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SCIENTIFIC REPORTS | (2020) 10:16666 |
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Estimating soil water retention
for wide ranges of pressure
head and bulk density based
on a fractional bulk density concept
Huihui Sun1, Jaehoon Lee1, Xijuan Chen2 & Jie Zhuang1,3*
Soil water retention determines plant water availability and contaminant transport processes in the
subsurface environment. However, it is usually dicult to measure soil water retention characteristics.
In this study, an analytical model based on a fractional bulk density (FBD) concept was presented for
estimating soil water retention curves. The concept allows partitioning of soil pore space according
to the relative contribution of certain size fractions of particles to the change in total pore space. The
input parameters of the model are particle size distribution (PSD), bulk density, and residual water
content at water pressure head of 15,000 cm. The model was tested on 30 sets of water retention
data obtained from various types of soils that cover wide ranges of soil texture from clay to sand and
soil bulk density from 0.33 g/cm3 to 1.65 g/cm3. Results showed that the FBD model was eective
for all soil textures and bulk densities. The estimation was more sensitive to the changes in soil bulk
density and residual water content than PSD parameters. The proposed model provides an easy way
to evaluate the impacts of soil bulk density on water conservation in soils that are manipulated by
mechanical operation.
Modeling of water ow and chemical movement in unsaturated soils has been emphasized by soil scientists and
hydrologists for dierent purposes, such as evaluations of root water uptake, soil erosion, and groundwater pol-
lution risk. However, high variability and complexity of soil texture in natural eld make direct measurements
of soil hydraulic properties costly and time-consuming. It is desirable to utilize readily available information,
such as soil texture and bulk density, to estimate soil hydraulic properties1–3. is kind of approach benets the
development of computationally ecient methods for evaluating soil hydraulic heterogeneity in watershed or
agricultural eld while ensuring the economic feasibility of eld investigation eorts within acceptable accuracy.
To date, many modeling eorts have been made to relate soil texture (expressed as particle size distribution), soil
structural properties, bulk density, and/or organic matter content to soil water retention4–7. Soil water retention
was estimated using multiple regression, neural network analyses, and other methods8–14. However, the applicabil-
ity and accuracy of the models are more or less unsatisfactory. Several prediction models were derived on global
soil hydraulic datasets, such as applying the Miller-Miller scaling approach to the soil dataset of SoilGrids1km
to provide a global consistent soil hydraulic parameterization15, but some of them possess a high correlation to
particular soil types and thereby may not be suitable for other soils16–18.
An important advancement in using soil particle size distribution to derive a soil water retention character-
istic was the development of a physical empirical model by Arya and Paris19,20, Later, Haverkamp and Parlange21
proposed a similar model by combining physical hypotheses with empirical representations and tested the model
on sandy soil. Tyler and Wheatcra22 interpreted the empirical scaling parameter α in the Arya and Paris model
as being equivalent to the fractal dimension of a tortuous fractal pore system. However, Arya etal.20 argued that
the fractal scaling was limited in estimating water retention characteristics in the complex soil matrix. In the
optimized model of Arya etal.20, three methods were proposed for calculating the scaling parameter α, but the
calculation still involved empirical component to some extent, making the model sometimes relatively dicult
OPEN
USA. Key Laboratory of Pollution Ecology and Environmental Engineering, Institute of Applied Ecology, Chinese
Center for Environmental Biotechnology, The University of
*email: jzhuang@utk.edu
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for broad application. e physical basis of the model of Arya and Paris19 or Arya etal.20 is weakened by the
assumption that the void ratio of bulk sample is equivalent to the void ratio of individual particle size class.
To improve the mathematical description of physical relations between soil particles and soil pores, we assume
that dierent fractions of soil particles may make dierent contributions to the total porosity or volumetric
water content in the bulk soils and that soil pore volume and associated bulk density are specic for particle size
fractions. is line of thinking might help derive a better physical model for mathematical estimation of soil
water characteristics. erefore, the objective of this work was to apply a fractional bulk density (FBD) concept
to the development of a soil water retention model that is eective for all soil textures and a wide range of soil
bulk density.
Results
Estimation accuracy. Model estimation of water retention characteristics for some soils is presented in
Fig.1. e results indicate that the new procedure was in good agreement with the measured data for most of
the soil textures except for sand in the range of water pressure head from 15cm to 15,000cm, which covers the
entire range of available water content. Table1 shows comparisons of the coecient of determination (R2), root
mean square error (RMSE), and t value of Student’s t distribution between the FBD model and the curve tting
using the Campbell model23, which was extended from the similar media concept24. e Campbell model is
expressed as
where ψe is air-entry water potential, θs is saturated volumetric water content, and q can be obtained using
In the equation, Dgi is the diameter of the ith particle-size fractions, and Mi is the cumulative mass percent-
age of the ≤ Dgi particles.
RMSE values were computed from soil water contents measured and estimated as described in the section
of methods. Table1 shows that the mean value of RMSE of the FBD model was 0.032 cm3/cm3 while that of the
Campbell model was 0.024 cm3/cm3. is result was acceptable because the Campbell model used the meas-
ured data to t ψe. e R2 values also supported the acceptability of the FBD model compared to the Campbell
model. According to the t values, the FBD model results had no signicant dierence and systematic bias from
the measurements for 25 out of the 30 soils. Figure2 shows an overall comparison between the water contents
(1)
θ
=
ψe
ψ
1
q
θ
s
(2)
q
=
Mi
ln Dgi
2
−
Miln Dgi
2
0.5
Figure1. Water retention characteristics measured (circle) and estimated (line) using the fractional bulk
density (FBD) model for eight dierent soil textures.
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Table 1. Statistical comparison of soil water contents estimated by the fractional bulk density (FBD) model
and tted by the Campbell model23. t is the value of Student’s t distribution, and the critical values of t0.05 for 04,
08, 09, 10, 15, 18 and 30 degrees of freedom are 2.776, 2.306, 2.262, 2.228, 2.131, 2.101 and 2.042, respectively;
R2 is determination coecient; RMSE is root mean square errors (cm3/cm3); n is the number of measured pairs
of water content and pressure head.
Soil No.
t R2RMSE
nFBD model Campbell model FBD model Campbell model FBD model Campbell model
01 1.600 −0.508 0.662 0.507 0.097 0.117 16
02 −1.316 −0.288 0.939 0.888 0.039 0.040 9
03 −1.924 −0.213 0.982 0.958 0.011 0.014 5
04 −0.123 −0.308 0.968 0.932 0.012 0.015 5
05 −0.191 0.115 0.962 0.900 0.059 0.060 10
06 −1.374 −0.331 0.951 0.922 0.068 0.039 5
07 −1.948 −0.123 0.908 0.922 0.027 0.018 5
08 −2.411 0.125 0.985 0.942 0.020 0.031 9
09 0.532 −0.270 0.954 0.923 0.017 0.024 5
10 1.345 0.179 0.980 0.998 0.042 0.003 9
11 0.144 0.164 0.977 0.995 0.028 0.007 9
12 0.266 0.070 0.995 0.964 0.020 0.020 9
13 10.840 −0.819 0.938 0.899 0.029 0.010 6
14 3.572 3.885 0.951 0.956 0.023 0.024 28
15 −0.883 −0.128 0.932 0.980 0.028 0.013 10
16 −0.209 −0.163 0.876 0.877 0.062 0.057 16
17 −0.794 −0.056 0.908 0.958 0.027 0.014 5
18 −0.441 −0.227 0.957 0.967 0.019 0.014 10
19 −0.256 −0.065 0.891 0.936 0.027 0.017 5
20 1.153 −0.253 0.892 0.920 0.032 0.023 10
21 1.932 −0.156 0.908 0.928 0.030 0.020 10
22 −5.494 −0.218 0.945 0.881 0.035 0.024 20
23 −3.341 −0.067 0.922 0.966 0.056 0.015 5
24 −1.421 −0.061 0.939 0.971 0.023 0.012 5
25 −1.558 −0.077 0.938 0.970 0.023 0.013 5
26 1.707 −0.075 0.953 0.969 0.022 0.010 10
27 −0.108 −0.052 0.916 0.963 0.020 0.013 5
28 −0.346 −0.113 0.958 0.956 0.014 0.014 9
29 −0.026 0.016 0.968 0.969 0.014 0.012 9
30 −1.823 −0.032 0.952 0.965 0.021 0.013 5
Mean 0.375 −0.009 0.934 0.931 0.032 0.024
Figure2. Comparison of measured and estimated volumetric water content using the fractional bulk density
(FBD) model for 30 soils with ranges of soil texture from clay to sand and bulk density from 0.33 to 1.65g/cm3.
e circle represents measured values, and the line denotes a 1:1 agreement.
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measured and estimated by the FBD model for the 30 soils. e values coalesced to the 1:1 line with the RMSE
being 0.041 cm3/cm3. is RMSE value was larger than the average in Table1. e discrepancy was due that
dierent methods were used for averaging the RMSE values for individual soils and all soils. Mayr and Jarvis25
presented pedotransfer functions to estimate soil water retention parameters of the Brooks–Corey model. e
resulting mean RMSE value was 0.043 cm3/cm3 for the dependent dataset and 0.048 cm3/cm3 for the independent
dataset. Tomasella etal.26 derived a pedotransfer function to predict the water retention parameters of the van
Genuchten equation. e mean RMSE values ranged from 0.038 cm3/cm3 to 0.058 cm3/cm3. Our model compared
favorably with these pedotransfer functions in terms of mean RMSE values. It could thus be concluded that the
FBD model behaved overall well, except for Acolian sandy soil (Soil #01). For sandy soil, the relatively poor
capture of the rapid change of water content was attributed to the limitation of applicability of capillary law (i.e.,
Young–Laplace equation) to sandy media and existence of macropores that might reduce the pore continuity8.
e continuity of soil pores was the dominant factor that aected the performance of our proposed model.
e FBD model also had relatively larger estimation errors for soils originated from ash parent materials (e.g.,
Soil #05, 14, 16, and 22) than for other soils (Table1). is was due likely to the oversimplication of soil particle
size distribution as a sigmoid curve, whereas the particle arrangement of soils developed from ash parent materi-
als was actually very complex (i.e., non-sigmoid). e less accurate prediction for sandy soils relative to the other
soil textures suggested that the sigmoid-shape assumption of particle size distribution might be arbitrary, despite
it was well applied to the particle systems of other soil textures. We infer that the sigmoid-type distribution was
more applicable to the soils with a broader range of particle sizes, which demonstrated a lognormal distribution
of particle fractions27,28. Soil aggregates with hierarchical pore structure have dual-porosity system. Dual-porosity
assumes that porous medium consists of two interacting regions, one associated with the macropore or fracture
system and the other comprising micropores inside soil material. Bimodal pore size distributions are frequently
observed in dual-porosity soil29. e water retention estimated with the FBD model for a wide range of water
pressure head (15–15,000cm) should thus be a sum of the eects of macropores and micropores30. e sigmoid-
type distribution should be more suitable for hierarchical soil aggregates than for less structured soils, such as
sandy soil whose pore system was simply dominated by primary particles. erefore, the FBD model might not
perform very well against the soils if their particle sizes have a narrow range.
Sensitivity analysis of model parameters. We performed a sensitivity analysis to identify input param-
eters that most strongly aected the model behavior and to determine the required precision of the key param-
eters. e parameters included in the sensitivity test were saturated water content (θs), residual water content
(θr), rate coecient (λ) of Logistic-type model for particle size distribution, and particle size distribution index
(ε). e value of each parameter was assumed to increase or decrease by 20% of its actual value since its measure-
ment error could be up to 20% according to our experience in eld survey. By taking Soil #22 as an example, the
test was implemented to monitor the change in the estimated soil water content caused by changing the value of
one parameter at a time while others remained constant. e sensitivity analysis not only showed the inuence
patterns of the parameters on the model behavior but also ranked the parameters in terms of the magnitude of
inuences. Figure3 shows that θs and θr had similarly large impacts on the model estimation. In comparison,
λ and ε played less roles in dening the model performance, but their accuracy was still very important for the
estimation accuracy. e sensitivity analysis provided insights into the behavior of the FBD model (Eq.20) and
Figure3. A sensitivity analysis on the parameters of the fractional bulk density (FBD) model (Eq.20). e
analysis was based on a sandy clay soil (Andisols, Soil #22 in Table2). θs, θr, ε, and λ refer to volumetric saturated
water content, volumetric residual water content at a pressure head of 15,000cm water, particle size distribution
index, and rate coecient in Eq.(15) for particle size distribution, respectively.
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supported the notion that parameter values may have physical meanings no matter in whatever ways the related
parameters are structured into a model.
Discussion
Particle size distribution forms a common descriptor of natural soils. It has been used routinely as one of the
inputs to estimate some of soil physical properties, for example, water retention characteristic31–33, bulk density34,
and hydraulic conductivity35–37. In this study, two parameters, rate coecient (λ in Eq.15) of the Logistic-type
model for particle size distribution and particle size distribution index (ε in Eq.13), were employed to translate
particle size distribution to soil water retention characteristic. However, two parameterization issues should be
mentioned for broadening model applicability. One is the estimation of λ in the case that the upper size limit of
the particle size distribution is 1,000μm for some soils while it is 2,000μm for other soils. In order to perform
a consistent comparison among all soils, the particle size distribution with the upper limit of 2,000μm was
normalized to that with the upper limit of 1,000μm using a normalization formula,
where Mi and Mi’ are measured and normalized percentage content of particles with sizes smaller than or equal
to the ith particle size, respectively. M1,000 denotes the mass percentage of particles with a diameter smaller than
or equal to 1,000μm. e other issue is pertinent to the calculation of ε. It involved three particle sizes (D10, D40,
and D60) below which the mass percentage of particles is 10%, 40%, and 60%, respectively. It is easy to identify
D60 but sometimes relatively dicult to nd D10 and D40. In some soils, the mass of particles with sizes smaller
than or equal to the measured lower limit size (e.g., 1μm or 2μm) was larger than 10%. In this case, an expo-
nential equation, which was obtained by tting the relation between the cumulative mass percentage and the
corresponding particle sizes, was used to extrapolate for estimating D10. To minimize the deviations arising from
the extrapolation, we used 50μm as the upper size limit of the particle size distribution.
ere is no doubt that particle assembling and resulting pore characteristics play important roles in regulating
physical, chemical, and biological functions of soils at various scales. e FBD model was generally based on the
assumption that the sizes of soil particles and the density of their packing are the primary determinants of the
pore size and pore volume. is, however, may not be the case under some conditions. Aggregation of primary
particles into secondary and tertiary particles, root channels, and microcracks would account for a fraction of
the pore volume with pore sizes not determined by the size distribution of primary particles. e abundance of
such pores considerably determines the extent of deviation of prediction. erefore, it is important to incorporate
information of soil structure into soil hydraulic modeling if possible38. Soil structure is a non-negligible factor
for accurate estimation of soil hydraulic properties using pedotransfer functions39,40. But this work is dicult to
initiate because soil structure information (e.g., soil aggregate size distribution) is mostly unavailable compared
to soil basic properties (e.g., particle size distribution, organic matter content, and bulk density). Insuciency
of identication of soil structure indices precludes the inclusion of soil structure characteristics into soil water
retention modeling. Relevant eorts have been made in some large-scale models that consider soil structure.
For instance, Fatichi etal.41 proposed to assess the impact of soil structure on global climate using an Ocean-
Land–Atmosphere Model (OLAM). Although the model in this study does not explicitly include a structural
component, in the FDB model we assume that soil bulk density could indirectly bring the inuence of soil
structure into the estimation of soil water retention.
Soil water retention characteristics were estimated using the FBD model from particle size distribution, bulk
density, and measured residual water content. e starting point was the similarity of curve shapes between
cumulative particle size distribution and soil water retention characteristics. Similarly, Arya and Paris19 and
Haverkamp and Parlange21 used a simple equation to derive a set of soil water content according to the mass
fraction of soil particles, and then a series of expressions were employed to regulate soil water pressure head to
pair with measured soil water content. e FBD model adopted an opposite approach. A set of water pressure
head from 15cm to 15,000cm were derived using a simple expression as Eq.(19), and then soil water contents
were estimated with Eq.(8) to match the derived water pressure head. Eventually, an analytical model (Eq.20)
was obtained. In the FBD model, the water retention function included a residual water content in relation to
the maximum water pressure head (15,000cm) and the parameter (b) of soil pore size distribution. Similarly, the
residual water content was considered in the van Genuchten model42 or Brooks and Corey model43. However,
Campbell23 described soil water retention curve by assuming there was no residual water content. An advantage
of the Campbell equation is its excellent tting capability. us, we evaluated the performance of the FBD model
by comparing it to the Campbell model in this study.
e selection of a Logistic-type equation for the model formulation was mainly due to the consideration that
particle size distribution and pore size distribution in most soils were approximately lognormal27,44–46. e logistic
growth equation generated a curve that tended towards an exponential form at low values and a power form at
high values, with a power index smaller than 1. is characteristic implicitly included the consideration that the
drainage of water in small pores at large suction was usually expected to be more impaired than the release of
water from large pores at small suctions47,48.
Conclusions
An analytical model, which is based on a fractional bulk density concept, was presented for estimating soil water
retention for the entire range of water pressure head that determines water availability. e proposed model was
tested using 30 sets of soil water retention data measured for various textures of soils that had a wide range of
soil bulk density from 0.33g/cm3 to 1.65g/cm3. Results showed that the proposed model could convert readily
(3)
M
′
i=
100M
i
M1,000
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available soil physical properties into soil retention curves in very good agreement with the measurements, and
the model was applicable to soils with limited data of soil particle size distribution at small loss of estimation
accuracy in the middle portion of water retention curves of sandy soils. Sensitivity analyses revealed that satu-
rated and residual water contents were two parameters of high sensitivity for accurate estimation of the water
retention curves. e agreement between the estimated and measured results supported the concept underly-
ing the FBD model. e modeling followed a process of conceptual partitioning of pore space according to the
relative contribution of certain sizes of particles to the change in pore space. In addition, the model assumed a
sigmoid curve of water retention characteristic for most soils. However, these assumptions need further veri-
cation by considering the physical reality of soils and potential improvements and extensions. Compared to
subsurface soils, larger deviations should be expected for surface soil materials where aggregation, cracking, and
root eects may be pronounced. Further tests of the model application to other soils (e.g., Vertisols, Aridisols,
and salt aected soils) and evaluation of the eects of water hysteresis, soil aggregation, and swelling-shrinkage
behaviors might reveal the weaknesses of the FBD model and help identify additional variables needed for model
improvement.
Material and methods
Fractional bulk density concept. e rst assumption is that soil particles with dierent sizes contribute
to dierent porosities and water holding capacities in bulk soil. Based on a non-similar media concept (NSMC)
dened by Miyazaki49, soil bulk density (ρb) is dened as
where M is the mass of a given soil, V is the volume of bulk soil, ρs is soil particle density, and S and d are char-
acteristic lengths of solid phase and pore space, respectively. e parameter τ is a shape factor of the solid phase,
dened as the ratio of the substantial volume of solid phase to the volume S3. e value of τ is 1.0 for a cube and
π/6 for a sphere. As pointed out by Miyazaki49, these characteristic lengths are not directly measurable but are
representative lengths in the sense of the characteristic length in a similar media concept (SMC). Following the
approach of NSMC represented by Eq.(4), we conceptually dened the volume of bulk soil as
where mi and ρbi are the solid mass and equivalent bulk density of the ith size fraction of soil particles, respec-
tively. In this study, diameters of the rst particle fraction and the last one were assumed to be 1µm and 1000µm,
respectively8. is equation suggests that dierent particle size fractions are associated with dierent equivalent
bulk densities due to dierent contributions of particle arrangement to soil pore space. As a result, the particles
with the same size fraction could have dierent equivalent bulk densities in soils with dierent textures or aer
the soil particles are rearranged (e.g., compaction). Figure4 provides a diagrammatic representation of such
fractional bulk density concept for the variation of soil pore volume with soil particle assemblage.
Calculation of volumetric water content. For a specic soil, Eq.(5) means
where Vpi(≤ Di) denotes the volume of the pores with diameter ≤ Di generated by soil particles with diametes ≤ Dgi
in unit volume of soil. Mi is the cumulative mass percentage of the ≤ Dgi particles. Since the pore volume has
the maximum value for a given bulk soil and the cumulative distribution of pore volume could be generally
hypothesized as a sigmoid curve for most of the natural soils44,45, we formulated Eq.(6) using a lognormal
Logistic equation,
where Vpmax is the maximum cumulative volume of pores pertinent to the particles smaller than or equal to the
maximum diameter (Dgmax) in unit volume of soil. In fact, here Vpmax is equal to the total porosity (φT) of soil. Vpi
(≤ Dgi) is the volume of the pores produced by ≤ Dgi particles in unit volume of soil, and bi is a varying parameter
of increase in cumulative pore volume with an increment of Dgi. By assuming a complete saturation of soil pore
space, Eq.(7) changes into
where θs is saturated volumetric water content calculated with
(4)
ρ
b=
M
V
=τρs
S
S+d3
(5)
V
=
n
i=1mi
ρb
=
m1
ρb1
+
m2
ρb2
+···
mn
ρbn
(6)
Vpi
(≤D
i
)=f
D
gi
,M
i
(7)
V
pi
≤Dgi
=
V
pmax
1
+κ
Dgi
bi
(8)
θ
i
≤Dgi
=
θ
s
1+κ
D
gi
bi
(9)
θ
s=
0.9ϕT,ρb<1
ϕ
T
,ρ
b
≥
1
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In the above equations, ρbis measured soil bulk density, and ρs is soil particle density (2.65g/cm3). e empiri-
cal parameter κ in Eqs. (7) and (8) is dened as
where θr is measured residual water content. In this study, θr is set as the volumetric water content at water pres-
sure head of 15,000cm. e empirical parameter bi is dened as
with ε, a particle size distribution index, calculated with
where D10, D40, and D60 represent the particle diameters below which the cumulative mass percentages of soil
particles are 10%, 40%, and 60%, respectively.
e parameter ωi is coecient for soil particles of the ith size fraction, with a range of value between θr/θs
and 1.0. By incorporating soil physical properties, ωi can be estimated with
where g is regulation coecient (1.0–1.2). We set it to be 1.2 in this study. λ is the ratio coecient of particle size
distribution tted using the lognormal Logistic model,
(10)
ϕ
T=
ρ
s
−ρ
b
ρs
(11)
κ
=
θ
s
−θ
r
θr
(12)
b
i=
ǫ
3
log
θ
s−
ω
i
θ
s
κωiθs
(13)
ε
=
(D40)
2
D10D60
(14)
ω
i=
g
1+κ
lnD
gi
(15)
M
i=
M
T
1+ηD
gi
Figure4. Diagrammatic representation of the fractional bulk density (FBD) model. V and ρb are the volume of
bulk soil and the bulk density of whole soil, respectively. mi, and ρbi refer to the solid mass and equivalent bulk
density associated with the ith particle-size fractions, respectively.
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where MT represent the total mass percentage of all sizes of soil particles, and η is a tting parameter. We set
MT = 101 in Eq.(15) for best t of the particle size distribution. In this study, this continuous function was gener-
ated from the discrete data pairs of Dgi and Mi at cutting particle diameters of 1,000, 750, 500, 400, 350, 300, 250,
200, 150, 100, 50, 30, 15, 7.5, 5, 3, 2, and 1μm. Considering the dierence in the upper limits of particle sizes
associated with existing datasets of Dgi and Mi, the particle size distribution with the upper limit of 2,000μm
for the Acolian sandy soil and volcanic ash soils in Table2 was normalized to the case with the upper limit of
1,000μm using Eq.(3).
Calculation of water pressure head. To estimate the capillary tube or pore diameter (Di in µm), which
was composed of particles with the size of Dgi (µm), Arya and Paris19 developed an expression
where α is the empirical scaling parameter varying between 1.35 and 1.40 in their original model19, but was
thought to vary with soil particle size in the optimized model of Arya etal.20. In Tyler and Wheatcra’s model22 α
is the fractal dimension of the pore. e parameter e is the void rate of entire soil and assumed unchanging with
particle size. However, according to Eqs. (5) and (6), e in Eq.(16) should vary with particle size and be replaced by
ei, which depends on soil particle sizes. ni is the number of particles in the ith size fraction with a particle diameter
(16)
D
i=Dgi
2
3
en(1−α)
i
0.5
Table 2. Physical properties of soils used in the study. ρb is bulk density (g/cm3); θr is residual water content
(cm3/cm3) at 15,000cm water pressure head; ε is particle size distribution index. eSoil water retention
data of uvo-aquic soil, red earth, humid-thermo ferralitic, purplish soil, meadow soil, and yellow earth were
measured with pressure membrane apparatus51,52. e soil water retention data of black soil, chernozem soil,
cinnamon soil, brown earth, and albic soil were obtained using the suction and pressure plate method50. e
soil water retention data of volcanic ash soil and Acolian sandy soil were measured using the suction and
pressure plate method53–55.
No Soil USDA soil taxonomy Texture
Particle percentage
ρbθrεSource < 2μm < 20μm
01 Acolian sandy soil Entisols Sand 0.11 0.53 1.65 0.024 1.37 53
02 Meadow soil Inceptisols Sandy loam 6.04 35.20 1.38 0.039 1.38 51
03 Fluvo-aquic soil Inceptisols Sandy loam 9.51 38.01 1.33 0.055 1.82 52
04 Fluvo-aquic soil Inceptisols Sandy loam 10.20 33.20 1.27 0.051 1.87 52
05 Volcanic ash soil Andisols Sandy loam 10.22 35.00 0.33 0.199 3.09 55
06 Fluvo-aquic soil Inceptisols Sandy loam 13.55 45.60 1.27 0.062 1.75 52
07 Fluvo-aquic soil Inceptisols Loam 10.76 42.40 1.32 0.088 1.65 52
08 Meadow soil Inceptisols Loam 13.27 44.37 1.28 0.054 1.74 51
09 Fluvo-aquic soil Inceptisols Loam 13.40 47.88 1.32 0.059 2.48 52
10 Purplish soil Inceptisols Loam 16.32 48.04 1.30 0.092 1.58 51
11 Yellow earth Inceptisols Silt clay loam 27.35 73.87 1.29 0.108 1.61 51
12 Meadow soil Inceptisols Clay loam 22.09 47.32 1.29 0.082 1.95 51
13 Fluvo-aquic soil Inceptisols Clay loam 28.86 58.39 1.28 0.159 2.21 52
14 Volcanic ash soil Andisols Clay loam 28.01 65.00 0.80 0.370 1.73 53
15 Chernozem soil Mollisols Sandy clay 30.14 48.56 1.24 0.148 4.57 50
16 Volcanic ash soil Andisols Sandy clay 34.56 45.60 0.70 0.263 1.57 54
17 Fluvo-aquic soil Inceptisols Sandy clay 36.22 76.05 1.29 0.185 2.15 52
18 Brown earth Alsols Sandy clay 36.77 54.36 1.29 0.142 3.85 50
19 Fluvo-aquic soil Inceptisols Sandy clay 40.02 73.30 1.28 0.195 2.31 52
20 Cinnamon soil Alsols Sandy clay 40.12 59.37 1.19 0.138 3.74 50
21 Black soil Mollisols Sandy clay 42.18 59.34 1.15 0.186 3.44 50
22 Volcanic ash soil Andisols Sandy clay 45.37 63.28 0.82 0.385 3.14 53
23 Fluvo-aquic soil Inceptisols Silty clay 34.20 73.98 1.31 0.148 2.10 52
24 Fluvo-aquic soil Inceptisols Silty clay 33.31 78.73 1.30 0.161 2.12 52
25 Fluvo-aquic soil Inceptisols Silty clay 33.56 79.44 1.35 0.169 2.17 52
26 Albic soil Spodosols Clay 52.76 77.60 1.16 0.230 1.66 50
27 Fluvo-aquic soil Inceptisols Clay 56.05 89.82 1.25 0.283 2.76 52
28 Red earth Ultisols Clay 58.88 79.26 1.22 0.195 1.03 51
29 Humid-thermo ferralitic Oxisols Clay 72.57 85.60 1.15 0.225 1.05 51
30 Fluvo-aquic soil Inceptisols Clay 68.81 98.02 1.08 0.303 2.04 52
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(Dgi in μm), assuming that the particles are spherical and that the entire pore volume formed by assemblage of
the particles in this class is represented by a single cylindrical pore. e equation for calculating ni is given as19
where mi is the mass of particles in the ith size fraction of particles. Assuming that soil water has a zero contact
angle and a surface tension of 0.075N/m at 25°C, the minimum diameter of soil pore (Dmin) was taken to be
0.2µm in this study, which is equivalent to the water pressure head of 15,000cm according to Young–Laplace
equation. We set this minimum pore size to correspond the minimum particle size (Dgmin = 1.0µm). e FBD
model might thus not apply well to porous media with pores smaller than 0.2μm. As a result, Eq.(16) can be
simplied into the following equation.
e equivalent capillary pressure (ψi in cm) corresponding to the ith particle size fraction can be calculated
using
In Eq.(19), the maximum water pressure head (ψr = 15,000cm) corresponds to θr and Dgmin (1μm). e
minimum water pressure head (ψ0 = 15cm) corresponds to θs and Dgmax (1,000μm). ese assumptions were
arbitrary and might not be appropriate for some soil types. But these values were used in the study because they
approximated the practical range of measurements well.
The resulting model of soil water retention. Equations8 and 19 formulate a FBD-based model for
estimation of soil water retention curve. To simplify the computation, we incorporated the two equations into
the following analytical form,
with the parameter b obtained using
In Eq.(21), a water pressure head of 15,000.1cm is employed to consecutively predict the soil water content
until the water pressure head of 15,000cm.
Soil dataset. Evaluation of the applicability of the proposed modeling procedure required datasets that
included soil bulk density, residual water content, and soil particle size distribution covering three particle
diameters (D10, D40, and D60) below which the cumulative mass fractions of particles were 10%, 40%, and 60%,
respectively. In addition, measured water content and water pressure head were required for the actual reten-
tion curve in order to compare with the result of the FBD model. In this study, the soil water retention data of
30 dierent soils, measured by Yu etal.50, Chen and Wang51, Zhang and Miao52, Liu and Amemiya53, Hayano
etal.54, and Yabashi etal.55 were used for model verication (Table2). e data covered soils in China (such as
black soil, chernozem soil, cinnamon soil, brown earth, uvo-aquic soil, albic soil, red earth, humid-thermo
ferralitic, purplish soil, meadow soil, and yellow earth) and soils in Japan (such as volcanic ash soil and acolian
sandy soil). e USDA soil taxonomy of these soils was provided in Table2. e 30 soils ranged in texture from
clay to sand and in bulk density from 0.33g/cm3 to 1.65g/cm3, which covered a much wider range of soil bulk
density than many of the existing models or pedotransfer functions56–59. Particle size fractions (Dgi) were chosen
as the upper limit of the diameters between successive sieve sizes. For the data set in which particle density was
not determined, 2.65g/cm3 was used.
Statistical parameters for model verication
Four statistical properties, R2, RMSE, mean residual error (ME), and t value were calculated to determine the
accuracy of the FBD model. e R2 values were computed at the same value of ψ, with the values of θ measured
and estimated by the FBD model (Eq.20). RMSE and ME were obtained, respectively, by
(17)
n
i=
6m
i
ρsπD3
gi
×10
12
(18)
Di=0.2Dgi
(19)
ψ
i=
3000
D
i
=
15000
D
gi
(20)
θ
=
θ
s
1+
θs−θr
θ
r
15,000
ψ
b
(21)
b
=ǫ
3log
(θs−θr)[ln(
15,000.1
ψ)]
−(g−1)θr
g(θ
s
−θr)
(22)
RMSE
=
1
n
(θest −θmea)2
0.5
(23)
ME
=
1
n
n
i=1
(θest −θmea
)
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where θmea was measured soil water content, θest was soil water content estimated with the FBD model, and n was
the number of measured pairs of water content and pressure head. With the assumption of normal distribution
and independence of dierences between the water contents measured and estimated by the FBD model, t was
calculated with
when calculated |t| was larger than t0.05 (the critical value of the Student’s t distribution for P = 0.05 and n−1
degrees of freedom), the dierences between the measured and estimated water contents were statistically sig-
nicant. If t < 0, soil water contents were underestimated and vice versa. us, t was a measure for the systematic
bias in the estimation. Values of t close to zero indicated that the measured and estimated soil water contents
were not dierent systematically from each other or, equivalently, that there was no consistent bias. Values of t
that diered greatly from zero indicated the presence of systematic bias. RMSE was a measure for the scatter of
the data points around the 1:1 line. Low RMSE values indicated less scatter. Low RMSE values also implied low
ME. Regarding the result that t was low while RMSE was high, it could be explained that negative and positive
deviations distributed more evenly on the two sides of 1:1 line.
Received: 29 April 2020; Accepted: 22 September 2020
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Acknowledgements
is work was initially supported by Japan Society for the Promotion of Science (JSPS) [Grant number P97470]
and later by the AgResearch Program of the University of Tennessee, Knoxville, USA.
Author contributions
Authors H.S. and J.Z. designed and tested the model. H.S. prepared the original dra. J.Z., X.C. and J.L. reviewed
and edited the manuscript. J.Z. supervised the research. All authors participated in improving the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to J.Z.
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