HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-2725-2017Saturated hydraulic conductivity model computed from bimodal water retention
curves for a range of New Zealand soilsPollaccoJoseph Alexander Paulpollacco.water@gmail.comWebbTrevorMcNeillStephenhttps://orcid.org/0000-0003-1288-4387HuWeihttps://orcid.org/0000-0002-5911-178XCarrickSamHewittAllanLilburneLindaLandcare Research, P.O. Box 69040, Lincoln 7640, New ZealandNew Zealand Institute for Plant & Food Research Limited, Private
Bag 4704, Christchurch 8140, New ZealandJoseph Alexander Paul Pollacco (pollacco.water@gmail.com)9June20172162725273730November201621December201628March201729March2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/2725/2017/hess-21-2725-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/2725/2017/hess-21-2725-2017.pdf
Descriptions of soil hydraulic properties, such as the soil moisture retention
curve, θ(h), and saturated hydraulic conductivities, Ks, are a prerequisite for hydrological models. Since the
measurement of Ks is expensive, it is frequently derived from
statistical pedotransfer functions (PTFs). Because it is usually more difficult to describe
Ks than θ(h) from pedotransfer functions, Pollacco et al. (2013)
developed a physical unimodal model to compute Ks solely from hydraulic
parameters derived from the Kosugi θ(h). This unimodal Ks model,
which is based on a unimodal Kosugi soil pore-size distribution, was
developed by combining the approach of Hagen–Poiseuille with Darcy's law and
by introducing three tortuosity parameters. We report here on (1) the
suitability of the Pollacco unimodal Ks model to predict Ks for a
range of New Zealand soils from the New Zealand soil database (S-map) and (2) further adaptations to this model to
adapt it to dual-porosity structured soils by
computing the soil water flux through a continuous function of an improved
bimodal pore-size distribution. The improved bimodal Ks model was tested
with a New Zealand data set derived from historical measurements of
Ks and θ(h) for a range of soils derived from sandstone and
siltstone. The Ks data were collected using a small core size of 10 cm
diameter, causing large uncertainty in replicate measurements. Predictions
of Ks were further improved by distinguishing topsoils from subsoil.
Nevertheless, as expected, stratifying the data with soil texture only
slightly improved the predictions of the physical Ks models because the
Ks model is based on pore-size distribution and the calibrated
parameters were obtained within the physically feasible range. The
improvements made to the unimodal Ks model by using the new bimodal
Ks model are modest when compared to the unimodal model, which is
explained by the poor accuracy of measured total porosity. Nevertheless, the
new bimodal model provides an acceptable fit to the observed data. The study
highlights the importance of improving Ks measurements with larger
cores.
Introduction
Modelling of the water budget, irrigation, and nutrient and contaminant
transport through the unsaturated zone requires accurate soil moisture
retention, θ(h), and unsaturated hydraulic conductivity, K(θ),
curves. The considerable time and cost involved in measuring θ(h) and
K(θ) directly for a range of soils mean that the information for specific
soils of interest is often not available (Webb, 2003). Therefore, these
curves are generally retrieved from pedotransfer functions (PTFs), which are
statistical relationships that generate lower-precision estimates of
physical properties of interest based on many rapid and inexpensive
measurements (e.g. Balland and Pollacco, 2008; Pollacco, 2008; Anderson and
Bouma, 1973; Webb, 2003; Cichota et al., 2013).
The S-map database (Lilburne et al., 2012; Landcare Research, 2015) provides
soil maps for the most intensively used land in New Zealand and is being
gradually extended to give national coverage. S-map provides data for
extensively used soil models, such as the soil nutrient model OVERSEER and
the daily simulation model APSIM used by agricultural scientists. McNeill et al. (2012) used the New Zealand National Soils Database to derive PTFs to
estimate θ(h) at five tensions from morphological data of soils mapped
in S-map. One of the current weaknesses of S-map is a lack of capacity to
estimate K(θ). Building on the work of Griffiths et al. (1999), Webb (2003) showed that morphologic descriptors for New Zealand soils can be used
to predict Ks. However, the predictions of Ks were found to be too
coarse for application to the wide range of soils within S-map. Therefore,
Cichota et al. (2013) tested published statistical PTFs developed in Europe
and the USA to predict θ(h) and K(θ) for a range of New Zealand
soils. They combined the best two or three PTFs to construct ensemble PTFs.
They considered the ensemble PTF for θ(h) to be a reasonable fit, but
the ensemble PTF for estimating Ks exhibited large scatter and was not
as reliable. The poor performance when estimating Ks was possibly due to
the absence of any measurements of pore-size distribution in their physical
predictors (Watt and Griffiths, 1988; McKenzie and Jacquier, 1997; Chapuis,
2004; Mbonimpa et al., 2002) and also to the large uncertainties in the
measurements from small cores (McKenzie and Cresswell, 2002; Anderson and
Bouma, 1973). Consequently, there is an urgent need in New Zealand to
develop a physically based Ks model which is based on pore-size
distribution.
Since PTFs developed to characterize θ(h) are more reliable than PTFs
to characterize K(θ) (e.g. Balland and Pollacco, 2008; Cichota et al.,
2013), Pollacco et al. (2013) developed a new physical model that predicts
unimodal Ks solely from hydraulic parameters derived from the
Kosugi (1996) θ(h). The Ks model is derived by combining the
Hagen–Poiseuille and Darcy laws (Sutera and Skalak, 1993) and by incorporating three
semi-empirical tortuosity parameters. The model is based on the soil
pore-size distribution and has been successfully validated using the
European HYPRES (Wösten et al., 1998, 1999; Lilly et
al., 2008) and the UNSODA databases (Leij et al., 1999; Schaap and van
Genuchten, 2006) but has not yet been applied to New Zealand soils. Most
New Zealand soils are considered to be structured, with two-stage drainage
(Carrick et al., 2010; McLeod et al., 2008) and bimodal pore-size
distribution (e.g. Durner, 1994). Romano and Nasta (2016) showed by using
the HYDRUS-1D package that large errors arise in the computation of the
water fluxes if unimodal θ(h) and K(θ) are used in structured
soils. We therefore propose to improve the unimodal Pollacco et al. (2013)
Ks model so that it can predict Ks for structured soils with
bimodal porosity.
Measured Ks values are widely recognized as one of the most variable soil
attributes (McKenzie and Cresswell, 2002; Carrick, 2009). This is also
recognized for New Zealand soils, both due to the high variability over
short distances in soil parent material, age, depth, and texture, as well as
strong macropore development with preferential macropore flow recognized as
the norm rather than the exception in New Zealand soils (Webb et al., 2000;
Carrick, 2009; McLeod et al., 2008). The measurement variability is also
expected to increase as the sampling diameter decreases because small cores
provide an unrealistic representation of the abundance and connectivity of
macropores (McKenzie and Cresswell, 2002; Anderson and Bouma, 1973).
McKenzie and Cresswell (2002) suggest that the standard Australian
laboratory measurements should use cores with a minimum diameter of 25 cm and
length of 20 cm. In New Zealand, Ks has been obtained by using small cores,
commonly with 10 cm diameter and 7.5 cm length. This has contributed to very
high variability in measured Ks (Webb et al., 2000).
The objectives of this research were to
test the suitability of the unimodal Pollacco et al. (2013) Ks model
to predict Ks from New Zealand soils,
develop a Ks bimodal model that makes predictions in structured soils
solely from hydraulic parameters derived from the Kosugi θ(h),
derive the uncertainties of the predictions of the Ks bimodal
model, and
provide recommendations on the critical data sets that are required to
improve the S-map database in New Zealand.
BackgroundKosugi unimodal water retention and unsaturated hydraulic conductivity
curve
There are a number of closed-form unimodal expressions in the literature
that compute the soil moisture retention curve θ(h) and the unsaturated
hydraulic conductivity K(θ) curves, such as the commonly used van
Genuchten (1980) and Brooks and Corey (1964) curves. We selected the
physically based Kosugi (1996) closed-form unimodal log-normal function
expression of θ(h) and K(θ) because its parameters are
theoretically sound and relate to the soil pore-size distribution (Hayashi
et al., 2009). Soils have a large variation in pore radius, r, which follows
a log-normal probability density function. The unimodal Kosugi log-normal
probability density function of pore radius (r) is often written in the
following form:
dθdr=θs-θrrσ2πexp-lnr/rm22σ2,
where θr and θs (cm3 cm-3) are the residual and
saturated water contents, rm (cm) is the median pore radius, and σ
(-) denotes the standard deviation of ln(r).
Let Se denote the effective saturation, defining Ser=θ-θr/θr-θs such that 0 ≤Se≤ 1.
Integrating Eq. (1) from 0 to r yields the unimodal water retention curve as a function of r:
Se(r)=12erfclnrm-lnrσ2,
with
r=rmexperfc-12Seσ2,
where erfc is the complementary error function.
The Young–Laplace capillary equation relates the soil-pore radius, r, to the
equivalent matric suction head, h (cm), at which the pore is filled or drained (i.e. r=Y/h, where Y= 0.149 cm2). Kosugi's unimodal moisture retention curve θuni(h) can be
written in terms of Se:
Se(h)=12erfclnh-lnhmσ2,
where hm (cm) is the median metric head.
The unimodal Kosugi unsaturated hydraulic conductivity function K(θ) is
written as
K(Se)=KsSe12erfcerfc-12Se+σ22,
where Ks (cm day-1) is the saturated hydraulic conductivity.
θs is computed from the total porosity, ϕ, which is deduced from
bulk density (ρb) and soil particle density (ρp) as follows:
ϕ=1-ρbρp.
Due to air entrapment, θs seldom reaches saturation of the
total pore space ϕ (Carrick et al., 2011). Therefore, to take into
account the fact that not all pores are connected, we perform the following
correction of ϕ with α in the range [0.9, 1]:
θs=αϕ.
It is accepted that α= 0.95 (Rogowski, 1971; Pollacco et al.,
2013; Haverkamp et al., 2005; Leij et al., 2005), but in this study the
optimal α was found to be 0.98, since using a value of 0.95 resulted
in several soil samples with θ5 (θ measured at 5 kPa)
greater than θs, which is not physically plausible. This was
due to the inaccuracy of measuring ϕ (discussed in Sect. 4.1).
The feasible range of the Kosugi hydraulic parameters is summarized in Table 1.
The hm and σ feasible range is taken from Pollacco et al. (2013), who combined data from the HYPRES (Wösten et al., 1998, 1999;
Lilly et al., 2008) and UNSODA (Leij et al., 1999;
Schaap and van Genuchten, 2006) databases.
Feasible range of the Kosugi parameters and θ5 (which
is θ measured at 5 kPa).
θsθrlog10hmσ(cm3 cm-3)(cm3 cm-3)(cm)(–)Minθ50.01.230.8Max0.600.205.424.0Pollacco unimodal saturated hydraulic conductivity model
The saturated hydraulic conductivity model, Ks_uni (Pollacco et al., 2013), computes
Ks from the Kosugi parameters θs, θr,
σ, and hm (or rm). Ks_uni is based on the
pore-size distribution (Eq. 1) and the tortuosity of the pores.
Ks_uni was derived by adopting the method of Childs and
Collisgeorge (1950) and modelling the soil water flux through a continuous
function of Kosugi (1996) pore-size distribution. This was performed by
combining the Hagen–Poiseuille equation (Sutera and Skalak, 1993)
with Darcy's law and introducing the connectivity and tortuosity parameters
τ1, τ2 of Fatt and Dykstra (1951) and τ3 of
Vervoort and Cattle (2003). Ks_uni is computed as
Ks_uni=C1-τ1θs-θr11-τ3∫01r21-τ2dSe,
with C=18ρwgη for water at 20 ∘C, density of water ρw=0.998 g cm-3, acceleration due to gravity g=980.66 cm s-2, dynamic
viscosity of water η=0.0102 g cm-1 s-1, and C is a constant
equal to 1.03663 × 109 cm day-1.
Integrating with Se instead of r avoids the complication of finding the
minimum and maximum values of r. Isolating r of Eq. (2b) and replacing it in
Eq. (7) gives
Ks_uniSe=C1-τ1θs-θr11-τ3∫01Y/hmexperfc-12Seσ221-τ2dSe
or
Ks_uni=C1-τ1θs-θr11-τ3∫01rmexperfc-12Seσ221-τ2dSe,
and rm=Y/hm (Young–Laplace capillary equation)
where τ1, τ2, τ3 are tortuosity parameters
[0–1).
Description of the tortuosity parameters.
TortuosityDescriptionτ1This takes into account the increased path length due to crookedness of the path. When τ1=0, the flow path is perfectly straight down. When τ1 increases, the flow path is no longer straight but meanders.τ2This theoretically represents the shape of a microscopic capillary tube. The τ2 parameter is used to estimate restrictions in flow rate due to variations in pore diameter and pore shape. When τ2=0, the shape of the capillary tube is perfectly cylindrical. When τ2 increases, the tube becomes less perfectly cylindrical, which causes lower connectivity.τ3High-porosity soils tend to have large effective pores, θs-θr, which tend to be more connected than soils with smaller effective pores, which have more dead ends. When τ3=0, the connectivity is the same between high- and low-porosity soils. When τ3 increases, the connectivity of the soil increases (Vervoort and Cattle, 2003; Pollacco et al., 2013). Pollacco et al. (2013) found τ3 to be the least sensitive parameter.
If tortuosity were not included (τ1, τ2, τ3= 0), the pore-size distribution model would mimic the permeability of a
bundle of straight capillary tubes. Vervoort and Cattle (2003)
state “In reality soils are much more complex, with twisted
and crooked pores, dead-ending or connecting to other pores. This means that
there is a need to scale the permeability from the capillary tube model to
include increased path length due to crookedness of the path (tortuosity) or
lack of connection between points in the soil (connectivity)”.
Soils that are poorly connected and have highly crooked pathways
theoretically have τ1, τ2, τ3≈
0.9. Further explanation of tortuosity is provided in Table 2.
Romano bimodal water retention curve
New Zealand soils are predominantly well structured, with two-stage drainage
(Carrick et al., 2010; McLeod et al., 2008) and therefore have a bimodal
pore-size distribution (e.g. Durner, 1994). As Ks_uni is
based on a unimodal curve, θuni(h), the proposed bimodal model,
Ks_bim, should be based on a bimodal θbim(h) curve.
Borgesen et al. (2006) showed that structured soils have both matrix
(inter-aggregate) pore spaces and macropore (intra-aggregate) pore spaces. Thus, when
the pores are initially saturated, such as (r > Rmac) or (h < Hmac), the flow is considered macropore flow, and when the soil is
desaturated, such as (r < Rmac) or (h > Hmac), the flow is considered
matrix flow, as shown in Fig. 1. Rmac is the theoretical pore-size r that delimits
macropore and matrix flow, and Hmac is the theoretical pressure that
delimits macropore and matrix flow. To model bimodal pore-size distribution,
Durner (1994) superposes two unimodal pore-size distributions by using an
empirical weighting factor, W, which partitions the volumetric percentage of
macropore and matrix pores. Recently, Romano et al. (2011) proposed the
following Kosugi bimodal θbim_rom(h) distribution:
θbim_rom(h)=θs-θrWerfclnh-lnhm_macσ_mac2+1-Werfclnh-lnhmσ2+θr,
where θs, hm_mac, and σ_mac are, respectively, the saturated water content, the
median pore radius, and the standard deviation of ln(h) of the macropore
domain; θr, hm, and σ are parameters of the matrix
domain; and W is a constant in the range [0, 1).
A typical Kosugi θbim(r) (Eq. 10a) and
θbim_mat(r) (Eq. 10b) with the matrix and macropore domains
and the positions of θs , θs_mac,
θr, rm, rm_mac, and Rmac
shown.
Theoretical development of novel bimodal saturated hydraulic
conductivity
We report on further adaptations to the physical model of Pollacco et al. (2013)
to suit it to dual-porosity structured soils, which are common in New
Zealand, solely from Kosugi hydraulic parameters describing θ(h). This
involves
rewriting the Romano bimodal θ(h) (Sect. 3.1) and
developing a novel bimodal Ks model based on the modified bimodal θ(h) (Sect. 3.2).
Modified Romano bimodal water retention curve
We propose a modified version of θbim_rom(h) (Eq. 9) that does not use the empirical parameter W. Our modified function,
θbim(h), is plotted in Fig. 1 and is computed as
θbim(h)=θbim_mat(h)+θbim_mac(h)θbim_mat(h)=θs_mac-θrerfclnh-lnhmσ2+θrθbim_mac(h)=θs-θs_macerfclnh-lnhm_macσ_mac2,
where θs_mac is the saturated water content that theoretically
differentiates macropore and matrix domains.
The shape of θbim(h) is identical to that of θbim_rom(h), but the advantage of θbim(h) is that
it uses the physical parameter θs_mac instead of
the empirical parameter W, and θs_mac (≤θs) is more easily parameterized than W particularly when there
are no available data in the macropore domain. When we do not have data in
the macropore domain, θs_mac is determined by
fitting the hydraulic parameters θs_mac,θr, hm, and σ of θbim_mat(h)
(Eq. 11) solely in the matrix range (r < Rmac or
h>Hmac). Fig. 1 shows that Rmac and θs_mac
delimit the matrix and the macropore domains and that rm of the Kosugi
model is the inflection point of θbim_mat(h) and
rm_mac is the inflection point of θbim_mac(h).
Novel bimodal saturated hydraulic conductivity model
Using θbim(h), we propose a new bimodal Ks_bim that is derived following Ks_uni (Eq. 7) but for
which we add a macropore domain:
Ks_bim=Ks_bim_mat+Ks_bim_macKs_bim_mat=C∫011-τ1θs_mac-θr11-τ3rmatrix21-τ2dSeKs_bim_mac=C∫011-τ1_macθs-θs_mac11-τ3_macrmacropore21-τ2_macdSe,
where rmacropore is r≥Rmac and rmatrix is r < Rmac.
The rmatrix of Eq. (14) is derived from Eq. (2b):
rmatrix=rmexperfc-12Seσ2,
and rmacropore is computed similarly as
rmacropore=rm_macexperfc-12Seσ_mac2.
We introduced rmatrix (Eq. 16) and rmacropore (Eq. 17) into
Ks_bim (Eq. 13), giving the equation for
Ks_bim:
Ks_bim=C∫011-τ1θs_mac-θr11-τ3rmexperfc-12Seσ221-τ2+1-τ11_macθs-θs_mac11-τ3_macrm_macexperfc-12Seσ_mac221-τ2_macdSe
or
Ks_bim=C∫011-τ1θs_mac-θr11-τ3Yhmexperfc-12Seσ221-τ2+1-τ11_macθs-θs_mac11-τ3_macYhm_macexperfc-12Seσ_mac221-τ2_macdSe.
In Eq. (19), rm_mac is replaced by
Y/hm_mac and rm is replaced by Y/hm. Note that the
bimodal Ks model requires that the flow in the macropore domain obeys
the Buckingham–Darcy law. Therefore, this model's performance may be
restricted in cases of non-Darcy flow, such as non-laminar and turbulent
flow, which may occur in large macropores.
Theoretical constraints of the Ks_bim model.
ConstraintExplanationθs≥θs_mac≫θrSelf-explanatory0 < σmac≤ 1.5To avoid any unnecessary overlap of θbim with θbim_mat1 > τ1 > τ1_mac≥ 0Flow in the macropore domain (larger pores) is expected to be straighter than in the matrix domain (smaller pores) due to reduced crookedness of the path1 > τ2 > τ2_mac≥ 0It is expected that the shape of the “microscopic capillary tube” of the macropore domain (larger pores) is more perfectly cylindrical than in the matrix domain (smaller pores)1 > τ3 > τ3_mac≥ 0The macropore domain has larger pores, and therefore it is assumed that the pores are better connected than the matrix pores
In this study, σ_mac is not derived from measured
θ(h) because measured data in the macropore domain are not always
available, and so it will be treated as a fitting parameter. As discussed
above, θs_mac, θr, σ, and
hm are optimized with θuni(h) measurement points only in the
matrix range (r < Rmac or h > Hmac), which means
that θs is not included in the observation data. In summary, Ks_bim requires optimization of
the parameters τ1, τ2,
τ3, and τ1_mac, τ2_mac,
τ3_mac, hm_mac, and σ_mac (if no data are available
in the macropore domain). The theoretically feasible range of the parameters
of Ks_bim is shown in Table 3.
One of the limitations of the New Zealand data set is that it has no θ(h) data points in the macropore domain. The closest data point near saturation
is θ(h= 50 cm), which is in the matrix pore space. Carrick et al. (2010) found that Hmac ranges from 5 to 15 cm, with an average
Hmac= 10 cm, which corresponds to a circular pore radius of
Rmac= 0.0149 cm (e.g. Jarvis, 2007; Jarvis and Messing, 1995; Messing
and Jarvis, 1993). Therefore, to reduce the number of optimized parameters
we make the following assumption:
hm_mac=expln(Hmac)Pm_mac,
where Pm_mac is a fitting parameter greater than 1. We found the
fitted value of Pm_mac was 2.0; however, this fitted parameter
was very broadly determined. The cause might be that we are optimizing
σ_mac, and therefore hm_mac and
σ_mac might be linked. Linked parameters (Pollacco et al.,
2008a, b, 2009) mean that there is an infinite combination of sets of linked
parameters hm_mac and σ_mac which produces
values of objective function close to that obtained with the optimal
parameter set and for which there exists a continuous relationship between
hm_mac and σ_mac. Further research needs to
determine if having more data in the macropore domain would reduce the cause
of non-uniqueness. To illustrate hm_mac, the equivalent
rm_mac point is shown in Fig. 1, where rm_mac is
the inflection point of the macropore domain. Figure 1 also shows that the
matrix and the macropore domains meet at Rmac (Hmac).
The soil data used in this study were sourced from two data sets. In the
first data set (Canterbury Regional Study; Table 4), soils were derived from
eight soil series on the post-glacial and glacial alluvial fan surfaces of
the Canterbury Plains (Webb et al., 2000). The soils varied from shallow,
well-drained silt loam soils to deep, poorly drained clay loam soils. The
second data set was derived from the Soil Water Assessment and Measurement
Programme to physically characterize key soils throughout New Zealand in the
1980s. Soils selected from this data set are listed by region in Table 4 and
were selected from soils formed from sediments derived from indurated
sandstone rocks, because this is the most common parent material for soils
in New Zealand and has a reasonably representative number of soils analysed
for physical properties.
The cores for particle size analysis and measurement of θ(h) had
diameters which ranged from 5.5 to 10 cm diameter and height which
varied from 5 to 6 cm. The 5 and 10 kPa measurements of the θ(h) were
derived using the suction table method as per Dane and Topp (2002),
following the NZ Soil Bureau laboratory method (Gradwell, 1972). The 20 to 1500 kPa
of the θ(h) were measured using the pressure plate method as per
Dane and Topp (2002), following the NZ Soil Bureau method (Gradwell, 1972).
The laboratory analysis for particle size followed Gradwell (1972).
The total porosity, ϕ, described in Eq. (5) contains uncertainties
from the measurement methods, where ϕ is derived from separate
measurements of particle density and bulk density, rather than being
directly measured. The uncertainty in ϕ measurements appeared to
have reduced the demonstrated benefits of using Ks_bim
instead of Ks_uni, which strongly relies on
ϕα-θs_mac and may have caused
the optimal α to be 0.98 and not the commonly accepted value of 0.95
(Rogowski, 1971; Pollacco et al., 2013; Haverkamp et al., 2005; Leij et al.,
2005).
The Ks data used were collected and processed at a time when the best
field practices in New Zealand were still being explored. Ks was derived
using constant-head Mariotte devices (1 cm head) from three to six cores (10 cm diameter and 7.5 cm thickness) for each horizon. The log10 scale
value of the standard error of the replicates of the measurements is shown
in Fig. 2, which shows large uncertainty in the measurements (up to 3 orders of magnitude). This uncertainty is due to
measurements of θ(h) and Ks being taken on different cores, which
caused some mismatch between θ(h) and Ks, resulting in 16 outliers
that negatively influenced the overall fit of the Ks model having to be
removed from the data set;
side wall leakage of some cores, which led to Ks values that were too high
(Carrick, 2009), resulting in six samples with unusually high Ks having
to be removed from the data set;
misreporting low Ks since the measurements of Ks were halted when
conductivity was less than 0.1 cm day-1, resulting in four samples with
low Ks having to be removed from the data set; and
small core samples, which led to considerable variability in the
absence/presence of structured cracks caused by roots or worm burrows
(McKenzie and Cresswell, 2002; Anderson and Bouma, 1973) that were evident
in dyed samples; we therefore removed measuredKs replicates that were
too high and showed evidence of macropore abundance by having values of
θs-θs_mac>0.05.
We therefore selected 235/262 samples (90 %) and removed only 27 outliers,
which is minimal compared, for instance, to the UNSODA (Leij et al., 1999;
Schaap and van Genuchten, 2006) and HYPRES databases (Wösten et al.,
1998, 1999; Lilly et al., 2008), which are used for the
development of PTFs such as the ROSETTA PTF (Patil and Rajput, 2009; Rubio,
2008; Young, 2009), and which were found to contain a large number of
outliers. Using these databases, Pollacco et al. (2013) selected only
73/318 soils (23 %), which complied with strict selection criteria prior to
modelling.
Uncertainty of the standard error of the observed Ks in
topsoil and subsoil. The lines in the box show upper and lower quartiles,
the median (red), and mean (green). Whiskers show values within 1.5 times
the quartile spread; values outside this range are shown as plotted points.
Note that the Ks observations in the topsoils have greater variability
than in the subsoil layers (Fig. 2). This is because topsoils are more
disturbed by anthropogenic disturbance and biological activity. Therefore,
the topsoils also have a greater abundance of macropores and therefore are
more prone to error when the sampling is performed with a small core size
that does not contain a representative volume of the macropore network.
Inverse modelling and goodness of fit
The parameterization of the model was performed in two consecutive steps:
Optimization of θs_mac, θr,
hm, and σ of the unimodal Kosugi θbim_mat(h) (Eq. 11) was performed by matching observed and simulated θ(h) in the range h < Hmac (as discussed, θs is not
included in the observation data since we did not have data in the macropore
domain). The feasible ranges of the Kosugi parameters are described in Table 1.
Optimization of the τ1 , τ2 , τ3 of theKs_uni model (Eq. 8) and τ1_mac , τ2_mac , τ3_mac , σ_mac parameters of the Ks_bim models
(Eq. 14), where the physical feasible ranges of the tortuosity parameters
are described in Table 3.
The inverse modelling was performed in MATLAB using AMALGAM, which is a
robust global optimization algorithm (http://faculty.sites.uci.edu/jasper/sample/) (e.g. ter Braak and Vrugt,
2008). For each step, we minimized the objective functions described below.
The objective function, OFθ, used to parameterize Kosugi's
θ(h) at the following pressure points (5, 10, 20, 40, 50, 100, 1500 kPa), is described by
OFθ=∑i=1i=Nθθsimhi,pθ-θobshiPower,
where the subscripts sim and obs indicate simulated and observed values, respectively.
Pθ is the set of predicted parameters (θs_mac, θr,hm, σ) and
Power is the power of the objective function. The computation of
Ks_bim requires θ(h) to be accurate near saturation,
when the drainage is mostly from large pores, and to achieve this balance we
found by trial and error that best results are achieved when Power= 6.
The parameters of Ks_uni and Ks_bim
models were optimized by minimizing the following objective function
OFks:
OFks=∑j=1j=NkslnKs_simpks-lnKs_obs2,
where the subscripts sim and obs indicate simulated and observed values, respectively.
Pks is the vector of the unknown parameters. The log transformation of
OFks puts more emphasis on the lower Ks and therefore reduces the
bias towards larger conductivity (e.g. van Genuchten et al., 1991; Pollacco
et al., 2011). Also, the log transformation considers that the uncertainty
in measured unsaturated hydraulic conductivity increases as K(θ)
increases.
The goodness of fit between simulated (Ks_uni or
Ks_bim) and observed Ks was computed by the
RMSElog10:
RMSElog10=∑j=1j=Nkslog10Ks_sim-log10Ks_obs2N,
where N is the number of data points.
The following transformation was necessary to scale the parameters to enable
the global optimization to converge to a solution:
τ1=1-10-T1,
where T1 is a transformed tortuosity τ1. Introducing Eq. (19)
into Ks_bim Eq. (14) gives
Ks_bim=C∫0110-T1θs_mac-θr11-τ3Yhmexperfc-12Seσ221-τ2+10-T1_macθs-θs_mac11-τ3_macYhm_macexperfc-12Seσ_mac221-τ2macdSe.
Results and discussion
We report on (1) the suitability of the Ks_uni model
(developed with European and American data sets, Pollacco et al., 2013) to
predict Ks for New Zealand soils experiencing large uncertainties, as
shown in Fig. 2; (2) improvements made by stratifying the data with texture
and topsoil/subsoil; and (3) enhancements made by using the bimodal
Ks_bim instead of the unimodal Ks_uni.
Improvement made by stratifying with texture and topsoil/subsoil
It was expected that stratifying with texture and topsoil/subsoil (layers)
should improve the predictions of Ks to only a modest degree. This is
because Ks_bim and Ks_uni are
physically based models that are based on pore-size distribution, and
therefore stratifying with soil texture or topsoil/subsoil is not likely to
provide extra information. For instance, Arya and Paris (1981) showed that
there is a strong relationship between pore-size distribution and the
particle-size distribution, and therefore adding soil texture information
should not improve the model.
The RMSElog10 reported by using Ks_bim and
Ks_uni models, by stratifying the data with/without
texture and layers.
Data stratification withRMSElog10Ks_uniKs_bimKs_bim-Ks_uniAll data combined0.5830.5600.023Loam and clay (texture)0.5770.5430.034Topsoil and subsoil (layers)0.4500.4300.020
As expected, no significant improvements were made by stratifying with soil
texture compared with a model that groups all texture classes (loam and
clay) and layers (topsoil and subsoil) (overall improvement of 3 %) (Table 5). However, a significant improvement was made by stratifying by layer
(topsoil and subsoil) (overall improvement of 23 %), and therefore the
remaining results are presented by stratifying by layer. These results are
obtained because topsoils have higher macropores and a smaller tortuous path
than that in subsoil, as demonstrated by τ1_top > τ1_sub or
T1_top < T1_sub, τ2_top > τ2_sub, τ3_top > τ3_sub
(Table 6). It is
important to note that tortuosity decreases as τ gets closer to 1.
Optimal tortuosity parameters of Ks_uni and
Ks_bim.
NRMSElog10T1τ2τ3T1_macτ2_macτ3_macσ_macKs_bimTopsoil510.2325.0070.9690.7874.7340.5110.0410.322Subsoil1810.4716.4440.8590.4083.9730.6420.7291.272Ks_uniTopsoil510.2595.8590.9670.530––––Subsoil1810.4916.4840.8540.316––––Improvement made by using Ks_bim instead of
Ks_uni
Figure 3 shows an acceptable fit between Ks_bim and
Ks_obs (RMSElog10= 0.450 cm day-1),
recognizing that the observations contain large uncertainties since the
measurements were taken by using small cores (Sect. 4.1). The overall
improvement made by using Ks_bim is somewhat modest
(5 % for all soils). As expected, the reasonable improvement is greater
for topsoil containing higher macroporosity (12 % improvement) than for
subsoil (4 % improvement) (Table 6). This is because topsoil has higher
macropore θmac(θs-θs_mac) (Table 7) caused by earthworm channels, fissures, roots, and tillage
than subsoil. The RMSElog10 of Ks_uni for subsoil
is 0.47 cm day-1 (Table 6), which is slightly worse compared to the
RMSElog10 of 0.420 cm day-1 when using UNSODA and HYPRES data sets
(Pollacco et al., 2013).
Descriptive statistics of the optimized θmac
(θs-θs_mac),θs ,
hm, and σ Kosugi hydraulic parameters. The bar represents the
average value, SD is the standard deviation, and N the number of measurement
points.
Plot between Ks_obs against
Ks_bim and Ks_uni for topsoil and
subsoil. The dotted line refers to the 1:1 line.
The reason Ks_bim shows smaller-than-expected
improvements compared to Ks_uni requires further
investigation and testing with a data set containing fewer uncertainties.
One plausible explanation is that Ks_bim is highly
sensitive to θs, computed from total porosity ϕ (Eq. 6),
which had inherent measurement uncertainties (Sect. 4.1). In addition, the
possible existence of non-Darcy flow in large biological pores may decrease
the outperformance of the bimodal model over the unimodal model.
Optimal tortuosity parameters
The optimal tortuosity parameters of Ks_bim and
Ks_uni (Table 6) show that the optimal parameters are
within the physically feasible limits, except for τ3_mac parameters of the subsoil, which are greater than τ3. This is
understandable because Pollacco et al. (2013) found τ3 not to be a
very sensitive parameter. As expected, T1_mac is smaller
than T1 (τ1_mac > τ1),
which suggests that the tortuosity parameters have a physical meaning.
The estimated value of the unimodal T1 parameter Ks_uni derived from the UNSODA and HYPRES data sets (T1= 0.1)
(Pollacco et al., 2013) is very different from the value estimated in this
present study (T1= 6.5). Cichota et al. (2013) also reported that
PTFs developed in Europe and the USA were not applicable to New Zealand. The
reasons why these PTFs are not directly applicable to New Zealand require
further investigation.
Uncertainty of the bimodal saturated hydraulic conductivity model
predictions
The practical application of the bimodal saturated hydraulic conductivity
model, Ks_bim, to New Zealand soils requires a model for
the uncertainty of the resultant predictions, since it is then possible to
attach a value for the uncertainty of future predictions of Ks. In a
conventional parametric statistical model, the uncertainty model follows
from the structure of the fitting model itself. In the present work,
Ks is estimated using an inverse model and this has no associated
functional uncertainty model. For this reason, the uncertainty is derived
empirically by fitting a relationship between the transformed residuals of
the model (the log-transformed measured Ks minus the log-transformed
estimated Ks) as a function of the log-transformed estimated Ks.
Although the uncertainty model could be derived from all the soils in the
study, this process results in a pooled estimate for uncertainty (e.g.
aggregated root mean square error). However, it has been observed that
topsoils and subsoils have different uncertainty behaviour for the estimated
Ks, so it is desirable to include an indicator variable to determine
whether the soil is a topsoil or not. In explicit form,
log10Ksobs-log10Kssim=a1L+a0+ϵ,
where a0 and a1 are fitting constants, L is an indicator
variable specifying whether the soil is a topsoil (value of 1) or a subsoil
(value of 0), and ϵ is the uncertainty distribution. The distribution
of the uncertainty ϵ could take a number of forms, but there is no
obvious choice, except that one might expect the distribution central
measure to be unbiased. To avoid an explicit distribution assumption, we
fitted a conditional quantile model (Koenker, 2005) for the transformed
residuals, based on the τ quantile, where τ=0.5 corresponds to
the conditional median, and τ=0.025 and τ=0.975 correspond,
respectively, to the 2.5 and 97.5 % quantiles and thus together
describe the 95 % containment interval of the residuals.
The conditional quantile model Eq. (25) was fitted using τ=0.5,0.025 and 0.975 (Table 8). The results suggest a strong dependence of the
scale of the residuals on whether the soil is a topsoil or not, but the size
of the 95 % residual containment interval is not dependent on the
simulated Ks. Notably, the confidence interval for the fitted median
(τ=0.5) quantile model suggests that the uncertainty distribution
median is unbiased; thus, predictions from Ks_bim show no
propensity for bias, which is a desirable result.
Summary of the quantile regression fit of the log-transformed
residuals.
Another way to illustrate the uncertainty model is to plot the observed
log10Ks_obs against the estimated logKs_bim, with the fitted median, lower, and upper 95 %
quantile lines, as shown in Fig. 4. The width of the 95 % containment
interval for the residuals is narrower (i.e. the predictions appear to be
more accurate) for topsoils. The quantile estimates for the conditional
median of both topsoil and subsoil are also shown in Fig. 4, with the shaded
region showing the 95 % confidence interval of the median estimate. The
shaded region covers the 1:1 line in Fig. 4, and thus there is no
compelling evidence that the median residual distribution is biased.
Error of Ks_bim plotted against
Ks_obs for topsoil and subsoil. The solid line refers to
the median line for each group, the dashed line refers to the upper or lower
95 % confidence interval lines, the dotted line refers to the 1:1
correspondence line, and the shaded region is the 95 % confidence interval
of the median estimate.
Recommended future work to improve the New Zealand soil database
A key outcome of this research will be to provide direction for future field
studies to quantify soil water movement attributes of New Zealand soils and
to prioritize which measurements will have the greatest value to reduce the
uncertainty in modelling of the soil moisture retention and hydraulic
conductivity relationships. Recommendations are to
evaluate the spatial representativeness of the current soil physics data set
and undertake more measurements of hydraulic conductivity and soil water
retention on key soils;
use larger cores for measurements of hydraulic conductivity;
take measurements of the moisture retention curve and saturated hydraulic
conductivity on the same sample;
provide more accurate measurements of total porosity;
conduct near-saturation measurements of θ(h) and K(θ) to better
characterize the macropore domain, which is responsible for preferential
flow behaviour; and
make more accurate measurements on slowly permeable soils (< 1 cm day-1), which are important for management purposes but are not well
represented in the current databases.
Conclusions
We report here on further adaptations to the saturated hydraulic
conductivity unimodal model to suit it to dual-porosity structured soils, by
computing the soil water flux through a continuous function of a modified
version of the Romano et al. (2011) θ(h) dual pore-size distribution. The
shape of the Romano θ(h) distribution is identical to the modified
θ(h), but the advantage of the developed bimodal θ(h) is that it
is more easily parameterized when no data are available in the macropore
domain.
The stratification of the data with texture only (loam or clay) slightly
improved the predictions of the Ks model, which is based on pore-size
distribution. This gives us confidence that the Ks model is accounting
for the effect of these physical parameters on Ks. A significant
improvement was made by separating topsoils from subsoils. The improvements
are higher for the topsoil, which has higher macroporosity caused by roots
and tillage compared to subsoils. The reason why a model with no
stratification is not sufficient is unclear and requires further
investigation.
The improvements made by using the developed bimodal Ks_bim (Eq. 20)
compared to the unimodal Ks_uni (Eq. 8) are
modest overall, but, as expected, greater for topsoils having larger
macroporosity. Nevertheless, an acceptable fit between
Ks_bim and Ks_obs was obtained when
due recognition was given to the high variability in the measured data. We
expect Ks_bim to provide greater improvement in Ks
predictions if more θ(h) measurements are made at tensions near
saturation and if measurements are made on larger cores and with more
accurate measurements of porosity.
The data are part of the New Zealand soil
databases, available at http://smap.landcareresearch.co.nz/ and https://soils.landcareresearch.co.nz/.
The authors declare that they have no conflict of interest.
Acknowledgements
We are grateful to Leah Kearns and Ray Prebble, who improved the
readability of the manuscript, and to the anonymous reviewers who significantly
improved the clarity of the manuscript. We also thank Ian Lynn
for his input on soil heterogeneity. This project was funded by Landcare
Research core funding, through the New Zealand Ministry of Business,
Innovation and Employment.
Edited by: N. Romano
Reviewed by: three anonymous referees
ReferencesAnderson, J. L. and Bouma, J.: Relationships between saturated hydraulic
conductivity and morphometric data of an argillic horizon, Soil Sci. Soc.
Am. J., 37, 408–413, 10.2136/sssaj1973.03615995003700030029x,
1973.Arya, L. M. and Paris, J. F.: A physicoempirical model to predict the soil
moisture characteristic from particle-size distribution and bulk density
data, Soil Sci. Soc. Am. J., 45, 1023–1030,
10.2136/sssaj1981.03615995004500060004x, 1981.
Balland, V. and Pollacco, J. A. P.: Modeling soil hydraulic properties for
a wide range of soil conditions, Ecol. Model., 219, 300–316, 2008.Borgesen, C. D., Jacobsen, O. H., Hansen, S., and Schaap, M. G.: Soil
hydraulic properties near saturation, an improved conductivity model,
J. Hydrol., 324, 40–50, 10.1016/j.jhydrol.2005.09.014, 2006.
Brooks, R. H. and Corey, A. T.: Hydraulic properties of porous media,
Hydrol. Pap., 3, 1964.
Carrick, S.: The dynamic interplay of mechanisms governing infiltration into
structured and layered soil columns, PhD, Lincoln University, Lincoln, 2009.Carrick, S., Almond, P., Buchan, G., and Smith, N.: In situ characterization
of hydraulic conductivities of individual soil profile layers during
infiltration over long time periods, Eur. J. Soil Sci., 61,
1056–1069, 10.1111/j.1365-2389.2010.01271.x, 2010.Carrick, S., Buchan, G., Almond, P., and Smith, N.: Atypical early-time
infiltration into a structured soil near field capacity: the dynamic
interplay between sorptivity, hydrophobicity, and air encapsulation,
Geoderma, 160, 579–589, 10.1016/j.geoderma.2010.11.006, 2011.Chapuis, R. P.: Predicting the saturated hydraulic conductivity of sand and
gravel using effective diameter and void ratio, Can. Geotech. J., 41,
787–795, 10.1139/t04-022, 2004.Childs, E. C. and Collisgeorge, N.: The permeability of porous materials,
Proc. R. Soc. Lon. Ser-A, 201, 392–405, 10.1098/rspa.1950.0068, 1950.Cichota, R., Vogeler, I., Snow, V. O., and Webb, T. H.: Ensemble
pedotransfer functions to derive hydraulic properties for New Zealand soils,
Soil Research, 51, 94–111, 10.1071/sr12338, 2013.
Dane, J. H. and Topp, G. C.: Methods of Soil Analysis, Part 4. Physical
Methods, Soil Science Society of America Book Series No. 5, Madison, WI,
USA, 692–698, 2002.Durner, W.: Hydraulic conductivity estimation for soils with heterogeneous
pore structure, Water Resour. Res., 30, 211–223,
10.1029/93wr02676, 1994.
Fatt, I. and Dykstra, H.: Relative permeability studies, T. Am. I. Min.
Met. Eng., 192, 249–256, 1951.
Gradwell, M. W.: Methods for physical analysis of soils, in: New Zealand Soil
Bureau Scientific Report No. 10C, 1972.Griffiths, E., Webb, T. H., Watt, J. P. C., and Singleton, P. L.:
Development of soil morphological descriptors to improve field estimation of
hydraulic conductivity, Aust. J. Soil Res., 37, 971–982,
10.1071/sr98066, 1999.Haverkamp, R., Leij, F. J., Fuentes, C., Sciortino, A., and Ross, P. J.:
Soil water retention: I. Introduction of a shape index, Soil Sci. Soc. Am. J., 69, 1881–1890, 10.2136/sssaj2004.0225, 2005.
Hayashi, Y., Kosugi, K., and Mizuyama, T.: Soil water retention curves
characterization of a natural forested hillslope using a scaling technique
based on a lognormal pore-size distribution, Soil Sci. Soc. Am. J., 73, 55–64, 2009.Jarvis, N. J.: A review of non-equilibrium water flow and solute transport
in soil macropores: principles, controlling factors and consequences for
water quality, Eur. J. Soil Sci., 58, 523–546,
10.1111/j.1365-2389.2007.00915.x, 2007.
Jarvis, N. J. and Messing, I.: Near-saturated hydraulic conductivity in
soils of contrasting texture measured by tension infiltrometers, Soil Sci. Soc. Am. J., 59, 27–34, 1995.
Koenker, R.: Quantile Regression, Cambridge University Press, New York,
2005.Kosugi, K.: Lognormal distribution model for unsaturated soil hydraulic
properties, Water Resour. Res., 32, 2697–2703, 10.1029/96wr01776,
1996.Landcare Research: S-map – New Zealand's national soil layer, available at: http://smap.landcareresearch.co.nz, 2015.
Leij, F. J., Alves, W. J., van Genuchten, M. T., and Williams, J. R.: The
UNSODA unsaturated soil hydraulic database, in: Proceedings of the
International Workshop on Characterization and Measurement of the Hydraulic
Properties of Unsaturated Porous Media, 1269–1281, 1999.Leij, F. J., Haverkamp, R., Fuentes, C., Zatarain, F., and Ross, P. J.: Soil
water retention: II. Derivation and application of shape index, Soil Sci. Soc. Am. J., 69, 1891–1901, 10.2136/sssaj2004.0226,
2005.Lilburne, L. R., Hewitt, A. E., and Webb, T. W.: Soil and informatics
science combine to develop S-map: a new generation soil information system
for New Zealand, Geoderma, 170, 232–238,
10.1016/j.geoderma.2011.11.012, 2012.Lilly, A., Nemes, A., Rawls, W. J., and Pachepsky, Y. A.: Probabilistic
approach to the identification of input variables to estimate hydraulic
conductivity, Soil Sci. Soc. Am. J., 72, 16–24,
10.2136/sssaj2006.0391, 2008.Mbonimpa, M., Aubertin, M., Chapuis, R. P., and Bussière, B.: Practical
pedotransfer functions for estimating the saturated hydraulic conductivity,
Geotechnical and Geological Engineering, 20, 235–259,
10.1023/A:1016046214724, 2002.McKenzie, N. and Jacquier, D.: Improving the field estimation of saturated
hydraulic conductivity in soil survey, Aust. J. Soil Res.,
35, 803–825, 10.1071/s96093, 1997.
McKenzie, N. J. and Cresswell, H. P.: Field sampling, in: Soil Physical
Measurement and Interpretation for Land Evaluation, CSIRO, Collingwood,
Victoria, 2002.McLeod, M., Aislabie, J., Ryburn, J., and McGill, A.: Regionalizing
potential for microbial bypass flow through New Zealand soils, J. Environ. Qual., 37, 1959–1967, 10.2134/jeq2007.0572, 2008.
McNeill, S., Webb, T., and Lilburne, L.: Analysis of soil hydrological
properties using S-map data, Landcare Research report, 977, 2012.
Messing, I. and Jarvis, N. J.: Temporal variation in the hydraulic
conductivity of a tilled clay soil as measured by tension infiltrometers,
J. Soil Sci., 44, 11–24, 1993.Patil, N. G. and Rajput, G. S.: Evaluation of water retention functions and
computer program “ROSETTA” in predicting soil water characteristics of
seasonally impounded shrink-swell soils, J. Irrig. Drain. E.-ASCE, 135, 286–294, 10.1061/(asce)ir.1943-4774.0000007,
2009.
Pollacco, J. A. P.: A generally applicable pedotransfer function that
estimates field capacity and permanent wilting point from soil texture and
bulk density, Can. J. Soil Sci., 88, 761–774, 2008.
Pollacco, J. A. P., Ugdale, J. M., Saugier, B., Angulo-Jaramillo, R., and
Braud,
I.: A Linking Test to reduce the number of hydraulic parameters
necessary to simulate groundwater recharge in unsaturated soils, Adv. Water
Resour., 31, 355–369, 2008a.Pollacco, J. A. P., Saugier, B., Angulo-Jaramillo, R., and Braud I.:
A Linking Test that establishes if groundwater recharge can be determined by
optimising vegetation parameters against soil moisture, Annals of Forest
Science, 65, 702, 10.1051/forest:2008046, 2008b.
Pollacco, J. A. P. and Angulo-Jaramillo, R.: A Linking Test that investigates
the feasibility of inverse modelling: Application to a simple rainfall
interception model for Mt. Gambier, southeast South Australia, Hydrol.
Process., 23, 2023–2032, 2009.Pollacco, J. A. P., Nasta, P., Ugalde, J. M. S., Angulo-Jaramillo, R.,
Lassabatere, L., Mohanty, B. P., and Romano, N.: Reduction of Feasible Parameter
Space of the Inverted Soil Hydraulic Parameter Sets for Kosugi Model, Soil Sci.,
178, 267–280, 10.1097/SS.0b013e3182a2da21, 2013.Rogowski, A. S.: Watershed physics – model of soil moisture characteristic,
Water Resour. Res., 7, 1575–1582, 10.1029/WR007i006p01575,
1971.Romano, N. and Nasta, P.: How effective is bimodal soil hydraulic
characterization? Functional evaluations for predictions of soil water
balance, Eur. J. Soil Sci., 67, 523–535, 10.1111/ejss.12354, 2016.Romano, N., Nasta, P., Severino, G., and Hopmans, J. W.: Using Bimodal
Lognormal Functions to Describe Soil Hydraulic Properties, Soil Sci. Soc.
Am. J., 75, 468–480, 10.2136/sssaj2010.0084, 2011.Rubio, C. M.: Applicability of site-specific pedotransfer functions and
ROSETTA model for the estimation of dynamic soil hydraulic properties under
different vegetation covers, J. Soil. Sediment., 8, 137–145,
10.1065/jss2008.03.281, 2008.Schaap, M. G. and van Genuchten, M. T.: A modified Mualem-van Genuchten
formulation for improved description of the hydraulic conductivity near
saturation, Vadose Zone J., 5, 27–34, 2006.
Sutera, S. P. and Skalak, R.: The History of Poiseuille's Law, Annu. Rev. Fluid
Mech., 25, 1–20, 10.1146/annurev.fl.25.010193.000245, 1993.
ter Braak, C. J. F. and Vrugt, J. A.: Differential evolution Markov chain
with snooker updater and fewer chains, Stat. Comput., 4,
435–446, 2008.
van Genuchten, M. T.: Closed-form equation for predicting the hydraulic
conductivity of unsaturated soils, Soil Sci. Soc. Am. J.,
44, 892–898, 1980.
van Genuchten, M. T., Leij, F. J., and Yates, S. R.: The RETC code for
quantifying the hydraulic functions of unsaturated soils, The RETC Code for
Quantifying the Hydraulic Functions of Unsaturated Soils, US Department of
Agriculture, Agricultural Research Service, 1991.
Vervoort, R. W. and Cattle, S. R.: Linking hydraulic conductivity and
tortuosity parameters to pore space geometry and pore-size distribution,
J. Hydrol., 272, 36–49, 2003.
Watt, J. P. C. and Griffiths, E.: Correlation of hydraulic conductivity
measurements with other physical properties New Zealand, New-Zealand Soil
Bureau Commentaries, 1983, 198–201, 1988.Webb, T. H., Claydon, J. J., and Harris, S. R.: Quantifying variability of
soil physical properties within soil series to address modern land-use
issues on the Canterbury plains, New Zealand, Aust. J. Soil Res., 38, 1115–1129, 10.1071/sr99091, 2000.Webb, T. H.: Identification of functional horizons to predict physical
properties for soils from alluvium in Canterbury, New Zealand, Aust. J. Soil Res., 41, 1005–1019, 10.1071/sr01077, 2003.
Wösten, J. H. M., Lilly, A., Nemes, A., and Le Bas, C.: Final report on
the EU funded project using existing soil data to derive hydraulic
parameters for simulation models in environmental studies and in land use
planning, DLO Winand Staring Centre, Wageningen, the Netherlands, 1998.
Wösten, J. H. M., Lilly, A., Nemes, A., and Le Bas, C.: Development and
use of a database of hydraulic properties of European soils, Geoderma, 90,
169–185, 1999.
Young, C. D.: Overview of ROSETTA for estimation of soil hydraulic
parameters using support vector machines, Korean J. Soil Sci.
& Fertilizer, 42, 18–23, 2009.