draft · 114 dimensional analysis of pwp response in a vegetated infinite slope 115 review of an...

Draft

Dimensional analysis of pore-water pressure response in a vegetated infinite slope

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2018-0272.R2

Manuscript Type: Article

Date Submitted by the Author: 29-Sep-2018

Complete List of Authors: Feng, Song; Fuzhou University, College of Civil Engineering; Hong Kong University of Science and Technology, Department of Civil and Environmental EngineeringLiu, Hong Wei; Fuzhou University, College of Environment and Resources; Hong Kong University of Science and Technology, Department of Civil and Environmental EngineeringNg, C.W.W.; the Hong Kong University of Science and Technology, Civil and environmental department

Keyword: Dimensional analysis; dimensionless number; pore-water pressure; vegetation; infinite slope

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Issue? :Not applicable (regular submission)

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Canadian Geotechnical Journal

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1

1

2 Dimensional analysis of pore-water pressure response in a vegetated

3 infinite slope

4 S. Feng, H.W. Liu* and C. W. W. Ng

5 Name: Dr Song Feng

6 Title: Lecturer

7 Affiliation: College of Civil Engineering, Fuzhou University, China; Formerly, Postdoctoral 8 Fellow, Department of Civil and Environmental Engineering, Hong Kong University of Science 9 and Technology, Hong Kong, China

10 Address: College of Civil Engineering, Fuzhou University, China

11 E-mail: [email protected]

12

13 Name: Dr. Hong Wei Liu* (Corresponding author)

14 Title: Lecturer

15 Affiliation: College of Environment and Resources, Fuzhou University, China; Formerly, 16 Doctoral Research Student, Department of Civil and Environmental Engineering, Hong Kong 17 University of Science and Technology, Hong Kong, China

18 Address: College of Environment and Resources, Fuzhou University, Fuzhou City, Fujian 19 Province, China


21 Tel: +86 159 2296 6328

22

23 Name: Dr Charles Wang Wai Ng

24 Title: Chair Professor of Civil and Environmental Engineering

25 Affiliation: Department of Civil and Environmental Engineering, Hong Kong University of 26 Science and Technology

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mailto:[email protected]

mailto:[email protected]

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27 Address: Department of Civil and Environmental Engineering, Hong Kong University of 28 Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China


30

31 Abstract

32 Pore-water pressure (PWP) induced by root water uptake has been usually investigated by

33 individual physical quantities. Limited dimensional analysis has been available for investigating

34 PWP response in vegetated slope. In this study, dimensional analysis was conducted to explore

35 dimensionless numbers controlling PWP distributions in vegetated slope. Three dimensionless

36 numbers governing unsaturated seepage were proposed, including capillary effect number (CN;

37 describing the relative importance of water flow driven by PWP gradient over that driven by

38 gravity), root water uptake number (RN; representing effects of root water uptake) and water

39 transfer-storage ratio (WR; ratio of water transfer to water storage rate). Dimensionless

40 relationships were further proposed to estimate PWP and root influence zone in vegetated slope.

41 Then analytical parametric studies were conducted to study effects of RN, CN and WR on PWP

42 distributions. Thereafter, the proposed relationships were validated by published field and

43 centrifuge tests. During drying period, effects of root water uptake on PWP and root influence

44 zone become more significant as CN decreases or RN increases. During wetting period, the larger

45 the WR, the deeper the wetting front moves and more reduction of negative PWP occurs. The

46 proposed dimensionless relationships can determine PWP and root influence zone in vegetated

47 soil reasonably.

48

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49 Key words: Dimensional analysis; dimensionless number; pore-water pressure; vegetation;

50 infinite slope

51

52

53 Introduction

54 Root water uptake affects soil hydrology and stability of civil engineering infrastructures,

55 including slopes, landfill covers and embankments (Liu et al. 2016; Smethurst et al. 2006, 2015;

56 Sinnathamby et al. 2013). Root water uptake reduces pore-water pressure (PWP) inside and outside

57 root zone, and its influence zone could be up to about five times of the root depth (Ni et al. 2018).

58 The reduction in PWP by root water uptake results in lower water permeability but higher shear

59 strength (Simon and Collison 2002; Gonzalez-Ollauri and Mickovski 2014; Indraratna et al. 2006).

60 As a result, rainfall infiltration would be reduced, while shallow slope stability (i.e. up to depth of

61 2 m) could be enhanced (Simon and Collison 2002; Lu and Godt 2013; Fell et al. 2005).

62

63 Although the influence of hydraulic parameters of soil on water infiltration and PWP distributions

64 in bare ground/slope have been widely studied (Green and Ampt 1911; Zhan and Ng 2004; Kasim

65 et al. 1998), the controlling parameters of PWP distributions in vegetated slope are not fully

66 understood. Zhan and Ng (2004) carried out analytical analysis and found that PWP distributions

67 are mainly governed by both and , rather than the individual value of each parameter / sq k /sk

68 (where q is rainfall intensity, is desaturation coefficient describing the rate of decrease in water

69 content as negative PWP (i.e., matric suction) increases, ks is saturated permeability). However,

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70 root water uptake was ignored in their study. Numerical simulations were conducted to investigate

71 variations in PWP distributions due to root water uptake in a vegetated slope by Nyambayo and

72 Potts (2010), who found that the maximum negative PWP decreases as root depth increases for a

73 given amount of root water uptake. Fatahi et al. (2009) conducted comprehensive numerical

74 parametric studies of the individual parameters affecting PWP distributions induced by root water

75 uptake, including transpiration rate, soil saturated permeability, etc. Nevertheless, only flat

76 vegetated ground was considered (i.e., ignoring the effect of the slope angle). More recently, Ng

77 et al. (2015) derived analytical solutions for calculating PWP in an infinite vegetated slope. All

78 the above mentioned valuable work only partially investigated the individual physical quantities

79 affecting PWP response in the vegetated soil, while rare study has been conducted on the combined

80 effects of relevant parameters. On one hand, the combined parameters could help reduce the groups

81 of independent variables, the number of which is frequently large due to the highly nonlinearity

82 and variability of hydraulic properties of soil and root. On the other hand, the combined parameters

83 in terms of dimensionless number could be useful for designing physical model (e.g., centrifuge

84 test), as they are not affected by the specific value of individual physical quantity and are scale-

85 independent. Dimensional analysis provides an effective method to investigate the combined

86 effects of different parameters by developing dimensionless numbers, which could help to reduce

87 the number of independent variables and to design dimensionally valid models of many kinds

88 (Butterfield 1999). However, very few dimensional analysis and dimensionless numbers have been

89 available for investigating PWP response induced by root water uptake in vegetated slope.

90

91 Controlling parameters are usually explored either by using analytical solutions or by conducting

92 numerical simulations. Since the governing equation of unsaturated seepage is highly non-linear,

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93 analytical solutions of PWP distributions can only be derived under certain assumptions and

94 specified boundary and initial conditions (Philip 1957; Srivastava and Yeh 1991; Zhan et al. 2013).

95 However, analytical solutions have unique advantages over numerical simulations. For example,

96 analytical solutions can provide insights on selection of controlling parameters of PWP

97 distributions. This is particularly useful when combined with dimensional analysis, due to the

98 explicit relationships among different parameters provided by analytical solutions. Analytical

99 solutions have been derived to investigate PWP distributions induced by root water uptake (Raats

100 1974; Yuan and Lu 2005; Ng et al. 2015). In this study, the analytical solutions derived by Ng et

101 al. (2015) were selected for dimensional analysis. Note that the solutions consider almost all the

102 key parameters affecting PWP distributions in vegetated slope, including root depth,

103 evapotranspiration rate, soil hydraulic properties (e.g., unsaturated water permeability), and

104 antecedent and subsequent rainfall/evaporation, etc.

105

106 This study aims to conduct dimensional analysis to investigate controlling parameters of PWP

107 response in vegetated slope. Dimensionless numbers governing PWP response in vegetated slope

108 were proposed. Then dimensionless relationships were proposed for estimating PWP and root

109 influence zone in vegetated slope based on the proposed dimensionless numbers. Thereafter,

110 parametric studies were performed to investigate the influences of these dimensionless numbers

111 on PWP distributions in vegetated slope. Finally, the proposed dimensionless relationships were

112 verified by field and centrifuge tests.

113

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114 Dimensional analysis of PWP response in a vegetated infinite slope

115 Review of an analytical model for PWP response in a vegetated infinite slope

116 The analytical model developed by Ng et al. (2015) was adopted and is briefly introduced. A

117 vegetated infinite slope of angle is shown in Fig.1. Infiltration/transpiration is considered at the

118 slope surface, while the water table is located at the bottom of the slope and is assumed to be fixed

119 in position. During rainfall, the model ignores the transpiration rate due to high relative humidity.

120 Moreover, surface runoff is ignored, i.e., the applied infiltration should be calculated as the

121 difference between rainfall and surface runoff. During drying, the applied transpiration rate should

122 be calculated/measured separately. The unsaturated seepage could be simplified as a one-

123 dimensional flow perpendicular to the slope surface by assuming that iso-bars are parallel to the

124 slope surface. Thus, the governing equation of unsaturated seepage in this vegetated infinite slope

125 could be expressed as follows:

126 (1)2

' ' '1'2 '

( )cos (z ) H(z ) s rp

s

k k kT g Lz z k t

127 where k represents the water permeability of soil (m s-1); is the perpendicular distance to the 'z

128 slope bottom (m; see Fig. 1); is the desaturation coefficient of soil (m-1; details are shown later

129 in Eq.(4)); is the slope angle (degree); represents the transpiration rate (m s-1); is the pT '1L

130 thickness of the domain outside root zone (m; Fig. 1); and are the residual and saturated r s

131 volumetric water contents (dimensionless), respectively; represents the saturated water sk

132 permeability of soil (m s-1); is time (s); and (dimensionless) is expressed as followst ' '1H(z )L

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133 (2) ' ' ' '1 1 2' '

1 ' '1

1 ( )H( )

0 0

L z L Lz L

z L

134 where is the thickness of root zone (m; Fig. 1).'2L

135

136 describes the effect of root architecture (m-1). In this study, the uniform root architecture is '(z )g

137 considered due to its simplicity and is defined as follows:

138 (3)''

2

1(z )gL

139 In practice, the uniform root can be found for a grass species named vetiver, which is frequently

140 used in most parts of Asia (e.g., China, Thailand and Malaysia) for ecological restoration and

141 enhancement of man-made slope stability (Liu 2017). It is noted that the PWP distribution is

142 affected by different root architectures (Ng et al. 2015), including an exponential root distribution

143 (where the root density decreases exponentially with depth) and a triangular distribution (a linear

144 decrease with depth). According to the analytical studies conducted by Ng et al. (2015), the PWP

145 distributions between exponential and triangular root distributions are similar, both induce larger

146 negative PWP by about 16 kPa than the uniform root for a given transpiration rate (i.e., 6.6 mm/day)

147 during drying period. Due to the assumption of uniform root architecture and hence constant

148 transpiration rate with depth, effects of root architecture can not be considered for this study.

149

150 The water permeability function and soil water retention curve (SWRC) are described respectively,

151 as follows (Gardner 1958):

inside root zone

outside root zone

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152 (4) expsk k

153 (5) ( ) expw r s r

154 where is PWP head (m). It is noted that when the soil is near saturated, Eqs. (4) and (5) may not

155 be very applicable, due to the two equations usually underestimate volumetric water content and

156 unsaturated water permeability, respectively (Zhan et al. 2013). However, they have been adopted

157 for deriving analytical solutions for unsaturated seepage analysis due to mathematical simplicity

158 (Gardner 1958; Srivastave and Yeh 1991; Yuan and Lu 2005; Zhan et al. 2013). In addition, the

159 hysteresis effect during wetting and drying cycles is ignored, thus a constant is adopted for both

160 drying and wetting. Moreover, it has been recognized that the soil hydraulic properties could be

161 altered by the presence of roots in the soil (Glendinning et al. 2009; Li and Ghodrati 1994;

162 Khalilzad et al. 2014; Jotisankasa and Sirirattanachat 2017; Vergani and Graf 2016). However,

163 there are no conclusive findings, since some research reported a reduction in the saturated

164 permeability ( ), while some showed the opposite. According to numerical simulations of Ni et sk

165 al. (2018), the root-induced reduction in could help preserve suction in the vegetated slope after sk

166 rainfall, while a larger results in a lower suction due to more rainfall infiltration. Due to the sk

167 assumption of negligible effect of root on soil hydraulic properties, the variation in the PWP

168 distribution caused by root-induced alteration of soil hydraulic properties can not be considered

169 by this study. For the above-mentioned analytical model (Eqs. (1)-(5)), Ng et al. (2015) validated

170 the derived analytical solutions against numerical simulations, which were conducted by a

171 commercial finite element software named COMSOL (COMSOL 4.3b 2013).

172

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173 Development of dimensionless numbers for PWP distributions in a vegetated infinite

174 slope

175 In order to develop dimensionless numbers, the following dimensionless variables are defined:

176 (6)'

'

zzL

177 (7)rs

kkk

178 (8)'2'

LrL

179 where is the thickness of the slope (i.e., ; see Fig. 1; unit: m); is the normalized 'L ' ' '1 2L L L z

180 by (dimensionless); is the relative permeability (dimensionless); and r is the root depth 'z 'L rk

181 ratio (dimensionless).

182

183 Substituting Eqs. (6), (7) and (8) into Eq. (1) and rearranging yields the normalized governing

184 equation of water flow in the vegetated infinite slope with the uniform root:

185 (9) 2 '

' '12'

1 1 ( )Hcos cos cos

pr r s r r

s s

Tk k L kz LL k r k tzz

186

187 On the left handside of Eq. (9), the first and second terms are related to water flow driven by PWP

188 gradient and gravity, respectively, while the third term is related to root water uptake.

189

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190 Based on Eq. (9), three dimensionless numbers can be proposed as follows:

191 Capillary effect number (CN)

192 Capillary effect number (CN) is defined as follows:

193 (10) '

1cos

CNL

194 CN describes the relative importance of water flow driven by PWP gradient ( in Eq. 2

2'

1cos

rkL z

195 (9)) over that driven by gravity ( in Eq. (9)). The relative importance of water flow driven by rkz

196 PWP gradient becomes more significant with CN. For a fixed , CN is affected by the desaturation 'L

197 coefficient and the slope angle (Eq. (10)). Generally. the greater the sand content, the larger

198 the value of . Correspondingly, the unsaturated water permeability decreases more quickly as

199 the negative PWP increases (Zhan and Ng 2004). According to Philip (1969), the typical value of

200 is about 1 m-1, and the rang of values 0.2-5 m-1 covers most soils. The range of 0.1 to 1 m-1

201 covers most of the published values (Morrison and Szecsody 1985). Hence, for a given water

202 table depth ( in Fig. 1) and a slope angle , ranges from to , 'L '

1cos

CNL

'

1cosL '

10cosL

203 considering . 0.1 1

204

205 Water transfer-storage ratio (WR)

206 Water transfer-storage ratio (WR) is defined as follows:

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207

''

cos cos( )( ) [ ]

water flow driven by gravity component perpendicular to the slope of water storage change across

s s

s rs r

'

k t kWRLL

tcharacteristic

characteristic rate L

208 (11)

209 In Eq. (11), represents the saturated water flow driven by gravity component cossk

210 perpendicular to slope, which is chosen as the characteristic water flow, while denotes '( )s r L

t

211 the characteristic rate of water storage change across . represents the maximum water 'L ( )s r

212 storage capacity of a given soil. Generally, the larger the pore sizes, the larger the value of ( )s r

213 (Vanapalli et al. 1998).

214

215 Root water uptake number (RN)

216 Root water uptake number (RN) is defined as follows:

217 (12)cos

root water uptake water flow due to gravity component perpendicular to slope

p

s

TRN

krate of

characteristic

218 RN describes the relative importance of root water uptake (i.e., ) over water flow driven by pT

219 gravity component perpendicular to slope. The larger the RN, the more significant influence of

220 root water uptake. For a given slope angle ( ), ranges from 10-3 m/s (sand) to 10-9 m/s (clay), sk

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221 while can range from 0 mm/day (rainy period) and 8 mm/day (sunny period; Leung and Ng pT

222 2013). Hence, for most soils, RN ranges from 0 to for a given slope angle of . 90

cos

223

224 It is noted that the combined parameters in terms of dimensionless numbers could be arranged in

225 other alternative forms. For instance, the effects of slope angle (i.e., ) would be isolated from cos

226 RN and CN as an individual dimensionless parameter.

227

228 Estimating variations in PWP due to root water uptake based on dimensionless numbers

229 For the vegetated infinite slope (Fig. 1), the negative PWP at the surface is the largest at drying

230 steady state (Ng et al. 2015), when the negative PWPs enable a time independent flow from the

231 (fixed position) water table given a certain imposed root water uptake. Based on the proposed

232 dimensionless numbers, it is possible to estimate the negative PWP by using the dimensionless

233 numbers (i.e., CN and RN) at drying steady state.

234

235 The steady-state PWP head ( ; unit: m) at slope surface can be calculated by using the ' 'z L

236 analytical solutions for the uniform root architecture derived by Ng et al. (2015):

237 ( ) (13)' '' '

ln r z Lz L

k

' ' 1r z Lk

238 In Eq. (13), represents the relative permeability at slope surface (dimensionless) and is ' 'r z Lk

239 expressed as follows:

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240 (14)' '

0

' ' ,

1exp 11 exp( )r z L r veg z L

s

qCNk k

CN k

241 where represents rainfall (positive value) or evaporation (negative value) at slope surface (m/s), 0q

242 which is ignored in this study (i.e., m/s) in order to isolate influence of root water uptake 0 0q

243 on PWP distributions; and represents the relative permeability affected by root water ' ',r veg z Lk

244 uptake at slope surface (dimensionless) and is expressed as follows:

245 (15)' '

' ''2

, ''22

'

[ ]1exp( ) exp( )r veg z L

RN L CN Lk CN LLL

CN CN L

246 On the righthand side of Eq. (14), the first, second and third terms describe variations in relative

247 permeability due to the effects of water table, rainfall (or evaporation) and root water uptake,

248 respectively.

249

250 Based on Eq. (14), the PWP induced by root water uptake is affected by water table (i.e.,

251 ). Hence, the PWP in vegetated slope is normalized by that induced by water table 1exp( )CN

252 (i.e., hydrostatic PWP). A normalized PWP ( ) is proposed as follows:

253 (16)r

b

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254 where is the PWP head induced by root water uptake at a particular depth (m); and is r 'z b

255 the hydrostatic PWP head in an infinite slope at a particular depth (m), which is determined as 'z

256 follows:

257 (17)' cosb z

258 Based on Eq. (15), effects of root water uptake are mainly affected by RN and CN. A new

259 dimensionless number , representing a ratio of the effect of the root water uptake and that of the

260 water table, is proposed as follows (details of derivation can be found in Appendix):

261 (18)1

1

exp( )=1

CNRNCN r

262 As shown in Eq. (18), considers the combined effects of RN, CN and r on . The dimensionless

263 relationship between and is later established based on the calculated results using Eq. (13).

264 Thereafter, PWP induced by root water uptake in vegetated slope could be estimated (see details

265 below).

266

267 Estimating the influence zone of root water uptake based on dimensionless numbers

268 Root water uptake affects PWP distributions inside and outside root zone. Based on Ng et al.

269 (2015), the PWP distributions induced by root water uptake for the uniform root architecture

270 outside root zone can be expressed for steady state condition, as follows:

271 (19)0 exp 1

1 ln{exp exp 1 }s

zqCNz zRN

CN k CN

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272 On the right-hand side of Eq. (19), the first, second and third terms describe the effects of water

273 table, rainfall/evaporation and root water uptake on PWP distributions, respectively. Based on the

274 third term in Eq. (19), the PWP induced by root water uptake outside root zone at a particular depth

275 depends on RN and CN. z

276

277 In Eq. (19), the first term on the righthand side is positive, while the third term is negative since

278 root water uptake decreases PWP. Hence, by comparing the negative value of the third term to the

279 first term, the following dimensionless number is defined for estimating the influence depth of root

280 water uptake:

281 (20)

'

'

1 exp coscos

exp cos

= exp 1

p

s

Tz

kz

zRNCN

282 Based on Eq. (20), effects of root water uptake become less prominent as decreases along depth.

283 The valve value of is defined as (dimensionless), below which the effects of root water valv

284 uptake are negligible. Based on Eq. (20), could be approximated as proportional to the ratio of

285 RN to CN. Hence, for a given value of RN and CN may be determined as follows:valv

286 (21)0 0

0valv valv

valvvalv

RN CNRN CN

287 where is the reference valve value of for a given value of and (dimensionless). 0valv 0

valvRN 0valvCN

288 The determination of is given later. 0valv

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289

290 According to Eq. (20), corresponding to can be determined as follows:z valv

291 (22)ln(1 )valv

valvz CNRN

292 Hence, when , the effect of root water uptake is negligible. valv

z z

293 Based on Eq. (22), the normalized influence depth of root water uptake by root depth can be

294 determined as follows:

295 (23)1 ln(1 )1

valvCNz RNRIFr r

296

297 Introductory analysis for parametric study

298 Steady state

299 With the defined dimensionless numbers (Eqs. (10) to (12)), the normalized governing equation

300 (Eq. (9)) at steady state is rearranged as follows:

301 (24) 2

' '12

1 H 0r rk kCN RN z Lrzz

302 Hence, for a given root architecture (i.e., the uniform root) and root depth ratio (i.e., ), the r

303 variations in steady-state PWP distributions due to root water uptake depend on RN and CN only.

304

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305 Transient state

306 With the defined dimensionless numbers, the normalized governing equation (Eq. (9)) in vegetated

307 slope with the uniform root architecture at transient state can be rewritten as:

308 (25) 2

' '12

1 Hr r rk k kCN RN z Lr WRzz

309 It can be deduced from Eq. (25) that, for a given root architecture (i.e., the uniform root) and a root

310 depth ratio r, the PWP distributions in the vegetated slope at the transient state depend on CN, RN

311 and WR.

312

313 Analytical scheme of parametric study

314 An infinite vegetated slope is shown in Fig. 1. The slope angle is 40o and the uniform root is

315 considered. The vertical depth of root is 0.5 m ( ). The water table is located at the bottom of the 2L

316 slope (i.e., H0=5 m). Table 1 shows the parametric analyses. Since CN is affected by the

317 desaturation coefficient and the slope angle for a fixed slope thickness (Eq. (10)), the first 'L

318 and second series of analytical experiments aim to investigate the effects of CN, which is varied

319 by controlling the desaturation coefficient and the slope angle respectively. The third and

320 fourth series of analytical experiments aim to study the effects of RN, which is controlled by sk

321 and respectively. The fifth series of analytical experiment is designed for studying effects of pT

322 WR on PWP response under rainfall at both transient and steady state with varied CN (0.7 and 0.3

323 in series 5 in Table 1).

324

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325 Analysis of the calculated results

326 In the following discussion, the calculated results of the parametric study are presented in terms of

327 the normalized PWP by hydrostatic pressure. The related PWP values can be found in Figs. S1-S4

328 in the supplementary document.

329 Effects of CN on PWP distributions

330 Fig. 2(a) shows the normalized PWP profiles in the vegetated slope with CN controlled by the

331 desaturation coefficient (series 1 in Table 1). The effects of root water uptake on normalized

332 PWP becomes more significant for a smaller CN (i.e., increases). This is attributed to that for a

333 constant , unsaturated water permeability becomes lower for a larger at a given negative sk

334 PWP. For a given amount of root water uptake (i.e., the same RN), according to Darcy’s law, the

335 lower the water permeability, the larger hydraulic gradient is needed to generate the same water

336 flux equaling root water uptake at steady state. Hence the soil with a larger has a larger negative

337 PWP.

338

339 Fig.2(b) shows the effects of CN, controlled by the slope angle (series 2 in Table 1), on the

340 normalized PWP distributions in the vegetated slope. The normalized PWP reduces as the slope

341 angle increases, since the overall water flux outside root zone is upward and equals the applied

342 root water uptake (i.e., ) based on mass balance. As the water flux caused by gravity (i.e., pT

343 in Eq. (1)) is downward and increases as the slope angle decreases, a larger PWP 'cos kz

344 gradient is needed to generate the same overall upward water flux following Darcy’s law. Although

345 the case of the slope angle of 60°has the largest RN of 0.04 (see series 2 in Table 1), its negative

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346 PWP is the lowest. It is because CN for the slope angle of 60° is the largest, leading to more

347 significant effects of water flow driven by PWP gradient ( in Eq. (24)) than that of root 2

2rkCN

z

348 water uptake (i.e., in Eq. (24)). As a result, the negative PWP reduces. ' '1

1 HRN z Lr

349

350 These results illustrate that, the negative PWP induced by root water uptake becomes larger for a

351 smaller CN (i.e., by increasing or decreasing slope angle ) with a given RN. In practice,

352 becomes larger for a given type of soil with more uniform distribution of grain size (Zhan and Ng

353 2004). Moreover, influence of root water uptake on PWP distributions and thus slope stability is

354 more prominent in gentle slope, such as sloping landfill cover, where the maximum slope angle is

355 18o based on design guideline (CAE 2000; Feng et al. 2017).

356

357 Effects of RN on PWP distributions

358 Fig. 3 shows the influence of RN, controlled by the saturated permeability (series 3 in Table 1), on

359 the steady-state PWP distributions in the vegetated slope. Generally, a larger negative PWP

360 induced by root water uptake is observed for a larger RN (i.e., a lower ). For the case with the sk

361 largest RN of 0.3, the negative PWP at the surface of vegetated slope is more than twice the

362 hydrostatic PWP. In contrast, for the case with RN of 0.03, the negative PWP induced by vegetation

363 is slightly larger than hydrostatic PWP. When RN further reduces to 0.003, effects of root water

364 uptake are almost negligible. The reason is that under a given root water uptake (i.e., the same ), pT

365 a larger hydraulic gradient is needed to induce the same water flux for a lower water permeability.

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366

367 Fig. 4(a) shows that for a given CN, the value of RN (controlled by ; series 4a in Table 1) p sT k

368 determines the PWP distributions induced by root water uptake. As expected, both the negative

369 PWP and the influence zone of root water uptake increase as RN increases from 0.01 to 0.1. For

370 the two cases with different values of and but the same value of (the cases with pT sk p sT k

371 identical RN of 0.1, but varied ks of 0.2×10-6 m/s and 2×10-6 m/s respectively), the negative PWP

372 distributions are the same. This is expected since these two cases have the same CN (i.e., the same

373 α) and RN, leading to the same governing equation (Eq. (24)). Hence the calculated results are the

374 same.

375

376 Fig. 4(b) shows the effects of RN (controlled by ; series 4b in Table 1) on the PWP p sT k

377 distributions with CN of 0.5. Similar trend can be observed as that in Fig. 4(a). But the induced

378 negative PWPs are larger than that in Fig. 4(a) for each case with the same RN. This is due to a

379 reduction of water flow driven by PWP gradient (i.e., in Eq. (24)) as CN decreases. As a 2

2rkCN

z

380 result, a larger PWP gradient is needed to compensate the same amount of root water uptake at a

381 given RN, leading to a larger negative PWP.

382

383 These results suggest that soils with a relative small would help amplify the impact of sk

384 vegetation on PWP distribution for a given root architecture (i.e., uniform root) and root depth. In

385 practice, increasing the degree of compaction (DOC) would decreases , leading to larger sk

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386 negative PWP and thus enhanced slope stability. However, the vegetation growth would be

387 constrained as DOC increases (Unger and Kaspar 1994; Kozlowski 1999). Hence, the engineers

388 are supposed to strike a balance between these two opposing effects to select a proper DOC in

389 man-made slope to exploit the benefits of vegetation on enhancing shallow slope stability.

390

391 Effects of WR on PWP distributions

392 Figs. 5(a) and 5(b) shows effects of WR on PWP distributions during rainfall with a constant CN

393 of 0.7 (cases 1a to 2d in series 5 in Table 1) for ( ) of 0.2 and 0.4, respectively. The initial s - r

394 PWP distributions for ( ) of 0.2 (Fig. 5(a)) and 0.4 (Fig. 5(b)) are the same as expected, since s - r

395 the initial conditions are based on PWP profiles at drying steady state and are independent of

396 ( ) (see Eq. (24)). For the case with ( ) of 0.2 after 12 hours of rainfall (Fig. 5(a)) and that s - r s - r

397 with ( ) of 0.4 after 24 hours of rainfall (Fig. 5(b)), WR is the same as 0.09. The PWP profiles s - r

398 for these two cases are identical, which is expected due to the same CN, RN (i.e., RN=0 during

399 rainfall) and WR, leading to the same governing equation (see Eq. (25)). After 24 hours of rainfall,

400 compared with the case with ( ) of 0.4 (Fig. 5(b)), the one with ( ) of 0.2 (Fig. 5(a)) has s - r s - r

401 deeper wetting front and lower negative PWP. It is because the latter case has a lower water storage

402 capacity (i.e., a lower value of ( )). Hence more water flow through soil compared with stored s - r

403 water, leading to deeper wetting front. These results are consistent with the Green and Ampt’s

404 model (1911), from which it can be deduced that wetting front advances more quickly as water

405 storage capacity decreases. After 300 hours of rainfall, WR reaches 2 (Fig. 5(a)) and 1 (Fig. 5(b))

406 for the cases with ( ) of 0.2 and 0.4, respectively. The PWP profiles of these two cases are s - r

407 close to the wetting steady-state results. It should be noted that since steady-state PWP profile is

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408 independent of water storage capacity (Eq. (24)), the wetting steady state lines for both ( ) of s - r

409 0.2 and 0.4 are the same. It may conclude that when WR is larger than certain value (i.e., 1), the

410 PWP profile nearly reaches the steady state, hence the PWP profile could be estimated by steady

411 state analysis rather than transient analysis for saving time.

412

413 Figs. 5(c) and 5(d) shows the PWP response during rainfall with CN of 0.3 for ( ) of 0.2 and s - r

414 0.4, respectively. With the same WR, Fig.5(c) shows shallower wetting front and larger negative

415 PWP reduction, comparing with Fig. 5(a). It is attributed to that CN of Fig. 5(c) is lower than that

416 of Fig. 5(a), indicating lower water flow driven by PWP gradient. Hence with the same WR and

417 water storage capacity, more water is stored under the same rainfall infiltration with lower CN.

418 Correspondingly, a larger reduction of negative PWP and a shallower wetting front can be found.

419 Similar trend can be observed by comparing Fig. 5(d) with Fig. 5(b). These results illustrate that

420 effects of WR on PWP response during transient state mainly depends on CN. Under a given CN,

421 the wetting front advances deeper and the normalized PWP reduces more, for a larger WR during

422 rainfall. As CN reduces with a given WR, the wetting front moves shallower, but the reduction in

423 negative PWP becomes larger due to reduced water transfer in soils. Moreover, the wetting steady

424 state line is governed by CN. The PWP profile at wetting steady state become steeper for a smaller

425 CN. This is due to water flow in the soil is the same as the applied rainfall infiltration at the wetting

426 steady state. As CN decreases, the influence of PWP gradient on water flow becomes less

427 prominent (Eq. (24)), leading to more significant effects of water flow driven by gravity. Hence,

428 PWP gradient reduces (i.e., PWP profile becomes steeper).

429

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430 Dimensionless relationships for estimating PWP and root influence

431 zone in vegetated slope

432 Dimensionless relationship of negative PWP induced by root water uptake

433 For a given root architecture, the influence of root water uptake on PWP depends on RN, CN and

434 r at steady state (Eq. (24)). The combined effects of RN, CN and r on PWP distributions may be

435 described by η (Eq. (18)). The effects of root water uptake on PWP are generally more significant

436 for a larger η. For example, η decreases with increasing r (i.e., a deeper root depth) or CN, leading

437 to less significant effects of root water uptake on PWP (see Fig. 2(a)). Meanwhile, as both RN and

438 CN are affected by the slope angle β, η becomes smaller for a larger slope angle β. Thereafter, the

439 influence of root water uptake on PWP becomes less prominent (see Fig. 2(b)). Moreover, η

440 increases with RN (i.e., either a larger Tp or a smaller ks cosβ), leading to more significant effects

441 of root water uptake on PWP (see Fig. 4).

442

443 Fig. 6 shows the influence of on the calculated normalized PWP at the surface and root depth

444 of the vegetated slope. The PWP profiles at drying steady state are adopted as the calculated results

445 (i.e., Figs 2 to 5). Generally, the normalized negative PWP increases with , due to more

446 significant effect of root water uptake on PWP profiles. Quadratic functions are found to fit well

447 relationships between and the normalized negative PWP. The fitted functions maybe useful for

448 preliminary estimation of the negative PWP (i.e., based on Eq. (16): ) induced by root r b

449 water uptake with different weather conditions (i.e., ), soil properties and root characteristic (i.e., pT

450 root depth).

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451

452 Dimensionless relationship of influence zone of root water uptake

453 Both PWP profiles inside and outside root zone are affected by root water uptake. It would be

454 useful to estimate root influence zone based on the proposed dimensionless relationship in Eq. (23).

455 Fig. 7 (a) shows the PWP profiles for the uniform rooted slope with RN of 0.03 (the second and

456 third cases in series 1 in Table 1). Note that hydrostatic pressure is the same for CN of 0.7 and 0.3

457 for bare soil, since the hydrostatic pressure at the depth of is determined as (where 'z ' coswz

458 is the unit weight of water (N/m3); Ng et al., 2015). The root influence zone is determined as w

459 the depth where the difference of PWP between vegetated soil and bare soil (hydrostatic pressure)

460 is more than 1kPa. For CN of 0.7 and 0.3, the root influence zone is determined as three and six

461 times of root depth, respectively. Fig. 7(b) shows the determination of root influence zone by ξ

462 (Eq. (20)). ξ increases near slope surface. Based on the root influence zone determined in Fig. 7(a),

463 the valve value of ξ is determined as 0.05 (i.e., Eq. (21), ), 0 0 00.05, 0.03, 0.3valv valv valvRN CN

464 below which effects of root water uptake is negligible. Hence, the root influence zone for a given

465 RN and CN could be determined by Eq. (23).

466

467 Validation of the dimensionless relationships for estimating PWP and root influence

468 zone

469 The proposed relationships in Fig. 6 for estimating negative PWP and the dimensionless

470 relationship of influence zone of root water uptake (Eq. (23)) were validated by measurements of

471 two field tests (Ni et al. 2018; Woon 2013) and two sets of centrifuge tests (Ng et al. 2016a; 2017).

472 The selected field and centrifuge tests are the rare cases that provide all the necessary information

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473 for both plants and soils. The same soil type (completely decomposed granite (CDG), classified as

474 silty sand) was adopted in all the selected cases. Schefflera heptaphylla and Cynodon dactylon

475 were adopted in the field tests conducted by Ni et al. (2018) and Woon (2013), respectively. Both

476 field tests consist of drying period (i.e., no rainfall). During testing, matric suction from depths of

477 0.01–0.5 m was monitored. The measured root architectures of Schefflera heptaphylla and

478 Cynodon dactylon can be approximated as the parabolic root (i.e., the root density takes a parabolic

479 form with depth, with a maxima between the ground surface and root tip)) and the uniform root,

480 respectively. According to Ng et al. (2015), the PWP distributions induced by the parabolic and

481 the uniform roots are similar, hence the uniform root was adopted for validation of the proposed

482 relationships for root influence zone and PWP.

483

484 In the two sets of centrifuge tests by Ng et al. (2016a) and Ng et al. (2017), an artificial root system

485 and real branch cuttings were used respectively, for simulating the influence of root water uptake

486 on PWP in the centrifuge tests. In these two centrifuge tests, tap root was adopted, which consists

487 of one primary vertical root with several branched secondary roots near its tip. Hence, it can be

488 approximated as the uniform root (Ng et al. 2016a). Both centrifuge tests started with a simulation

489 of transpiration until there was negligible change of negative PWP profiles along depth.

490

491 As for the input parameters in Eqs. (4) and (5), measured SWRCs reported by Ni et al. (2018) and

492 Ng et al. (2016a; 2017) were adopted for the respective cases. The SWRC measured by Leung et

493 al. (2015) was adopted for the selected case by Woon (2013). It is noted that these two cases used

494 the same soil type (i.e., CDG) and similar DOC around 90%. Hence, the measured SWRC by

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495 Leung et al. (2015) could be a reasonable approximation for the selected case by Woon (2013).

496 Fig. 8(a) shows the measured SWRCs of the soil used for the selected cases. By fitting the

497 measured SWRCs based on Eq. (5), the parameters of , and were calibrated as shown in s r

498 Table 2. By using these calibrated parameters and the measured reported in each selected case, sk

499 unsaturated permeability functions were estimated by Eq. (4) and are shown in Fig. 8(b). The

500 simulated transpiration rate of 1 mm/day as measured by Ng et al. (2016a) was used for the two

501 centrifuge tests. In the field test reported by Ni et al. (2018), the measured transpiration rate was

502 4.8 mm/day. The transpiration rate for the field test of Woon (2013) was estimated as 3 mm/day,

503 based on the measured evapotranspiration rate by Leung and Ng (2013). Note that Leung and Ng

504 (2013) and Woon (2013) conducted field test in the similar period (i.e., June) and location (i.e.,

505 Hong Kong). Thus, the measured transpiration rate by Leung and Ng (2013) was adopted as a

506 reasonable approximation. The measured slope angle, water table depth and root depth for each

507 case are summarized in Table 2.

508

509 Fig. 9 shows the comparisons between the measured and the estimated PWP using the relationship

510 between η and the normalized PWP (Fig. 6). Generally, the estimated values of PWP are consistent

511 with the measurements. The maximum difference between measured and estimated PWP is about

512 18 kPa, corresponding to the field test conducted by Woon (2013). This could be probably

513 attributed to the ignorance of the effects of roots on soil hydraulic properties (i.e., water

514 permeability; Ni et al. 2018; Liu et al. 2018). The analytical parametric studies carried out by Liu

515 et al. (2018) show that PWP would be underestimated during drying period, when ignoring the

516 reduction of water permeability due to the occupancy of soil pores by root. For the selected

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517 centrifuge test conducted by Ng et al. (2017), the PWP is underestimated by the proposed method

518 by about 17 kPa. This is likely due to the steady state assumption made in the proposed

519 relationships. Hence, the amount of root water uptake was overestimated, leading to an

520 underestimation of PWP. As for the field test of Ni et al. (2018), the estimated PWP is larger than

521 the measurement by about 10 kPa at the soil surface. This indicates that the amount of root water

522 uptake near soil surface would be underestimated, probably due to the assumption of uniform root

523 architecture and hence constant transpiration along root depth. The results demonstrate that the

524 proposed relationships in Fig. 6 can predict PWP distributions in vegetated soil reasonably.

525

526 Fig. 10 shows the comparisons between the measured and the calculated influence depths of root

527 water uptake using RIF (Eq. (23)). Generally, the measured and calculated results are consistent.

528 The maximum difference between measured and calculated root influence depths is about 1 m,

529 corresponding to the field test conducted by Woon (2013). This is probably due to the steady state

530 assumption adopted in the proposed method, leading to deeper drying front propagation and hence

531 larger root influence depth. The results illustrate that the proposed RIF (Eq. (23)) can predict root

532 influence depth reasonably.

533

534 Summary and conclusions

535 Dimensional analysis of PWP distributions in a vegetated infinite slope was conducted. Three

536 dimensionless numbers were proposed, including (i) capillary effect number (CN; Eq. (10)),

537 representing the relative importance of water flow driven by PWP gradient over that driven by

538 gravity; (ii) water transfer-storage ratio (WR; Eq. (11)), representing the relative importance of

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539 water transfer over water storage at transient state; and (iii) root water uptake number (RN; Eq.

540 (12)), describing the relative importance of root water uptake (i.e., ) over water flow. Then pT

541 dimensionless relationships were proposed for estimating PWP and root influence zone in

542 vegetated slope. Moreover, the effects of the newly proposed dimensionless numbers on PWP

543 distributions in a vegetated infinite slope were investigated by conducting parametric studies.

544 Finally, the proposed dimensionless relationships for estimating PWP and root influence zone were

545 validated against published field and centrifuge tests.

546

547 At drying steady state, PWP profiles in a vegetated infinite slope are governed by CN and RN only

548 for given boundary conditions. When RN is less than 0.02, effects of root water uptake on PWP

549 distributions might be ignored. The combined effects of CN and RN on PWP distributions and root

550 influence zone could be described by (Eq. (18)) and (Eq. (20)), respectively. Generally, the

551 negative PWP induced by root water uptake increases with . As increases, the root influence

552 zone becomes deeper.

553

554 At transient state, PWP response in a vegetated infinite slope depends on CN, RN and WR. During

555 transient wetting period (i.e., rainfall), effects of RN are negligible due to high humidity and thus

556 low transpiration. Under such case and given boundary conditions, PWP response depends on WR

557 and CN only. For a given CN and the same initial condition, the larger the WR, the deeper the

558 wetting front moves and more reduction of negative PWP occurs. Generally, when WR is larger

559 than 1, PWP profile is close to the wetting steady-state results. Hence, the PWP may be predicted

560 by the steady state simulation for saving time. For a given WR, as CN decreases, the wetting front

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561 moves shallower but more negative PWP reduction occurs, due to less water flow and more water

562 storage in soils.

563

564 Compared with measured field and centrifuge tests data, the proposed dimensionless relationships

565 are capable of determining PWP and root influence zone in vegetated soil reasonably. The steady

566 state assumption is made in the proposed relationships. It is noted that steady state conditions

567 seldom exist in the field, due to the variation in the climate condition. Nevertheless, the proposed

568 relationships could be useful (i) for estimating the potential range of negative PWP and depth of

569 root influence zone under drying condition in the field; (ii) for evaluating the time needed to reach

570 steady state condition; and (iii) for designing centrifuge tests where long periods of uniform

571 climate could be usually imposed. Hence, these proposed dimensionless relationships provide a

572 simple and valid method for estimating PWP distributions in vegetated slope.

573

574 Acknowledgements

575 The authors would like to acknowledge the financial supports provided by the National Natural

576 Science Foundation of China (51808125), the Research Grants Council (RGC) of the Hong Kong

577 Special Administrative Region (T22-603/15N), and Fuzhou University in China (510597/GXRC-

578 18029). The authors are grateful to Dr. Anthony Kwan Leung for fruitful discussions about the

579 challenging problem that was addressed in this study. The authors would also like to express

580 sincere gratitude to the two anonymous reviewers for their constructive comments and suggestions.

581

582

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583 Notation

584 CN Capillary effect number (dimensionless)

585 CN value corresponding to (dimensionless)0valvCN 0

valv

586 DOC degree of compaction (%)

587 the shape function describing root architecture (m-1)'(z )g

588 (dimensionless) ' '1H( )z L

' ' ' '1 1 2' '

1 ' '1

1 ( )H( )

0 0

L z L Lz L

z L

589 the water permeability of soil (m s-1)k

590 the relative water permeability (dimensionless)rk

591 the saturated water permeability (m s-1)sk

592 the water permeability at the slope bottom (m s-1)' 0zk

593 the relative permeability at slope surface (dimensionless)' 'r z Lk

594 the relative permeability induced by root water uptake at slope surface ' ',r veg z Lk

595 (dimensionless)

596 the thickness of the domain outside root zone (Fig. 1; m)'1L

597 the thickness of root zone (Fig. 1; m)'2L

598 the thickness of slope (i.e., ; m; see Fig. 1)'L ' ' '1 2L L L

inside root zone

outside root zone

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599 rainfall (positive value) or evaporation (negative value) at slope surface (m s-1)0q

600 RN Root water uptake number (dimensionless)

601 RN value corresponding to (dimensionless)0valvRN 0

valv

602 RIF the normalized influence depth of root water uptake by root depth

603 (dimensionless)

604 r root depth ratio ( ) (dimensionless)'2'

LrL

605 SWRC soil water retention curve

606 the transpiration rate (m s-1)pT

607 t time (s)

608 WR Water transfer-storage ratio (dimensionless)

609 the perpendicular distance to the slope bottom (m; see Fig. 1) 'z

610 the normalized by (dimensionless)z 'z 'L

611 the normalized z corresponding to (dimensionless)valv

z valv

612 the desaturation coefficient of soil (m-1)

613 the slope angle (degree)

614 the unit weight of water (N/m3)w

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615 dimensionless number defined as (dimensionless)1

1

exp( )=1

CNRNCN r

616 the saturated volumetric water content (dimensionless)s

617 the residual volumetric water content (dimensionless)r

618 ξ dimensionless number defined as (dimensionless)= exp 1zRNCN

619 the valve value of at RN and CN (dimensionless)valv

620 the reference valve value of at and (dimensionless)0valv 0

valvRN 0valvCN

621 PWP head (m)

622 the steady-state PWP head at slope surface (dimensionless)' 'z L

623 the normalized PWP by hydrostatic pressure (dimensionless)

624 PWP head in vegetated slope (m)r

625 hydrostatic PWP head in an infinite slope (m)b

626

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735 rainfall infiltration into an unsaturated infinite slope and its application to slope stability

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738

739 List of Figures

740 Figure. 1. Schematic diagram showing a vegetated infinite slope (after Ng et al., 2015)

741 Figure. 2. Effects of CN on PWP distributions for uniform rooted slope: (a) CN controlled by

742 desaturation coefficient ; (b) CN controlled by slope angle

743 Figure. 3. Effects of RN (controlled by saturated permeability ks) on PWP distributions in

744 uniform rooted slope with CN of 0.7

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745 Figure. 4. Effects of RN (controlled by the ratio of transpiration rate to saturated permeability

746 (Tp/ks)) on PWP distributions in uniform rooted slope with varied CN: (a) CN of 0.7; (b) CN of

747 0.5

748 Figure. 5. Effects of water transfer-storage ratio (WR) on PWP distributions in uniform rooted

749 slope with varied CN: (a) CN of 0.7 (θs- θr=0.2); (b) CN of 0.7 (θs-θr=0.4); (c) CN of 0.3 (θs-

750 θr=0.2); (d) CN of 0.3 (θs- θr=0.4)

751 Figure. 6. Effects of η (Eq. (18)) on normalized PWP at the surface of uniform rooted slope

752 Figure. 7. Determination of root influence zone for uniform rooted slope with root water

753 uptake number (RN) of 0.03: (a) determined by difference of PWP between vegetated soil and

754 hydrostatic pressure; (b) determined by ξ (Eq. (20))

755 Figure. 8. Soil hydrology used for method validation: (a) measured and fitted SWRCs; (b)

756 fitted permeability functions

757 Figure. 9. Comparisons between measured and estimated PWP based on the relationship

758 between η and normalized PWP

759 Figure. 10. Comparisons between measured and estimated influence depth of root water uptake

760 based on RIF (Eq. (23))

761

762 List of Tables

763 Table 1 Summary of parametric studies

764 Table 2 Summary of input parameters for method validation

Page 38 of 52



Draft

1

H0 Root zone

L2

L1

Figure. 1. Schematic diagram showing a vegetated infinite slope (after Ng et al. 2015)

q0

'1L

'2L

'L

'z

Page 39 of 52



Draft

2

1 1.05 1.1 1.15 1.20.0

3.0

6.0

9.0

CN=0.3

CN=0.7

CN=3

Normalized pore-wate pressure by hydrostatic pressure ( )

Nor

mal

ized

dep

th b

y ro

ot d

epth

CN=0.3; α=1.0m-1

CN=0.7; α=0.5m-1

CN=3; α=0.1m-1

Root zone

(a)

1 1.05 1.1 1.15 1.20.0

3.0

6.0

9.0

CN=0.6; β=30°

CN=0.7;β= 40°

CN=1;β=60°

Normalized pore-wate pressure by hydrostatic pressure ( )

Nor

mal

ized

dep

th b

y ro

ot d

epth

Root zone

(b)

Figure. 2. Effects of CN on PWP distributions for uniform rooted slope: (a) CN controlled by desaturation coefficient ; (b) CN controlled by slope angle

(b)

CN=0.6; β=30o

CN=0.7; β=40o

CN=1; β=60o

Page 40 of 52



Draft

3

1 1.5 2 2.50.0

3.0

6.0

9.0

RN=0.26

RN=0.026

RN=0.0026

Normalized pore-water pressure by hydrostatic pressure ( )N

orm

aliz

ed d

epth

by

root

dep

th

Figure. 3. Effects of RN (controlled by saturated permeability ks) on PWP distributions in uniform rooted slope with CN of 0.7

RN=0.3, ks=0.2×10-6 m/s

RN=0.03, ks=2×10-6 m/s

RN=0.003, ks=20×10-6 m/s

Root zone

Page 41 of 52



Draft

4

1 1.1 1.2 1.3 1.4 1.5 1.6 1.70.0

3.0

6.0

9.0

ks=0.2×10-6 m/s RN=0.131

ks=2×10-6 m/s RN=0.013

ks=2×10-6 m/s RN=0.026

ks=2×10-6 m/s RN=0.052

ks=2×10-6 m/s RN=0.131

Normalized pore-water pressure by hydrostatic pressure ( )N

orm

aliz

ed d

epth

by

root

dep

th

(a)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.70.0

3.0

6.0

9.0

ks=0.2×10-6 m/s RN=0.131

ks=2×10-6 m/s RN=0.013

ks=2×10-6 m/s RN=0.026

ks=4×10-6 m/s RN=0.052

ks=2×10-6 m/s RN=0.131

Normalized pore-water pressure by hydrostatic pressure ( )

Nor

mal

ized

dep

th b

y ro

ot d

epth

(b)

Figure. 4. Effects of RN (controlled by the ratio of transpiration rate to saturated permeability (Tp/ks) ) on PWP distributions in uniform rooted slope with varied CN: (a) CN of 0.7; (b) CN of 0.5

Root zone

Root zone

RN=0.1 (ks=0.2×10-6 m/s; Tp=2×10-8 m/s)

RN=0.01 (ks=2×10-6 m/s; Tp=2×10-8 m/s)

RN=0.03 (ks=2×10-6 m/s; Tp=4×10-8 m/s)

RN=0.05 (ks=2×10-6 m/s; Tp=8×10-8 m/s)

RN=0.1 (ks=2×10-6 m/s; Tp=20×10-8 m/s)

RN=0.1 (ks=0.2×10-6 m/s; Tp=2×10-8 m/s)

RN=0.01 (ks=2×10-6 m/s; Tp=2×10-8 m/s)

RN=0.03 (ks=2×10-6 m/s; Tp=4×10-8 m/s)

RN=0.05 (ks=2×10-6 m/s; Tp=8×10-8 m/s)

RN=0.1 (ks=2×10-6 m/s; Tp=20×10-8 m/s)

Page 42 of 52



Draft

5

0 0.5 1 1.5 20

3

6

9

WR=0 (initial)

WR=0.09 (12h)

WR=0.2 (24h)

WR=2 (300h)

Wetting steady state

Normalized pore-water pressure by hydrostatic pressure ( ) N

orm

aliz

ed d

epth

by

root

dep

th

Root zone

0 0.5 1 1.5 20

3

6

9

WR=0 (initial)

WR=0.04 (12h)

WR=0.09 (24h)

WR=1(300h)



Nor

mal

ized

dep

th b

y ro

ot d

epth

Root zone

(a) (b)

0 0.5 1 1.5 20

3

6

9

WR=0 (initial)

WR=0.09 (12h)

WR=0.2 (24h)

WR=2 (300h)



Nor

mal

ized

dep

th b

y ro

ot d

epth

Root zone

0 0.5 1 1.5 20

3

6

9

WR=0 (initial)

WR=0.09 (12h)

WR=0.2 (24h)

WR=2 (300h)



Nor

mal

ized

dep

th b

y ro

ot d

epth

Root zone

(c) (d)

Figure. 5. Effects of water transfer-storage ratio (WR) on PWP distributions in uniform rooted slope with varied CN: (a) CN of 0.7 (θs- θr=0.2); (b) CN of 0.7 (θs-θr=0.4); (c) CN of 0.3 (θs-θr=0.2); (d) CN of 0.3 (θs- θr=0.4)

Page 43 of 52



Draft

6

0 0.2 0.4 0.6 0.8 11.0

1.5

2.0

2.5Calculated-slope surfaceCalculated-root depthFitted-slope surfaceFitted-root depth

η

Nor

mal

ized

max

imum

neg

ativ

e po

re-w

ater

pr

essu

re (

)

Figure. 6. Effects of η (Eq. (18)) on normalized PWP at the surface of uniform rooted slope

Fitted function at slope surface

Fitted function at root depth

2

2

0.9942 0.0448 1.04340.991R

2

2

0.7219 0.1346 1.03620.9554R

Page 44 of 52



Draft

7

(a)

(b)

Figure. 7. Determination of root influence zone for uniform rooted slope with root water uptake

number (RN) of 0.03: (a) determined by difference of PWP between vegetated soil and hydrostatic pressure; (b) determined by ξ (Eq. (20))

-4.00E+01 -2.00E+01 0.00E+00 2.00E+010.0

3.0

6.0

9.0

Hydrostatic

CN=0.68

CN=0.34

Pore-water pressure (kPa)

Nor

mal

ized

dep

th b

y ro

ot d

epth

Root influence zone

Root influence zone

0 0.1 0.2 0.3 0.4 0.50

3

6

9

CN=1.468 (a=0.5)

CN=2.936 (a=1)

ξ

Nor

mal

ized

dep

th b

y ro

ot d

epth

Root influence zone Root influence

zone

0 0.05valv

CN=0.7

CN=0.3

CN=0.7

CN=0.3

Page 45 of 52



Draft

8

0.1 1.0 10.0 100.0

0

5

10

15

20

25

30

35

40

45

Measured-Ni et al. (2018)Fitted-Ni et al. (2018)Measured-Ng et al. (2017)Fitted-Ng et al. (2017)Measured-Ng et al. (2016a)Fitted-Ng et al. (2016a)Measured-Leung et al. (2015)Fitted-Leung et al. (2015)

Vol

umet

ric w

ater

con

tent

(%)

Matric suction (kPa)

(a)

0.1 1 10 1001E-12

1E-11

1E-10

1E-09

1E-08

1E-07

1E-06

1E-05

Fitted-Ni et al. (2018)Fitted-Ng et al. (2018)Fitted-Ng et al. (2016a)Fitted-Leung et al. (2015)

Perm

eabi

lity

(m/s

)

Matric suction (kPa)

(b)

Figure. 8. Soil hydrology used for method validation: (a) measured and fitted SWRCs; (b) fitted permeability functions

Measured ks=1.22*10-6 m/s



Page 46 of 52



Draft

9

Figure. 9. Comparisons between measured and estimated PWP based on the relationship between η and normalized PWP

Woon (2013)

-100 -80 -60 -40 -20 0-100

-80

-60

-40

-20

0At surface-Ng et al. (2016a)At root depth-Ng et al. (2016a)At root depth-Ng et al. (2017)At root depth-Woon (2013)At root depth-Ni et al. (2018)At surface-Ni et al. (2018)

Calculated pore-water pressure (kPa)

Mea

sure

d po

re-w

ater

pre

ssur

e (k

Pa)

1:1 line

Centrifuge test

Field test

Page 47 of 52



Draft

10

0 1 2 3 4 50

1

2

3

4

5Ng et al. (2016a) Ng et al. (2017)Woon (2013)

Calculated root influence zone (m)

Mea

sure

d ro

ot in

fluen

ce z

one

(m)

1:1 line

Figure 10. Comparisons between measured and estimated influence depth of root water uptake based on RIF (Eq. (23))

Page 48 of 52



Draft

1

Table 1 Summary of parametric studies

Series Case number

ks (10-6 m/s)

a (m-1)

β (degree) θs-θr

Tp (10-8 m/s)

Rainfall (10-6 m/s)

Rainfall duration (hr) CN RN WR Solution method

1 0.1 32 0.5 0.713 1

400.3

0.03

1 30 0.6 0.022 40 0.7 0.032a

3

2

60 1 0.041 0.2 0.32 2 0.0333 20

0.5

4

0.70.003

1 0.2 2 0.12 2 2 0.013 2 4 0.034 2 8 0.05

4a

5 2

0.5

20

0.7

0.11 0.2 2 0.12 2 2 0.013 2 4 0.034 2 8 0.05

4b

5 2

0.65

0.4

20

0 Not applicable

0.5

0.1

Not applicable Steady state (drying)

1a 12 0.091b 24 0.21c 300 2

Transient state (wetting)

1d

0.2

infinite Not applicable Steady state (wetting)2a 12 0.042b 24 0.092c 300 1


2d

0.5

0.4

infinite

0.7

Not applicable Steady state(wetting)3a 12 0.093b 24 0.23c 300 2


3d

0.2

infinite Not applicable Steady state (wetting)4a 12 0.044b 24 0.094c 300 1


5b

4d

2

1

40

0.4

4 2

infinite

0.3

0.03

Not applicable Steady state (wetting)

Note: (a) The perpendicular depth of the slope (L’) is 3.8 m for different slope angles. (b) The initial condition before rainfall is obtained as pore-water pressure profiles at drying steady state with RN 0.03. During rainfall, root water uptake is ignored (i.e., RN=0). To simplify the problem, 40% of rainfall is assumed as surface runoff (Ng et al. 2015).

Page 49 of 52



Draft

2

Table 2 Summary of input parameters for method validation

Case slope angle (degree)

water table depth (m)

root depth (m)

Tp (mm/day)

α (m-1)

ks (m/s) θs θr

Ng et al. (2016a)a 45 3 0.75 1 1 1.00×10-7 0.41 0.15Ng et al. (2017)b 40 3.5 1 1 1 1.00×10-7 0.36 0.14Ni et al. (2018)c 0 4.5 0.2 4.8 0.7 1.22×10-6 0.36 0.15Woon (2013)d 0 2.7 0.07 3 0.7 1.50×10-7 0.41 0.15

Note: (a) Based on test results of measured pore-water pressure along depth after simulated transpiration for slope models supported by tap root (uniform root).

(b) Based on measured pore-water pressure along depth after 1st drying by the simulated transpiration.

(c) Based on measured pore-water pressure along depth before ponding for the field studies reported by Ng et al. (2016b).

(d) Based on measured pore-water pressure profiles for grassed plots (G100-F) after 6 days of drying from 3-June to 9-June 2012 (see Fig. 4.5 in Woon (2013)).

Page 50 of 52



Draft

1

1 Appendix A1

2 When CN is small, in Eq. (15) is much smaller than the other two terms. Hence Eq. '2

1exp( )

L

CN

3 (15) can be further simplified as:

4 (A1)

' ', ''22'

' '2 2

1 1[ ]1cos cos exp( )

1 1 [ ]cos cos exp( cos )

r veg z L

RNkLL

CN LRNL L

5 Using Taylor expansion

6 (A2)' '2 2

1 1exp( cos ) 1 cosL L

7 Substituting Eq. (A2) into Eq. (A1) yields:

8 (A3)

' ', ' '2 2

1 1[ ]cos cos (1 cos )

(1 )

r veg z L

RNkL L

RNr

CN

9 The influence of root water uptake on PWP profiles may be evaluated by comparing with the

10 effects of water table (i.e., in Eq. (14)): 1exp( )CN

Page 51 of 52



Draft

2

11 (A4)

1

1

1

1

1(1 )

exp( )exp( ) =(1 )

RNCN rCNCNRN

CN r

Page 52 of 52



draft · 114 dimensional analysis of pwp response in a vegetated infinite slope 115 review of an...

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