# An Insight into the Projection Characteristics of the Soil-Water Retention Surface

^{1}

^{2}

^{*}

## Abstract

**:**

_{r}) with suction (s) and void ratio (e), is of crucial importance for understanding and modeling the hydro-mechanical behavior of unsaturated soils. As a 3D surface in the S

_{r}–e–s space, the SWRS can be projected onto the constant S

_{r}, constant s, and constant e planes to form three different 2D projections, which is essential for establishing the SWRS and understanding its various characteristics. This paper presents a series of investigations on the various characteristics of the three SWRS projections. For the S

_{r}–s and S

_{r}–e relationships, (i) a tangential approximation approach is proposed to quantitatively capture their asymptotes, and (ii) a new criterion is presented to distinguish the low and high suction ranges within which these two relationships exhibit different features. On the other hand, a modified expression is introduced to better capture the characteristics of the s–e relationships. The various projection characteristics and the proposed approaches are validated using a wide set of experimental data from the literature. Studies presented in this paper are useful for the rational interpretation of the SWRS and the hydro-mechanical coupling behavior of unsaturated soils.

## 1. Introduction

_{r}and the suction s. The mechanical behavior refers to the volumetric strains caused by various external stresses. Mechanical constitutive models through the use of net stress and suction cannot describe the dependence of mechanical behavior on the degree of saturation [12,13]. Similarly, hydraulic constitutive models (such as SWRC models) cannot accurately reflect the effect of stress–strain behavior on the degree of saturation [14,15,16,17]. In other words, the hydraulic behavior and mechanical behavior of unsaturated soils are inherently coupled because the volumetric change caused by external stress modifies the SWRC simultaneously [3,5] and the change in the S

_{r}due to s also influences soil’s skeleton stress and therefore the stress–strain behavior [18,19,20,21,22,23,24].

_{r}relationship as a 3D surface (i.e., soil-water retention surface, SWRS). This SWRS model can describe the irreversible changes of degree of saturation caused by the hydraulic (i.e., wetting and drying) and mechanical (i.e., confining pressure and shearing stress) behaviors and can be effectively used in the numerical modelling of coupled flow-deformation problems. More recently, Tarantino [32] developed a SWRC model that is similar but simpler than Gallipoli et al.’s [26] model. Gallipoli [33] improved the Gallipoli et al. [26] model to predict the hysteretic response of soils during both drying and wetting cycles at constant e and compression and swelling cycles at constant s, which is virtually the projection characteristics of the SWRS. Ghasemzadeh et al. [11] established a hysteretic SWRC model based on the power function proposed by Gallipoli et al. [26].

_{r}) remain outstanding in the current literature, irrespective of their significant importance. The SWRS has three different projection scenarios: projection at constant void ratio, e, and projection at constant suction, s and projection at constant degree of saturation, S

_{r}. On the constant e plane, projections are a series of s–S

_{r}relationships (i.e., SWRC). On the constant s plane, projections are a series of e–S

_{r}relationships which reflect the changes in S

_{r}due to the mechanical stress induced variation in e. In other words, the e–S

_{r}relationships can reflect the influence of mechanical behavior on hydraulic behavior. On the constant S

_{r}plane, projections are a series of s–e relationships indicating the dependence of mechanical behavior on hydraulic behavior. The s–S

_{r}and e–S

_{r}relationships are measurable using pressure plate and unsaturated triaxial tests, while the s–e relationship cannot be directly obtained.

_{r}planes, three independent 2-D equations are formulated based on the SWRS model proposed by Gallipoli et al. [26]. The specific characteristics of the projections are discussed in detail. Modifications and improvements are also introduced with respect to (i) capturing the asymptotes and distinguishing the different behaviors in the high and low suction ranges for the s–S

_{r}and e–S

_{r}relationships and (ii) describing the s–e relationships.

## 2. Projections of the SWRS

_{r}= 0.80, respectively.

- (i)
- The SWRCs: the S
_{r}versus s plot at constant e (referred to as plane e). - (ii)
- The S
_{r}versus e plot where s is constant (referred as plane s). It is denoted as the hydro-mechanical coupling curves (HMCCs) in this paper. - (iii)
- The s versus e plot at constant S
_{r}(referred to as plane S_{r}). It is commonly referred to as the retention consolidation curves (RCCs).

_{r}versus s plot; SWRC) has an increasing negative tangent with decreasing S

_{r}(defined in Equation (3) and shown in Figure 2a using data from Salager et al. [36]), and the tangent is equal to −mn (note that mn > 0) as S

_{r}tends to zero. Similarly, Equation (2) on the plane s (i.e., S

_{r}versus e plot; HMCC) has an increasing positive tangent with decreasing S

_{r}and the tangent is equal to −mnψ (note that mnψ > 0) as S

_{r}tends to zero (defined in Equation (4) and Figure 2b);

_{r}space when s and e tend to infinity and S

_{r}tends to zero. Asymptotes expression of Equation (2) was defined as Equation (5);

_{w}= eS

_{r}) and the product mnψ = 1, the log-planar asymptote (i.e., Equation (7)) is independent of void ratio.

## 3. Characteristics of the Projections of the SWRS

#### 3.1. Soil-Water Retention Curves (SWRCs)

_{0}= initial void ratio at saturation; ψ

_{e}, ω

_{e}, m

_{e}, and n

_{e}are model parameters for plane e. Equation (8) converges to the van Genuchten [16] expression:

_{e}/(e

_{0})

^{ψ}

^{e}, indicating the air-entry suction; n

_{e}is a parameter related to pore-size distribution; m

_{e}is a parameter related to the overall symmetry of the SWRC.

_{r}tends to m

_{e}n

_{e}logα, and (ii) when logS

_{r}tends to zero; logs tends to logα

_{0}.

_{e}, n

_{e}in Equation (9) and λ

_{e}(slope of asymptotes, see Figure 2a) and κ

_{e}(horizontal intercept of asymptotes, see Figure 2a) for the four soils at different initial void ratios are summarized. The product m

_{e}n

_{e}is close to the absolute value of λ

_{e}and the value of κ

_{e}is close to logα. These two observations are consistent with the assumptions proposed by Gallipoli [33]. Therefore, relationships m

_{e}n

_{e}≅ −λ

_{e}and κ

_{e}= logα are applied in the later sections to determine the position of asymptotes.

_{e}/(e

_{0})

^{ψ}

^{e}yields logα = logω

_{e}− ψ

_{e}loge

_{0}. The ψ

_{e}is the slope of the logα–loge

_{0}relationship and logω

_{e}is the vertical intercept of the logα–loge

_{0}line at loge

_{0}= 0. Figure 4 shows the logα–loge

_{0}relationships for the four soils. The results in Figure 4 are consistent with the conclusion of Stange and Horn [43], who found linear relationships between loge

_{0}and logα.

_{r}plane. Similarly, HMCC at different suction can be obtained by a rigid translation from a reference HMCC along the e axis in the loge–logS

_{r}plane. A rigid translation implies that all asymptotes of the SWRCs at different initial void ratios are parallel to each other and their m

_{e}n

_{e}values are identical. In other words, the horizon distance between two asymptotes at different initial void ratios is constant and equals to the horizon distance between the intercepts of the two asymptotes at logs = 0. This is consistent with the postulation proposed by Nuth and Laloui [28] that the SWRC has an intrinsic shape at constant e

_{0}and this intrinsic curve were parallel to the SWRCs at all constant values of e

_{0}.

_{1}and e

_{2}(e

_{2}< e

_{1}), α

_{1}, m

_{1}, n

_{1}and α

_{2}, m

_{2}, n

_{2}are their parameter values of Equation (9), and λ

_{1}, κ

_{1}and λ

_{2}, κ

_{2}are the respective slope and horizontal intercept of asymptotes. Two coordinate points are required for determining the asymptotes: one point is A

_{1}(logα

_{1}, 0) or A

_{2}(logα

_{2}, 0) which are easily obtained; the other point can be indirectly obtained from the SWRC, referred to as matching point. According to m

_{1}n

_{1}≅ −λ

_{1}and m

_{2}n

_{2}≅ −λ

_{2}, the matching point must satisfy the condition that the absolute value of the slope calculated by the two points (i.e., point A and the matching point) is close to the value of the product m

_{1}n

_{1}(or m

_{2}n

_{2}). In order to obtain the matching point, an iteration-based tangential approximation method can be used, which is detailed below.

_{1}is the vertical projection of point A

_{1}on the corresponding SWRC (see Figure 5). Substituting the abscissa value of point A

_{1}into Equation (2), point B

_{1}can be obtained, namely (logα

_{1}, m

_{1}log(1/2)). A tangent line of the SWRC at point B

_{1}can be conveniently defined using the slope of the SWRC defined in Equation (3) and point B

_{1}(i.e., logS

_{r}= −1/2⋅m

_{1}n

_{1}⋅(logs − logα

_{1}) + m

_{1}log(1/2)). With this tangent line and the asymptote of the SWRC passing point A

_{1}(defined by Equation (10)), point C

_{1}can be obtained at the intersection (see Fig 5) and its coordination is (logα

_{1}− 1/n

_{1}·log(1/4), m

_{1}log(1/4)). The first iteration (A

_{1}→B

_{1}→C

_{1}) ends here. Taking point C

_{1}as the starting point (similar to A

_{1}), one can continue with the next iteration using the same approach detailed for first iteration. This procedure is repeated until a matching point is found. The matching point is the point where the starting point overlaps with its projection on the SWRC, and typically can be determined within seven iterations. The matching point obtained after seven iterations is denoted as point H

_{1}(logα

_{1}+ 1.5382/n

_{1}, −1.5506m

_{1}). The slope calculated by both points A

_{1}and H

_{1}is about equal to −1.008m

_{1}n

_{1}, meeting the necessary condition of m

_{1}n

_{1}≅ −λ

_{1}. Therefore, the asymptotes can be well captured by points A

_{1}and H

_{1}. The proposed procedure is an objective and simple procedure to determine the asymptote. It is assumed that any points on the SWRC after point H

_{1}belong to the asymptote.

_{1}and H

_{1}is reasonable. The purpose of the proposed calibration is to ensure that the value of the product m

_{1}n

_{1}estimated from linear best-fitting of experimental data (Equation (10)) is consistent with a logarithmic planar behavior over the experimental data range. Gallipoli [33] suggested that the values of m

_{1}and n

_{1}are considered acceptable if logS

_{r}< −m

_{1}over the experimental asymptotic range (m

_{1}and n

_{1}are obtained from the product m

_{1}n

_{1}value using the relationship m

_{1}= 1 − 1/n

_{1}proposed by van Genuchten [16]); otherwise, they have to be recalibrated by imposing logS

_{r,max}= −m

_{1}, where S

_{r,max}is the maximum experimental value of the degree of saturation. Considering that logS

_{r}= −1.5506m

_{1}< −m

_{1}for the matching point H

_{1}(logα

_{1}+ 1.5382/n

_{1}, −1.5506m

_{1}), the asymptote computed by points A

_{1}and H

_{1}using the proposed method satisfies the criterion logS

_{r}< −m

_{1}suggested by Gallipoli [33] and captures the logarithmic planar behavior over the experimental range.

_{1}and e

_{2}, the horizontal projection of point H

_{1}on the SWRC at e

_{2}can be obtained and is denoted as R

_{1}(logα

_{2}+ 1.5506m

_{1}/m

_{2}n

_{2}, −1.5506m

_{1}). The horizontal distance between point H

_{1}and point R

_{1}in log-log scale is (see Figure 5):

_{2}/α

_{1}) when m

_{1}n

_{1}is close to m

_{2}n

_{2}. In this case, asymptotes of two main drying curves at different values of e

_{1}and e

_{2}are parallel. It can be seen from Table 2 that the product m

_{e}n

_{e}of the same soil are almost the same at different values of the initial void ratio for some soils (such as silt sand, compacted till and Ca-bentonite). Therefore, their main drying curves are parallel.

_{e}n

_{e}and α

_{e}, if m

_{e}n

_{e}values are not close. An example is shown to highlight this scenario using data of a silty sand (soil properties are summarized in Table 1 and m

_{e}n

_{e}and α

_{e}values are summarized in Table 2). Assume its two SWRCs at e

_{0}= 0.68 and 1.01 are available and used as reference curves, while the SWRC at e

_{0}= 0.86 is used to provide comparisons between the predictions and measurements. Considering the linear relationships between loge

_{0}and logα, the α value of the SWRC at e

_{0}= 0.86 can be obtained from the linear relationship defined by loge

_{0}and logα of the reference SWRCs at e

_{0}= 0.68 and 1.01. As shown in Figure 6a, for SWRCs at e

_{0}= 0.86, the predicted curve (solid line) obtained by a rigid translation of the reference SWRC at e

_{0}= 1.01 shows better agreements with measurements than the predicted curve translated from the reference SWRC at e

_{0}= 0.68 (dash line). It can be seen from the summarized information in Table 2 that the deviation from logα values of the SWRCs at e

_{0}= 0.86 to the logα values of the reference SWRCs at e

_{0}= 1.01 and e

_{0}= 0.68 are 0.251 and 0.367, respectively. This means that the smaller deviation of logα is, the higher the accuracy of the prediction is. In addition, m

_{e}n

_{e}values of reference SWRC at e

_{0}= 1.01 are also closer to the m

_{e}n

_{e}values of SWRC at e

_{0}= 0.86, which contributes to a better prediction.

_{1}, e

_{2}, and e

_{3}, respectively, for further explanation. mn values of SWRCs at e

_{1}and e

_{2}are close but different from that of SWRCs at e

_{3}. When mn values of two SWRCs (e.g., SWRCs at e

_{1}and at e

_{2}) are close, their horizon distance d is mainly controlled by their logα values. On the contrary, when mn values of two SWRCs (e.g., SWRCs at e

_{2}and at e

_{3}) are different, their horizon distance D is controlled by both mn and logα. This is also reflected in Equation (11). The equal-length arrows in Figure 6b indicates the horizontal distance between two SWRCs can be determined by their logα only (arrows with solid lines indicate shifting from SWRC at e

_{1}to SWRC at e

_{2}and arrows with dash lines indicate shifting from SWRC at e

_{2}to SWRC at e

_{3}). As can be seen from Figure 6b, the horizontal shifting works well for SWRCs at e

_{1}and at e

_{2}as the arrows reach the SWRC at e

_{2}. On the contrary, such horizontal shifting introduces errors for SWRCs at e

_{2}and at e

_{3}as the arrows do not always stop at the SWRC at e

_{3}.

_{e}n

_{e}in Table 2 are not always identical for different values of initial void ratios for some soils (such as tailing sand and sandy loam). To further validate the curve shifting method, Figure 7 shows predicted SWRCs obtained by a rigid translation of the same reference SWRC (e

_{0}= 0.802) for a tailing sand. Some differences in predictions and the measurements can be observed from the summarized information in Figure 7. These differences may be attributed to the two factors; (1) the difference in mn values, and (2) the difference in logα or e

_{0}values. For the former, deviations in the mn values of the predicted SWRCs (at e

_{0}= 0.746 and e

_{0}= 0.695) and the reference SWRC (at e

_{0}= 0.802) are about 0.13 and 0.30 (see Table 2). Similarly, for the latter, deviations in the e values of the predicted SWRCs (at e

_{0}= 0.746 and e

_{0}= 0.695) and the reference SWRC (at e

_{0}= 0.802) are 0.056 and 0.107. Predictions using SWRC at e

_{0}= 0.746 as reference curve is slightly better than that using SWRC at e

_{0}= 0.695 as a reference curve due to the smaller deviation in the mn and e

_{0}values.

_{0}are small, predicted SWRCs (e

_{0}= 0.984 and 1.175) obtained by a rigid translation of the same reference SWRC (e

_{ref}= 1.193) show a good agreement with the experimental data. Therefore, three aspects deserve attention when predicting SWRCs at constant e

_{0}using the curve shifting method: (1) at least two sets of SWRCs with different initial void ratios experimental data must be known to estimate the parameter α values of Equation (9) and the linear relationships between loge

_{0}and logα; (2) the reference curve has to be fitted by Equation (9) prior to translation because of typical limitations of the experimental data; (3) the difference between the e

_{0}values of the predicted SWRCs and reference SWRCs should be as small as possible.

#### 3.2. Hydro-Mechanical Coupling Curves (HMCCs)

_{con}= constant suction; ψ

_{s}, ω

_{s}, m

_{s}, and n

_{s}are model parameters for the constant suction condition.

_{s}/s

_{con})/ψ

_{s}. Equation (13) is the asymptote of Equation (2) on the plane s. When loge tends to zero, logS

_{r}tends to m

_{s}n

_{s}ψ

_{s}·logβ. Meanwhile, loge tends to logβ when logS

_{r}tends to zero. Figure 9 shows the evolution of experimental HMCCs in loge–logS

_{r}plane at different constant s using data obtained from Salager et al. [36] and Sun and Sun [42]. The results indicate that HMCCs move to the left-hand side along the e axis with increasing suction.

_{s}, n

_{s}and λ

_{s}, κ

_{s}for four soils used in this study at different constant suctions are summarized in Table 3. The λ

_{s}is the slope of asymptotes and κ

_{s}is horizontal intercept of asymptotes (see Figure 2b). It can be observed from Table 3 that: (i) the product m

_{s}n

_{s}ψ

_{s}is close to the absolute value of λ

_{s}, namely m

_{s}n

_{s}ψ

_{s}≅ −λ

_{s}and (ii) κ

_{s}is approximately equal to logβ, namely logβ ≅ κ

_{s}. Note that CDG and CDE in Table 3 denote different stress paths for the constant suction, respectively. Unlike the logs–logα relationships shown in Figure 4 which are linear, the logs

_{con}–logβ relationships are bi-linear.

_{s}/s

_{con})/ψ

_{s}is less effective in fitting the data over the entire suction range. Due to this reason, Equation (14) is proposed to separately describe the logs

_{con}-ogβ relationships in the low and high suction ranges.

_{L}, ψ

_{H}, ω

_{L}, ω

_{H}, β

_{L}, β

_{H}are parameters. In addition, ψ

_{L}and ω

_{L}(or ψ

_{H}and ω

_{H}) have a clear physical meaning as they are associated with the slope and intercept of the straight line interpolating experimental data in the logs

_{con}–logβ plane at low or high suctions.

_{w}space [33]. Tarantino [32] presented a model similar to the Gallipoli [26] expression. This model satisfies the condition of mnψ = 1, and the precondition for this model is that the suction has to be located in the high suction range. Therefore, constant suctions imposed on the HMCC can be considered as high suctions when the product m

_{s}n

_{s}ψ

_{s}= 1. It is important to note that the product m

_{s}n

_{s}ψ

_{s}is not always equal to 1 (see Table 3, for instance the case for the sand-bentonite and expansive silty-clay). For this reason, the imposed constant suctions will be regarded as low suctions when 0 < mnψ < 1. Combining with this conclusion and several sets of data related to silty sand (i.e., s

_{con}and logβ) in Table 3, a bilinear relationship exists in the logs

_{con}–logβ plane over the full suction range (see Figure 10a). It is evident for the tested silty sand that 100 kPa can be considered as the critical suction value for distinguishing low suctions from high suctions. This conclusion is consistent with the results of Salager et al. [34,36] obtained from a graphical approach that above 100 kPa suction, SWRCs with different initial void ratios can be regarded as an overlapping curve on the constant e plane. Figure 10b shows that there is a well-defined linear relationship between logs

_{con}and logβ at low suctions for sand-bentonite and expansive silty clay.

_{s}n

_{s}ψ

_{s}departs significantly from the absolute value of λ

_{s}. A novel method is suggested in the present study to assure m

_{s}n

_{s}ψ

_{s}≅ −λ

_{s}:

_{s}n

_{s}ψ

_{s}> 1 and not equal to −λ

_{s}, m

_{s}n

_{s}ψ

_{s}is assumed equal to 1 and relationship m

_{s}= 1/(n

_{s}ψ

_{s}) is substituted into Equation (12) to fit the experimental data. This results in a new set of m

_{s}, n

_{s}, and ψ

_{s}values. If the new product m

_{s}n

_{s}ψ

_{s}is close to −λ

_{s}, then the m

_{s}, n

_{s}, and ψ

_{s}values are deemed suitable. If still m

_{s}n

_{s}ψ

_{s}significantly departs from −λ

_{s}, additional calibration is needed. In this case, the new m

_{s}, n

_{s}, and ψ

_{s}values are used to plot the HMCC and the λ

_{s}of the plotted HMCC is determined (this updated λ

_{s}is denoted as λ

_{s}

^{*}). Substitute relationship m

_{s}= λ

_{s}

^{*}/(n

_{s}ψ

_{s}) into Equation (12) again to update m

_{s}, n

_{s}, and ψ

_{s}values until m

_{s}n

_{s}ψ

_{s}≅ −λ

_{s}is achieved. The final m

_{s}, n

_{s}, and ψ

_{s}values satisfying m

_{s}n

_{s}ψ

_{s}≅ −λ

_{s}are deemed acceptable.

_{s}n

_{s}ψ

_{s}< 1 and m

_{s}n

_{s}ψ

_{s}≠ −λ

_{s}, as a first step it is assumed equal to −λ

_{s}and then substituted into Equation (12). The subsequent processing is the same as for the m

_{s}n

_{s}ψ

_{s}> 1 case, which was detailed in the earlier step (i).

_{s}n

_{s}ψ

_{s}. The product m

_{s}n

_{s}ψ

_{s}is approximately equal to −λ

_{s}after three iterations for the data of Table 3 (bold fonts). In addition, Equation (12) with new parameter values provides a good match with experimental data. Hence, the calibration method proposed in this paper is reasonable.

_{i}n

_{i}ψ

_{i}and logβ

_{i}are the parameter values of Equation (13) corresponding to different constant suctions s

_{i}(i = 1, 2, 3, 4). Dash lines stand for asymptotes of HMCCs (solid lines) on a logarithmic scale. All HMCCs at high suctions (i.e., m

_{s}n

_{s}ψ

_{s}= 1) can be obtained by a rigid translation of the same graph in the loge–logS

_{r}plane [33]. On the contrary, the values of the product m

_{s}n

_{s}ψ

_{s}at low suctions (0 < m

_{s}n

_{s}ψ

_{s}< 1) are different. Due to this reason, HMCCs at high suctions may not be obtained by extending this rigid translation technique from a reference HMCC at low suctions. In other words, rigid translation is only feasible in high suction range or in low suction range, separately. Rigid translation from high suction range to low suction range or vice versa is not reliable. Figure 12 shows predicted HMCCs (at s = 10

^{4}kPa and 10

^{5}kPa) obtained by a rigid translation of two reference HMCCs (s

_{ref}= 10

^{3}kPa and 10 kPa) for a silty sand. The translation between the prediction HMCC and the reference HMCC is evaluated by term: |logβ

_{ref}− logβ

_{pre}| = |log(s

_{pre}/s

_{ref})/ψ|. The values of ψ can be obtained by Equation (14) and (15), where logβ

_{i}and s

_{i}are given in Table 3 (ψ

_{L}= 0.085 and ψ

_{H}= 0.320). It is evident that using the reference HMCC at low suctions to predict HMCCs at high suctions can lead to deviations that are significant from the experimental data.

_{1}and s

_{2}. For a given pair of low suction values of s

_{1}

^{′}and s

_{2}

^{′}(s

_{2}

^{′}< s

_{1}

^{′}), the constant horizontal distance between the two asymptotes is presented in Equation (16):

_{1}

^{′}, ψ

_{1}

^{′}, m

_{1}

^{′}, n

_{1}

^{′}and ω

_{2}

^{′}, ψ

_{2}

^{′}, m

_{2}

^{′}, n

_{2}

^{′}are two sets of parameter values of Equation (12) corresponding to HMCCs at s

_{1}and s

_{2}, respectively. β

_{1}

^{′}= (ω

_{1}

^{′}/s

_{1})

^{1/}

^{ψ}

^{1’}and β

_{2}

^{′}= (ω

_{2}

^{′}/s

_{2})

^{1/}

^{ψ}

^{2’}. Δloge can be considered approximately equal to log(β

_{2}

^{′}/β

_{1}

^{′}) when m

_{1}

^{′}n

_{1}

^{′}ψ

_{1}

^{′}is close to m

_{2}

^{′}n

_{2}

^{′}ψ

_{2}

^{′}. To simplify and facilitate the application, it is convenient to assume that ω

_{1}

^{′}= ω

_{2}

^{′}= ω

_{L}and ψ

_{1}

^{′}= ψ

_{2}

^{′}= ψ

_{L}for low suction ranges, the horizontal distance between the predicted HMCC and the reference HMCC is Δloge = log(s

_{1}/s

_{2})/ψ

_{L}. A trial calculation method is introduced to improve prediction accuracy of the parameter ψ

_{s}, in addition to Equation (14) which can be directly used to determine ψ

_{s}.

_{L}. Different ψ

_{L}values can be tried to obtain the Δloge and therefore translate the HMCC at s = 300 kPa to HMCC at s = 600 kPa. A suitable ψ

_{L}value is obtained when the translated HMCC at s = 600 kPa fits the measurements of the HMCC at s = 600 kPa. The HMCCs with different constant suctions obtained by a rigid translation of the reference HMCC (s = 300 kPa) and compared with experimental data (see Figure 13) to check validity of the proposed method. The results show predicted HMCCs obtained from the proposed method match well with experimental data. Therefore, the curve shifting method proposed by Gallipoli [33] is also feasible at low suctions.

#### 3.3. Retention Consolidation Curves (RCCs)

_{rcon}= constant degree of saturation; ψ

_{s}

_{a}, ω

_{s}

_{a}, m

_{s}

_{a}, and n

_{s}

_{a}are model parameters. The derivation of RCCs in the logs–loge plane at S

_{rcon}is obtained as

_{r}(i.e., RCC) from experimental studies. In order to represent the RCC, the SWRS (i.e., Equation (1)) shall be determined from measurements of the SWRCs at different void ratios or HMCCs at different suctions first and then set S

_{r}to constant to obtain the RCC. Salager et al. [36] measured SWRCs of a clayey silt sand. The five sets of experimental data are fitted by Equation (1) with best-fit parameter values ψ = 4.180, ω = 11.335, n = 0.686, and m = 0.565. RCCs at different constant S

_{r}are obtained by substituting these parameter values into Equation (17). As shown in Figure 14, RCCs are linear in the logs–loge plane, and their slope equals to −1/ψ

_{sa}. The RCCs move towards the left-hand direction along the s axis with the increase in constant S

_{r}.

_{r}= 1 and s = 0.1 kPa in Figure 14a).

_{1}, b

_{1}, c

_{1}, d

_{1}, and χ are empirical parameters. The values of a

_{1}, b

_{1}, c

_{1}, d

_{1}, and χ are 1000, 400, 0.466, 2.896, and 0.21, respectively, which was suggested by Salager et al. [34].

_{0}, its initial suctions and corresponding void ratios at different constant S

_{r}can be obtained from the simultaneous solution of Equation (17) and Equation (19). As a result, these initial suctions and corresponding void ratios would form a curve, which is called the modified curve in this study. As shown in Figure 14b, the short-dashed line is a modified curve that shows initial states of unsaturated soil specimens at different constant S

_{r}. The modified curve shows that the initial suction is increasing with the reducing constant S

_{r}. For different e

_{0}(i.e., 0.44, 0.68, and 1.01) in Figure 14b, the initial suction is decreasing as e

_{0}increases at equal S

_{r}. Such a behavior is expected because at equal S

_{r}, the water content is increasing as initial void ratio increases based on S

_{r}= G

_{s}w/e

_{0}, resulting in the decrease of suction. Hence, modified curves proposed in this study are reasonable.

## 4. Conclusions

_{r}planes in order to have an insight into the hydromechanical behavior of unsaturated soils. The details are summarized below:

- (1)
- The SWRCs tend to move towards the left-hand direction along the s axis on a log-log scale with the increase in initial e. When the gap of initial e values between the predicted SWRCs and reference SWRCs is as small as possible, SWRCs at different initial e can be obtained by a rigid translation of a reference SWRC alone the s axis in the logs-logS
_{r}plane. - (2)
- Similarly, the HMCCs and the RCCs move towards the left-hand direction along the e axis on a log-log scale with the increase in s and S
_{r}, respectively. HMCCs at high suctions cannot be obtained from a rigid translation of a reference HMCC at low suctions. The constant suctions imposed on the HMCCs are suggested to be high suctions when the mnψ = 1 and low suctions when 0 < mnψ < 1 (i.e., m, n, and ψ are the Gallipoli et al. [26] model parameters). - (3)
- The RCC equation proposed is capable of describing the relationship between e and s on the constant S
_{r}plane. The modified RCCs show that the initial suction increases with the reducing constant S_{r}. Moreover, the initial suction is reducing as initial e increases at equal values of S_{r}.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Main drying curves with respective asymptotes (data from Salager et al. [36]): (

**a**) on the plane e (log-log scale), (

**b**) on the plane s (log-log scale).

**Figure 6.**(

**a**) Measured and predicted SWRC for silty sand at the initial void ratio, e

_{0}= 0.86; (

**b**) Influence of reference SWRCs on predicted SWRCs.

**Figure 10.**Linear relationship between in the logs–loge plane at constant suction with data from Table 3: (

**a**) at full suction range; (

**b**) at low suctions.

**Figure 11.**Schematic diagram of the hydro-mechanical coupling curves (HMCCs) with different constant suctions at low suctions and high suctions.

**Figure 14.**Retention consolidation curves at different constant degree of saturation at (data from Salager et al. [36]): (

**a**) initial void ratio e

_{0}= 0.44; (

**b**) different initial void ratios along with modifying lines.

Soil Name | w_{L} (%) | w_{p} (%) | Sand (%) | Silt (%) | Clay (%) | USCS | Reference |
---|---|---|---|---|---|---|---|

Silty sand | 25 | 14.5 | 72 | 18 | 10 | CL | Salager et al. [36] |

Compacted till | 35.5 | 16.8 | 28 | 42 | 30 | CL | Vanapalli et al. [37] |

Ca-Bentonite | 99 | 41 | n/a | n/a | n/a | CH | Sun et al. [38] |

Tailing sand | n/a | n/a | 30.1 | 55.7 | 14.2 | ML | Aubertin et al. [39] |

Sandy loam | n/a | n/a | 54 | 35 | 11 | SM | Laliberte et al. [40] |

Sand-Bentonite | 473.9 | 26.6 | n/a | n/a | n/a | n/a | Sun & Sun [41] |

Expansive Silty-Clay | 50 | 31 | 3 | 48 | 39 | CL | Zhan [42] |

Nonexpansive-Clay | 49 | 22 | 0 | 50 | 50 | CL | Sun et al. [20] |

Soil Type | e_{0} | m_{e} | n_{e} | logα | κ_{e} | m_{e}n_{e} | λ_{e} |
---|---|---|---|---|---|---|---|

Silty sand (Salager et al. [36]) | 0.680 | 0.542 | 0.671 | 1.835 | 1.803 | 0.364 | −0.362 |

0.860 | 0.792 | 0.446 | 1.547 | 1.585 | 0.353 | −0.364 | |

1.010 | 0.710 | 0.488 | 1.217 | 1.326 | 0.347 | −0.357 | |

Compacted till (Vanapalli et al. [37]) | 0.444 | 0.135 | 1.358 | 2.438 | 2.376 | 0.184 | −0.176 |

0.474 | 0.185 | 1.061 | 2.359 | 2.253 | 0.196 | −0.184 | |

0.514 | 0.222 | 0.878 | 2.157 | 2.009 | 0.195 | −0.179 | |

0.517 | 0.137 | 1.161 | 1.824 | 1.824 | 0.159 | −0.158 | |

Ca-Bentonite (Sun et al. [38]) | 0.940 | 0.253 | 1.162 | 3.135 | 3.138 | 0.293 | −0.291 |

1.126 | 0.207 | 1.261 | 2.673 | 2.803 | 0.261 | −0.264 | |

1.765 | 0.213 | 1.271 | 1.969 | 2.040 | 0.271 | −0.276 | |

Tailing sand (Aubertin et al. [39]) | 0.695 | 0.809 | 1.116 | 2.025 | 1.905 | 0.904 | −0.831 |

0.746 | 0.605 | 1.279 | 1.847 | 1.761 | 0.774 | −0.727 | |

0.802 | 0.479 | 1.272 | 1.655 | 1.598 | 0.609 | −0.588 | |

Sandy loam (Laliberte [40]) | 0.845 | 0.115 | 11.417 | 0.754 | 0.749 | 1.317 | −1.284 |

0.984 | 0.064 | 14.239 | 0.596 | 0.596 | 0.913 | −0.913 | |

1.075 | 0.042 | 20.014 | 0.492 | 0.492 | 0.840 | −0.840 | |

1.193 | 0.031 | 30.311 | 0.437 | 0.437 | 0.930 | −0.930 |

Soil Type | s_{con} (kPa) | m_{s} (10^{−2}) | n_{s} | logβ | m_{s}n_{s}ψ_{s} | λ_{s} | κ_{s} |
---|---|---|---|---|---|---|---|

Silty-Sand (Salager et al. [36]) | 1 | 0.358 | 7.456 | −0.224 | 0.173 | −0.173 | −0.224 |

10 | 6.65 | 2.383 | −0.266 | 0.680 | −0.675 | −0.268 | |

100 | 11.7 | 2.729 | −0.394 | 1.028 | −1.027 | −0.395 | |

1000 | 7.06 | 3.654 | −0.621 | 1.001 | −1.001 | −0.621 | |

10,000 | 16.4e | 4.234 | −0.895 | 1.002 | −1.002 | −0.895 | |

100,000 | 33.2 | 1.674 | −1.260 | 1.005 | −1.005 | −1.261 | |

Sand-Bentonite (Sun & Sun [41]) | 300 | 2.68 | 4.446 | −0.595 | 0.620 | −0.620 | −0.595 |

600 | 6.65 | 2.383 | −0.669 | 0.700 | −0.669 | −0.700 | |

1200 | 4.77 | 3.373 | −0.905 | 0.533 | −0.905 | −0.533 | |

1500 | 5.66 | 2.643 | −1.076 | 0.451 | −0.452 | −1.076 | |

Silty-Clay (Zhan [42]) | 25 | 1.22 | 5.491 | −0.240 | 0.749 | −0.749 | −0.240 |

50 | 5.79 | 3.461 | −0.284 | 0.786 | −0.781 | −0.285 | |

100 | 6.62 | 3.491 | −0.339 | 0.676 | −0.668 | −0.341 | |

200 | 7.45 | 1.882 | −0.390 | 0.625 | −0.621 | −0.393 | |

Nonexpansive-Clay (Sun et al. [7]) | 98 | 1.17 | 3.609 | −0.203 | 0.804 | −0.804 | −0.203 |

147(CDG) | 2.88 | 4.737 | −0.314 | 0.606 | −0.606 | −0.314 | |

147(CDE) | 3.42 | 4.716 | −0.223 | 0.848 | −0.848 | −0.223 | |

196 | 2.31 | 2.926 | −0.247 | 0.924 | −0.924 | −0.247 | |

245 | 8.49 | 3.017 | −0.333 | 0.785 | −0.784 | −0.333 |

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Ye, Y.-x.; Zou, W.-l.; Han, Z.
An Insight into the Projection Characteristics of the Soil-Water Retention Surface. *Water* **2018**, *10*, 1717.
https://doi.org/10.3390/w10121717

**AMA Style**

Ye Y-x, Zou W-l, Han Z.
An Insight into the Projection Characteristics of the Soil-Water Retention Surface. *Water*. 2018; 10(12):1717.
https://doi.org/10.3390/w10121717

**Chicago/Turabian Style**

Ye, Yun-xue, Wei-lie Zou, and Zhong Han.
2018. "An Insight into the Projection Characteristics of the Soil-Water Retention Surface" *Water* 10, no. 12: 1717.
https://doi.org/10.3390/w10121717