Fractional bulk density concept
The first assumption is that soil particles with different sizes contribute to different porosities and water holding capacities in bulk soil. Based on a non-similar media concept (NSMC) defined by Miyazaki49, soil bulk density (ρb) is defined as
$$rho _{{{b}}} = frac{{{M}}}{{{V}}} = tau rho _{{{s}}} left( {frac{{{S}}}{{{{S}} + {{d}}}}} right)^{3}$$
(4)
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 characteristic lengths of solid phase and pore space, respectively. The parameter τ is a shape factor of the solid phase, defined as the ratio of the substantial volume of solid phase to the volume S3. The 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 defined the volume of bulk soil as
$$V = frac{{mathop sum nolimits_{{{i } = { 1}}}^{{{n}}} {{m}}_{{{i}}} }}{{{rho }_{{{b}}} }} = frac{{{{m}}_{{1}} }}{{{rho }_{{{{b1}}}} }}{ + }frac{{{{m}}_{{2}} }}{{{rho }_{{{{b2}}}} }}{ + } cdot{mkern -4mu}cdot{mkern -4mu}cdot frac{{{{m}}_{{{n}}} }}{{{rho }_{{{{bn}}}} }}$$
(5)
where mi and ρbi are the solid mass and equivalent bulk density of the ith size fraction of soil particles, respectively. In this study, diameters of the first particle fraction and the last one were assumed to be 1 µm and 1000 µm, respectively8. This equation suggests that different particle size fractions are associated with different equivalent bulk densities due to different contributions of particle arrangement to soil pore space. As a result, the particles with the same size fraction could have different equivalent bulk densities in soils with different textures or after the soil particles are rearranged (e.g., compaction). Figure 4 provides a diagrammatic representation of such fractional bulk density concept for the variation of soil pore volume with soil particle assemblage.
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.
Calculation of volumetric water content
For a specific soil, Eq. (5) means
$${{V}}_{{{{pi}}}} left( { le {{D}}_{{{i}}} } right){{ } = { f}}left( {{{D}}_{{{{gi}}}} {{, M}}_{{{i}}} } right)$$
(6)
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,
$${{V}}_{{{{pi}}}} left( { le {{D}}_{{{{gi}}}} } right) = frac{{{{V}}_{{{{pmax}}}} }}{{1 + kappa left( {{{D}}_{{{{gi}}}} } right)^{{{{b}}_{{{i}}} }} }}$$
(7)
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
$$theta_{{{i}}} left( { le {{D}}_{{{{gi}}}} } right) = frac{{theta_{{{s}}} }}{{1 + kappa left( {{{D}}_{{{{gi}}}} } right)^{{{{b}}_{{{i}}} }} }}$$
(8)
where θs is saturated volumetric water content calculated with
$$theta_{{{s}}} = left{ {begin{array}{*{20}l} {0.9varphi _{{{T}}} ,} hfill & {~rho _{{{b}}} < 1} hfill {varphi _{{{T}}} ,} hfill & {~~rho _{{{b}}} ge 1} hfill end{array} } right.$$
(9)
$$varphi _{{{T}}} = frac{{rho _{{{s}}} – rho _{{{b}}} }}{{rho _{{{s}}} }}$$
(10)
In the above equations, ρbis measured soil bulk density, and ρs is soil particle density (2.65 g/cm3). The empirical parameter κ in Eqs. (7) and (8) is defined as
$${{kappa}} = frac{{theta_{{{s}}} – theta_{{{r}}} }}{{theta_{{{r}}} }}$$
(11)
where θr is measured residual water content. In this study, θr is set as the volumetric water content at water pressure head of 15,000 cm. The empirical parameter bi is defined as
$${{b}}_{{{i}}} = frac{{epsilon }}{{3}}{log}left( {frac{{{theta }_{{{s}}} {{ – omega }}_{{{i}}} {theta }_{{{s}}} }}{{{{kappa omega }}_{{{i}}} {theta }_{{{s}}} }}} right)$$
(12)
with ε, a particle size distribution index, calculated with
$${varepsilon }; = ;frac{{left( {{{D}}_{{{40}}} } right)^{{2}} }}{{{{D}}_{{{10}}} {{D}}_{{{60}}} }}$$
(13)
where D10, D40, and D60 represent the particle diameters below which the cumulative mass percentages of soil particles are 10%, 40%, and 60%, respectively.
The parameter ωi is coefficient 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
$${omega }_{{{i}}} = frac{{{g}}}{{{{1 + kappa }}left( {{{lnD}}_{{{{gi}}}} } right)^{{lambda}} }}$$
(14)
where g is regulation coefficient (1.0–1.2). We set it to be 1.2 in this study. λ is the ratio coefficient of particle size distribution fitted using the lognormal Logistic model,
$$M_{i} = frac{{M_{T} }}{{1 + eta D_{{gi}} ^{lambda } }}$$
(15)
where MT represent the total mass percentage of all sizes of soil particles, and η is a fitting parameter. We set MT = 101 in Eq. (15) for best fit of the particle size distribution. In this study, this continuous function was generated 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 1 μm. Considering the difference in the upper limits of particle sizes associated with existing datasets of Dgi and Mi, the particle size distribution with the upper limit of 2,000 μm for the Acolian sandy soil and volcanic ash soils in Table 2 was normalized to the case with the upper limit of 1,000 μm 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
$${{D}}_{{{i}}} {{ } = { D}}_{{{{gi}}}} left[ {frac{{2}}{{3}}{{en}}_{{{i}}}^{{{{(1 – alpha )}}}} } right]^{{{0}{{.5}}}}$$
(16)
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 et al.20. In Tyler and Wheatcraft’s model22α is the fractal dimension of the pore. The 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 (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. The equation for calculating ni is given as19
$$n_{i} = frac{{6m_{i} }}{{rho_{s} pi D_{gi}^{3} }} times 10^{12}$$
(17)
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 0.075 N/m at 25 °C, the minimum diameter of soil pore (Dmin) was taken to be 0.2 µm in this study, which is equivalent to the water pressure head of 15,000 cm according to Young–Laplace equation. We set this minimum pore size to correspond the minimum particle size (Dgmin = 1.0 µm). The FBD model might thus not apply well to porous media with pores smaller than 0.2 μm. As a result, Eq. (16) can be simplified into the following equation.
$${{D}}_{{{i}}} { = 0}{{.2D}}_{{{{gi}}}}$$
(18)
The equivalent capillary pressure (ψi in cm) corresponding to the ith particle size fraction can be calculated using
$$psi_{{{i}}} = frac{{{3000}}}{{{{D}}_{{{i}}} }} = frac{{{15000}}}{{{{D}}_{{{{gi}}}} }}$$
(19)
In Eq. (19), the maximum water pressure head (ψr = 15,000 cm) corresponds to θr and Dgmin (1 μm). The minimum water pressure head (ψ0 = 15 cm) corresponds to θs and Dgmax (1,000 μm). These 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
Equations 8 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,
$${theta }; = ;frac{{{theta }_{{{s}}} }}{{{1 + }left( {frac{{{theta }_{{{s}}} – {theta }_{{{r}}} }}{{{theta }_{{{r}}} }}} right)left( {frac{{15,000}}{{psi }}} right)^{{{b}}} }}$$
(20)
with the parameter b obtained using
$${{b}}; = ;frac{{epsilon }}{{3}}{log}left{ {frac{{{{(theta }}_{{{s}}} – {theta }_{{{r}}} {{)[ln(}}frac{{{15,000}{{.1}}}}{{psi }}{)]}^{{lambda }} – {{(g}} – {{1)theta }}_{{{r}}} }}{{{{g(theta }}_{{{s}}} – {theta }_{{{r}}} {)}}}} right}$$
(21)
In Eq. (21), a water pressure head of 15,000.1 cm is employed to consecutively predict the soil water content until the water pressure head of 15,000 cm.
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 retention curve in order to compare with the result of the FBD model. In this study, the soil water retention data of 30 different soils, measured by Yu et al.50, Chen and Wang51, Zhang and Miao52, Liu and Amemiya53, Hayano et al.54, and Yabashi et al.55 were used for model verification (Table 2). The data covered soils in China (such as black soil, chernozem soil, cinnamon soil, brown earth, fluvo-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). The USDA soil taxonomy of these soils was provided in Table 2. The 30 soils ranged in texture from clay to sand and in bulk density from 0.33 g/cm3 to 1.65 g/cm3, which covered a much wider range of soil bulk density than many of the existing models or pedotransfer functions56,57,58,59. 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, 2.65 g/cm3 was used.
Source: Ecology - nature.com
