses are the percentages of the real Ks.