Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters

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Article Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters

Ji-Peng Wang 1 , Pei-Zhi Zhuang 2,*, Ji-Yuan Luan 1, Tai-Heng Liu 1, Yi-Ran Tan 1 and Jiong Zhang 1,* 1 School of Civil Engineering, Shandong University, Jinan 250061, China 2 School of Civil Engineering, University of Leeds, Leeds LS2 9JT, UK * Correspondence: [email protected] (P.-Z.Z.); [email protected] (J.Z.)

Received: 26 July 2019; Accepted: 28 August 2019; Published: 31 August 2019

Abstract: Estimation of unsaturated hydraulic conductivity could benefit many engineering or research problems such as water flow in the vadose zone, unsaturated seepage and capillary barriers for underground waste isolation. The unsaturated hydraulic conductivity of a soil is related to its saturated hydraulic conductivity value as well as its water retention behaviour. By following the first author’s previous work, the saturated hydraulic conductivity and water retention curve (WRC) of sandy soils can be estimated from their basic gradation parameters. In this paper, we further suggest the applicable range of the estimation method is for soils with d10 > 0.02 mm and Cu < 20, in which d10 is the grain diameter corresponding to 10% passing and Cu is the coefficient of uniformity (Cu = d60/d10). The estimation method is also modified to consider the porosity variation effect. Then the proposed method is applied to predict unsaturated hydraulic conductivity properties of different sandy soils and also compared with laboratory and field test results. The comparison shows that the newly developed estimation method, which predicts the relative permeability of unsaturated sands from basic grain size parameters and porosity, generally has a fair agreement with measured data. It also indicates that the air-entry value is mainly relative to the mean grain size and porosity value change from the intrinsic value. The rate of permeability decline with suction is mainly associated with grain size polydispersity.

Keywords: hydraulic conductivity; unsaturated granular soil; relative permeability; grain size distribution; porosity

1. Introduction Unsaturated hydraulic conductivity (or permeability) is an important parameter for the study of water ﬂow in the vadose zone, unsaturated seepage process, underground waste isolation etc. However, measuring unsaturated hydraulic conductivity values of diﬀerent sediments could be time- and cost-consuming. An estimation method of unsaturated hydraulic conductivity could be useful for early design and research processes. Normally, unsaturated hydraulic conductivity is regarded as a parameter associated with its saturated hydraulic conductivity and the soil water retention curve (WRC, the relationship between suction and degree of saturation) [1–3]. For sandy soils, it is also widely accepted that using grain size distribution parameters can estimate its saturated hydraulic conductivity [4–8] and water retention curve [9–12]. Therefore, the unsaturated conductivity properties of sandy soils may also be primarily predicted from their particle size distributions. The recent work of Wang et al. [8,12] can be extended for estimating unsaturated conductivity properties, for example, the relative permeability. By using the dimensional analysis method and combining with regression analysis on a database, Wang’s model [8] has been developed to predict hydraulic conductivity values of saturated sandy soils

Water 2019, 11, 1826; doi:10.3390/w11091826 www.mdpi.com/journal/water Water 2019, 11, 1826 2 of 16 from grain size parameters. The method shows the best prediction accuracy among the classic methods. Moreover, they have also developed a method to estimate water retention behaviour of sandy soils after van Genuchten’s closed-form equation [12], which is also related to the unsaturated permeability. In this paper, we will clarify the applicable soil type (range of particle size distributions) for Wang’s estimation methods by assuming that the methods are more suitable for sandy soils with unimodal pore-size distributions, which has not been covered by the previous studies. Further verification of Wang’s model [8] for saturated hydraulic conductivity predictions will be carried out on sands beyond the dataset used for the model development. Besides soil gradation parameters, as an important new contribution we will also consider the porosity variation effect on the air-entry value of relative permeability and water retention curve, which will further improve the model accuracy. Then, by embedding the van Genuchten’s closed-form equation, the relative permeability of unsaturated sandy soils can be calculated from d60 (60% passing grain size), coefficient uniformity Cu and porosity φ. Model validations will be carried out based on laboratory and field test results. The effect of key gradation parameters and porosity variations on unsaturated hydraulic conductivity properties will also be discussed.

2. Estimation of Saturated Hydraulic Conductivity

2.1. Estimation Equations Based on Grain Size Parameters To predict the unsaturated hydraulic conductivity of a soil, the hydraulic conductivity at the fully saturated condition is required. Generally, the unsaturated hydraulic conductivity properties of unsaturated soils are predicted or estimated based on their saturated hydraulic conductivity values, such as: Ku = KrK, (1) where Ku is the unsaturated hydraulic conductivity, K is the hydraulic conductivity at the fully saturated condition and Kr is the relative coefficient of permeability, which is a function of suction or degree of saturation (the function should also be associated with the soil pore structures). It is well known that the saturated hydraulic conductivity of sandy soils can be estimated from its grain size parameters. There are several typical and widely used equations, the most well-known equation being the Hazen equation [4], in which the saturated hydraulic conductivity is expressed as a proportional relationship to the squared value of a characteristic particle size (usually d10): g K = C d2 , (2) H v 10 where g is the gravitational acceleration (m/s2) and ν is the fluid kinematic viscosity (m2/s) (ν = 6 2 4 0.89 10 m /s at 25 ◦ C for water). Empirically, CH is a unitless coefficient about 6.54 10 [13]. × − × − Meter can be used as the length unit in this equation to keep the unit consistency (to obtain hydraulic conductivity value in m/s). Furthermore, the effect of particle size uniformity is considered in another equation proposed by Beyer [5], which can be written as: g 500 2 K = CB log d10, (3) v Cu

4 in which the empirical coefficient CB is 6 10− (unitless) and the coefficient of uniformity Cu (unitless) × d60 is the ratio of grain size at 60% passing and grain size at 10% passing (Cu = ). Moreover, the d10 Kozeny–Carman model [6,14–16] is another classical model with the porosity effect embedded:

3 g φ 2 K = CK d10, (4) v (1 φ)2 − Water 2019, 11, 1826 3 of 16

where the empirical coeﬃcient CK is 1/180 determined by ﬂow in capillary tubes or beds of spheres and φ is the porosity. Empirically, Chapuis [7] proposed an equation to estimate saturated hydraulic conductivity based on d10 and void ratio e. In the international unit system (SI), it can be expressed as:

3 !0.7825 6 2 e K = 0.024622 10− d (5) 10 1 e − More recently, Wang et al. [8] analyzed the relationship between saturated hydraulic conductivity and particle size distribution for sandy soils by using dimensional analysis. They found that grain size uniformity coeﬃcient and a dimensionless group term, expressed by gravitational acceleration and a characteristic grain size d60, are the main two determinative parameters for estimating hydraulic conductivity. In Wang’s equation [8], K can be expressed as:

  1   1 gd3 − gd3 − a g 2  60  g 2  60  K = CWCu d log  CW d log  , (6) ν 60 10 ν2  ≈ ν 10 10 ν2 

3 where CW = 2.9 10 and a 2. The above equation has no extra parameters comparing with other × − ≈ − classic methods. Wang et al. [8] have proved that the above equation has higher accuracy especially for soils with hydraulic conductivity values ranging from 2 10 5 m/s to 2 10 3 m/s. From the × − × − dimensional analysis, the above equation already counts as part of the porosity effect as porosity has an intrinsic correlation with particle size uniformity Cu. Following Vukovic and Soro [17], the intrinsic porosity of a sand is a function of its Cu empirically as: Cu φ0 = ω 1 + β , (7) where ω and β are unitless constants. After a database consisting of 431 unlithified sediment samples [18] the two parameters are fitted as ω = 0.2 and β = 0.93 by Wang et al. [12]. However, the model in Equation (7) didn’t consider the porosity variation of a particular sample due to its micro-structure arrangement. Then, a modified model was proposed in which the porosity difference between the current value and its intrinsic value is considered as:

   1   gd3 −   a 60   g 2 K = CWCu log  + b∆φ d , (8)   10 ν2   ν 60 where b is a ﬁtting parameter and ∆φ = φ φ in which φ is the current porosity and φ is the intrinsic − 0 0 porosity calculated by Equation (7).

2.2. Applicability and Validity of the Estimation Equation It has been proved in the literature [8] that the new models in Equations (6) and (8) have better performance than the classic models. The new models in Equations (6) and (8) are developed semi-empirically based on a database of 431 sandy soils [18]. It is required to further clarify the applicability and validity of the new models on more soils beyond the database. As the new models are proposed based on the Roasas database, the validity of the model should be restricted by a certain range of soil types. The key particle size parameters of this database is as the following range: 0.05 mm

Water 2019, 11, x FOR PEER REVIEW 4 of 16 the horizontal axis is the model predicted values and the vertical axis is the measured values. If the datasetdataset is is c closerloser to to the the equality equality line, line, that that means means the the slope slope of of the the regression regression line line is is closer closer to to 1 1 then then the the performanceperformance of of the the model model is is better. better. It It can can be be seen seen that, that, if if the the particle particle size size range range is is not not considered considered in in thethe model model application, application, the the overall overall prediction prediction is is not not ac accuratecurate enough enough for for all all models models (although (although W Wang’sang’s secondsecond model model is is the the best) best)..

(a) (b)

(e) (f)

FigureFigure 1. 1. PredictionPrediction on on saturated saturated hydraulic hydraulic conductivity conductivity of di ﬀ oferent different models models on soils on from soils the UNSODAfrom the UNSODAdatabase. (a database.) Hazen model; (a) Hazen (b) Beyer model; model; (b) Beyer (c) Kozeny–Carman model; (c) Kozeny model;–Carman (d) Chapuis model; model; (d) ( Chapuise) model model;of Equation (e) model (6); ( fof) model Equation of Equation (6); (f) model (8). of Equation (8).

The prediction errors of the different models in Figure 1 could be induced by the fine content in some soils. The introduced estimation models in section 2.1 are mostly based on granular soils without aggregation effect, which may mainly have single peak pore size distributions. However, if

Water 2019, 11, 1826 5 of 16

The prediction errors of the different models in Figure1 could be induced by the fine content in some soils. The introduced estimation models in Section 2.1 are mostly based on granular soils without Water 2019, 11, x FOR PEER REVIEW 5 of 16 aggregation effect, which may mainly have single peak pore size distributions. However, if the sandy thesoils sandy have soils a certain have a amount certain ofamount fine particles of fine likeparticles clay, aggregationlike clay, aggregation and cementation and cementation effect may leadeffect the maysoil lead to a the dual soil pore to structurea dual pore (sketch structure can be(sketch seen incan Figure be seen2). Asin discussedFigure 2). byAs somediscussed authors by [some20,21 ], authorssilty and [20,21 clayey], sil soilsty and may clayey have soils bimodal may shapehave bimodal pore size shape distributions, pore size which distributions, can not be which covered canby not the bemodels covered introduced by the models above. introduced This means above. that theThis models means presented that the models above are presented more suitable above forare granular more suitablesoils or for sands granular with soils unimodal or sands pore with size unimodal distribution. pore Soils size with distribution. other shapes Soils of with pore other size distributionshapes of porecurve size should distribution be excluded curve should when applying be excluded the hydraulicwhen applying conductivity the hydraulic models. conductivity models.

FigureFigure 2. 2.SketchSketch of of the the fine fine contentcontent effeffectect which which may may change change the porethe pore size distributionsize distribution to a dual-structure. to a dual- structure. For soils with very fine particles which could induce dual-structure pore size, the characteristic grainFor sizesoilsd 10withcould very be fine very particles small and which the could coeffi cientinduce of dual uniformity-structureCu poremay size, become the relativelycharacteristic large. grainHere, size we 푑 restrict10 could the be twovery parameters small and the as dcoefficient10 > 0.02 mm of uniformity and Cu < 20, 퐶푢 asmay according become to relatively the particle large. size Here,range we of restrict the Rosas the databasetwo parameters [18]. Soils as are푑10 selected > 0.02mm from and the 퐶 UNSODA푢 < 20, as databaseaccording according to the particle to the size above rangesoil gradationof the Rosas range. database Then, the[18]. model Soils performancesare selected from are comparedthe UNSODA again database based on according the filtered to soils the in aboveFigure soil3. gradation In Figure3 c,d,frange. only Then, soils the with model available performances porosity orare void compared ratio values again are based compared. on the filtered It can be soilsseen in thatFigure for 3. sandy In Figure soils 3(c,d,f) from the only UNSODA soils with database available with porosityd10 > 0.02 or void mm ratio and Cvaluesu < 20 are all compared. models have It muchcan be better seen predictionthat for sandy accuracy. soils from Therefore, the UNSODA to predict database the saturated with hydraulic 푑10 > 0.02mm conductivity and 퐶푢 of < granular 20 all modelssoils based have on much particle better size prediction distribution, accuracy. the diff erent Therefore, estimation to predict methods the may saturated be more hydraulic suitable for conductivitygranular soils of withgranular a unimodal soils based shape on pore particle size distribution.size distribution, To restrict the different the application estimation of the methods models to maysandy be more soils withinsuitable a for particular granular particle soils with size distributiona unimodal shape range (pored10 > size0.02 distribution. mm and Cu To< 20 restrict at the the same applicationtime) is recommended. of the models to sandy soils within a particular particle size distribution range (푑10 > 0.02mm and 퐶푢 < 20 at the same time) is recommended.

(a) (b)

Water 2019, 11, x FOR PEER REVIEW 5 of 16 the sandy soils have a certain amount of fine particles like clay, aggregation and cementation effect may lead the soil to a dual pore structure (sketch can be seen in Figure 2). As discussed by some authors [20,21], silty and clayey soils may have bimodal shape pore size distributions, which can not be covered by the models introduced above. This means that the models presented above are more suitable for granular soils or sands with unimodal pore size distribution. Soils with other shapes of pore size distribution curve should be excluded when applying the hydraulic conductivity models.

Figure 2. Sketch of the fine content effect which may change the pore size distribution to a dual- structure.

For soils with very fine particles which could induce dual-structure pore size, the characteristic grain size 푑10 could be very small and the coefficient of uniformity 퐶푢 may become relatively large. Here, we restrict the two parameters as 푑10 > 0.02mm and 퐶푢 < 20, as according to the particle size range of the Rosas database [18]. Soils are selected from the UNSODA database according to the above soil gradation range. Then, the model performances are compared again based on the filtered soils in Figure 3. In Figure 3(c,d,f) only soils with available porosity or void ratio values are compared.

It can be seen that for sandy soils from the UNSODA database with 푑10 > 0.02mm and 퐶푢 < 20 all models have much better prediction accuracy. Therefore, to predict the saturated hydraulic conductivity of granular soils based on particle size distribution, the different estimation methods may be more suitable for granular soils with a unimodal shape pore size distribution. To restrict the applicationWater 2019, 11 ,of 1826 the models to sandy soils within a particular particle size distribution range (푑610 of 16> 0.02mm and 퐶푢 < 20 at the same time) is recommended.

Water 2019, 11, x FOR PEER REVIEW 6 of 16 (a) (b)

(e) (f)

Figure 3. Prediction of different models on sandy soils with 푑10 > 0.02mm and 퐶푢 < 20 from the Figure 3. Prediction of diﬀerent models on sandy soils with d10 > 0.02 mm and Cu < 20 from the UNSODAUNSODA database. (a)) Hazen Hazen model; model; (b ()b Beyer) Beyer model; model; (c) Kozeny–Carman (c) Kozeny–Carman model; model; (d) Chapuis (d) Chapuis model; model;(e) model (e) model of Equation of Equation (6); (f) model(6); (f) model of Equation of Equation (8). (8).

3. Prediction of Unsaturated Relative Permeability

3.1. Van Genuchten’s Closed-Form Equation

The relative permeability (퐾푟) in Equation (1) is normally regarded as a function of degree of saturation or suction. In the classic Mualem’s equation [22], the relative permeability is expressed as:

S 2 e  −1dS 0 e KSre= 1 (9) −1  dSe 0 , in which 푆푒 is the effective degree of saturation and 훹 is suction. Normally, 푆푒 is determined as 푟푒푠 푟푒푠 푟푒푠 푆푒 = ( 푆푟 − 푆푟 )⁄(1 − 푆푟 ) in which 푆푟 is degree of saturation and 푆푟 is the residual degree of saturation. According to van Genuchten’s closed-form equation [2], the effective degree of saturation has the following relationship with suction 훹 and an air-entry value related parameter 훼:

n m  Se =+1  (10)  , 1 in which n and m are model parameters. For sandy soils, m is suggested to be equal to − 1, therefore: 푛

Water 2019, 11, 1826 7 of 16

3. Prediction of Unsaturated Relative Permeability

3.1. Van Genuchten’s Closed-Form Equation

The relative permeability (Kr) in Equation (1) is normally regarded as a function of degree of saturation or suction. In the classic Mualem’s equation [22], the relative permeability is expressed as:

2 R Se 1  p  ψ− dSe  K = S  0  , (9) r e R 1   1dS  0 ψ− e in which Se is the eﬀective degree of saturation and Ψ is suction. Normally, Se is determined res res res as Se = (Sr S )/(1 S ) in which Sr is degree of saturation and S is the residual degree of − r − r r saturation. According to van Genuchten’s closed-form equation [2], the eﬀective degree of saturation has the following relationship with suction Ψ and an air-entry value related parameter α:

!n!m ψ Se = 1 + , (10) α in which n and m are model parameters. For sandy soils, m is suggested to be equal to 1 1, therefore: n −

!n! 1 1 ψ n − Se = 1 + (11) α

By substituting the above equation into Mualem’s equation (Equation (9)), we can obtain the following equation: " 1 #2 p n 1 n Kr = Se 1 1 Se 1 n − (12) − − − And by implementing the van Genuchten’s closed form equation, the relative permeability can be rewritten as a function of Ψ and α:

1 n  1 2 " !n# −  !n 1" !n# 1 ψ 2n  ψ − ψ n −  K = 1 + 1 1 +  (13) r   α  − α α 

3.2. Prediction of Van Genuchten’s Parameters from Particle Size Distribution In literature, there are a number of mathematical models to predict the water retention behaviour of an unsaturated soil from its particle size distribution, which is usually called pedotransfer functions. One of the most recent models is proposed by [12] using a semi-physical (based on dimensional analysis) and semi-empirical approach after van Genuchten’s closed-form equation. The authors have proved that for sandy soils it generally has better performance than other typical empirical models, especially when the grain size is more uniform. In this model, the following type of equation can be employed to estimate the parameter n in van Genuchten’s equation: C n = 1 + 1, (14) log10 Cu in which Cu is the coeﬃcient of uniformity and C1 is a model constant, which is suggested to be αd60 approximated as 1.07. For the parameter α, it is found that the dimensionless term γ is approximately a constant C2 which means α is inversely proportional to the mean particle size. Then α can be written as: C2γ α = (15) d60 Water 2019, 11, 1826 8 of 16

And C2 is suggested to be around 12.07. By combining the estimation of α and n with the closed-form equation of relative permeability in Equation (13), the hydraulic conductivity properties of unsaturated sandy soils can also be estimated from the gradation parameters.

3.3. Effect of Porosity Variation on the Air Entry Value Referring back to Equation (7), it can be seen that there is an average or intrinsic porosity for a sandy soil which can be estimated from its grain size uniformity coefficient. However, it is just a general form relationship between key grain size distribution and porosity. It does not consider the variation of the pore structures (due to its fabric or stress conditions) which may change the initial porosity of a sandy soil with a particular particle size distribution. Here we propose a new model to include the effect of porosity variation, which has not been considered in the original Wang’s equation. Hu et al. [23] proved that the deformation-induced void ratio change is positively correlated to the logarithm scale difference of mean pore size. Following the same spirit, as the air-entry value is associated with the mean pore size, we may also propose that the porosity change of a granular soil from its intrinsic value (φ φ ) can lead to a logarithm scale difference in air-entry value parameter − 0 α as: (ln α lnα ) (φ φ ), (16) − 0 ∝ − 0 in which α0 represents the air-entry value parameter when φ = φ0. By introducing a parameter ξ to the correlation, it may be expressed as:

ln α lnα = ξ(φ φ ) (17) − 0 − 0 There were 70 soils samples in the original analysis of Wang et al. [12] and porosity values of 18 sands out of the 70 are available (and 17 of the 18 sands have d10 > 0.02 mm and Cu < 20). Table1 demonstrates the gradation parameters and the water retention curve coeﬃcients (best ﬁtted from van Genutchten’s model) of these sands. Figure4 shows the relationship between φ φ and ln α ln α . − 0 − 0 It can be seen that they are generally in a linear relationship and the parameter can be taken as ξ = 4.7 − (used in Section4). The above equation can then be reformatted as:

ξ(φ φ0) α = α0e − , (18) in which e is the natural constant. By employing Equation (15) to estimate α0, it can be written as:

C2γ ξ(φ φ ) α = e − 0 (19) d60

This equation estimates the air-entry value of the water retention behaviour based on not only gradation parameters but also the relative density or porosity variation. Therefore, the relative permeability of a granular soil can be calculated by Equation (13) with parameters of n and α being estimated from its gradation and porosity in Equations (14) and (19) (which also gives the water retention curve). The performance of the corrected estimation equation on the air-entry parameter is demonstrated in Figure5 based on the 18 sands. Figure5a is the prediction performance of the original estimation method in Equation (15) and Figure5b shows the results of the corrected model in Equation (19) which considers the variation of porosity for the same sand. The horizontal axis is the predicted α value by the estimation models and the vertical axis is the measured α based on the experimental values (best ﬁtted by van Genuchten’s model). It can be seen that the R2 value is increased by a ﬁtted ξ = 1.61. − Water 2019, 11, x FOR PEER REVIEW 8 of 16 correlated to the logarithm scale difference of mean pore size. Following the same spirit, as the air- entry value is associated with the mean pore size, we may also propose that the porosity change of a granular soil from its intrinsic value (휙 − 휙0) can lead to a logarithm scale difference in air-entry valueWater parameter2019, 11, 1826 훼 as: 9 of 16

(ln -ln 00)  (  -  ) , (16) Table 1. Soil gradation parameters, porosity values and best ﬁtted water retention curve (WRC) in whichparameters 훼0 represents of α and then for air the-entry 18 sandy value soils. parameter when 휙 = 휙0. By introducing a parameter 휉 to the correlation, it may be expressed as: Fitted Sample d10 d30 d60 Goodness of Fitting lnCu -ln  =φ   -  Parameters ID * (mm) (mm) (mm) 00( ) (17) α (kPa) n SSE RMSE R2 There1011 were0.00946 70 soils 0.10699 samples 0.15511 in the original19.743 analysis 0.43 of Wang 0.43 et al. 2.75 [12] and 0.016 porosity 0.047 values of 0.989 18 sands1014 out of 0.02078the 70 are 0.11454 available 0.16293 (and 179.350 of the 18 0.45sands have 0.94 푑10 > 2.660.02mm 0.007 and 퐶푢 0.027< 20). Table 0.995 1 demonstrates1461 0.21825 the gradation 0.30949 parameters 0.43887 and2.395 the water 0.37 retention 9.47 curve 3.70 coefficients 0.050 (best 0.085 fitted from 0.933 van Genutchten’s1462 0.12691 model) 0.23000 of these 0.30867 sands. F2.818igure 4 shows 0.43 the relationship 5.70 3.43 between 0.037 휙 − 휙 0.068 and ln 0.955훼 − 1463 0.12733 0.23915 0.31552 2.846 0.40 6.21 3.65 0.025 0.0560 0.970 ln 훼0.1464 It can be0.10089 seen that 0.14356 they are 0.20548 generally2.552 in a linear 0.37 relationship 6.15 and 3.13 the parameter 0.049 can 0.078 be taken 0.950 as 휉 = −14654.7 (used0.02491 in Section 0.07375 4). The 0.10463 above equation5.000 can 0.38 then be 2.08 reformatted 1.88 as: 0.005 0.025 0.996 1466 0.05631 0.07855 0.09897 2.034 0.41 5.44 4.56 0.009 0.034 0.993 ( − 0 ) 1467 0.02932 0.20852 0.31649 13.299= 0 0.31e , 2.13 1.58 0.011 0.037 0.989(18) 3330 0.04041 0.20388 0.28925 8.526 0.42 1.62 1.65 0.024 0.069 0.971 in which3331 푒 is0.11858 the natural 0.22780 constant. 0.29709 By employing2.861 Equation 0.44 ( 4.5315) to estimate 2.58 훼0 0.026, it can be 0.072 written 0.975as: 3332 0.20284 0.25656 0.32451 1.799 0.43 7.78 3.48 0.014 0.054 0.987 C  3340 0.12617 0.18315 0.26612 2.5492 0.46( − 0 ) 3.95 2.26 0.086 0.055 0.973  = e (19) 4523 0.12133 0.16532 0.21988 2.106 0.41 8.83 7.04 0.072 0.081 0.969 d60 4650 0.07221 0.23130 0.31953 5.201 0.38 2.20 2.01 0.032 0.037 0.992 T4651his equation0.08383 estimates 0.22687 the 0.32525 air-entry4.646 value of 0.38the water 1.95 retention 2.01 behaviour 0.029 based 0.036 on not only 0.992 gradation4660 parameters0.06469 0.21709 but also 0.30134 the relative5.488 density 0.46 or porosity 0.45 variation. 1.48 Therefore, 0.036 0.039 the relative 0.986 4661 0.07221 0.22944 0.31132 5.022 0.43 0.79 1.74 0.015 0.026 0.995 permeability of a granular soil can be calculated by Equation (13) with parameters of 푛 and 훼 being 2 estimatedSSE: from sum of its square gradation errors. RMSE: and porosity root-mean-square in Equations error. R : ( coe14ﬃ) cientand of(19 determination.) (which also *: numbered gives the IDs water are from the UNSODA database. retention curve).

FigureFigure 4. 4.EffectEﬀect of ofinitial initial porosity porosity on on air air-entry-entry value: value: relationship relationship between between porosity porosity variation variation and and logarithmlogarithm scale scale air air-entry-entry value value difference diﬀerence..

The performance of the corrected estimation equation on the air-entry parameter is demonstrated in Figure 5 based on the 18 sands. Figure 5a is the prediction performance of the original estimation method in Equation (15) and Figure 5b shows the results of the corrected model in Equation (19) which considers the variation of porosity for the same sand. The horizontal axis is the predicted 훼 value by the estimation models and the vertical axis is the measured 훼 based on the

Water 2019, 11, x FOR PEER REVIEW 9 of 16

experimentalWater 2019, 11, 1826 values (best fitted by van Genuchten’s model). It can be seen that the 푅2 value10 of is 16 increased by a fitted 휉 = −1.61.

(a) (b)

FigureFigure 5. 5. PredictionPrediction performance performance of parameter α훼by by the the original original method method and and the the corrected corrected method. method. (a ) (originala) original method method (Equation (Equation (15)); (15) ();b )(b corrected) corrected method method (Equation (Equation (19)). (19)).

4. VerificationTable 1. Soil of gradation the Estimation parameters, Model porosity values and best fitted water retention curve (WRC) parameters of 훼 and 푛 for the 18 sandy soils. 4.1. The Effect of Porosity on Predictions of Water Retention Curve (WRC) and Relative Permeability Sample 풅ퟏퟎ 풅ퟑퟎ 풅ퟔퟎ Fitted Parameters Goodness of Fitting 푪풖 ϕ ID* The(mm)proposed (mm) model (mm) can then be applied to different휶 (kPa) sediments 풏 with variousSSE initialRMSE porosity R2 1011 values0.00946 to have 0.10699 a further verification.0.15511 19.743 In the UNSODA0.43 database,0.43 there2.75 are three0.016 typical sandy0.047 soils0.989 1014 with0.02078 different initial0.11454 porosities.0.16293 They9.350 are the Wagram0.45 sand0.94 (ID: 1140,2.66 1141, 1142),0.007 the Berlin0.027 coarse0.995 1461 sand0.21825 (ID: 1460, 0.30949 1461) and0.43887 the Berlin medium2.395 sand0.37 (ID: 1462,9.47 1463), which3.70 have0.050 not been used0.085 for the0.933 1462 calibration0.12691 process0.23000 in the previous0.30867 section.2.818 The three0.43 typical5.70 sands are also3.43 widely0.037 used in other0.068 studies 0.955 1463 of water0.12733 retention 0.23915 and hydraulic0.31552 conductivity2.846 0.40 behaviours 6.21 in the literature3.65 [230.025,24 ]. Soil0.056 gradation 0.970 1464 parameters0.10089 and0.14356 best-fitted 0.20548 water retention2.552 curve0.37 coe fficients6.15 of these3.13 sandy soils0.049 are summarised0.078 in0.950 1465 0.02491 0.07375 0.10463 5.000 0.38 2.08 1.88 0.005 0.025 0.996 Table2. For each sand, as the soil gradations are similar, we used the average gradation parameters for 1466 0.05631 0.07855 0.09897 2.034 0.41 5.44 4.56 0.009 0.034 0.993 1467 the hydraulic0.02932 property0.20852 estimations0.31649 and13.299 regard porosity0.31 variation2.13 as the1.58 main controlling0.011 0.037 parameter. 0.989 3330 The van0.04041 Genuchten’s 0.20388 model0.28925 parameter 8.526α is estimated0.42 by Equation1.62 (19)1.65 in which both0.024 the soil0.069 gradation 0.971 3331 effect0.11858 and porosity0.22780 variation 0.29709 eff ect are2.861 considered 0.44 (ξ = 4.74.53is used in2.58 Section 4 0.026as it is directly0.072 fitted 0.975 − 3332 from0.20284 the diagram). 0.25656 The van0.32451 Genuchten’s 1.799 model0.43 parameter 7.78n is predicted3.48 by Equation0.014 (14)0.054 which is0.987 3340 only0.12617 related to the0.18315 coe fficient0.26612 of uniformity2.549 Cu. The0.46 Wagram3.95 sand has2.26 only water0.086 retention 0.055 curve data.0.973 4523 Therefore,0.12133 we firstly0.16532compare 0.21988 the measured2.106 and0.41 predicted water8.83 retention7.04 curve0.072 inFigure 60.081 in which 0.969 4650 the points0.07221 are measured0.23130 results0.31953 by experiments5.201 and0.38 the lines2.20 are predicted2.01 by the0.032 model. It can0.037 be seen0.992 4651 0.08383 0.22687 0.32525 4.646 0.38 1.95 2.01 0.029 0.036 0.992 from Figure6 that model predictions agree well with experimental results. With porosity decrease, 4660 0.06469 0.21709 0.30134 5.488 0.46 0.45 1.48 0.036 0.039 0.986 4661 the water0.07221 retention 0.22944 curve 0.31132 is shifted to5.022 the right and0.43 the air-entry0.79 value1.74 becomes 0.015 higher. The0.026 modified 0.995 modelSSE: fairly sum presents of square the errors effect. ofRMSE: porosity root- change.mean-square error. R2: coefficient of determination. *: numbered IDs are from the UNSODA database. Table 2. Soil gradation parameters, porosity values and best fitted WRC parameters (α and n) for the 4. Verificationthree sandy of soils the which Estimation are employed Model for the model verification. Fitted Sample d10 d30 d60 Goodness of Fitting Soil Type * Cu φ Parameters 4.1. The Effect of Porosity IDon **Predictions(mm) of(mm) Water(mm) Retention Curve (WRC) and Relative Permeability α (kPa) n SSE RMSE R2 The proposed model1140 can then be applied to different0.428 sediments 3.752 with 3.657 various 0.011 initial 0.031 porosity 0.995 Wagram sand 11410.051 0.147 0.25 4.9 0.336 4.318 3.340 0.024 0.044 0.989 values to have a further1142 verification. In the UNSODA database, 0.272 4.889there are 2.881 three 0.023 typical 0.043 sandy 0.989 soils 1460 0.297 5.510 8.236 3.950 0.703 0.444 with differentBerlin coarse initial sand porosities. 0.217They are 0.308 the 0.522Wagram2.4 sand (ID: 1140, 1141, 1142), the Berlin coarse sand (ID: 1460, 1461) and1461 the Berlin medium sand (ID: 1462, 0.373 1463), 3.123 which 3.702 have 0.050not been 0.085 used 0.933 for the Berlin medium 1462 0.43 3.233 3.424 0.037 0.068 0.955 0.127 0.235 0.360 2.8 calibration sandprocess in the1463 previous section. The three typical 0.399 sands 3.514 are 3.654 also widely 0.025 0.056used in 0.970 other studiesSSE: of sum water of square retenti errors.on RMSE: and hydraulic root-mean-square conductivity error. R2: behaviours coefficient of determination. in the literature *: Soil gradation[23,24]. Soil gradationparameters parameters are average and values best for-fitted each soil. water **: numbered retention IDs are curve from the coefficients UNSODAdatabase. of these sandy soils are

Water 2019, 11, x FOR PEER REVIEW 10 of 16

summarised in Table 2. For each sand, as the soil gradations are similar, we used the average gradation parameters for the hydraulic property estimations and regard porosity variation as the main controlling parameter. The van Genuchten’s model parameter 훼 is estimated by Equation (19) in which both the soil gradation effect and porosity variation effect are considered (휉 = −4.7 is used in Section 4 as it is directly fitted from the diagram). The van Genuchten’s model parameter 푛 is

predicted by Equation (14) which is only related to the coefficient of uniformity 퐶푢. The Wagram sand has only water retention curve data. Therefore, we firstly compare the measured and predicted water retention curve in Figure 6 in which the points are measured results by experiments and the lines are predicted by the model. It can be seen from Figure 6 that model predictions agree well with experimental results. With porosity decrease, the water retention curve is shifted to the right and the air-entry value becomes higher. The modified model fairly presents the effect of porosity change.

Table 2. Soil gradation parameters, porosity values and best fitted WRC parameters (훼 and 푛) for the three sandy soils which are employed for the model verification.

Soil Sample 풅ퟏퟎ 풅ퟑퟎ 풅ퟔퟎ Fitted Parameters Goodness of Fitting 푪풖 ϕ Type* ID** (mm) (mm) (mm) 휶 (kPa) 풏 SSE RMSE R2 1140 0.428 3.752 3.657 0.011 0.031 0.995 Wagram 1141 0.051 0.147 0.25 4.9 0.336 4.318 3.340 0.024 0.044 0.989 sand 1142 0.272 4.889 2.881 0.023 0.043 0.989 Berlin 1460 0.297 5.510 8.236 3.950 0.703 0.444 coarse 0.217 0.308 0.522 2.4 1461 0.373 3.123 3.702 0.050 0.085 0.933 sand Berlin 1462 0.43 3.233 3.424 0.037 0.068 0.955 medium 0.127 0.235 0.360 2.8 1463 0.399 3.514 3.654 0.025 0.056 0.970 sand SSE: sum of square errors. RMSE: root-mean-square error. R2: coefficient of determination. *: Soil

Watergradation2019, 11, 1826 parameters are average values for each soil. **: numbered IDs are from the UNSODA11 of 16 database.

decrease φ

FigureFigure 6. 6. MeasuredMeasured and and predicted predicted water water retention retention behaviour behaviour of of the the Wagram Wagram sand. sand.

Furthermore,Furthermore, both both of of the the water water retention retention curve curve prediction prediction model model (Equation (Equationss ( (11),11), ( (14)14) and and ( (19))19)) andand the the relative relative permeability permeability prediction prediction model model (Equation (Equationss ( (13),13), ( (14)14) and and ( (19))19)) are are applied applied to to Berlin Berlin coarsecoarse sands sands and and Berlin Berlin medium medium sand sand as these as sandsthese have sands both have water both retention water curve retention and unsaturated curve and unsaturatedhydraulic conductivity hydraulic conductivity data. The model data. performance The model performance is demonstrated is demonstrated in Figure7. Figure in Figure7a compares 7. Figure 7athe compares estimated the water estimated retention water behaviour retention and behaviour measured and results measured of Berlin results coarse of sands. Berlin It coarse can be sand seens. that It for the sample with φ = 0.373 the model fits the experimental measurements in the drying path and for the sample with φ = 0.297 the model catches the basic trend. Figure7b presents model predictions of relative permeability of Berlin coarse sands. The predicted curve coincides with the laboratory data and it shows that a lower porosity leads to a higher air-entry value. It also indicates that the estimation methods normally have higher accuracy when the suction is relatively low (with a high degree of saturation). As introduced by Wang et al. [25], when the degree of saturation is lower, the morphology of liquid-air interfaces could be much more complex and more related to grain shape parameters besides grain size distribution. This partially explains the error when suction is relatively higher. The model estimations of water retention curves and relative permeability for Berlin medium sand are presented in Figure7c–d. The estimation models have good agreement with the experimental results as the measured and predicted air-entry values are similar. It also indicates that the model prediction performance for medium sands in Figure7c–d is better (as for Berlin coarse sand with φ = 0.297 there is an over-estimation of parameter n). Comparisons between measured (best fitted) model parameters of α and n and predicted values by Equations (14) and (19) are also depicted in Figure8. It can be seen that the predictions normally have good agreement with measured values except the Berlin coarse sand with φ = 0.297 (ID 1460) in which α and n are underestimated. This could because of the experiment error or inclusion of some fine contents in the sample. Water 2019, 11, x FOR PEER REVIEW 11 of 16

can be seen that for the sample with 휙 = 0.373 the model fits the experimental measurements in the drying path and for the sample with 휙 = 0.297 the model catches the basic trend. Figure 7b presents model predictions of relative permeability of Berlin coarse sands. The predicted curve coincides with the laboratory data and it shows that a lower porosity leads to a higher air-entry value. It also indicates that the estimation methods normally have higher accuracy when the suction is relatively low (with a high degree of saturation). As introduced by Wang et al. [25], when the degree of saturation is lower, the morphology of liquid-air interfaces could be much more complex and more related to grain shape parameters besides grain size distribution. This partially explains the error when suction is relatively higher. The model estimations of water retention curves and relative permeability for Berlin medium sand are presented in Figure 7c–d. The estimation models have good agreement with the experimental results as the measured and predicted air-entry values are similar. It also indicates that the model prediction performance for medium sands in Figure 7c–d is better (as for Berlin coarse sand with 휙 = 0.297 there is an over-estimation of parameter 푛). Comparisons between measured (best fitted) model parameters of 훼 and 푛 and predicted values by Equations (14) and (19) are also depicted in Figure 8. It can be seen that the predictions normally have good agreement with measured values except the Berlin coarse sand with 휙 = 0.297 (ID 1460) in which 훼 and 푛 are underestimated. This could because of the experiment error or inclusion of some fine Water 2019, 11, 1826 12 of 16 contents in the sample.

(a) (b)

FigureFigure 7. 7.Prediction Prediction performanceperformance of coarse and medium Berlin Berlin sands. sands. (a ()a )water water retention retention curve curve of of Berlin coarse sand; (b) relative permeability of Berlin coarse sand; (c) water retention curve of Berlin Berlin coarse sand; (b) relative permeability of Berlin coarse sand; (c) water retention curve of Berlin medium sand; (d) relative permeability of Berlin medium sand. Watermedium 2019, 11 sand;, x FOR ( dPEER) relative REVIEW permeability of Berlin medium sand. 12 of 16

(a) (b)

FigureFigure 8. 8.Comparison Comparison between between predicted predicted andand measuredmeasured parameters for for Berlin Berlin sands. sands. (a ()a parameter) parameter α; 훼; (b) parameter 푛. (b) parameter n.

4.2.4.2. Veriﬁcation Verificatio onn on a a Set Set of of Field FieldTest Test DataData byby InstantaneousInstantaneous P Proﬁlerofile M Methodethod TheThe model model constants constants in in the the estimationestimation model are are determined determined based based on onthe the database database (the (the 18 sandy 18 sandy soilssoils in Table in Table1). The 1). The applicability applicability of the of proposed the proposed method method beyond beyond the database the database should should also also be proved. be proved. Here, we extend the application of the proposed estimation equations to other experimental Here, we extend the application of the proposed estimation equations to other experimental data. data. In the UNSODA database, the field measured unsaturated hydraulic conductivity values of some sandy soils are available and the data have not been used for model calibration in Section 3. Among the different field measurements, the instantaneous profile method [26] is the most widely employed one which can be applied both in situ and in the laboratory [27]. Therefore, we carried out further model validation on these sandy soils. As we suggested in Section 2, the estimation model is

more suitable for sandy soils within the range of 푑10 > 0.02mm and 퐶푢 < 20 and it does not consider the hydraulic hysteresis effect. Therefore, sediments which have been measured by the instantaneous profile method in the drying path within the above grain size distribution range should be chosen for this verification. After checking throughout the UNSODA database, sandy soils with ID numbers 1014, 1023, 1024, 1241, 2105, 3134, 3162, 3163 and 3164 are eligible for these conditions. Soil gradation parameters of these sandy soils are summarised in Table 3.

Table 3. Soil gradation parameters of sandy soils tested by the instantaneous profile method.

Sample 풅 풅 풅 풅 풅 ퟏퟎ ퟑퟎ ퟓퟎ ퟔퟎ ퟗퟎ 푪 ID (mm) (mm) (mm) (mm) (mm) 풖 1014 0.021 0.115 0.163 0.194 0.469 9.35 1023 0.125 0.555 0.713 0.808 1.473 6.48 1024 0.115 0.515 0.665 0.755 1.347 6.59 1241 0.237 0.415 0.598 0.689 1.252 2.90 2105 0.022 0.106 0.161 0.198 0.430 8.83 3134 0.107 0.163 0.250 0.289 0.444 2.70 3162 0.031 0.085 0.132 0.160 0.358 5.11 3163 0.051 0.087 0.129 0.154 0.289 2.99 3164 0.052 0.091 0.135 0.161 0.338 3.09

Water 2019, 11, 1826 13 of 16

In the UNSODA database, the field measured unsaturated hydraulic conductivity values of some sandy soils are available and the data have not been used for model calibration in Section3. Among the different field measurements, the instantaneous profile method [26] is the most widely employed one which can be applied both in situ and in the laboratory [27]. Therefore, we carried out further model validation on these sandy soils. As we suggested in Section2, the estimation model is more suitable for sandy soils within the range of d10 > 0.02 mm and Cu < 20 and it does not consider the hydraulic hysteresis effect. Therefore, sediments which have been measured by the instantaneous profile method in the drying path within the above grain size distribution range should be chosen for this verification. After checking throughout the UNSODA database, sandy soils with ID numbers 1014, 1023, 1024, 1241, 2105, 3134, 3162, 3163 and 3164 are eligible for these conditions. Soil gradation parameters of these sandy soils are summarised in Table3.

Table 3. Soil gradation parameters of sandy soils tested by the instantaneous proﬁle method.

d d d d d Sample ID 10 30 50 60 90 C (mm) (mm) (mm) (mm) (mm) u 1014 0.021 0.115 0.163 0.194 0.469 9.35 1023 0.125 0.555 0.713 0.808 1.473 6.48 1024 0.115 0.515 0.665 0.755 1.347 6.59 1241 0.237 0.415 0.598 0.689 1.252 2.90 2105 0.022 0.106 0.161 0.198 0.430 8.83 3134 0.107 0.163 0.250 0.289 0.444 2.70 3162 0.031 0.085 0.132 0.160 0.358 5.11 3163 0.051 0.087 0.129 0.154 0.289 2.99 3164 0.052 0.091 0.135 0.161 0.338 3.09

Figure9 depicts the original and corrected model predictions for the sand with ID number 1014. The experimental results are also demonstrated as circles for comparison. In the original estimation method, the air-entry value parameter is estimated based on its grain size distribution (the intrinsic porosity is 0.302 by Equation (7)). However, the measured porosity of this sand is 0.45. It can be seen that without considering the porosity variation, the original uncorrected model (Equation (15)) has an overestimation of the air-entry value which leads to a higher relative permeability. However, by applying the corrected model, the model performance is significantly enhanced as the prediction line becomes much closer to the measured data points (the parameter α is reduced from 2.53 kPa to 1.76 kPa). Comparisons between the corrected model prediction and measured relative permeability by the instantaneous profile method for other sands in Table3 are presented in Figure 10. There are two sediments in each subfigure. Figure 10a shows experimental measurements of sediments 1023 and 1024. These two sediments have similar grain size distribution parameters and porosity values. Therefore, their relative permeability curves are closed to each other. It can be seen that the proposed model fairly coincides with the measurements. The slope of the relative permeability curves are almost parallel with the measured curve and the predicted air-entry parameter α is around 1kPa and the measured value is between 1 kPa and 2 kPa. In Figure 10b,c, results of 1241, 2105 and 3134, 3162 are presented respectively. It can be seen that in general model predictions match the experimental results. The air-entry parameters are basically similar to the measured data. The slopes of the predited relative permeability curves are also agrees the measurement except the number 2105 sand which under-estimates parameter n (n 3 in the experiment and n = 2.13 in the prediction). The comparisons ≈ also indicate that a higher value of Cu, which means a wider particle size distribution, will lead the sample to have a higher relative permeability at the same suction. The relative permeability decrease rate with suction is lower with larger Cu. Similar to Figure 10a, the two sands in Figure 10d, with ID numbers 3163 and 3164, have similar particle size distributions. Therefore, they have similar relative permeability curves and the model predictions again fit the experiments well. The comparisons Water 2019, 11, 1826 14 of 16 Water 2019, 11, x FOR PEER REVIEW 13 of 16 in Figuresapplying9 and the corrected10 prove model, that the the proposed model performance estimation is significantly method has enhanced good applicability as the prediction to di linefferent experimentalbecomes much measurements. closer to the measured data points (the parameter α is reduced from 2.53 kPa to 1.76 kPa).

Figure 9. Comparison of model performance between using the original estimation of 훼 (Equation Figure 9. Comparison of model performance between using the original estimation of α (Equation (15)) (15)) and using the corrected estimation of 훼 (Equation (19)) (experimental measurements are based and using the corrected estimation of α (Equation (19)) (experimental measurements are based on the on the instantaneous profile method). Waterinstantaneous 2019, 11, x FOR proﬁle PEER method).REVIEW 14 of 16 Comparisons between the corrected model prediction and measured relative permeability by the instantaneous profile method for other sands in Table 3 are presented in Figure 10. There are two sediments in each subfigure. Figure 10a shows experimental measurements of sediments 1023 and 1024. These two sediments have similar grain size distribution parameters and porosity values. Therefore, their relative permeability curves are closed to each other. It can be seen that the proposed model fairly coincides with the measurements. The slope of the relative permeability curves are almost parallel with the measured curve and the predicted air-entry parameter α is around 1kPa and the measured value is between 1 kPa and 2 kPa. In Figure 10b,c, results of 1241, 2105 and 3134, 3162 are presented respectively. It can be seen that in general model predictions match the experimental results. The air-entry parameters are basically similar to the measured data. The slopes of the predited relative permeability curves are also agrees the measurement except the number 2105 sand which under-estimates parameter 푛 ( 푛 ≈ 3 in the experiment and 푛 = 2.13 in the prediction). The

comparisons also indicate that a higher value of 퐶푢, which means a wider particle size distribution, will lead the sample to(a ) have a higher relative permeability at the same(b) suction. The relative

permeability decrease rate with suction is lower with larger 퐶푢. Similar to Figure 10a, the two sands in Figure 10d, with ID numbers 3163 and 3164, have similar particle size distributions. Therefore, they have similar relative permeability curves and the model predictions again fit the experiments well. The comparisons in Figure 9 and Figure 10 prove that the proposed estimation method has good applicability to different experimental measurements.

FigureFigure 10. 10.Comparisons Comparisons between between the the corrected corrected modelmodel predictions and and instantaneous instantaneous profile proﬁle method method measuredmeasured results results of relativeof relative permeability permeability of of sandssands in the UNSODA UNSODA database database.. (a) ( asands) sands 1023 1023 and and 1024; 1024; (b) sands 1241 and 2105; (c) sands 3134 and 3162; (d) sands 3163 and 3164. (b) sands 1241 and 2105; (c) sands 3134 and 3162; (d) sands 3163 and 3164. 5. Conclusions Based on the estimation method proposed by Wang et al. [8], the saturated hydraulic conductivity of sandy soils can be estimated from basic soil gradation parameters. Further verification of this model has been carried out in this paper on more sandy soils. It shows that the model has better performance than the classic models [4,5,7,14,16]. Further discussions also imply that the model is only suitable for sandy soils with unimodal pore-size distributions. We suggest that

the grain size distribution should satisfy 푑10 > 0.02mm and 퐶푢 < 20 at the same time when applying the estimation method in [8]. Furthermore, in this study, an estimation model to predict relative

permeability of unsaturated sandy soils from basic soil gradation parameters (푑10 and 퐶푢) with the variation of initial porosity also being considered. In this model, the slope of the relative permeability

curve is associated with 퐶푢 and the air-entry value is associated with 푑60 and soil porosity. Verification of the proposed relative permeability estimation method is carried out on different sands with relative permeability values measured in the laboratory and in the field. This indicates the proposed method has a fair performance, which can be employed as a primary estimation method in future studies and applications dealing with permeability properties of unsaturated sands.

Author Contributions: Conceptualization, J.-P. Wang; Methodology, J.-P. Wang, P.-Z. Zhuang; Validation, J.-Y. Luan; Formal Analysis, T.-H. Liu, Y.-R. Tan; Investigation, J.-Y. Luan, T.-H. Liu, Writing—Original Draft Preparation, J.-P. Wang; Writing—Review and Editing, P.-Z. Zhuang; Supervision, J.-P. Wang, J. Zhang; Funding, J.-P. Wang, J. Zhang.

Water 2019, 11, 1826 15 of 16

5. Conclusions Based on the estimation method proposed by Wang et al. [8], the saturated hydraulic conductivity of sandy soils can be estimated from basic soil gradation parameters. Further verification of this model has been carried out in this paper on more sandy soils. It shows that the model has better performance than the classic models [4,5,7,14,16]. Further discussions also imply that the model is only suitable for sandy soils with unimodal pore-size distributions. We suggest that the grain size distribution should satisfy d10 > 0.02 mm and Cu < 20 at the same time when applying the estimation method in [8]. Furthermore, in this study, an estimation model to predict relative permeability of unsaturated sandy soils from basic soil gradation parameters (d10 and Cu) with the variation of initial porosity also being considered. In this model, the slope of the relative permeability curve is associated with Cu and the air-entry value is associated with d60 and soil porosity. Verification of the proposed relative permeability estimation method is carried out on different sands with relative permeability values measured in the laboratory and in the field. This indicates the proposed method has a fair performance, which can be employed as a primary estimation method in future studies and applications dealing with permeability properties of unsaturated sands.

Author Contributions: Conceptualization, J.-P.W.; Methodology, J.-P.W., P.-Z.Z.; Validation, J.-Y.L.; Formal Analysis, T.-H.L., Y.-R.T.; Investigation, J.-Y.L., T.-H.L., Writing—Original Draft Preparation, J.-P.W.; Writing—Review and Editing, P.-Z.Z.; Supervision, J.-P.W., J.Z.; Funding, J.-P.W., J.Z. Funding: National Natural Science Foundation of China (Grant no. 51909139), Taishan Scholar Program of Shandong Province, China (Award no. tsqn201812009) and Shandong Provincial Natural Science Foundation, China (Nos. 2017GSF22101 and ZR2018MEE046). Acknowledgments: The first author acknowledge the Taishan Scholar Program of Shandong Province, China and Qilu Young Scholar Program of Shandong University. Conflicts of Interest: The authors declare no conflict of interest.

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