A novel framework for deriving water retention behavior of multimodal unsaturated soils based on pore size distribution data

Abstract

The soil water retention curve (SWRC) strongly influences the hydro-mechanical properties of unsaturated soils. It plays a decisive role in geotechnical and geo-environmental applications in the vadose zone. This paper advances a novel framework to derive the water retention behavior of multimodal deformable soils based on the pore size distribution (PSD) measurements. The multiple effects of suction on the soil pore structure and total volume during SWRC tests are considered. The complete picture of soil microstructure is quantitatively described by the void ratio (for the pore volume) and a newly defined microstructural state parameter (for pore size distribution) from a probabilistic multimodal PSD model. Assuming a reversible microstructure evolution, a unique PSD surface for wetting and drying links the SWRC and PSD curves in the pore radius-suction-probability space. A closed-form water retention expression is obtained, facilitating the model's implementation in particle applications. The model is validated using the water retention data of four different soil types, showing a strong consistency between the measurement and the reproduced curve. The proposed method provides new insights into the pore structure evolution, the water retention behavior and the relationship between them for multimodal deformable soils.

1 Introduction

The soil water retention curve (SWRC) describing the relationship between the soil suction and water content (gravimetric water content $w$, degree of saturation ${S}_{r}$ or volumetric water content $\theta$) is dominated by soil pore structure. SWRC and soil microstructure play a decisive role in the unsaturated soil applications, as they significantly influence the hydraulic (e.g., unsaturated permeability [23, 31, 41, 44]) and mechanical properties (e.g., the Bishop's effective stress parameter $\chi$ [3, 29], shear strength [56], compressibility [45]) of unsaturated soils.

Experimental methods for determining soil water retention curves are tedious, expensive, and time-consuming, primarily due to the relatively low hydraulic conductivity of unsaturated soils. To address practical scenarios when direct measured data is scarce or even unavailable, predictive SWRC models [23, 54, 55, 63], commonly referred to as pedotransfer functions, have been devised. These models established relationships between SWRC parameters and other soil properties. Over the past decades, they have been successfully applied in geotechnical and geo-environmental engineering applications in the vadose zone.

Conventionally, the soil pore structure is regarded as a bundle of capillary tubes with a circular cross section. The pore radius of the capillary channel in the soil associated with a given suction level $s$ is determined by the Young–Laplace equation:

$$ r = \frac{C}{s} = \frac{{2T\,{\text{cos}}\,\theta }}{s} $$

(1)

Here, $C$ is a constant equal to $2T\text{cos}\theta$, $T$ is the surface tension of water (0.072 N/m at 25 °C), and $\theta$ is the contact angle between soil and water [16, 36]. For the sake of simplicity, it is feasible to assume $\theta$ ≈ 0° for desiccation [55]. The pores smaller than or equal to this value $r$ are filled with water at a given suction level $s$. Assuming that the pore structure remains constant during the SWRC tests, the dependency of the water content (in terms of the effective degree of saturation ${S}_{\text{e}}$) on the soil suction $s$ can be derived from the integration of pore size distribution function:

$$ S_{{\text{e}}} = \frac{{S_{{\text{r}}} - S_{{{\text{r}},{\text{res}}}} }}{{1 - S_{{{\text{r}},{\text{res}}}} }} = \mathop \int \nolimits_{0}^{r = C/s} f\left( \xi \right)d\xi $$

(2)

where $\xi$ is a dummy variable of integration representing pore radius, $f\left(\xi \right)$ is the pore size distribution (PSD) function, and ${S}_{\text{r},\text{res}}$ is the residual water content. The procedure in Eqs. (1) and (2) is referred to as the "direct transformation method" [67]. Inversely, soil pore size distribution can be derived from SWRC when assuming a rigid pore structure [52].

In the past decades, many empirical unimodal SWRC equations have been advanced, assuming $f\left(\xi \right)$ obeys normal distribution [39], lognormal distribution [30], or an empirical unimodal distribution [6, 17, 21]. However, numerous research literature has shown that the pore structure of compacted clayey soils or mixed soils consists of interaggregate (macro-) and intraaggregate (micro-) porosities [11, 22,23,, 27] or even more than two sub-porosities [25,26,, 51, 64]. A multimodal pore structure might cause a multimodal SWRC that the conventional unimodal SWRC functions cannot accurately describe. For this case, a multimodal pore size distribution function $f\left(r\right)$ is needed in Eq. (2).

Simms and Yanful [58] pointed out that the "direct transformation method" oversimplifies the relationship between the PSD and the water retention behavior of deformable soils, as the pore structure may significantly vary in response to water content and suction changes. That is why the directly transformed SWRC from a PSD data set differs from the measured SWRC for deformable soils [51, 5, 13]. Simms and Yanful [59] measured the microstructures of four fine-grained soils using MIP tests. They reported that the pore size distribution curve significantly changed with suction during the SWRC tests. For bimodal soils under constant load, the contribution of suction to the effective stress acting on the inter- and intraaggregate porosities differs [3], resulting in the pore structure depending on suction or water content. Monroy et al. [42] observed the microstructural transformation of London Clay (in oedometer with under a vertical load of 7 kPa) from an aggregate structure at the as-compacted state (high suction) to a matrix structure at the saturation state (zero suction). Cai et al. [7] measured different pore size distributions of the compacted Guilin lateritic soil at different suction levels during wetting and drying paths. The evolving microstructure implies that one obtains different SWRCs for a given soil when substituting the pore size distribution curves $f\left(r\right)$ into Eq. (2). This means that the water retention behavior predicted by the "direct transformation method" (Eqs. (1) and (2)) is not "characteristic" for deformable soils.

Very few attempts have been made to derive the SWRC of multimodal deformable soils from their PSD measurements. Simms and Yanful [59] proposed a network approach considering the suction-induced shrinkage and pore trapping inferred from the PSD curves. This approach was later adopted by Zhang and Li [70] to predict the SWRCs of different soils varying from gravel to clay. Nevertheless, it seems that the predicted SWRCs do not match well with their measurements. Della Vecchia et al. [13] incorporated an empirical correlation between the aggregate porosity and water content to account for the evolutionary fabric in a bimodal pore size distribution expression to model the water retention behavior of compacted soils with different activity of clay fractions. Nonetheless, a closed-from SWRC expression, which facilitates the implementation of the model in practical applications, cannot be derived from this approach.

This paper aims to establish a framework to interpret the relationship between the evolving microstructure and the water retention behavior for deformable soils. To overcome the limitations of the "direct transformation method", the proposed method employs a probabilistic multimodal pore size distribution expression and considers the multiple effects of suction on the pore structure and the overall volume during SWRC tests.

2 Theory

2.1 A newly defined microstructural state parameter

Recently, Yan and Cudmani [69] developed a general multimodal pore size distribution model based on probability theory for porous medium. Following Scheidegger [53] and Juang and Holtz [28], the pore radius denoted to a point in the void space is defined as the radius of the largest sphere containing this point within the void space (i.e., the sphere cannot cross the soil particle). Under this framework, the pore radius is a continuous random variable depending on the position of the denoted point. According to probability theory, we have

$$\int_{0}^{+\infty }f\left(r\right)\,\text{dr}=1.$$

(3)

Here, $f\left(r\right)$ is the pore size density function, and the term $f\left(r\right)\text{dr}$ represents the frequency of the occurrence of a sphere with a radius $r\to r+\text{dr}$ in the void space.

For a general case (i.e., a multimodal soil), the overall soil microstructure might be regarded as an assembly of N individual monomodal sub-porosities [15, 52, 68]. The overall pore size density function $f\left(r\right)$ is acquired by a linear superposition of the PSD functions of sub-porosities ${f}_{i}\left(r\right)$:

$$f\left(r\right)=\sum_{i=1}^{N}{R}_{i}{f}_{i}\,\left(r\right).$$

(4)

Here, ${R}_{i}$ represents the volumetric fraction of each sub-porosity, which fulfills the condition

$$\sum_{i=1}^{N}{R}_{i}=1.$$

(5)

In addition, the function ${f}_{i}\left(r\right)$ must fulfill the condition

$$\int_{0}^{+\infty }{f}_{i}\left(r\right)\,\text{dr}=1.$$

(6)

Considering the requirement in Eq. (6), a proper monomodal pore size density function derived from van Genuchten model [21] (with the constraint m + 1/n = 1) is employed

$$ f_{i} \left( r \right) = \frac{{m_{i} \;\left( {r_{i}^{{\text{f}}} /r} \right)^{{\frac{1}{{1 - m_{i} }}}} }}{{r\left( {1 - m_{i} } \right)\,\left[ {1 + \left( {r_{i}^{{\text{f}}} /r} \right)^{{\frac{1}{{1 - m_{i} }}}} } \right]^{{m_{i} + 1}} }} $$

(7)

where ${m}_{i}$ is the pore size spectrum number ($0<{m}_{i}<1$), and ${r}_{i}^{f}$ is a reference pore radius proportional to the maximum and mean (or dominant) pore size of the $i$-th sub-porosity. Substituting Eq. (7) in Eq. (4), a general multimodal pore size density function gives

$$f\left(r\right)=\sum_{i=1}^{N}{R}_{i}\frac{{m}_{i}{\left({r}_{i}^{\text{f}}/r\right)}^{\frac{1}{1-{m}_{i}}}}{{r\left(1-{m}_{i}\right)\left[1+{\left({r}_{i}^{\text{f}}/r\right)}^{\frac{1}{1-{m}_{i}}}\right]}^{{m}_{i}+1}}$$

(8)

which fulfills the condition in Eq. (3).

The cumulative pore size distribution curve yields

$$F\left(r\right)=\int_{0}^{r}f\left(r\right)\,\text{dr}=\sum_{i=1}^{N}{R}_{i}{\left[1+{\left(\frac{{r}_{i}^{\text{f}}}{r}\right)}^{\frac{1}{1-{m}_{i}}}\right]}^{-{m}_{i}}$$

(9)

The cumulative pore size distribution curve measured by mercury intrusion porosimetry (MIP) tests can be expressed as [69]:

$${V}_{\text{cum}}\left(r\right)=\frac{{e}_{\text{MIP}}}{{\rho }_{\text{s}}}\int_{r}^{+\infty }f\left(r\right)\,\text{dr}=\frac{{e}_{\text{MIP}}}{{\rho }_{\text{s}}}\left\{1-\sum_{i=1}^{N}\,{R}_{i}{\left[1+{\left(\frac{{r}_{i}^{\text{f}}}{r}\right)}^{\frac{1}{1-{m}_{i}}}\right]}^{-{m}_{i}}\right\}$$

(10)

Here, ${V}_{\text{cum}}\left(r\right)$ is the mercury intrusion volume in [cm³/g], ${e}_{\text{MIP}}$ is the void ratio before MIP tests, and ${\rho }_{\text{s}}$ is soil particle density. The parameters in the pore size distribution model possess a clear physical meaning. A simple parameter calibration method using MIP test data was introduced by Yan and Cudmani [69].

Under this framework, the complete picture of soil pore structure is quantitatively described by two independent state variables, the void ratio (for the overall volume) and a new microstructure parameter (for the pore size distribution). From the probabilistic multimodal pore size distribution model [Eq. (8)], a proper microstructural state variable ${\varvec{\Omega}}$ is defined as

$${\varvec{\Omega}}={\Omega }_{k}=\left\{{R}_{1}, {r}_{1}^{\text{f}},{m}_{1},\dots ,{R}_{N}, {r}_{N}^{\text{f}},{m}_{N}\right\}$$

(11)

Here, the microstructural state variable ${\varvec{\Omega}}$ (or ${\Omega }_{k}$ with $1\le k\le 3N$) is a vector with 3 N entries characterizing a general N-modal microstructure. Then, the multimodal pore size density function in Eq. (8) can be rewritten as

$$f\left(r\right)=f\left({\Omega }_{k},\text{r}\right).$$

(12)

2.2 Modeling the microstructural evolution

Soil microstructure is affected by various aspects, for example, sample preparation method (e.g., compaction energy and water content, initial dry density) [10, 33, 65], mechanical loading (e.g., pre-consolidation pressure and current stress state)[11, 22], hydraulic loading (e.g., wetting–drying cycles) [7, 8, 42] and even chemical (e.g., salinity of pore water) [37] and thermal loading [12, 24]. This study considers only the microstructural evolution along hydro-mechanical loading paths.

Delage and Lefebvre [11] observed that the interaggregate porosity in Champlain clay was significantly reduced with increasing pre-consolidation pressure; while, the intraaggregate porosity remained unchanged. Similar findings have also been reported for other deformable soils under proctor compaction [32, 51], isotropic compression [22], and oedometric compression [47]. In unimodal soils, the mean pores size was suggested to be proportional to mean net stress [25,49,, 46, 61]. These experimental studies imply that soil microstructure depends on the current stress state and the loading history.

Soil microstructure also alters with suction in wetting and drying cycles. Figure 1 shows the microstructural evolution of the compacted London Clay during wetting (Fig. 1a) and that of the compacted Guilin lateritic soil during wetting and drying (Fig. 1b). These experimental results reveal the fact that the pore structure is suction-dependent during the SWRC test. On the other hand, different mechanisms dominate the microstructural behavior in the two soils. London Clay (Fig. 1a) demonstrated distinguished bimodal characteristics at the as-compacted state (s = 996 kPa). The decrease in suction along a wetting path resulted in the expansion of aggregates and the degradation of interaggregate porosity, which vanished at the saturated state. In Guilin lateritic soil (Fig. 1b), the intraaggregate porosity remained almost unchanged, and the pore size of interaggregate pores increased with suction, which might be attributed to the "rearrangement of the aggregates" during the SWRC test.

Fig. 1

Microstructure evolution in deformable soils a Expansion of aggregates in London Clay (data from Monroy et al. [42]) b Rearrangement of aggregates in Guilin lateritic soil (data from Cai et al. [7])

Hydration on the unsaturated soil subjected to high external loading might cause "wetting collapse" phenomenon (Alonso et al. [2]), leading to irreversible volumetric deformation and changes in pore structure. For example, Della Vecchia [14] reported a significant reduction of macro-pores of a compacted Boom Clay sample saturated at high vertical stress of 600 kPa.

Based on the experimental evidence, a general constitutive law is proposed to parameterize the microstructural evolution under hydro-mechanical loading paths:

$${\varvec{\Omega}}={\Omega }_{k}\left({\sigma }_{ij}^{\text{net}},{p}_{0}^{*},s\right)$$

(13)

Here, ${\sigma }_{\text{ij}}^{\text{net}}$ is the net stress tensor, and ${p}_{0}^{*}$ is the pre-consolidation stress at the saturation state, representing the loading history associated with the irreversible volumetric deformation of the unsaturated soil [2]. Equation (13) means that all the entries in ${\varvec{\Omega}}$ depend on the current net stress, loading history and suction level. For an arbitrary set of ${\Omega }_{k}$ at a given stress state and suction level, following condition is held:

$$\int_{0}^{+\infty }f\left({\Omega }_{k},r\right)\,\text{dr}=1, \forall {\Omega }_{k}$$

(14)

2.3 Mechanical and hydraulic wetting

Tarantino [62] pointed out that the increase in the degree of saturation could originate from two aspects: (i) an increase in the water volume (e.g., by decreasing the suction level), which is referred to as "hydraulic wetting"; (ii) decrease in the void ratio (e.g., by increasing the mean net stress) which is referred to as "mechanical wetting". The proposed framework regarding the microstructural evolution along hydro-mechanical loading path can interpret this essential feature of the water retention behavior.

Substituting Eq. (13) into Eq. (2), we obtain

$${S}_{e}=\int_{0}^{C/s}f\left({\Omega }_{k}\left({\sigma }_{ij}^{\text{net}},{p}_{0}^{*},s\right),r\right)\,\text{dr}=F\left({\Omega }_{k}\left({\sigma }_{ij}^{\text{net}},{p}_{0}^{*},s\right),\frac{C}{s}\right)$$

(15)

Equation (15) describes a general relationship between the effective degree of saturation and soil suction based on a probabilistic multimodal pore size distribution concept. The model indicates that the soil suction, the current stress state, and the loading history dominate soil water content. From Eq. (15), the infinitesimal change of effective degree of saturation can be expressed by

$$ {\text{d}}S_{e} = \underbrace {{ - \frac{C}{{s^{2} }}f\left( {\Omega_{k} \left( {\sigma_{ij}^{{{\text{net}}}} ,\sigma_{0}^{*} ,s} \right),\frac{C}{s}} \right){\text{d}}s}}_{{T^{1} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{\partial s}{\text{d}}s}}_{{T^{2} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \mathop \sum \limits_{i,j = 1}^{3} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{{\partial \sigma_{ij}^{{{\text{net}}}} }}{\text{d}}\sigma_{ij}^{{{\text{net}}}} }}_{{T^{3} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{{\partial p_{0}^{*} }}{\text{d}}p_{0}^{*} }}_{{T^{4} }} $$

(16)

In Eq. (16), the term ${T}^{1}$ represents the change of ${S}_{e}$ induced by a suction increment with the assumption of a constant microstructure during the SWRC test; the term ${T}^{2}$ describes the change of ${S}_{e}$ due to the suction-induced microstructural variation; the terms ${T}^{3}$ and ${T}^{4}$ are the infinitesimal change of ${S}_{e}$ resulting from the change in the current stress state and the loading history of the soil, respectively.

Let us consider a soil sample subjected to mechanical loading at a constant suction level, Eq. (16) degrades to

$$ {\text{d}}S_{e} = \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \mathop \sum \limits_{i,j = 1}^{3} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{{\partial \sigma_{ij}^{{{\text{net}}}} }}{\text{d}}\sigma_{ij}^{{{\text{net}}}} }}_{{T^{3} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{{\partial p_{0}^{*} }}{\text{d}}p_{0}^{*} }}_{{T^{4} }} $$

(17)

which quantitatively describes the "mechanical wetting" according to Tarantino [62]. Figure 2a schematically demonstrates an infinitesimal increase in the degree of saturation in bimodal soil subjected to mechanical loading. At a constant suction level ${s}_{0}$, the upper integration bound ${r}_{0}$ in Eq. (2) is determined by the Young–Laplace equation. Assuming that the mechanical loading leads to the horizontal shifting of the interaggregate sub-curve from state A to state B, the infinitesimal increase in the effective degree of saturation can be represented by the shaded area, as shown in Fig. 2a.

Fig. 2

Schematic for the interpretation of a mechanical wetting b hydraulic wetting with evolving pore structure c hydraulic wetting with a rigid pore structure

When considering an unsaturated deformable soil subjected to hydraulic loading (change of suction level) at a constant net stress state, Eq. (16) degrades to

$$ {\text{d}}S_{e} = \underbrace {{ - \frac{C}{{s^{2} }}f\left( {\Omega_{k} \left( {p_{0}^{*} ,s} \right),\frac{C}{s}} \right){\text{d}}s}}_{{T^{1} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{\partial s}{\text{d}}s}}_{{T^{2} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{{\partial p_{0}^{*} }}{\text{d}}p_{0}^{*} }}_{{T^{4} }} $$

(18)

Here, $\text{d}{p}_{0}^{*}$ in term ${T}^{4}$ represents the suction-induced variation in pre-consolidation pressure (e.g., in the wetting–collapse process [2]). For instance, experimental evidence for the irreversible pore structure variation occurring in the interaggregate porosity associated with wetting–collapse in compacted loess has been recently reported by Ge et al. [20].

Since SWRCs are conventionally measured under zero or minor net stress (e.g., Pressure Plate, Tempe Pressure Cells, and Chilled-Mirror Hygrometers [16, 36]), it is reasonable to assume that the pre-consolidation pressure ${p}_{0}^{*}$ is not affected by suction during SWRC tests (i.e., no wetting collapse). Then, the term ${T}^{4}$ in Eq. (18) vanishes. The change in the effective degree of saturation induced by a suction change in deformable soils during SWRC tests gives

$$ {\text{d}}S_{e} = \underbrace {{ - \frac{C}{{s^{2} }}f\left( {\Omega_{k} \left( s \right),\frac{C}{s}} \right){\text{d}}s}}_{{T^{1} }} + \underbrace {{\mathop \sum \limits_{k = 1}^{3N} \frac{\partial F}{{\partial \Omega_{k} }}\frac{{\partial \Omega_{k} }}{\partial s}{\text{d}}s}}_{{T^{2} }} $$

(19)

The closed-form expression of Eq. (19) gives

$${S}_{e}=\int_{0}^{C/s}f\left({\Omega }_{k}\left(s\right),r\right)\,\text{dr}=F\left({\Omega }_{k}\left(s\right),\frac{C}{s}\right)$$

(20)

Equation (20) describes the SWRC derived from suction-dependent pore size distribution curves. As the effective degree of saturation increases during drying ($\text{d}s>0;\text{d}{S}_{e}\le 0$) and decreases during wetting ($\text{d}s<0;\text{d}{S}_{e}\ge 0$), following condition must be held

$$-\frac{C}{{s}^{2}}f\left({\Omega }_{k}\left(s\right),\frac{C}{s}\right)+\sum_{k=1}^{3N}\frac{\partial F}{\partial {\Omega }_{k}}\frac{\partial {\Omega }_{k}}{\partial s}\le 0$$

(21)

From Eq. (20), it is not hard to see that suction affects the degree of saturation twofold: on the one hand, suction directly influences the pore size distribution function (see Fig. 2b, the horizontal shifting of the sub-curve); On the other hand, suction changes the upper bound of the integration (see Fig. 2b, ${r}_{1}$ increases to ${r}_{2}$). In Fig. 2b, the shaded area represents the infinitesimal increase in the effective degree of saturation under the dual effect of suction during wetting, defined as "hydraulic wetting", according to Tarantino [62].

For granular soils (e.g., sand and gravel) with a rigid pore structure, the microstructural parameter ${\varvec{\Omega}}$ is constant during SWRC tests. Equation (19) degrades to

$$ {\text{d}}S_{e} = \underbrace {{ - \frac{C}{{s^{2} }}f\left( {\Omega_{k} ,\frac{C}{s}} \right){\text{d}}s}}_{{T^{1} }} $$

(22)

The increase in the effective degree of saturation during "hydraulic wetting" is solely attributed to the rise in the upper integration bound (i.e., decrease in suction), as schematically shown in Fig. 2c.

2.4 The link between the water retention curve and evolving pore structure

2.4.1 The pore size distribution surface

Based on the proposed framework (Eq. (20)), the water retention behavior of deformable soils can be derived from suction-dependent PSD curves associated with microstructural evolution during SWRC tests (i.e., characterized by ${\Omega }_{k}\left(s\right)$). The main drying curve gives

$${S}_{e}^{\text{d}}\left(s\right)=\int_{0}^{2T\text{cos}{\theta }_{\text{d}}/s}f\left({\Omega }_{k}\left(s\right),r\right)\,\text{dr}=F\left({\Omega }_{k}\left(s\right),\frac{{C}_{d}}{s}\right)$$

(23)

Here, ${C}_{d}$ is a constant equal to $2T\text{cos}{\theta }_{\text{d}}$, ${\theta }_{\text{d}}$ is the contact angle between soil and water during drying (${\theta }_{\text{d}}\approx$ 0°), and ${\Omega }_{k}\left(s\right)$ represents the suction-dependent microstructural parameter fulfilling the condition of Eq. (21).

For a better interpretation of Eq. (23), the transformation from PSD measurements to SWRC for deformable soils is visualized in the pore radius (r)—suction (s)—probability (F = S_e) space (see Fig. 3a). The PSD surface consisting of the cumulative PSD curves at different suction levels is expressed as:

Fig. 3

a Pore size distribution surface in the pore radius-suction-probability space b Water retention curves in the log s–S_e plane c Pore size distribution curves in the log r–F plane

$${S}_{e}^{\text{d}}\left(r,s\right)=F\left({\Omega }_{k}\left(s\right),r\right)$$

(24)

Comparing Eq. (23) and (24), the main drying curve can be equivalently expressed by

$${S}_{e}^{\text{d}}\left(s\right)=F\left({\Omega }_{k}\left(s\right),r\right){|}_{r={C}_{d}/s}$$

(25)

Equation (25) indicates that the main drying curve can be visualized as the cross section between the PSD surface and the capillary plane (i.e., $r={C}_{d}/s$) in the $r-s-F\left(={S}_{e}\right)$ space (see Fig. 3a).

Figure 3a shows an example of a PSD surface for Lias Clay [67] (adopted entries in ${\varvec{\Omega}}$: ${m}_{1}=0.47$; ${m}_{2}=0.41$; ${r}_{1}^{\text{f}}=5.0$ µm; ${r}_{2}^{\text{f}}=0.03$ µm; ${R}_{1}=0.22\times \left[1-\text{exp}\left(-0.002s\right)\right]$; ${R}_{2}=1-{R}_{1}$). The pore structure is unimodal at saturation state and exhibits a distinguished bimodal characteristic at the high suction range. On the one hand, we obtain the main drying curve by projecting the cross section between the PSD surface and the capillary plane on the conventional log s–S_e plane (see Fig. 3b). On the other hand, projecting the cross section on the log r–F plane yields the cumulative pore size distribution curve derived from SWRC (the dashed red curve in Fig. 3c). One sees that it differs from the actual PSD curve at any suction level (the solid grey curves). Traditionally, a constant PSD curve derived from SWRC data was used to calculate the unsaturated hydraulic conductivity [19, 21, 44]. The phenomenon in Fig. 3c reveals the shortcoming of the traditional approaches for predicting the unsaturated hydraulic conductivity for deformable soils. An advanced approach is required to incorporate the suction-dependent pore size distribution into unsaturated hydraulic conductivity functions.

2.4.2 Hysteresis effect

For the sake of simplicity, a reversible microstructural behavior during the SWRC tests (i.e., a unique PSD surface for wetting and drying) is assumed. The "hysteresis effect" of SWRC is attributed to different contact angles during wetting and drying. Similar to the drying curve, the main wetting curve is obtained by projecting the cross section between the PSD surface and the capillary plane for wetting on the log s–S_e plane (see Fig. 3a and b). It is interesting to note that the shape of the wetting curve differs from that of the drying curve. Another factor causing hysteresis is the "ink bottle" effect [1, 16, 36]. A part of pore air is entrapped in the soil during wetting, resulting in a partial saturation at zero suction.

Considering the change of contact angle and the "ink bottle" effect, the main wetting curve can be expressed as

$${S}_{e}^{\text{w}}={\beta }_{\text{ink}}\int_{0}^{2T\text{cos}{\theta }_{w}/s}f\left({\Omega }_{k}\left(s\right),r\right)\,\text{dr}={\beta }_{\text{ink}}F\left({\Omega }_{k}\left(s\right),\frac{{\beta }_{\theta }{C}_{d}}{s}\right)$$

(26)

Here, ${\beta }_{\theta }$ and ${\beta }_{\text{ink}}$ are two empirical parameters to describe the hysteresis effect. The parameter ${\beta }_{\theta }$ is defined as

$${\beta }_{\theta }=\frac{\text{cos}{\theta }_{d}}{\text{cos}{\theta }_{w}}\ge 1$$

(27)

where ${\theta }_{w}$ is the contact angle between soil and water during wetting. The parameter ${\beta }_{\text{ink}}$ ($0<{\beta }_{\text{ink}}\le 1$) is a scaling factor from the "ink bottle" effect. It can be approximated by the value of the effective degree of saturation at zero suction of a wetting curve. Figure 3a (in the r-s-F space) and 3b (on the log s–S_e plane) show an example of a main wetting curve with ${\beta }_{\theta }$ = 10 and ${\beta }_{\text{ink}}$ = 1.

2.4.3 The complete water retention model

The water retained in unsaturated soil comprises capillary and adsorptive water [26, 35, 36]. Equation (20) describes the capillary water dominated by soil pore size distribution. The specific surface of soil and mineralogy govern the adsorptive water, which is tightly bonded on the soil particle surface as a thin film by attractive forces (i.e., van der Waals forces and hydrogen bonds) between the soil particles and pore water. The volume of adsorptive water decreases with decreasing water vapor concentration [50]. For the aim of simplicity, it is assumed as a constant (i.e., the residual water content ${S}_{r}^{\text{res}}$) in this study. Considering the residual state of the soil (i.e., ${S}_{r}\to {S}_{r}^{\text{res}}$ as $s\to +\infty$), a complete water retention model in terms of degree of saturation gives:

$$ S_{r} \left( s \right) = \left\{ {\begin{array}{*{20}l} {\left( {1 - S_{r}^{{{\text{res}}}} } \right)F\left( {{\Omega }_{k} \left( s \right),\frac{{C_{d} }}{s}} \right) + S_{r}^{{{\text{res}}}} , \quad {\text{for drying}}} \hfill \\ {\left( {1 - S_{r}^{{{\text{res}}}} } \right)\beta_{{{\text{ink}}}} F\left( {{\Omega }_{k} \left( s \right),\frac{{\beta_{\theta } C_{d} }}{s}} \right) + S_{r}^{{{\text{res}}}} , \quad {\text{for wetting}}} \hfill \\ \end{array} } \right. $$

(28)

Expressing the SWRC regarding gravimetric water content $w$ (w-SWRC), the change in total volume must be considered [18, 40]. Denoting the shrinkage and swelling curves as a function of soil suction ${e}_{d}\left(s\right)$ and ${e}_{w}\left(s\right)$, respectively, the w-SWRC yields

$$ w\left( s \right) = \left\{ {\begin{array}{*{20}l} {\frac{{e_{d} \left( s \right)}}{{G_{s} }}\left[ {\left( {1 - S_{r}^{{{\text{res}}}} } \right)F\left( {{\Omega }_{k} \left( s \right),\frac{{C_{d} }}{s}} \right) + S_{r}^{{{\text{res}}}} } \right], \quad {\text{for drying}}} \hfill \\ {\frac{{e_{w} \left( s \right)}}{{G_{s} }}\left[ {\left( {1 - S_{r}^{{{\text{res}}}} } \right)\beta_{{{\text{ink}}}} F\left( {{\Omega }_{k} \left( s \right),\frac{{\beta_{\theta } C_{d} }}{s}} \right) + S_{r}^{{{\text{res}}}} } \right], \quad {\text{for wetting}}} \hfill \\ \end{array} } \right. $$

(29)

Equation (29) describes the link between the macroscopic gravimetric water content at a given suction level, the microscopic evolving pore structure, and the total soil volume during SWRC tests.

3 Applications

In this section, the proposed model is validated using the water retention data of four different types of soils. The residual water content is assumed to be zero (i.e., $S_{r}^{{{\text{res}}}} = 0$). It is found that the simplification does not significantly influence the prediction accuracy for the presented applications. Generally, 3 N suction-dependent entries in the microstructural parameter ${\varvec{\Omega}}$ are required, which may lead to difficulty in practical applications. Nevertheless, it will be shown that one or two suction-dependent entries are adequate to describe the microstructural evolution during SWRC tests in the presented applications.

3.1 Silty sand

Angerer [4] prepared two statically compacted low-plasticity silty sand (90.5% sand + 8.5% silt + 1% Clay, ${e}_{\text{min}}$ = 0.47, ${e}_{\text{max}}$ = 0.76, ${G}_{s}$ = 2.68) samples at the same relative density (${I}_{\text{d}}$ = 0.69) with different initial water contents (w = 3 and 10%). The cumulative pore size distribution curves of both samples were measured by MIP tests at the as-compacted state (see Fig. 4a). Afterward, the samples were saturated and then desiccated from the saturation state to measure the main drying curves (see Fig. 4b).

Fig. 4

a Measured and reproduced cumulative pore size distribution curves b measured and predicted main drying curves of a silty sand (data from [4])

The rigid pore structure (including the void ratio and microstructure) remains unchanged for granular soils during the SWRC tests. All the entries in the microstructural parameter ${\varvec{\Omega}}$ are constants. Considering the distinct bimodality of the pore structure (aggregates formed by the fine contents during compaction), a bimodal PSD expression (Eq. (8) with $N$ = 2) is substituted into Eq. (23) to predict the main drying curve. The parameters of the PSD model are listed in Table 1. They are calibrated using the MIP data of the soil samples (Fig. 4a). A convenient parameter calibration procedure for the probabilistic multimodal PSD model is described in detail by Yan and Cudmani [69].

Table 1 Parameters of the proposed model for different types of soils.

In Fig. 4b, the predicted SWRCs agree well with the SWRC measurements. The contact angle θ between soil and water was assumed to be equal to zero for drying. In the low suction range (i.e., $s$ < 10 kPa), capillary water is almost completely desiccated from interaggregate pores in a narrow suction range. An identical air entry suction (about 1 kPa) of the samples is correctly reproduced. In the relatively high suction range (i.e., 10 kPa < $s$  < 1 × 10⁶ kPa), adsorption mechanism governs the relationship between water content and soil suction. Consequently, the SWRC model, derived from pore size distribution evolution and capillary effect, exhibited a minor underestimation of water content. A distinct expression correlating soil suction with adsorptive water content becomes imperative to precisely characterize the dry end of the soil water retention curve [50, 66].

3.2 Lateritic soil

Cai et al. [7] conducted a comprehensive investigation on the water retention behavior of the statically compacted Guilin lateritic soil (${w}_{L}$ = 67.3%, ${w}_{p}$ = 38.1%, ${G}_{s}$ = 2.74) for a wide suction range from 0 to 287.6 MPa, including the main drying and wetting curves, the volumetric deformation, and microstructural evolution. The pore size distribution curves at five suction levels (50, 200, and 71,120 kPa from drying, 100 and 21,820 kPa from wetting) were detected by MIP tests. The experiments were performed on the samples with different initial void ratios, 1.3, 1.5, and 1.7 (denoted with CS 1.3, CS1.5, and CS1.7, respectively). In this section, the proposed model is validated by deriving the water retention curves regarding the degree of saturation and gravimetric water content for the sample CS1.5. The model validation based on the data of samples CS1.3 and CS1.7 will be shown in the Appendix.

3.2.1 Volumetric deformation

Fig. 5 demonstrates the volumetric deformation of the sample CS1.5 during the SWRC test. The soil was first desiccated from a saturated state to a high suction level of 287.6 MPa and then wetted to a low suction level of 50 kPa. Generally, the suction-induced volume change can be described in a differential equation [2, 57] or continuous closed-form expression [34, 38, 60]. In this study, the following curvilinear equation is proposed to simulate the volumetric deformation during the SWRC test:

Fig. 5

Volumetric deformation of the CS1.5 sample during drying and wetting (data from Cai et al. [7])

$$e\left(s\right)={e}_{0}-\frac{{e}_{0}-{e}_{\text{s}}}{{\left[1-\text{exp}\left(\zeta s\right)\right]}^{\eta }}$$

(30)

Here, ${e}_{\text{s}}$ and ${e}_{0}$ represent the void ratio at the shrinkage limit and saturation state, respectively; $\zeta <0$ and $\eta <0$ are two fitting parameters determining the shape and curvature of the shrinkage and swelling curves.

For the CS1.5 sample, the parameter ${e}_{0}$ is approximated by the maximum void ratio of the measured $e-s$ curve (${e}_{0}=1.50$ for drying and ${e}_{0}=1.40$ for wetting). A common ${e}_{\text{s}}$ for the shrinkage and swelling curve is approximated by the minimum void ratio at the highest suction level of 287.6 MPa (i.e., ${e}_{s}=1.26$, see Fig. 5). The reproduced $e-s$ curves for drying and wetting are demonstrated in Fig. 5, which match well with the measurements.

3.2.2 Microstructural evolution

Fig. 6 presents the mercury intrusion curves during drying and wetting, which show a distinct bimodal characteristic. These curves were attained by numerical integration of the original pore size distribution data published by Cai et al. [7]. A delimiting point was set at the turning point (r = 1 μm) of the transition range. The sub-curve representing interaggregate pores translated with suction, meaning that the interaggregate pore size increase with suction. The smaller interaggregate pores merged with larger ones (e.g., forming fissures in soil). On the other, the volume of interaggregate porosity remained unchanged (V_{interaggregate} = 0.2 g/cm³), implying the shrinkage of aggregates during drying. The slight decrease in the intraaggregate pore size significantly reduced the total volume due to their large number.

Fig. 6

Measured and reproduced mercury intrusion curves of the CS1.5 sample (data from Cai et al. [7])

The reproduced bimodal mercury intrusion curves (using Eq. (10) with $N$ = 2) showed a good agreement with the measurements. The model parameters are listed in Table 2. The maximum intrusion volume e_MIP (i.e., the void ratio before mercury intrusion) was determined from the soil shrinkage and swelling curves (see Fig. 5). These suction-dependent parameters comprised the microstructural parameter ${\varvec{\Omega}}$ and characterized the essential features of microstructural behavior during wetting and drying. The increasing parameter ${r}_{1}^{\text{f}}$ indicated the increase in the mean pore size in interaggregate region, leading to a horizontal translation of the interaggregate sub-curve. In addition, the mean interaggregate pore size increased monotonically with suction, irrespective of the drying or wetting process. This phenomenon supported the assumption of the reversible microstructural behavior during water retention tests (i.e., a unique PSD surface for wetting and drying).

Table 2 Parameters of the pore size distribution model for the CS1.5 sample

3.2.3 PSD surface

To determine the PSD surface of the CS1.5 sample, suction-dependent microstructural parameter ${\varvec{\Omega}}$ is attained based on the analysis of the parameters listed in Table 2. Three parameters ${r}_{1}^{\text{f}}$, ${R}_{1}$ and ${R}_{2}$ vary with suction. Figure 7a shows the nonlinear fitting equation characterizing the evolution of ${r}_{1}^{\text{f}}$ with suction, for which a plateau of ${r}_{1}^{\text{f}}$ is assumed at the high suction range. The volumetric fraction ${R}_{1}$ is the ratio of the intraaggregate pore volume ${e}_{M}$ (= ρ_sV_{interaggregate} = 0.55) to the total pore volume ${e}_{d}\left(s\right)$. The evolution of ${R}_{1}$ with suction matched the data points well (see Fig. 7b). The volumetric fraction of intraaggregate pores (${R}_{2}$) was solved from Eq. (5).

Fig. 7

a Evolution of the reference pore radius ${r}_{1}^{\text{f}}$ in the interaggregate porosity b Evolution of the volumetric fraction ${R}_{1}$ of the interaggregate porosity c The unique PSD surface in the r-s-F(= S_e) space for the CS1.5 sample (data from Cai et al. [7])

The entries in the microstructure parameter ${\varvec{\Omega}}$ characterizing the pore structure evolution of the CS1.5 sample are given in Table 1. Substituting the microstructural parameter ${\varvec{\Omega}}$ in Eq. (23) gives the main drying curve. Figure 7c demonstrates the PSD surface in the $r-s-F\left(={S}_{e}\right)$ space. The model predicted a quasi-unimodal pore structure at the saturation state and a distinct bimodal pore structure at the high suction levels. The main drying cure was visualized as the cross section of the PSD surface and the capillary plane for drying in the $r-s-F\left(={S}_{e}\right)$ space (see Fig. 7c).

3.2.4 Predicting water retention behavior

Assuming a reversible microstructural behavior during wetting and drying, Eq. (28) reproduces water retention curves in the conventional $\text{log}s-{S}_{r}$ plane (see Fig. 8a), showing a strong consistency with the measurements and a distinguishing bimodal characteristic. For the main wetting curve, the hysteresis parameters (${\beta }_{\text{ink}}=0.8$ and ${\beta }_{\theta }=1.1$) were approximated based on the experimental data. Figure 8b demonstrates w-SWRCs using Eq. (29). The model predicted an inclined line at the beginning of desiccation. At this stage, the soil remained saturated while the overall volume reduced due to the increase in the effective stress [48].

Fig. 8

a Measured and derived s-S_r relationship b measured and derived s-w relationship c measured and derived shrinkage and swelling curves for the CS1.5 sample (data from Cai et al. [7])

Fig. 8c shows the predicted shrinkage and swelling curves in the conventional $w-e$ plane. It is interesting to note that the curves exhibited a "double S-shape" characteristic resulting from the bimodal pore structure. The proposed model captured the feature and indicated four phases of the desiccation. Phase 1: For suction lower than air entry suction, pore water was discharged, and soil retained saturated (the part along the saturation line); Phase 2: Pore water was rapidly drained from interaggregate pores with a slight reduction in volume; Phase 3: The interaggregate pores were almost entirely drained. The intraaggregate porosity retained saturated as the air entry suction of aggregates was not reached. The overall soil volume decreased due to the increase in the effective stress. An apparent "collapse" phenomenon (i.e., a remarkable reduction in void ratio with a sight change in water content) in the w–e plane was observed at this transition stage; Phase 4: Suction surpassed the air entry suction of aggregates. The water in the intraaggregate pores was discharged, associated with a dramatic reduction in volume. In addition, the wetting path exhibited a similar "double S-shape" characteristic, which was captured by the proposed model.

For CS 1.3 and CS 1.7 samples, the predicted water retention curves and shrinkage and swelling curves are demonstrated in Appendix. The parameters and the suction-dependent entries in ${\varvec{\Omega}}$ are listed in Table 1.

3.3 Sandy loam

Cuisinier and Laloui [9] investigated the pore structure evolution of a sandy loam (${w}_{L}$ = 30%, ${w}_{p}$ = 18%, ${G}_{s}$ = 2.66) along a drying path. They measured the gravimetric water content and pore size distribution (using mercury intrusion porosimetry) of the soil at five suction levels, i.e., 0, 50, 100, 200 and 400 kPa. The mercury intrusion curves were attained by numerical integration of pore size distribution data from Cuisinier and Laloui [9] and simulated using Eq. (10) with $N$ = 2. It was found that solely the parameter ${r}_{2}^{\text{f}}$ decreases with increasing suction, implying the shrinkage of aggregates along the drying path. The constant entries and the suction-dependent ${r}_{2}^{\text{f}}$ in the microstructural parameter ${\varvec{\Omega}}$ are given in Table 1. Figure 9a shows the reproduced pore size distribution curves of the soil characterized by ${\varvec{\Omega}}$, showing good consistency with the measurements. The main drying curve was obtained by substituting the microstructural parameter in Eq. (23) (with θ_d = 0°). Since the volumetric deformation during the SWRC test was not given in Cuisinier and Laloui [9], the SWRC in terms of S_e was estimated by normalizing the gravimetric water content based data with the maximum gravimetric water content at saturation. As illustrated in Fig. 9b, the reproduced SWRC matched the measured data well for suctions below 400 kPa. Unfortunately, no measurements were taken for suctions exceeding 400 kPa. The slight underestimation of water content within the suction range of 200–400 kPa (as depicted in Fig. 9b) suggested a potential deviation between predictions and measurements in the high suction range. This observation indicated the model's constrained ability to accurately predict the dry end of SWRCs.

Fig. 9

a Comparison between the measured and reproduced mercury intrusion curves b Comparison between the measured and derived main drying curve of sandy loam (data from Cuisinier and Laloui [9])

3.4 London clay

Monroy et al. [42] investigated the microstructural evolution of compacted London Clay (${w}_{L}$ = 83%, ${w}_{p}$ = 29%, ${G}_{s}$ = 2.70) along a wetting path. Figure 10a demonstrates the pore size distribution curves measured by MIP tests at five suction levels 996 kPa (initial suction at as-compacted state), 470, 150, 40, and 0 kPa. As mentioned in Sect. 2.2, the microstructural behavior of the compacted London Clay was understood as "swelling of aggregates", leading to the increase in the parameter ${r}_{2}^{\text{f}}$ and the intraaggregate volumetric fraction ${R}_{2}$. The constant entries and the suction-dependent ${r}_{2}^{\text{f}}$ and ${R}_{2}$ in the microstructural parameter ${\varvec{\Omega}}$ are given in in Table 1. Figure 10a compares the measured and reproduced mercury intrusion curves characterized by the microstructural parameter ${\varvec{\Omega}}$ at the five suction levels. The interaggregate porosity decreased during wetting starting from the as-compacted state (s = 996 kPa) and vanished at saturation state (s = 0 kPa). This means the bimodal pore structure transformed into an unimodal pore structure along the wetting path.

Fig. 10

a Comparison between the measured and reproduced mercury intrusion curves b Comparison between the measured and derived main drying curve of London Clay (data from Monroy [43] and Monroy et al. [42])

The main drying curve is acquired using Eq. (23) (with θ_d = 0°) and compared to the SWRC data published in Monroy [43] in Fig. 10b. The model reproduced a distinct bimodal drying curve and indicated a relatively low air entry suction (ca. 5 kPa) for the clayey soil. This phenomenon might be attributed to the clay aggregates at low suction levels (see the distinct bimodal PSD at $s$ = 40 kPa). For suctions greater than 1 × 10³ kPa, the predicted water content was lower than the measurement. This observation again implied the model's restricted capacity to accurately predict the dry end of soil water retention curves (SWRCs), especially for clayey soils. Nevertheless, the quality of the reproduced SWRC is considered sufficient, particularly given its derivation from bimodal pore size distribution data.

4 Discussion: inter- and intraaggregate air entry suction

The proposed framework offers a new understanding of air entry suction (${s}_{\text{ae}}$) for bimodal deformable soils. Figure 11 depicts the evolving inter- and intraaggregate reference pore radius (${r}_{1}^{\text{f}}$ and ${r}_{2}^{\text{f}}$) of Guilin lateritic soil sample CS1.5 (see Sect. 3.2) in log s–log r plane. At low suction levels (e.g., s = 1 kPa in Fig. 11), ${r}_{1}^{\text{f}}$ was close to ${r}_{2}^{\text{f}}$, indicating a unimodal pore structure. The reference pore radii were underneath the Young–Laplace equation line. The soil was hence saturated. After that, ${r}_{1}^{\text{f}}$ significantly increased with suction due to the shrinkage and rearrangement of aggregates. At the cross point between the Young–Laplace equation line and the ${r}_{1}^{\text{f}}$ evolution curve, meniscus firstly formed as the water surface tension, pore-air- and pore-water-pressure were in equilibrium in the maximum pore of interaggregate porosity. This implied the onset of the desiccation in interaggregate porosity. Afterward, the water stored in interaggregate porosity was rapidly discharged due to increased suction and pore size. Eventually, ${r}_{1}^{\text{f}}$ reached a high value as fissures formed at oven-dry state (point C). This mechanism revealed the fact that the "air entry suction" observed in a SWRC measurement was not directly related to the maximum pore radius in soil but rather the reference pore radius at the equilibrium point. In addition, the desiccation in intraaggregate porosity started from the cross point between the Young–Laplace equation line and the ${r}_{2}^{\text{f}}$ evolution curve.

Fig. 11

Insights into the air entry values of the bimodal Guilin lateritic soil

5 Summary and conclusions

The water retention behavior of soils is crucial in water flow problems in the vadose zone. Conventionally, soil water retention curves are measured in the laboratory and used to derive pore size distribution and unsaturated hydraulic conductivity function. However, the water retention measurement of deformable soils does not represent their pore size distribution because the total volume and pore structure change with suction. This problem is more acute when considering the multimodality in soil microstructure.

This paper develops a feasible systematic approach to derive water retention behavior of deformable soils from their pore size distribution data. The microstructure parameter ${\varvec{\Omega}}$ characterizes a PSD surface in pore radius-suction-probability space, which is inferred from the mercury intrusion curves at different suction levels using a probabilistic PSD model. Projecting the cross section between the PSD surface and the capillary plane (i.e., Young–Laplace equation) on the soil suction–probability plane gives the water retention curve. The model validation relies on the water retention data of four soils, including silty sand, lateritic soil, sandy loam, and clay. The predicted water retention curves show strong consistency with the measurements.

The proposed framework provides new insights into the relationship between the pore size distribution and the water retention behavior for multimodal deformable soils. The following conclusions are drawn:

• Soil suction controls the water content, the overall volume, and the pore size distribution during wetting and drying. For unimodal soils, the change in pore size distribution is related to the maximum or mean pore size. For bimodal or multimodal soils, this problem becomes more complex. The contribution of suction to the effective stress acting on the inter- and intraaggregate porosities differs [3], resulting in the pore structure depending on suction.

• The measured water retention curve of deformable soils is a macroscopic observation of the soil suction-water content relationship resulting from the changing pore size distribution and total volume. The nature of the water retention curve is a subset of the hydraulic states at different suction levels or water contents [13]. Despite assuming a reversible pore structure evolution, the shape of the main drying curve might differ from that of the main wetting curve.

• Soil microstructure evolution characteristics strongly depend on soil material properties (e.g., plasticity and mineralogical constituents) and sample preparation methods (e.g., reconstitution and compaction). Even the modality number may change during wetting and drying. Thus, the specific microstructure evolution characteristics differ from soil types. The proposed method enables quantifying various microstructure evolution paths under a unique framework.

• Air entry suction determined from a water retention measurement is assumed to be inversely proportional to its maximum pore size in traditional water retention models, e.g., van Genuchten model. It is not related to the “maximum pore size” for deformable soils but results from their microstructure evolution and the capillary effect (i.e., Young–Laplace equation).

• The "pore size distribution" of multimodal deformable soils derived from the macroscopic water retention curve differs from the actual ones governed by their hydro-mechanical state. Incorporating this "pore size distribution" into conventional unsaturated hydraulic conductivity functions may result in significant errors [58]. An advanced approach is required to incorporate the suction-dependent pore size distribution into unsaturated hydraulic conductivity functions.

Appendix

(See Figs. 12 and 13)

Fig. 12

Sample CS1.3 of compacted Guilin lateritic soil a volumetric evolution during the SWRC tests b predicted water retention curves in terms of degree of saturation c predicted water retention curves in terms of gravimetric water content d predicted shrinkage and swelling curves (data from Cai et al. [7])

Fig. 13

Sample CS1.7 of compacted Guilin lateritic soil a volumetric evolution during the SWRC tests b predicted water retention curves in terms of degree of saturation c predicted water retention curves in terms of gravimetric water content d predicted shrinkage and swelling curves (data from Cai et al. [7])