Geoderma 435 (2023) 116511

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Repeated falling-head method for in situ measurements of saturated
hydraulic conductivity using a single cylinder
Mitsuhiro Inoue a, Hirotaka Saito b, *, Haruyuki Fujimaki a, Kosuke Noborio c, Koji Inosako d,Jiri Šimůnek eaArid Land Research Center, Tottori University, Tottori, Japan
Institute of Agricultural Science, Tokyo University of Agriculture & Technology, Tokyo, Japan
School of Agriculture, Meiji University, Kanagawa, JapandFaculty of Agriculture, Tottori University, Tottori, Japane
Department of Environmental Sciences, University of California, Riverside, CA, USAbc

A R T I C L E I N F O

A B S T R A C T

Handling Editor: Morgan Cristine L.S.

A simple, repeated falling-head (RFH) method, which uses an inexpensive portable cylinder to quickly determine
field-saturated hydraulic conductivity (Kfs), is presented. A cylinder of radius r, inserted vertically from the soil
surface to a depth d, is used as a water supply tank, through which water infiltrates into the soil. The falling-headtest is repeated without allowing the cylinder to be drained entirely to eliminate the effects of the initial soil
moisture conditions. Either change in the water level (H) in the water supply cylinder or strong linearity between
the infiltration flux (qs) and H can be used to determine Kfs directly or by multiplying the slope with an empiricalring-installation scaling length, Lg; thus, no soil-dependent variable is necessary. To obtain the Lg value forvarious conditions, we carried out numerical experiments using HYDRUS (2D/3D) with a recently implemented
reservoir boundary condition for six different soil textures, ranging from coarse soil (e.g., sand) to fine soil (e.g.,silty clay loam), and two different ranges of water levels, 15 ≤ H ≤ 25 cm and 2 ≤ H ≤ 6 cm. Numerical results
showed that the Lg has a strong linear dependence on d and r, where Lg increases as d and r increase. We propose
a new linear model to determine Lg with optimized coefficients for each H range. Numerical results showed nosignificant differences in the estimated Kfs after the second repetition, confirming that repeating the falling headtest twice is sufficient. The newly obtained Lg values were then applied to the experimental data obtained at
multiple fields having different soil textures. The Kfs values determined using the RFH method agreed well withthose measured using the field two-ponding depths steady-state method and/or the FH method in the lab. These
results demonstrate the reliability of the proposed RFH method.

Keywords:
Field-saturated hydraulic conductivity
Empirical ring-installation scaling length
HYDRUS
Numerical experiment
Drylands

1. Introduction
Hydraulic properties must be accurately determined to properly
manage soil and optimize irrigation strategies, particularly where water
resources are scarce and land degradation due to water erosion andsalinization is severe. The field-saturated hydraulic conductivity (Kfs) is
one of the most important soil hydraulic properties used in many hy­
drological (e.g., separating infiltration and runoff) and agricultural (e.g.,determining water requirements in irrigation systems) applications.
While several methods have been developed, many still require some
soil texture-dependent parameters or rather expensive devices. In
addition, since water is a precious resource in arid regions, there is alimit on the amount of water that can be used in experiments. There is,therefore, a great need to develop a quick, simple, and reliable method
using an inexpensive portable device that does not require much water
to determine Kfs in situ. Various steady-state and transient methods havebeen developed over the years. Steady-state methods include, forexample, using a single-cylinder pressure infiltrometer (Reynolds andElrick, 1990; Elrick and Reynolds, 1992; Reynolds et al., 2002; AnguloJaramillo et al., 2016). Transient methods include one-dimensionalfalling-head infiltration methods (Philip, 1992), early-time fallinghead infiltration methods (Elrick et al., 1995), single-ring infiltrometer
methods (Wu and Pan, 1997; Wu et al., 1999), simplified falling-head
infiltration methods (Bagarello et al., 2004; Bagarello and Sgroi, 2007;* Corresponding author.
E-mail address: hiros@cc.tuat.ac.jp (H. Saito).
https://doi.org/10.1016/j.geoderma.2023.116511
Received 10 December 2022; Received in revised form 3 April 2023; Accepted 7 May 2023
Available online 22 May 2023
0016-7061/© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).

M. Inoue et al.

Geoderma 435 (2023) 116511

Bagarello et al. 2013; Bagarello et al. 2014; Noborio et al., 2018),
Beerkan infiltration methods (Lassabatère et al., 2006; Bagarello et al.
2017), bottomless bucket falling-head methods (Nimmo et al., 2009), or
modified bottomless bucket methods (Mirus and Perkins, 2012), among
others.
A method based on the steady-state infiltration rate, measured for asingle water level (H) (the one-ponding depth approach) in a cylinder ofradius r inserted vertically from the soil surface to a depth d, requires asoil texture-dependent parameter (α*), defined as Kfs /ϕm, where ϕm isthe matric flux potential (Elrick and Reynolds, 1992). In contrast, thetwo-ponding depths (TPD) approach (Reynolds and Elrick, 1990) does
not require α* and measures steady-state infiltration fluxes, qs1 and qs2,
at two water levels, H1 and H2, respectively, and obtains Kfs using
Kfs = Gπr

(qs2 − qs1 )(H2 − H1 )means that the final estimate of Kfs depends on the duration of theexperiment. Kfs of the field soil may not be constant as soil structure canchange with time due to many factors, such as swelling and shrinkage
under wetting and drying. However, like many similar studies thatdeveloped in situ measurement methods, we developed the method
assuming that soil is rigid and homogenous so that Kfs is constant andtime-independent. Thus, it would be inconvenient to have timedependent Kfs. Additionally, although Nimmo et al. (2009) and
Noborio et al. (2018) used the RE coefficients to estimate Lg values, using
the coefficients derived for steady state conditions may be questionable
under transient conditions.
In this study, we propose a simple in-situ falling head test, a repeatedfalling head (RFH) test, in which the effect of initial soil wetness iseliminated by repeating the falling head multiple times, with a simple
data analysis that does not require any soil texture-dependent parame­
ters. The data analysis procedure requires the value of Lg following
Nimmo et al. (2009) and Noborio et al. (2018) but eliminates somedrawbacks of their analyses presented above. Details about the data
analysis procedure are presented in Materials and Methods. We cali­brated Lg under transient conditions for various initial water contents,
ranges of H, combinations of d and r, and soil types through numericalexperiments using the HYDRUS (2D/3D) software package with arecently implemented reservoir boundary condition (Šimůnek et al.,2018). We also conducted field and laboratory experiments to compare
Kfs obtained with RFH using the newly estimated Lg with the ones
measured with the TPD and falling head methods for soil cores from six
soils of different textures.

(1)

The dimensionless shape factor, G, accounts for complex interactions
among r, d, H, the three-dimensional soil capillarity effect, and gravity.
When G values are plotted as a function of d/r, they approximate astraight line with a slope (C1) and an intercept (C2) as follows:
( )
d
+ C2
G = C1(2)r
Reynolds and Elrick (1990) determined that C1 = 0.316 and C2 =0.184 (referred to as the RE coefficients throughout the manuscript)
using the numerical experiments for the steady-state, constant-head, andsingle-cylinder infiltration tests performed with cylinders having r = 5,
7.5, and 10 cm, and d = 3 and 5 cm. These coefficients were obtained byaveraging the G values for given H-r-d combinations for four different
soil types. According to them, the G values estimated using Eq. (2) areapplicable to 5 ≤ r ≤ 10 cm, 3 ≤ d ≤ 5 cm, and 5 ≤ H ≤ 25 cm. Although
the RE coefficients were validated only for a narrow range of d/r for
steady-state constant-head infiltration conditions, they have been
widely used, even for d/r outside of this range and for both steady andtransient methods (e.g., Youngs et al., 1995; Wu and Pan, 1997; Mirus
and Perkins, 2012; Bagarello et al., 2019). Compared with transientfalling head methods, steady-stage methods require longer measure­ment time to attain steady state conditions and an additional device,
such as a Marriott flask, to keep the water level constant.
Among many transient methods, the simplified falling-head method(Bagarello et al., 2004) is a simple method that uses a single cylinder but
requires soil texture-dependent parameter α*, in addition to values suchas the difference between the saturated and initial water contents, Δθ.
The Beerkan infiltration method (Lassabatère et al., 2006; Bagarello
et al., 2017) is a simple, inexpensive method that uses small cups as
cylinders through which water infiltrates into the soil to determine, forexample, a complete hydraulic characterization such as the hydraulicconductivity curve based on the relation by Brooks and Corey (1964).
However, the data analysis procedure requires the knowledge of soil
texture-dependent particle size distribution, which makes it difficult for
quick in-situ determination. The bottomless bucket falling-head method(Nimmo et al., 2009) is a simple method with a relatively simple data
analysis procedure. The formula derived for the constant head test isapplied to a very short duration of a falling head test where the rate ofchange in H equals the infiltration flux, qs. A soil texture-dependent
macroscopic capillary length (λ) (White and Sully, 1987) is stillrequired in addition to a ring-installation scaling length, Lg, that ac­
counts for complex interactions among r, d, H, the three-dimensional soil
capillarity effect, and gravity, similar to G used in the steady-state tests.
Since Noborio et al. (2018) followed in their falling head test the data
analysis of Nimmo et al. (2009), a value of Lg is still required, but insteadof using soil texture-dependent parameters α* or λ, they used the slope
between qs and H to estimate Kfs. Because qs was obtained by time dif­ferentiation of H that was approximated by a quadratic function of t, qs
became a linear function of time, and Kfs became time-dependent. This

2. Materials and methods
2.1. Procedure of repeated falling head method
As a variant of available transient falling head field tests to determine
Kfs, we propose, in this study, a repeated falling head (RFH) method, inwhich a cylinder of a radius r inserted vertically from the soil surface to adepth d is used. In RFH, the falling-head test is repeated without
allowing the cylinder to be drained entirely to eliminate the effects of
the initial soil moisture conditions. Detailed data analysis proceduresthat do not require soil texture-dependent parameters for RFH are given
below. To determine the number of necessary repetitions in RFH, field
experiments and numerical simulations were used.
2.2. Theory of repeated falling head method
Now, consider the situation of performing an in-situ falling head test,
where a cylinder of a radius r is inserted from the surface to a depth
d and the height of the water inside the column H [L]. As water in­
filtrates into the soil, H starts to drop. Assuming that the RE equation,
originally developed for the constant head test, can be applied to thevery short time period of the falling head test, the hydraulic conductivity
can be estimated from the following equation, as given by Nimmo et al.
(2009) and Noborio et al. (2018):()
dH
Lg + λ + H
qs = −
= Kfs
(3)
dt
Lg
This equation accounts for additional factors besides gravity, such as
sorption, horizontal spread of water below the column, and other factorsthat cause Kfs to systematically deviates from qs. Note that by replacing
Lg and λ with the column height and the bottom constant pressure head,respectively, this equation becomes equivalent to the Darcy equation
used to analyze the laboratory falling head test. The following data
analysis procedure of RFH relies on Eq. (3).
By integrating Eq. (3) from t = 0 to a given elapsed time t [T], during
which the water level drops from the initial level of H0 [L] to H, we get
the time evolution of H as follows:2

M. Inoue et al.

[
]
(
)
Kfs
H = H0 + Lg + λ exp −t − Lg − λLgGeoderma 435 (2023) 116511

Table 1
Parameter values in the van Genuchten-Mualem soil hydraulic functions pro­vided in the HYDRUS (2D/3D) software package for the six soils.

(4)

If Lg is given, Kfs can be directly obtained by fitting Eq. (4) to H-tdata.
By taking a derivative of Eq. (3) with respect to H, we can obtain thefollowing relationship.
Kfs = Lg

dqs
dH

Soil
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Silty clay loam

(5)

If Lg is given, the value of Kfs can be estimated based on Eq. (5) from
dqs/dH. This form is mathematically identical to that Noborio et al.
(2018) used to obtain Kfs. Nimmo et al. (2009) and Noborio et al. (2018)used parameter values proposed by Reynolds and Elrick (1990) for aconstant head test to determine Lg based on the linear model given as Eq.
(2). In this study, we propose a new linear model, which is more suitablefor in-situ falling head tests, through numerical experiments.
In practice, after obtaining Lg, there are mainly two approaches todetermining Kfs using data collected from RFH. One approach is, asmentioned above, to use Eq. (4) directly (Approach 1). Anotherapproach is to determine Kfs from Eq. (5), in which the value of dqs/dH isrequired. If H is monitored at a constant or approximately constant timeinterval, instead of applying Eq. (4), we may take the central differenceof H to obtain qs for each H. We can then obtain the slope dqs/dH withsimple linear regression. Kfs is finally obtained from Eq. (5) with Lg(Approach 2). Mathematically, this approach should give almost thesame result as the first approach, but does not require any curve fitting.
One can initially interpolate H-t data with a polynomial function if data
show large fluctuations or extremely uneven time intervals. Noborio
et al. (2018), for example, used a quadratic function to interpolate their
H-t data with the following equation:
H = A1 t 2 + A2 t + A3

m = 1−

α

n

l

Ks

cm3cm− 3

cm− 1

–

–

cm min− 1

0.045
0.057
0.065
0.078
0.067
0.089

0.43
0.41
0.41
0.43
0.45
0.43

0.145
0.124
0.075
0.036
0.02
0.01

2.68
2.28
1.89
1.56
1.41
1.23

0.5
0.5
0.5
0.5
0.5
0.5

0.495
0.24320.07370.01730.00750.00117

1
n

(7b)
(7c)

where θr [L3 L− 3] is the residual water content, θs [L3 L− 3] is the satu­
rated water content, Ks [L T− 1] is the saturated hydraulic conductivity, α[L− 1] is a shape coefficient, m, n, and l are dimensionless empirical co­efficients, h [L] is the soil water pressure head, θ [L3 L− 3] is the watercontent, and Se is the effective saturation [-].
The axisymmetrical simulation domain had a radius of 50 cm and a
depth of 60 cm. This domain size was selected to prevent the infiltration
front from reaching the domain’s boundaries for all soil types consideredin this study during the numerical experiments. The domain was dis­cretized into finite elements using the HYDRUS mesh generator with atargeted element size of 0.5 cm. The bottom of the simulation domain
was set as a free drainage boundary condition, along which the pressure
head gradients were set to zero. No-flow boundary conditions wereassigned to the sides of the domain and the soil surface, except inside of
the cylinder (similar to Wu and Pan, 1997; Šimůnek et al., 2018; Bag­
arello et al., 2019; Di Prima et al., 2019). The soil surface in the cylinder
was set to a reservoir boundary condition (Šimůnek et al., 2018), along
which pressure is dynamically updated in response to infiltration. The
pressure head at the soil surface corresponds to H.
We carried out 168 numerical experiments for various combinationsof H, r, d, and soil types. Four r (2, 4, 7.5, 14 cm), four d (3, 4, 5, 7 cm),and six soils were used for a high and wide range of H (15 ≤ H ≤ 25 cm
where H0 = 25 cm and H1 = 15 cm: denoted 15H25 hereafter), resulting
in a total of 96 cases. Three r (2, 4, 7.5 cm), four d (2, 3, 4, 5 cm), and six
soils were used for a low and narrow range of H (2 ≤ H ≤ 6 cm where H0= 6 cm and H1 = 2 cm; denoted 2H6 hereafter), resulting in a total of 72
cases. Most of these values were those employed in previous studies(Reynolds and Elrick, 1990; Di Prima et al., 2018; Noborio et al., 2018;
Shiraki et al., 2019) except for d = 2 cm, which was tested in our field
experiment. The choice of the H range depends on the available cylinderand other factors. Considering it should be easily carried, the cylinder
should be at most 30 cm long. We do not want to carry much water formeasurement in arid regions where water is scarce. Thus, a low andnarrow range is preferred. In this study, 15H26 is used as an example forthe former, while 2H6 is used for the latter. It should be noted that
because Lg depends on the used H range, one needs to perform numericalsimulations before measurements if different H ranges are used so thatappropriate Lg values can be determined.
The initial condition was set to either a uniform pressure head with agiven h or a hydrostatic equilibrium, having a pressure head of –100 cm
at the soil surface, depending on the purpose of each simulation. In thesimulations, as demonstrated in Fig. 1a and Fig. 2a, the initial H at thereservoir boundary was set to a value slightly higher than the highest
height of measurement, H0 (i.e., 27 cm for 15H25 and 7 cm for 2H6). As
the simulation proceeds, H at the reservoir boundary drops because of
infiltration. When H reaches a lower preset value (i.e., 13 cm for 15H25
and 1 cm for 2H6), H is returned to the initial H by applying a large flux
rate (i.e., adding water into a reservoir) at the boundary to repeat the(6)

2.3. Numerical experiments to obtain Lg values for various conditions
To determine Kfs using either Eq. (4) or (5), we need the Lg value. In
this study, the Lg values for various conditions were obtained throughnumerical simulations. The HYDRUS (2D/3D) software package with arecently implemented reservoir boundary condition (Šimůnek et al.,2018), which allows us to mimic the falling head process, was used forthe numerical experiments. We assume that the soil is homogeneous andisotropic, and is characterized by the default parameters (Carsel andParrish, 1988) of the van Genuchten-Mualem soil hydraulic functions(van Genuchten, 1980) provided in the HYDRUS (2D/3D) software
package for six soils ranging from fine to coarse textures: sand, loamy
sand, sandy loam, loam, silt loam, and silty clay loam according to the
USDA soil texture classification (Table 1). The van Genuchten-Mualem
soil hydraulic functions are given as:θ − θr− m= (1 + |αh|n )θs − θrθs

cm3cm− 3[)m ]2(K(h) = Ks Sel 1 − 1 − Se1/m

where A1 [LT− 2], A2 [LT− 1], and A3 [L] are empirical fitting parameters.
One of the drawbacks of this approach is that if dqs/dH is derived
analytically from Eq. (6), the final Kfs becomes time-dependent (see Eq.
(12) of Noborio et al. (2018)). Kfs of the field soil may not be constant assoil structure may change with time due to many factors, such as
swelling and shrinkage. However, in this study, like in many similar
studies, we assume the soil is rigid so that Kfs is not time-dependent.
Thus it is inconvenient to have time-dependent Kfs. To eliminate thetime dependency of Kfs, Noborio et al. (2018) suggested using half thetime from start to end instead of using the actual elapsed time. Instead ofusing an analytical solution, we can numerically obtain qs-H data fromthe interpolated H-t data simply by taking a slope, dH/dt (=qs), for given
H. After developing a qs-H relationship, simple linear regression can beused to obtain dqs/dH (Approach 3).

Se =

θr

(7a)

3

M. Inoue et al.

Geoderma 435 (2023) 116511

Fig.1. The numerical solution of (a) the time evolution of H and qs, and (b) a relationship between H and qs for a high and wide range of H (15H25), r = 7.5 cm, d =3 cm, and sandy loam. The hydrostatic equilibrium with a pressure head of –100 cm at the soil surface and dry condition (a constant h of − 5000 cm) were used as theinitial condition in (b).

Fig. 2. The numerical solution of (a) the time evolution of H and qs, and (b) a relationship between H and qs for a low and narrow range of H (2H6), r = 7.5 cm, d = 3
cm, and silty clay loam. The hydrostatic equilibrium with a pressure head of –100 cm at the soil surface and dry condition (a constant h of − 5000 cm) were used as
the initial condition in (b).

falling head process. Initially, the refill process was repeated four times
in each case to investigate if the effect of the initial condition could be
reduced as we repeated. The subsequent falling head periods were
denoted as Cycles 1, 2, 3, and 4. The number of repetitions can be
reduced if the results for successive repetitions are not significantly
different. In the numerical experiments, because HYDRUS outputs bothqs and H at the same print time, the slope dqs/dH can be directly ob­tained from linear regression between qs and H. The Lg value for eachcase was then obtained by solving Eq. (5) for Lg using Ks used in thesimulations. A multi-variable liner model is finally used to determine Lg
using d and r as independent variables for each range of H.
The accuracy of fitted Lg values in numerical experiments wasevaluated using the root mean square error (RMSE) defined as:√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
)2
1 ∑N (
Lg − Lgreg
RMSE =
i=1
N(8)where N is the total number of simulations, and Lgreg is Lg estimated using
determined linear coefficients. Since the true KS value is known in the
numerical experiments, the accuracy of Kfs estimated using Lgreg can bequantified using the relative error, Err (%), defined as⃒⃒⃒
Kfs ⃒⃒(9)
Err (%) = 100⃒⃒1 −Ks ⃒4

M. Inoue et al.

Geoderma 435 (2023) 116511

2.4. Field experiments

where B1 [LT− 1] and B2 [T− 1] are fitting parameters to estimate thesteady state qsi [LT− 1] for each of two water levels, Hi (i = 1, and 2), usedin Eq. (1). The TPD tests were carried out after the RFH tests using thesame cylinders without removing/reinstalling them. Since the TPD tests
were performed in this study purely for comparison, we do not intend to
discuss the validity of continuing measurements for long periods over
several days.

Field experiments are subject to various uncertainties and errors
associated with, for example, readings, an inclination of cylinders,temperature fluctuations, swelling of the soil, entrapped air, macro­pores, gaps between cylinder and soil, and disturbance of the soil surface
by poured water, which are difficult to replicate in numerical experi­ments. To evaluate the accuracy of RFH under actual field conditions,
we carried out RFH tests at six experimental sites with various soil
textures in Japan and compared the results with those using a common
two-ponding depth (TPD) approach and a laboratory falling head (FH)
method.
The six sites, as listed in Table 2, include a sandy field at the Arid
Land Research Center, Tottori University (ALRC, N 35◦ 32′ 06′′ , E
134◦ 12′ 42′′ ), three Faculty of Agriculture field plots, Tottori University(FATU, N 35◦ 30′ 54′′ , E 134◦ 10′ 13′′ ), a field at Tokyo University ofAgriculture and Technology (TUAT, N 35◦ 41′ 02′′ , E 139◦ 29′ 10′′ ), and afield at the Ikuta Campus of Meiji University (SAMU, N 35◦ 36′ 38′′ , E
139◦ 32′ 52′′ ). Table 2 summarizes the fractions of sand, silt, and clay,
bulk density, and soil classification according to the USDA classificationof soils at the experimental field sites. The r and d values used in theexperiments are also listed in Table 2. Steel cylinders with r = 5.5 and14 cm were used at the FATU site, while transparent acrylic cylinders
with scales were used at the other sites. When the H drop lasted several
days, cylinder tops were covered with a plastic sheet to prevent the ef­
fects of rain and evaporation. Gaps between the soil and the cylinder
formed during cylinder insertion were filled with in-situ soil using sticks.
Soil surface disturbance that occurred during water application was
minimized by placing a paper towel on the soil surface inside the cyl­
inder, which was later removed during the experiment. Cylinders werecovered with a heat insulation jacket with high reflectance to avoid
significant temperature effects when the measurement took a long time.
Because the H drop rate was fast at the ALRC and FATU-1 sites where thesoil texture was sand, changes in H were recorded by video images. Core
samples with a diameter of 5.1 cm and a height of 5 cm (resulting in 100
cm3 in volume) were taken at the ALRC and three FATU sites below thecylinders after the RFH tests were concluded to determine the Ks values
using a falling head method in the laboratory.
The TPD tests were carried out at the ALRC, FATU-1, and FATU-2
sites. In the TPD tests, a constant float valve was used in the watersupply container to mitigate the effects of temperature and atmosphericpressure fluctuations. Although experiments were continued until analmost constant flux was achieved, there is no standard protocol for
judging if a steady-state condition is reached. The time evolution of qs
was then fitted with:

3. Results and discussions
3.1. Numerical experiments to estimate Lg
Fig. 1 and 2 shows, as examples, H(t), qs(t), and qs(H) computed forsandy loam, 15H25, r = 7.5 cm, and d = 3 cm, and, for silty clay loam,
2H6, r = 7.5 cm, and d = 3 cm, respectively. Fig. 1b and Fig. 2b indicate
that, regardless of the soil type and H range, repeated cycles give almost
the same or similar qs(H) except for Cycle 1. Fig. 1b and Fig. 2b also
shows qs(H) for two different initial conditions: a hydrostatic equilib­
rium with a pressure head of –100 cm at the soil surface and a constant hof − 5000 cm. Fig. 3 shows the slope, dqs/dH, for six different soil texture
types, 15H25, r = 7.5 cm, and Cycles 2, 3, and 4, as a function of d. As
can be seen, dqs/dH values among different cycles are almost the same
regardless of the soil types and d. Same results were obtained for 2H6.
These results confirm that, except for Cycle 1, the initial condition’s
effect may be diminished and that repeating the falling-head experiment
twice is sufficient for RFH. By terminating the number of repetitions at
two, one can save considerable time and water compared with TPD
while minimizing the effect of the initial water content. The data from
Cycle 2 are then used further to obtain Lg. Fig. 4 depicts the Lg values forboth 15H25 and 2H6 as a function of d for 168 numerical experiments(96 for 15H25 and 72 for 2H6) under different combinations of radii andsoil textures. It shows that Lg depends linearly on d for all H ranges
regardless of the soil texture. It also shows that Lg increases as r in­creases, indicating that Lg is a function of both d and r. As Lg values
depend on the used H range, Lg was regressed using the following linear
model for each H range:
Lg = D1 d + D2 r + D3

There are three fitting parameters required in this model, while there
are only two parameters fitted to estimate G using the RE model (Eq.
(2)). Fitted D1 [-], D2 [-], and D3 [L] were 0.774, 0.294, and 0.663,
respectively, for 2H6, and those for 15H25 were 0.926, 0.427, and
0.446, respectively. Fitted lines for each r are also depicted in Fig. 4,
demonstrating that the proposed linear model (Eq. (11)) can be used to
predict Lg accurately. Fig. 4 also shows predicted Lg (grey lines) for each
H range using the RE coefficients for D1 (=0.993) and D2 (=0.578) with
D3 = 0, indicating that the RE coefficients overestimate Kfs under RFH.
The reliability of the linear model was indicated by small errors where
RMSE and Err for 2H6 were 0.155 and 2.61%, respectively, and 0.176
and 1.78% for 15H25, respectively. Note that these coefficients, unlike
α*, were valid regardless of the soil type.

(10)

qs = B1 exp (B2 t) + qsi

Table 2
The fractions of sand, silt, and clay, bulk density, and soil classification ac­
cording to the USDA classification of soils at the experimental field sites. The r
and d values employed in the RFH tests are also listed.
Exp.
Sites

Sand

Silt

Clay

(%)

(%)

(%)

ALRC
FATU1
FATU2

99
96

0.6
1

0.4
3

71

17

FATU3
TUAT

44

SAMU

Soil
textureBulkdensity

Employed cylinder(g cm− 3)r (cm)d (cm)

sand
sand

1.46
1.47

5.1
5.1, 7.5

4
4, 2

12

sandy
loam

1.18

2, 3, 4,
5

28

28

1.47

52.1

42.4

5.5

0.82

4.1, 7.5

5, 2

40

45

15

clay
loam
sandy
loam
loam

2.5, 4.1,
5.2, 5.5,
7.5, 142.5

0.63

4.1, 7.5

5, 2

(11)

3.2. Field experiments
All the Kfs and Ks values determined experimentally are, in theremainder of the paper, adjusted to those at 20 ◦ C (e.g., Hopmans andDane, 1986) to eliminate the effect of temperature following Klute(1965), in which the temperature dependence of water viscosity is used.
Some measurements took days to reach a steady state for the constanthead field infiltration experiments. In such cases, we adjusted Kfs using
an average temperature of the measurement duration. For the lab ex­
periments, the temperature of drained water was used to adjust the Ks
values.

75

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Geoderma 435 (2023) 116511

Fig. 3. Relationships between slope dqs/dH and insertion depth d for Cycles 2, 3, and 4 of the repeated falling-head test obtained for six soils using the cylinder withradius r = 7.5 cm and 2H6.

Fig. 4. The Lg values as a function of r and d for r = 7.5, 4, 2 cm, d = 2, 3, 4, 5 cm in case of 2H6, and for r = 14, 7.5, 4, 2 cm, d = 3, 4, 5, 7 cm in case of 15H25, lines
mean the fitting curve by Eq. (11).

estimated as 0.0247 cm min− 1. At FATU-2, it took about 120 min tocomplete two cycles, and the volume of water used was at least 1.3 L.
Based on our numerical experiments and some field experiments, weonly repeated the falling head process twice in RFH for the rest of theexperiments presented in this manuscript.

3.2.1. Effect of number of cycles
As an example, Fig. 5a depicts the time evolution of H observed for
2H6, r = 5.5 cm, and d = 3 cm at the FATU-2 site. In Fig. 5b, H is plottedagainst time, t’ [T], elapsed from the time t0i [T] when the water level
reached H = 6 cm during the falling head in each cycle, i. H(t’) curves
were almost identical except for Cycle 1, which also supports ourconclusion of using Cycle 2 to determine Kfs from the above numericalexperiments. A curve drawn in Fig. 5b is Eq. (4) fitted to data from Cycle
2, demonstrating that Eq. (4) can accurately describe H(t’). Using the Lgvalue computed with the coefficients obtained for 2H6, Kfs was

3.2.2. Comparison with the laboratory falling head and field Two-Ponding
depths methods
The TPD method was performed right after the RFH measurement
using the same cylinder without removing it at some experimental sites.
6M. Inoue et al.

Geoderma 435 (2023) 116511

Fig. 5. The H for a combination of 2H6, r = 5.5 cm, d = 3 cm, and FATU-2 (Table 2) (a) as a function of elapsed time t and (b) as a function of elapsed time t’ since thestart of each H = 6 cm falling head cycle (b).

in the infiltration flux qs for H1 and H2 are plotted in Fig. 6. By fitting Eq.
(10), qs1 = 0.875 cm min− 1 for H1, and qs2 = 1.075 cm min− 1 for H2 were
determined with the RMSE for both less than 0.006 cm min− 1. Using the
RE coefficients (C1 = 0.316, C2 = 0.184), we obtained G = 0.432 using
Eq. (2) for the TPD method. The Kfs value determined using Eq. (1) withr = 5.1 cm was then 0.198 cm min− 1, while Kfs determined by the RFH at
the FATU-1 site was 0.181 cm min− 1. At this site, after completing both
RFH and TPD tests, core sampling was performed at a depth of 5 cm inthe cylinder to determine the laboratory Ks, which resulted in 0.163 cm
min− 1. The relative deviation between Kfs determined by RFH and TPD
was less than 15%. It has to be noted that the total time required for the
TPD method was over 500 min (Fig. 6) because we had to perform theinfiltration tests twice for two water levels, which was 25 times longer
than the 20 min required by the RFH method at the FATU-1 site. To savetime, the test could be stopped when dqs/dt is, for example, less than halfof the initial slope and estimate the asymptote using Eq. (10), but results
may be less reliable.
Figs. 7 and 8 plot Kfs obtained using the data of Cycle 2 of the RFH
experiment based on the proposed data analysis for 15H25 and 2H6,
respectively, and those obtained using the standard laboratory FH andfield TPD methods for all experimental sites tested in this study. For
15H25, only Approach 1 was used for RFH, while for 2H6, all three
approaches were used. For all soils, from coarse to fine, most points fallclosely along a 1:1 line depicted as a solid line in Figs. 7 and 8, with amaximum deviation of less than 25%, which is considered “accurate”
based on Reynolds (2013), suggesting that the proposed method isapplicable to a variety of soil types. The average deviation from Ks ob­
tained using the FH method was 10.8%, while the average discrepancy
from the TPD-measured Kfs was 11.4%. Those deviations are acceptable,
given that Ks can vary by a few orders of magnitude, even for samples ofthe same soil texture. Fig. 8 shows that all three approaches result inalmost the same results suggesting that any approach can be taken. The
overall results indicate that the RFH method with newly determined Lg is
as reliable as the standard laboratory FH and field TPD methods. Given
the simpler experimental setup and analytical method, we suggest using
the RFH method in case Kfs needs to be determined in situ.

Table 3
Experimental conditions and results of the two-ponding depths (TPD) steadystate method.
Exp.Sites
ALRC
ALRC
FATU-1
FATU-1
FATU-2
FATU-2

r

d

H1

H2

qs1

qs2

Kfs

cm

cm

cm

cm

cm min− 1

cm min− 1

cm min− 1

5.1
5.1
5.1
7.5
7.5
5.1

4
4
4
4
4
4

16.5
17
16.6
12.8
11.7
10

24.7
24
21.5
23.3
24.3
20.5

6.4
4
0.875
2
0.00565
0.022

7.9
5.1
1.075
2.4
0.0072
0.024

1.265
1.087
0.282
0.316
0.001020.00132

Table 3 summarizes the experimental conditions of TPD performed atour experimental sites with the resulting Kfs. At ALRC and FATU-1 sites,
the TPD tests were performed twice with different water levels. For
example, at the FATU-1 site, the TPD method was performed withconstant water levels H1 = 16.6 cm and H2 = 21.5 cm for conditions witha radius r = 5.1 cm and an insertion depth d = 4 cm. Temporal changes

3.2.3. Choice of H range
The RFH method was performed at the TUAT and SAMU sites withtwo different H ranges, respectively. The test conditions are as follows:

Fig. 6. Changes in the infiltration flux qs during the TPD test at FATU-1 for two
different H with a curve given by Eq. (10) fit to each H to obtain the steady state
infiltration flux qs.
7

M. Inoue et al.

Geoderma 435 (2023) 116511(1) 15H25, r = 4.1 cm, and d = 5 cm, and (2) 2H6, r = 7.5 cm, and d = 5
cm. At the TUAT site, the obtained Kfs were (1) 0.102 and (2) 0.112 cm
min− 1, respectively, while at the SAMU site, they were (1) 0.799 and (2)0.823 cm min− 1, respectively. For both sites, relative differences be­
tween the two conditions are less than 9%. In both cases, the water used
was less than 2.5 L, and the measurement time was shorter by about onethird for 2H6 compared to 15H25. For sandy soils, 15H25 (or a similar H
range) is recommended because of the rapid H change, while a low andnarrow range, such as 2H6, should be adopted for clayey soils to shorten
measurement time.
4. Conclusions
In this study, we propose a repeated falling head method to deter­mine filed saturated hydraulic conductivity, Kfs. A ring-installation
scaling length, Lg, required in the data analysis procedure was cali­brated with numerical experiments using the HYDRUS (2D/3D) pro­gram. The following conclusions are drawn from this study bycomparing the RFH method with the standard laboratory falling head(FH) and field two-point depth (TPD) methods for various experimental
conditions and soil types.
1. Strong linearity between the water level, H, in the water supply
cylinder and the infiltration flux qs during the second cycle of the
RFH method, regardless of the initial soil wetness, allows us to adopt
the data analysis requiring Lg proposed by Nimmo et al. (2009) and
Noborio et al. (2018). The effect of the initial soil wetness can beeliminated after two repetitions by keeping the cylinder from being

Fig. 7. Comparison between Kfs determined using a repeated falling-head
method (RFH) and the two-ponding depth (TPD) approach, and Ks measured
using a laboratory falling-head method on soil cores for 15H25.

Fig. 8. Comparison between Kfs determined using a repeated falling-head method (RFH) and Ks measured using a laboratory falling-head method on soil coresfor 2H6.
8

M. Inoue et al.

Geoderma 435 (2023) 116511

completely drained. Thus, repeating the falling head experiment
twice is sufficient in the RFH method.
2. HYDRUS (2D/3D) with a newly implemented reservoir boundary
condition allows one to simulate the RFH method. Because thesaturated hydraulic conductivity is known in the simulations, the Lgvalues can be obtained for various conditions for the RFH method.
Numerical experiments show that Lg has a strong linear dependence
on d and r, where Lg increases as they increase and is not soil-texture
dependent. Multivariate linear regression analyses provided opti­
mized coefficients for each H range for a linear model proposed inthis study. One should employ different linear coefficients fordifferent H ranges.
3. The Kfs values determined for various soil types with the newly
determined Lg for two H ranges in the RFH method were comparable
to those obtained from the laboratory FH and field TPD methods,
confirming the reliability of the proposed method. The maximum
deviation between the proposed and standard methods’ results wasmuch less than 25%, which can be considered “accurate.”

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Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgments
This study was supported partly by Tottori University International
Platform for Dryland Research and Education, Joint Research Programof Arid Land Research Center, Tottori University (No. 03C2018), and
JSPS Grant-in-aid Scientific Research Program (20H03097, 23H02325).
The authors are grateful to former students Koichi Noguchi and Koki
Oikawa for their help and contribution to the experimental data pre­sented in this paper. We would like to thank Editage (www.editage.com)
for English language editing.
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