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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
Is the invited manuscript for consideration in a Special
Issue? :Not applicable (regular submission)
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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
20 E-mail: [email protected]
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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27 Address: Department of Civil and Environmental Engineering, Hong Kong University of 28 Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
29 E-mail: [email protected]
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
627 References
628 Butterfield, R. 1999. Dimensional analysis for geotechnical engineers. Géotechnique, 49(3), pp.
629 357–366.
Page 32 of 52
https://mc06.manuscriptcentral.com/cgj-pubs
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33
630 CAE 2000. Landfill Guidelines: Towards Sustainable Waste Management in New Zealand. Centre
631 for Advanced Engineering, University of Canterbury, Private Bag 4800, Christchurch, New
632 Zealand, pp. 45.
633 COMSOL 4.3b. 2013. COMSOL multiphysics reference manual. COMSOL, Inc., Stockholm,
634 Sweden.
635 Fatahi, B, Khabbaz, H., and Indraratna, B. 2009. Parametric studies on bioengineering effects of
636 tree root-based suction on ground behaviour. Ecological Engineering, 35(10), pp. 1415-1426.
637 Feddes, R.A., Kowalik, P.J., and Zaradny, H. 1978. Simulation of field water use and crop yield.
638 Centre for Agricultural Publishing and Documentation, Wageningen, Simulation Monographs,
639 pp. 194-209.
640 Fell, R., Ho, K.K., Lacasse, S., and Leroi, E. 2005. A framework for landslide risk assessment and
641 management. Landslide risk management, Taylor and Francis, London, pp. 3-25.
642 Feng, S., Ng, C.W.W., Leung, A.K., and Liu, H.W. 2017. Numerical modelling of methane
643 oxidation efficiency and coupled water-gas-heat reactive transfer in a sloping landfill cover.
644 Waste Management, 68, pp. 355-368.
645 Gardner, W.R. 1958. Some steady-state solutions of the unsaturated moisture flow equation with
646 application to evaporation from a water table. Soil Science, 85(4), pp. 228-232.
647 Glendinning, S., Loveridge, F., Starr-Keddle, R.E., Bransby, M.F., and Hughes, P.N. 2009. Role
648 of vegetation in sustainability of infrastructure slopes. Engineering Sustainability, 162(2), pp.
649 101–110.
650 González-Ollauri, A., and Mickovski, S.B. 2014. Integrated model for the hydro-mechanical
651 effects of vegetation against shallow landslides. EQA-International Journal of Environmental
652 Quality, 13(13), pp. 37-59.
Page 33 of 52
https://mc06.manuscriptcentral.com/cgj-pubs
Canadian Geotechnical Journal
Draft
34
653 Green, W.H., and Ampt, G.A. 1911. Study on soil physics: I. Flow of air and water in soils. The
654 Journal of Agricultural Science, 4, pp. 1–24.
655 Indraratna, B., Fatahi, B., and Khabbaz, H. 2006. Numerical analysis of matric suction effects of
656 tree roots. Proceedings of the ICE - Geotechnical Engineering, 159(2), pp. 77–90.
657 Jotisankasa, A., and Sirirattanachat, T. 2017. Effects of grass roots on soil-water retention curve
658 and permeability function. Canadian Geotechnical Journal, 54(11), pp. 1612-1622.
659 Khalilzad, M., Gabr, M.A., and Hynes, M.E. 2014. Effects of woody vegetation on seepage-
660 induced deformation and related limit state analysis of levees. International Journal of
661 Geomechanics, 14(2), pp. 302-312.
662 Kasim, F.B., Fredlund, D.G., and Gan, J.K.M. 1998. Effect of steady state rainfall on long term
663 matric suction conditions in slopes. Proceedings of the second international conference on
664 unsaturated soils, Beijing, China. Vol. 1, pp. 78–83.
665 Kozlowski, T.T. 1999. Soil compaction and growth of woody plants. Scandinavian Journal of
666 Forest Research, 14(6), pp. 596-619.
667 Leung, A.K., and Ng, C.W.W. 2013. Analyses of groundwater flow and plant evapotranspiration
668 in vegetated soil slope. Canadian Geotechnical Journal, 50(12), pp. 1204–1218.
669 Leung, A.K., Garg, A., and Ng, C.W.W. 2015. Effects of plant roots on soil-water retention and
670 induced suction in vegetated soil. Engineering Geology, 193, pp. 183-197.
671 Li Y., and Ghodrati, M. 1994. Preferential transport of nitrate through soil columns containing root
672 channels. Soil Science Society of American Journal, 58, pp. 653–659.
673 Liu, H. W. 2017. Effects of root architecture on pore-water pressure distributions and slope
674 stability: analytical and experimental study. PhD thesis, the Hong Kong University of Science
675 and Technology, Hong Kong, pp. 145.
Page 34 of 52
https://mc06.manuscriptcentral.com/cgj-pubs
Canadian Geotechnical Journal
Draft
35
676 Liu, H.W., Feng, S., and Ng, C.W.W. 2016. Analytical analysis of hydraulic effect of vegetation
677 on shallow slope stability with different root architectures. Computers and Geotechnics, 80, pp.
678 115-120.
679 Liu, H.W., Feng, S., Garg, A. and Ng, C.W.W. 2018. Analytical solutions of pore-water pressure
680 distributions in a vegetated multi-layered slope considering the effects of roots on water
681 permeability. Computers and Geotechnics, 102, pp. 252-261.
682 Lu, N., and Godt. J. 2008. Infinite slope stability under steady unsaturated seepage conditions.
683 Water Resources Research, 44(11), pp. 1-13.
684 Morrison, R., and Szecsody, J. 1985. Sleeve and casing lysimeters for soil pore water sampling.
685 Soil Science, 139, pp. 446–451.
686 Ng, C.W.W., Liu, H.W., and Feng, S. 2015. Analytical solutions for calculating pore water
687 pressure in an infinite unsaturated slope with different root architectures. Canadian
688 Geotechnical Journal, 52, pp.1981-1992.
689 Ng, C.W.W., Kamchoom, V., and Leung, A.K. 2016a. Centrifuge modelling of the effects of root
690 geometry on transpiration-induced suction and stability of vegetated slopes. Landslides, 13(5),
691 pp. 925-938.
692 Ng, C.W.W., Ni J.J., Leung, A.K., and Wang, Z.J. 2016b. A new and simple water retention model
693 for root-permeated soils. Géotechnique Letter, 6(1), pp.106–11.
694 Ng, C.W.W., Leung, A.K., Yu, R., and Kamchoom, V. 2017. Hydrological effects of live poles on
695 transient seepage in an unsaturated soil slope: centrifuge and numerical study. Journal of
696 Geotechnical and Geoenvironmental Engineering, 04016106.
697 Ni, J.J., Leung, A.K., Ng, C.W.W., and Shao, W. 2018. Modelling hydro-mechanical
698 reinforcements of plants to slope stability. Computers and Geotechnics, 95, pp. 99-109.
Page 35 of 52
https://mc06.manuscriptcentral.com/cgj-pubs
Canadian Geotechnical Journal
Draft
36
699 Nyambayo, V.P., and Potts, D.M. 2010. Numerical simulation of evapotranspiration using a root
700 water uptake model. Computers and Geotechnics, 37(1–2), pp. 175–186.
701 Philip, J.R. 1957. Theory of infiltration: 1. The infiltration equation and its solution. Soil Science,
702 83, pp. 345–357.
703 Philip, J.R. 1969. Theory of infiltration. Soil Science, 5, pp. 215–296.
704 Prasad, R. 1988. A linear root water uptake mode. Journal of Hydrology, 99(3), pp. 297–306.
705 Raats, P.A.C. 1974. Steady flows of water and salt in uniform soil profiles with plant roots. Soil
706 Science Society of America Journal, 38(5), pp. 717-722.
707 Sinnathamby, G., Phillips, D.H., Sivakumar, V., and Paksy, A. 2013. Landfill cap models under
708 simulated climate change precipitation: impacts of cracks and root growth. Géotechnique, 64
709 (2), pp. 95–107.
710 Simon, A., and Collison, A.J. 2002. Quantifying the mechanical and hydrologic effects of riparian
711 vegetation on streambank stability. Earth Surface Processes and Landforms, 27(5), pp. 527-546.
712 Smethurst, J.A., Clarke, D., and Powrie, W. 2006. Seasonal changes in pore water pressure in a
713 grass covered cut slope in London Clay. Géotechnique, 56 (8), pp. 523–537.
714 Smethurst, J.A., Briggs, K.M., Powrie, W., Ridley, A., and Butcher, D.J.E. 2015. Mechanical and
715 hydrological impacts of tree removal on a clay fill railway embankment. Géotechnique, 65(11),
716 pp. 869–882.
717 Srivastava, R., and Yeh, T.C.J. 1991. Analytical solutions for one-dimension, transient infiltration
718 toward the water table in homogeneous and layered soils. Water Resources Research, 27(5),
719 pp. 753–762.
720 Unger, P.W., and Kaspar, T.C. 1994. Soil compaction and root growth: a review. Agronomy
721 Journal, 86(5), pp. 759-766.
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722 Vanapalli, S.K., Sillers, W.S., and Fredlund, M.D. 1998. The meaning and relevance of residual
723 state to unsaturated soils. In Proceedings of the 51st Canadian Geotechnical Conference,
724 Edmonton, Alta, pp. 4-7.
725 Vergani, C., and Graf, F. 2016. Soil permeability, aggregate stability and root growth: a pot
726 experiment from a soil bioengineering perspective. Ecohydrology, 9(5), pp. 830-842.
727 Woon, K.X. 2013. Field and Laboratory Investigations of Bermuda Grass Induced Suction and
728 Distribution. Mphil thesis, Hong Kong University of Science and Technology, Hong Kong, pp.
729 67-79.
730 Yuan, F., and Lu, Z. 2005. Analytical Solutions for Vertical Flow in Unsaturated, Rooted Soils
731 with Variable Surface Fluxes. Vadose Zone Journal, 4(4), pp. 1210-1218.
732 Zhan, T.L., and Ng, C.W.W. 2004. Analytical analysis of rainfall infiltration mechanism in
733 unsaturated soils. International Journal of Geomechanics, 4(4), pp. 273-284.
734 Zhan, T.L., Jia, G.W., Chen, Y.M., Fredlund, D.G., and Li, H. 2013. An analytical solution for
735 rainfall infiltration into an unsaturated infinite slope and its application to slope stability
736 analysis. International Journal for Numerical and Analytical Methods in Geomechanics, 37(12),
737 pp. 1737-1760.
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
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H0 Root zone
L2
L1
Figure. 1. Schematic diagram showing a vegetated infinite slope (after Ng et al. 2015)
q0
'1L
'2L
'L
'z
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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
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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
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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)
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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)
Wetting steady state
Normalized pore-water pressure by hydrostatic pressure ( )
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)
Wetting steady state
Normalized pore-water pressure by hydrostatic pressure ( )
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)
Wetting steady state
Normalized pore-water pressure by hydrostatic pressure ( )
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)
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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
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(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
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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
Measured ks=1.50*10-7 m/s
Measured ks=1.00*10-7 m/s
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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
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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))
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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
Transient state (wetting)
2d
0.5
0.4
infinite
0.7
Not applicable Steady state(wetting)3a 12 0.093b 24 0.23c 300 2
Transient state (wetting)
3d
0.2
infinite Not applicable Steady state (wetting)4a 12 0.044b 24 0.094c 300 1
Transient state (wetting)
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).
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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)).
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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
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11 (A4)
1
1
1
1
1(1 )
exp( )exp( ) =(1 )
RNCN rCNCNRN
CN r
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