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Computers and Geotechnics

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Research Paper

Analytical solution for one-dimensional vertical electro-osmotic drainageunder unsaturated conditions

Liujiang Wanga, Yaoming Wanga,⁎, Sihong Liua, Zhongzhi Fub, Chaomin Shena, Weihai Yuanc

a College of Water Conservancy and Hydropower, Hohai University, Nanjing 210098, ChinabNanjing Hydraulic Research Institute, Nanjing 210029, Chinac College of Mechanic and Materials, Hohai University, Nanjing 210098, China

A R T I C L E I N F O

Keywords:Electro-osmosisDrainageUnsaturated soilAnalytical solution

A B S T R A C T

In this study, exponential functions were incorporated in a one-dimensional model to represent the relationshipsof hydraulic and electro-osmotic conductivities with matric suction and the soil-water characteristic curve.Analytical solutions for pore water pressure, water content and drainage were derived, and laboratory tests wereperformed to verify the effectiveness of these solutions. Finally, a parametric study indicated that the soil-watercharacteristics of the soil had a remarkable impact on the drainage behaviour of electro-osmosis under un-saturated conditions and that the proposed analytical solutions could be used to design electro-osmotic drainagesystems in unsaturated soils.

1. Introduction

Electro-osmosis is an electrically induced process in which porewater is driven from the anode to the cathode through the dissolvedelectrolytes, thus leading to the dewatering and consolidation of satu-rated soil or other porous media. Casagrande [1] used this phenomenonto consolidate soft soil and enhance its geotechnical properties for en-gineering purposes. Since then, electro-osmosis has been successfullyused for soft ground improvement, slope stabilisation, dewatering oftailings and sludge, pile foundation and soil remediation [2–12].However, problems such as electrode erosion, low efficiency, unevenconsolidation and unsatisfactory treatment results widely exist inpractical applications, and improvement methods such as polarity re-versal, application of alternating current, injection of a chemical solu-tion and combination with other consolidation methods are highlydesired by geotechnical engineers [13]. Recently, electro-osmosis hasbeen used in unsaturated soils to enhance the performance of com-pacted clay liners at polluted sites by electrokinetic barriers [14–16],improve the stability of partially saturated slopes [17–18], reduce theadhesion of excavated clay materials on steel surfaces of tunnel drivingmachines [19] and decrease the water content of over-wet subgrade fillor expansive soil [20–23].

On the basis of the assumption that the pore water flow resultingfrom the hydraulic gradient and the electrical gradient can be super-imposed linearly, the governing equation for electro-osmotic

consolidation was developed, and many analytical solutions were de-rived using different conditions to analyse the development of porewater pressure and the degree of consolidation [24–31]. Esrig [24] firstproposed a one-dimensional theoretical model and derived the solutionfor the excess pore water pressure under different boundary conditions.Wan and Mitchell [25] investigated the coupling effect of surchargepreloading and electro-osmotic consolidation. Following their pio-neering work, several analytical models were proposed for electro-os-motic consolidation, including the two-dimensional model in a verticalplane [26–27] and the two-dimensional model in a horizontal plane[28]. Considering that prefabricated vertical drain and electric verticaldrain are often installed in an equilateral triangular pattern in theground, axisymmetric models were developed and the correspondinganalytical solutions were derived [11,29]. Recently, the non-linearvariations in soil compressibility and hydraulic and electro-osmoticconductivities were incorporated into a one-dimensional model, andanalytical solutions for excess pore water pressure and degree of con-solidation were derived [30,31]. These mathematical analyses gener-ated significant knowledge pertaining to electro-osmotic consolidationand provided useful formulas for engineering design. However, all theseanalytical solutions were derived based on the assumption that the soilis fully saturated. The flow of pore water from the anode to the cathodechanges the soil from a saturated state to an unsaturated state duringelectro-osmosis, thus leading to a decrease in the degree of saturationand non-linear variations in soil properties such as hydraulic

https://doi.org/10.1016/j.compgeo.2018.09.011Received 22 June 2018; Received in revised form 14 September 2018; Accepted 15 September 2018

⁎ Corresponding author.E-mail addresses: 15850514642@163.com, wymhhu@163.com (Y. Wang).

Computers and Geotechnics 105 (2019) 27–36

0266-352X/ © 2018 Elsevier Ltd. All rights reserved.

T

conductivity and electro-osmotic conductivity [32–37]. Such variationsinevitably affect the development of pore water pressure and drainageduring electro-osmosis, and predictions from the existing analyticalsolutions with the assumption of fully saturated soil would be in-accurate. However, to the best of our knowledge, there is no existinganalytical solution for electro-osmotic drainage under unsaturatedconditions.

In this study, a one-dimensional model for electro-osmotic drainagein the vertical direction considering unsaturated effects was proposedon the basis of the conservation of fluid mass, Darcy’s law and theelectro-osmotic flow equation. Then, exponential functions were usedto represent the relationships of hydraulic and electro-osmotic con-ductivities with matric suction and the soil-water characteristic curve,and these were incorporated into the one-dimensional model. Theanalytical solutions for pore water pressure, water content and drainagewere derived, which were reasonably well comparable with the ex-perimental results. Finally, a parametric study was conducted to in-vestigate the influence of the soil-water characteristics on the drainagebehaviour of electro-osmosis under unsaturated conditions.

2. Theoretical analysis

Similar to previous studies, a schematic diagram of a one-dimen-sional model for electro-osmotic drainage was developed in this studyas shown in Fig. 1, with the anode on the bottom and the cathode on thetop [30,31]. The bottom boundary is impermeable, and the topboundary is permeable. As in previous studies, the z axis was directeddownwards [26,31]. Thus, the coordinate of any point in this one-di-mensional model can be represented using a positive z value, which isconsistent with the depth of the model, H. The following assumptionswere made to develop the analytical model for electro-osmotic drainageunder unsaturated conditions.

(1) The soil is homogeneous, and under unsaturated conditions, theelectrical properties of the soil mass are constant over time.

(2) The pore water and soil grain are incompressible, the deformationof the soil is neglected and the drainage of pore water occurs in thevertical direction.

(3) The velocity of pore water flow due to electro-osmosis is directlyproportional to the electrical gradient and can be superimposedlinearly due to the hydraulic gradient.

(4) The pore water flow caused by the thermal gradient and chemicalconcentration gradient is neglected.

(5) The differential equation describing electro-osmotic flow in un-saturated soil obeys the Richards’ equation.

(6) The pore air pressure in the soil is kept unchanged, and a constantatmospheric pressure is maintained.

For the configuration shown in Fig. 1, a voltage V= V0 is appliedbetween the electrodes. In this case, the electric field intensity is ex-pressed as follows:

= −∇ = −∂∂

E iV Vz (1)

where E is the intensity of the electric field (in V/m) and i is a unitvector in the z direction. The negative sign in Eq. (1) denotes that thedirection of the field intensity vector is from the anode (positive elec-trode) to the cathode (negative electrode). From assumption (1) thatthe electrical properties of the soil are homogeneous and constant overtime, the electric field can be approximated as uniform within the soilmass. Thus, the distribution of the electrical voltage gradient betweenelectrodes is assumed to be linear during the electrokinetic process.This assumption is in agreement with theory and the experimentalobservation at lower electric field intensities (typically, |E| < 100 V/m) [26]. In addition, considering the voltage loss at the electrode-soilcontact, the magnitude of the electric field intensity is usually lowerthan the total applied voltage divided by the electrode spacing, V0/H (His the height of the model), which has been observed in many labora-tory and field tests [39].

By using the electric field intensity expressed in Eq. (1) and on thebasis of the proportionality between the electrically induced velocity ofwater flow through the soil and the voltage gradient, the combined porewater flow induced by the hydraulic and electrical gradients duringelectro-osmosis can be described as follows [24]:

= −∂ −

∂− ∂

∂v k

γu γ z

zk V

z( )

zw

w

weo

(2)

where vz is the pore water flow in the vertical direction; γw is the unitweight of water; u is the pore water pressure (in such unsaturated si-tuation, the matric suction is equal to the negative pore water pressure);and kw and keo are the hydraulic and electro-osmotic conductivities ofunsaturated soil, respectively.

On the basis of the conservation of fluid mass and assumptions (3)and (5) and neglecting the deformation of unsaturated soil, the one-dimensional non-linear differential equation that describes water flowin unsaturated soil during electro-osmotic drainage is expressed asfollows:

∂∂

⎡⎣⎢

∂ −∂

+ ∂∂

⎤⎦⎥

= ∂∂

= ∂∂

∂∂z

kγ

u γ zz

k Vz

θ z tt

θu

ut

( ) ( , )w

w

weo

(3)

where θ is the volumetric water content, ∂θ/∂u is the storativity and t isthe time.

Both the hydraulic conductivity and the volumetric water content ofunsaturated soil are assumed to be functions of the matric suction [38]

=k u k e( )w sαu (4a)

= + −θ u θ θ θ e( ) ( )r s rαu (4b)

where θs and θr are the volumetric water content at saturation and theresidual volumetric water content, respectively; ks is the hydraulicconductivity at saturation and α is the desaturation coefficient thatrepresents the pore size distribution.

Similar to hydraulic conductivity under unsaturation conditions,electro-osmotic conductivity can also be defined as the product of theintrinsic conductivity, depending on the microstructural properties ofthe porous medium, and a non-dimensional relative conductivitycoefficient, which accounts for the effects of partial saturation on soilelectro-osmotic conduction properties. According to the experimentalresults obtained in electrokinetic filtration tests performed on un-saturated specimens [32,33], the exponential relationship between theelectro-osmotic conductivity and the matric suction is also satisfied.

=k u k e( )eo eβu (5)

where ke is the electro-osmotic conductivity at saturation and β is therelative electro-osmotic conductivity exponent. To facilitate the devel-opment of analytical solutions, the relative electro-osmotic conductivityexponent β is assumed to be equal to the desaturation coefficient α inFig. 1. One-dimensional model for electro-osmotic drainage.

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28

this study.Substituting Eq. (4) and Eq. (5) into Eq. (3), we can obtain

∂∂

⎡⎣⎢

∂ −∂

+ ∂∂

⎤⎦⎥

= − ∂∂z

kγ

eu γ z

zk e V

zα θ θ e u

t( )

( )s

w

αu we

αus r

αu

(6)

Then, introducing a dummy variable ξ as named by Esrig [24],

= +ξ u z t p z( , ) eo (7a)

= − = −p f γ kk

Eγ γeo eo we

sw w (7b)

where E is the absolute value of the field intensity vector E and feo is thecoefficient of electro-osmotic pressure, which physically corresponds tothe slope of the electro-osmotic pressure that yields a triangular dis-tribution of electro-osmotic pressure in the one-dimensional direction[24,26].

After substituting Eq. (7a) into Eq. (6), we can obtain

⎜ ⎟

∂∂

⎛⎝

∂∂

⎞⎠

−∂∂

=− ∂

∂zξz

e αpξz

eαγ θ θ

ke

ξt

( )αξeo

αξ w s r

s

αξ

(8)

According to Eq. (8), we introduce the Kirchhoff variable

=W eαξ (9)

so that Eq. (8) can be rearranged as follows:

∂∂

−∂∂

=− ∂

∂αpWz

Wz

γ θ θk p

Wt

1 ( )

eo

w s r

s eo

2

2 (10)

Substituting Z= αpeoz and T= αpeo2kst/γw(θs-θr) and then defininga new variable S(Z,T),

= −S W e·T Z4 2 (11)

Then, Eq. (10) can be further rearranged as

∂∂

= ∂∂

ST

SZ

2

2 (12)

By corresponding transformations, the boundary conditions in-troduced in Fig. 1 can be rewritten as follows:

⎧⎨⎩

= = ⇒ =

= − + = ⇒ + =∂∂

∂∂

∂∂( )

z u S e

z H k z k

0, 0

, 0 0

T

w zuγ eo

Vz

SZ

S

/4

2w (13)

Through similar transformations, the initial pore water pressure canalso be written as

= = ⇒ = +T u u S e0, iαu Z/2i (14)

where ui is the initial pore water pressure at t=0.Combining Eq. (12) with Eqs. (13) and (14) and using the method of

the separation of variables, u can be solved (details are given in Ap-pendix 1).

= − −

= − + ∑ − −+ =∞{ }( )

u z t S Z T Z T

e C β T β Z

( , ) {ln[ ( , )] /2 /4}

ln 1 ( , ) sin( )

α

αZ

L n n n nZα

Tα

1

12 1 2 4

T4 (15)

The functions in Eq. (15) are

∫ ∫=

+ ⎡⎣⎢

⎛⎝ +

− ⎞⎠

⎤⎦⎥

−

−

C β T B β e

A βe Z

Le β Z dZ dτ

( , ) ( )

1( ) 8 4

14

sin( )

n n n nβ T

n n

T β τ T Ln0

( )0

n

nτ

2

24

(16a)

∫= ⎛⎝

++

− ⎞⎠

+B βA β

e ZL

β Z dZ( ) 1( ) 2

1 sin( )n nn n

L αun0

Z/2s

(16b)

∫=A β β Z dZ( ) sin ( )n nL

n02

(16c)

in which the eigenvalue βn satisfies βncot(βnL)=−0.5 and L= αpeoH.

By substituting the obtained pore water pressure into the soil-watercharacteristic curve (expressed as shown in Eq. (3b), the volumetricwater content can be calculated, and then, the quantity of drainage atany time, Q(t), can be obtained as follows:

∫ ∫= − + −Q θ dz θ θ θ e dz[ ( ) ]H

iH

r s rαu z t

0 0( , )

(17)

where θi is the initial volumetric water content.The integral in Eq. (17) cannot be expressed directly by elementary

functions. Therefore, the Newton-Cotes formula is used to estimate itsvalue.

3. Experimental validation

Laboratory tests and large-scale experiments on electro-osmoticdrainage in unsaturated soil were conducted by Li and Zhuang et al.[40,41], respectively. However, these experiments were carried out tosimulate two-dimensional electro-osmotic drainage, and the corre-sponding calculation parameters were not tested. Hence, it is difficult toadopt these experimental results to validate the proposed analyticalsolutions. In this study, a laboratory test was conducted to verify thecapacity and calculation accuracy of a one-dimensional analytical so-lution for electro-osmotic drainage under unsaturated conditions. Softsoil used in the test was taken from Jiangxin Island, which is located inNanjing, Jiangsu Province, China. Before the model test, the geo-technical properties of the soft soil and corresponding parameters in-cluded in the analytical solutions were tested, as listed in Table 1. Then,the soil was oven dried, mixed with water at a water content of 40%and compacted into the test device in five layers according to thepredetermined void ratio in the range of 0.95–1.10. Finally, a plasticmembrane was laid on the surface of the soil sample to prevent theevaporation of pore water.

As shown in Fig. 2, the electro-osmotic drainage test was conductedusing a self-designed apparatus that was 200mm in diameter and550mm in height. An aluminium sheet was used as the electrode, withthe anode plate placed on the top of the soil sample and the porouscathode plate was placed on the bottom, allowing the drainage of porewater into a bottle placed on the electronic balance. To investigate thechanges in the water content at different places between the anode andthe cathode during electro-osmosis, four soil moisture sensors wereinstalled along the height of the soil sample at intervals of 150mm. Thesoil moisture sensor has a measurement range of 100% and a measuringaccuracy of± 3%, with an output capacity of 2 V and 20mA. In addi-tion, the other devices included a direct current (DC) power supply withan output capacity of 60 V and 5 A, electrical wiring and five potentialneedles. The needles were Φ2 mm wires that were insulated, except theoutermost 5mm of the wire, positioned at 150 or 190mm intervals.Needles E1 and E4 were inserted into the soil close to the anode and thecathode, respectively, which can measure the effective voltage appliedto the soil sample.

When the sample preparation was completed, the drainage valve atthe bottom of the test device was closed, and the whole system was

Table 1Geotechnical properties and calculation parameters of soft soil.

Property Value

Plastic limit, wP (%) 19.8Liquid limit, wL (%) 41Specific gravity of the solid, Gs 2.67Initial water content, w0 (%) 48.5Initial void ratio, e0 1.32Moisture at saturation, θs 0.52Residual moisture, θr 0.3Desaturation coefficient, α (kPa−1) 0.016Saturated hydraulic conductivity, ks (m/s) 9.7× 10−9

Saturated electro-osmotic conductivity, ke (m2·s−1·V−1) 5.0× 10−9

L. Wang et al. Computers and Geotechnics 105 (2019) 27–36

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allowed to stand for at least 2 days to reach equilibrium. Next, a DCvoltage of 25 V was applied continuously for 172 h. In the electro-os-motic drainage test, the drainage, water content and voltage weremonitored every 1–2 h and sometimes at long intervals such as at nighttime under restricted conditions.

During the test, the effective voltage applied to the soil sample(between E1 and E4) was found to decrease with time because of theincrease in the interfacial resistance with the duration of electro-os-mosis. Considering the increase in voltage loss with time, an averageeffective voltage of 14 V was substituted into the analytical solutions.Except the effective voltage, the other corresponding parameters usedin the analytical solutions are listed in Table 1, and the initial volu-metric water content (θi) along the height of soil sample was assumedto be uniform with a value of 0.5. Because the drainage direction in thetest was opposite to that in the theoretical model in Fig. 1, Eq. (7b) canbe rewritten as follows:

= +p kk

Eγ γeoe

sw w (18)

Fig. 3 presents the time behaviour of the electro-osmotic drainagebetween the measured and the calculated values. The calculated resultshowed a good agreement with the experimental observation, althoughthere was a little difference between the measured and calculated re-sults during the initial 80 h. The faster increase in the measured drai-nage in the early stages can be attributed to the actual applied effectivevoltage in this period being greater than the actual applied effective

voltage of the average effective voltage used in the analytical solutions.Fig. 4 shows the comparisons of the volumetric water content at M1,M2 and M4 between the measured and the calculated results. Becauseof the failure of the soil moisture sensor of M3, the data for the volu-metric water content at this point were not obtained. As shown inFig. 4, the decrease in the volumetric water content for the soil aroundthe anode was much larger than that near the cathode; this is consistentwith the findings of previous experimental studies [42,43]. The volu-metric water content for the soil near the cathode fluctuated throughoutthe test, although the overall trend showed a decrease. This phenom-enon could be attributed to the formation of a relatively impermeablesoil layer above the porous stone, which caused the pore water not toand accumulated near the cathode, which is different from thepermeable boundary defined in the mathematical model. In general, thecalculated volumetric water content is basically in good agreement withthe observations, especially for soils around the anode (M1 and M2).Hence, this good agreement between the calculated and measured va-lues indicates that the proposed analytical solutions can predict thebehaviour of electro-osmotic drainage under unsaturated conditions.

Fig. 2. One-dimensional apparatus for an electro-osmosis experiment (lengthunit: mm).

Fig. 3. Comparison of drainage between the calculated and experimentallymeasured values.

Fig. 4. Comparison of volumetric water content between the calculated andexperimentally measured values.

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4. Parametric study

To investigate the influence of the hydraulic conductivity and soil-water characteristic relationship on the time behaviour of the porewater pressure and drainage, a parametric study was conducted usingthe different values of the model parameters listed in Table 2. Theheight of the homogeneous soil was taken to be 1.0 m. The initial porewater pressure was assumed to be ui = 0. The effective voltage wasfixed at V0= 50 V. In the following sensitivity analysis, the influence ofthe ratio of the saturated electro-osmotic conductivity to the saturatedhydraulic conductivity (ke/ks), desaturation coefficient (α) and residualvolumetric water content (θr) were studied separately.

4.1. Effect of ke/ks

Fig. 5 shows the time variations in the average pore water pressuresand drainage that correspond to the three values of ke/ks (i.e., ke/ks = 0.1, 1.0 and 10), with other parameters fixed as follows:ke= 2.0× 10−9 m2·s−1·V−1, α=0.01 kPa−1, θs= 0.7, and θr = 0.2.As shown in Fig. 5(a), the value of ke/ks had a significant influence onthe development of the pore water pressure. The greater the value of ke/

ks, the greater was the value of the matric suction. The drainage alsoincreased with an increase in the value of ke/ks, as shown in Fig. 5(b).The variation in the drainage with ke/ks was different from that in thematric suction. When ke/ks increased from 0.1 to 1.0, the increase in thematric suction was much smaller than that when ke/ks increased from1.0 to 10. However, the increase in the drainage was much larger be-cause of the exponential relationship between the volumetric watercontent and the matric suction (the soil-water characteristic curve). Asthe matric suction increased within a relatively small range, the volu-metric water content decreased significantly. Nevertheless, when thematric suction was large and increased further, the volumetric watercontent of the soil decreased slightly and then tended to be invariable.Therefore, the drainage could be found to increase non-linearly withthe increase in ke/ks, especially for soil with a large α value. In thissituation, because electro-osmotic consolidation time would be pro-longed with the increase in ke/ks [29], a larger value of ke/ks is notsuitable for electro-osmosis. Overall, the determination of a suitablevalue of ke/ks for electro-osmosis should balance its effects on drainagevolume and time and consider the influence of the soil-water char-acteristic relationship.

4.2. Effect of α

Three values of the desaturation coefficient (α=0.005, 0.01 and0.05 kPa−1) were used to illustrate its effect on the time behaviour ofpore water pressure and drainage under electro-osmosis. As shown inFig. 6(a) and (b), the value of ke/ks was taken as 1.0. Fig. 6(a) shows thevariations in the average pore water pressure versus time for the soil

Table 2Parameters of one-dimensional analysis of electro-osmotic drainage.

ks (1× 10−9 m/s) ke (1× 10−9 m2 s−1 V−1) α (kPa−1) θs θr

0.2, 2.0 and 20 2.0 0.005, 0.01and 0.05

0.7 0.1, 0.2and 0.3

Fig. 5. Average pore water pressure and drainage versus time relationships fordifferent ke/ks ratios.

Fig. 6. Average pore water pressure and drainage versus time relationships fordifferent values of α.

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with α=0.005, 0.01 and 0.05 kPa−1. Because the changes in the hy-draulic and electro-osmotic conductivities with the matric suction wereassumed to be consistent in this study, the ratio of the hydraulic con-ductivity to electro-osmotic conductivity (keo/kw) during electro-os-mosis remained constant. Thus, the ultimate pore water pressures wereequal for soil with different α values. For the unsaturated soil, the de-saturation coefficient is an important parameter related to the particlesize distribution of the soil. A small value of α leads to a high retentioncapability for water in the soil, which inhibits the drainage of the porewater. In contrast, a large value of α means a low water retentioncapability, which promotes the drainage of the pore water. Therefore,the α value has a significant effect on the variations of pore waterpressure and drainage during electro-osmosis. The greater the α value,the faster is the increase in the drainage and the slower is the devel-opment of the matric suction. Fig. 6(b) shows the influence of the de-saturation coefficients on drainage under electro-osmosis. As expected,the drainage increased with the increase in α value. As the α value in-creased from 0.005 to 0.01 kPa−1, the drainage increased from 307.8 to393.4 L/m2 (increase of 95.6 L/m2). As the α value increased further to0.05 kPa−1, a further increase of 79.6 L/m2 was observed. Note that anon-linear relationship existed between the drainage and desaturationcoefficient (α), consistent with the soil-water characteristic curve, inwhich the relationship between the water content and the matric suc-tion is described by an exponential function. Regarding the power ofthe exponential function, changes in the α value within a relativelysmall range led to a considerable variation in the drainage. Therefore,considering that the main purpose of electro-osmotic drainage is toremove excessive water from unsaturated soils, it is preferable to havesoil with a relatively large α value.

4.3. Effect of θr

The residual volumetric water content is also an important para-meter in the soil-water characteristic curve, as shown in Eq. (4b). Threevalues of θr, 0.1, 0.2 and 0.3, were used to illustrate its effect while theother parameters remained the same. The value of ke/ks was taken as1.0, and the desaturation coefficient (α) was taken as 0.01. Fig. 7(a) and(b) show the development of pore water pressure and drainage withtime for soil with different θr values. Together with the desaturationcoefficient (α), the residual volumetric water content (θr) can reflect thewater retention capability of the soil. The larger the value of θr, thehigher is the water retention capability. Hence, for soil with a larger θr,the decrease in drainage is more remarkable, and the development ofpore water pressure is faster. As Fig. 7(b) shows, the drainage decreasedwith the increase in θr, and the relationship between the drainage andθr was linear. In addition, the drainage time (the time corresponding to90% of final drainage) also decreased with the increase in θr. As the θrvalue increased from 0.1 to 0.3, the drainage time decreased from 81 to54 days. Hence, the effects of the residual volumetric water content (θr)on the drainage volume and time should be balanced when designingan electro-osmotic drainage system in unsaturated soils.

5. Discussion

As mentioned above, the exponential coefficients for the hydraulicand electro-osmotic conductivities (α and β) were assumed to be equalin this study to facilitate the development of analytical solutions. In thiscase, the effects of non-linear variations in kw and keo were balanced byeach other; therefore, the analytical solution for the ultimate pore waterpressure simplifies to the equation proposed by Esrig [24] and Wan andMitchell [25] as follows:

= − +uk γ

kV z γ z

·( )ult

e w

sw (19)

Thus, the use of the same exponential coefficients for the hydraulicand electro-osmotic conductivities will have a large effect on the pre-diction of the pore water pressure. To study the effect of this choice,numerical simulations were performed considering the non-synchro-nous variations in kw and keo with the matric suction. The calculationmodel and parameters were the same as those in the above parametricstudy, with V0= 50 V, H=1m, ke= 1×10−9 m2 s−1 V−1, ke/ks = 1,θs = 0.7, θr = 0.2, and α=0.01 kPa−1. According to the previous ex-perimental studies by de Wet [32] and Gabrieli et al [33], with a de-crease in the degree of saturation, the decrease in the electro-osmoticconductivity was found to be smaller than that in the hydraulic con-ductivity, which indicates α> β in practice. Thus, in these simulations,the value of β was varied at 0.01, 0.009 and 0.008 kPa−1, while thevalue of α was fixed at 0.01 kPa−1. Fig. 8(a) and (b) show the devel-opment of pore water pressure and drainage over time for soil withdifferent β values. Fig. 8(a) shows that the difference in the pore waterpressure was obvious for different β values, which means that the use ofthe same exponential coefficients (β= α) may cause a large differencebetween the analytical solution and the experimental measurement of

Fig. 7. Average pore water pressure and drainage versus time relationships fordifferent values of θr.

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the pore water pressure. However, the difference in the drainage vo-lumes for different β values was minor, and the drainage volumes wereapproximately equal during the first 50 days, as shown in Fig. 8(b). Atthe final stage, the predication errors in the drainage were only 2.1%and 4.1% in the cases of α=0.01 kPa−1, β=0.009 kPa−1 and

α=0.01 kPa−1, β=0.008 kPa−1, respectively, based on the assump-tion of equal exponential coefficients. Fig. 9 shows the variations in thevolumetric water content with depth and time for different β values.The comparison result confirms that the assumption of equal ex-ponential coefficients has less influence on the predication of the watercontent. In sum, although a larger error will be caused by using theassumption of equal exponential coefficients to predict the pore waterpressure, the predicted results of the water content and the quantity ofdrainage are acceptable. Thus, in this study, the analytical solutionsdeveloped on the basis of the assumption of equal exponential coeffi-cients can reflect the behaviour of electro-osmotic drainage in un-saturated soils to some extent.

6. Summary and conclusions

In this study, a one-dimensional model for electro-osmotic drainagein the vertical direction under unsaturated conditions was proposed onthe basis of the law of fluid mass conservation, Darcy’s law and theelectro-osmotic flow equation. Exponential functions were used to re-present the relationships of hydraulic and electro-osmotic con-ductivities with soil suction and the soil-water characteristic curve. AKirchhoff variable and several dimensional variables were used to lin-earise the governing equation. Analytical solutions for the pore waterpressure, water content and drainage were derived and compared withthe results from a laboratory test for verification. Then, a parametricstudy was conducted to analyse the effect of the soil-water character-istic relationship on the behaviour of electro-osmotic drainage underunsaturated conditions.

The proposed analytical results agreed well with the experimentalobservations both for the water content and for the quantity of drai-nage; these results illustrate the accuracy of the proposed analyticalsolution. In addition, the calculated results demonstrated that the im-pact of the ratio of saturated electro-osmotic conductivity to saturatedhydraulic conductivity (ke/ks), desaturation coefficient (α) and residualvolumetric water content (θr) was significant on the development of thepore water pressure and drainage during electro-osmosis. More speci-fically, a larger ratio of the saturated electro-osmotic conductivity tosaturated hydraulic conductivity (ke/ks) resulted in a higher pore waterpressure. However, when the value of ke/ks increased beyond a certainvalue, the drainage increased slowly, but the drainage time increasedremarkably. Hence, the smaller the value of ke/ks, the higher the drai-nage. With the increase in the desaturation coefficient (α) and the de-crease in the residual volumetric water content (θr), the matric suctiondeveloped more slowly, and the quantity of the drainage increased.Moreover, the drainage increased non-linearly with an increase in the αvalue because the soil-water characteristic curve satisfied the ex-ponential relationship. According to this study, the influence of thewater retention capability of soil is significant and should be consideredin predictions of the drainage behaviour of electro-osmosis in un-saturated soils.

Acknowledgements

The authors greatly appreciate the guidance provided by ProfessorCaisheng Chen of Hohai University in the mathematical derivation.Financial support from the National Natural Science Foundation ofChina (Project No. 51509077) and the Fundamental Research Funds forthe Central Universities (Project No. 2016B03514) are gratefully ac-knowledged. The authors also appreciate the reviewers’ excellentcomments and suggestions, which helped to improve the quality of thispaper.

Fig. 8. Average pore water pressure and drainage versus time relationships fordifferent values of β.

Fig. 9. Comparisons of the distribution of volumetric water content for dif-ferent values of β.

L. Wang et al. Computers and Geotechnics 105 (2019) 27–36

33

Appendix 1

The governing equation is rearranged as follows:

∂∂

= ∂∂

ST

SZ

2

2 (11)

The boundary conditions for the problem are

⎧⎨⎩

= = ⇒ =

= − + = ⇒ + =∂∂

∂∂

∂∂( )

Z u S e

Z L k u x k u

0, 0

, ( ) ( ) 0 0

T

w zuγ eo

Vz

SZ

S

/4

2w (12)

and the initial condition can be set as

= = ⇒ = +T u u S e0, iαu Z/2i (13)

We define new variables V and M

= +S Z T N Z T M Z T( , ) ( , ) ( , ) (S1)

that satisfy the following equations:

⎧⎨⎩

= = +

= + = + + += = =

∂∂ =

=∂∂ =

=∂∂ =

=

e S N M

Z N M

| | |

0 | | |

TZ Z Z

ZZ Z L

Z LNZ Z L

Z LMZ Z L

Z L

/40 0 0

12

12

12 (S2)

Here, we let variable M satisfy the following equations:

⎧⎨⎩

=

+ ==

∂∂ =

=

M e

M

|

| 0Z

T

MZ Z L

Z L

0/4

12 (S3)

We construct a linear function to satisfy Eq. (S3), which can be written as

= −+

M Z T e ZL

e( , )2

T T/4 /4(S4)

Substituting Eqs. (S1) and (S4) into Eqs. (11) to (13), we have

∂∂

= ∂∂

−∂∂

NT

NZ

MT

2

2 (S5)

⎧

⎨⎪

⎩⎪

= − = + −=

+ =

= = =+

+

=∂∂ =

=

N Z M eN

V

| | | 1| 0

| 0

T T Tαu Z Z

L

ZNZ Z L

Z L

0 0 0/2

2

012

i

(S6)

Eq. (S5) is solved using the method of separation of variables with N given by

=N Z T P Z R T( , ) ( ) ( ) (S7)

where P is a function of Z only and R is a function of T only; thus, we have the following eigenvalue equations:

+ =d PdZ

λP 02

2 (S8)

⎧⎨⎩

=

+ ==

==

P

P

| 0

| 0Z

dPdZ Z L

Z L

012 (S9)

The solution of Eq. (S8) can be expressed as follows:

= +P Z M βZ M βZ( ) cos( ) sin( )1 2 (S10)

where M1, M2 and β are the integration constants from solving Eq. (S8). According to the boundary condition Eq. (S9), we have

⎧⎨⎩

== − = …

Mβ β L n

0cot( ) 0.5, 1, 2, 3n n

1

(S11)

Substituting Eqs. (S13) and (S14) into Eq. (S12), the following equation can be obtained:

∑==

∞

N Z T R T β Z( , ) ( ) sin( )n

n n1 (S12)

Substituting Eq. (S12) into Eq. (S5) yields

∑ ⎛⎝

+ ⎞⎠

= −∂∂

= ⎛⎝ +

− ⎞⎠=

∞ dRdT

β R β Z MT

ZL

esin( )8 4

14n

nn n n

T

1

2 /4

(S13)

L. Wang et al. Computers and Geotechnics 105 (2019) 27–36

34

Eq. (S13) can be rewritten as follows:

∫ ∫⎛⎝

+ ⎞⎠

= ⎛⎝ +

− ⎞⎠

dRdT

β R β Z dZ ZL

e β Z dZsin ( )8 4

14

sin( )nn n

Ln

L Tn

20

20

/4(S14)

Here, we define

∫=A β β Z dZ( ) sin ( )n nL

n02

(S15)

Substituting Eq. (S15) into Eq. (S14), we have

∫= ⎛⎝ +

− ⎞⎠

d R edT

eA β

ZL

e β Z dZ( )( ) 8 4

14

sin( )nβ T β T

n n

L Tn0

/4n n2 2

(S16)

Eq. (S16) can be further rearranged as

∫ ∫= + ⎡⎣⎢

⎛⎝ +

− ⎞⎠

⎤⎦⎥

− −R T R eA β

e ZL

e β Z dZ dτ( ) (0) 1( ) 8 4

14

sin( )n nβ T

n n

T β τ T Ln0

( )0

n nτ2 24

(S16)

According to the initial condition in Eq. (S6) and Eq. (S12), we have

∑++

− = =+

=

∞

e ZL

N Z R β Z2

1 ( , 0) (0) sin( )αu Z

nn n

/2

1

i

(S17)

The following equation can be further obtained from Eq. (S17):

∫ ∫⎛⎝

++

− ⎞⎠

= =+e ZL

β Z dZ R β Z dZ R A β2

1 sin( ) (0) sin ( ) (0) ( )L αu Z

n nL

n n n n0/2

02i

(S18)

Here, we define

∫= = ⎛⎝

++

− ⎞⎠

+B β RA β

e ZL

β Z dZ( ) (0) 1( ) 2

1 sin( )n n nn n

L αu Zn0

/2i

(S19)

Then, we introduce a new variable to simplify the solution,

∫ ∫= = + ⎡⎣⎢

⎛⎝ +

− ⎞⎠

⎤⎦⎥

− −C β T R T B β eA β

e ZL

e β Z dZ dτ( , ) ( ) ( ) 1( ) 8 4

14

sin( )n n n n nβ T

n n

T β τ T Ln0

( )0

n nτ2 24

(S20)

Eq. (S12) can be written as

∑==

∞

N Z T C β T β Z( , ) ( , ) sin( )n

n n n1 (S21)

Substituting Eqs. (S4) and (S21) into Eq. (S1), we finally have

∑= ⎛⎝

−+

⎞⎠

+=

∞

S Z T ZL

e C β T β Z( , ) 12

( , ) sin( )n

n n n1

T4

(S22)

Substituting Eq. (S22) into Eq. (11) yields

= − −

= − + ∑ − −+ =∞{ }( )

u z t S Z T Z T

e C β T β Z

( , ) {ln[ ( , )] /2 /4}

ln 1 ( , ) sin( )

α

αZ

L n n n nZα

Tα

1

12 1 2 4

T4 (S23)

References

[1] Casagrande L. Electro-osmosis in soil. Géotechnique 1948;1(3):159–77.[2] Casagrande L. Stabilization of soil by means of electro-osmotic state-of-art. J Boston

Civ Eng ASCE 1983;69(3):255–302.[3] Lo KY, Ho KS. Electroosmotic strength of soft sensitive clays. Can Geotech J

1991;28(1):62–73.[4] Reddy KR, Saichek RE. Effect of soil type on electrokinetic removal of phenanthrene

using surfactants and cosolvents. J Environ Eng 2002;129(4):336–46.[5] Glendinning S, Lamont-Black J, Jones CJFP. Treatment of sewage sludge using

electrokinetic geosynthetics. J Hazard Mater 2007;139(3):491–9.[6] Jones CJFP, Lamont-Black J, Glendinning S, et al. Recent research and applications

in the use of electro-kinetic geosynthetics. In: Proceedings of the 4th Europeangeosynthetics conference, vol. 4; 2008. p. 365–367.

[7] Chien SC, Ou CY, Wang MK. Injection of saline solutions to improve the electro-osmotic pressure and consolidation of foundation soil. Appl Clay Sci2009;44:218–24.

[8] Lamont-Black J, Weltman A. Electrokinetic strengthening and repair of slopes.Ground Eng 2010.

[9] Cameselle C, Reddy KR. Development and enhancement of electro-osmotic flow forthe removal of contaminants from soil. Electrochim Acta 2012;86:10–22.

[10] Hu LM, Wu WL, Wu H. Numerical model of electro-osmotic consolidation in clay.

Géotechnique 2012;62(6):537–41.[11] Wu H, Hu LM. Analytical solution for axisymmetric electro-osmotic consolidation.

Géotechnique 2013;63(12):1074–9.[12] Hu LM, Wu H. Mathematical mode of electro-osmotic consolidation for soft ground

improvement. Géotechnique 2014;64(2):155–64.[13] Ye ZJ. Discussion on some problems of electro-osmotic consolidation of soil. In: 4th

int conf on sustainable energy and environmental eng; 2015. p. 148–52.[14] Mitchell JK. Conduction phenomena: from theory to geotechnical practice.

Géotechnique 1991;41:229–340.[15] Yeung AT, Sadek SM, Mitchell JK. A new apparatus for the evaluation of electro-

kinetic processes in hazardous waste management. Geotech Test J 1992;15:207–16.[16] Yeung AT. Electrokinetic flow processes in porous media and their applications. In:

CORAPCIOGLUMY (Ed.), Advances in porous media 2; 1994. p. 309–95.[17] Zhuang YF, Wang Z, Chen L. Model test of slope reinforcement through electro-

osmosis and its numerical simulation based on energy analysis method. Rock andSoil Mechanics 2008;29(9):2409–14. [in Chinese].

[18] Jones CJFP, Lamont-Black J, Glendinning S. Electrokinetic geosynthetics in hy-draulic applications. Geotext Geomembr 2011;29(4):381–90.

[19] Spagnoli G, Klitzsch N, Fernandez-Steeger T, Feinendegen M, Real Rey A, Stanjek H,et al. Application of electro-osmosis to reduce the adhesion of clay during me-chanical tunnel driving. Environ Eng Geosci 2011;17(4):417–26.

[20] Wang SD, Zhang LH, Wu HJ, Duan X, Chen JH. Parameter design of over-wet soil filltreated by electro-osmosis. Chinese Journal of Geotechnical Engineering

L. Wang et al. Computers and Geotechnics 105 (2019) 27–36

35

2010;32(3):212–5. [in Chinese].[21] Wang NW, Zhang L, Xiu YJ, Jiao J. Electro-osmosis drainage experiment of un-

saturated clayed soil. Eng Mech 2013;30(S):191–4. [in Chinese].[22] Wu H, Hu LM. Microfabric change of electro-osmotic stabilized bentonite. Appl Clay

Sci 2014;101:503–9.[23] Wu H, Hu LM, Wen QB. Electro-osmotic enhancement of bentonite with reactive

and inert electrodes. Appl Clay Sci 2015;111:76–82.[24] Esrig MI. Pore pressures, consolidation and electrokinetics. J Soil Mech Found

Engng Div, ASCE 1968;94(4):899–922.[25] Wan TY, Mitchell JK. Electro-osmotic consolidation of soil. J Geotech Engng Div,

ASCE 1976;102(5):473–91.[26] Shang JQ. Electro-osmotic enhanced preloading consolidation via vertical drains.

Can Geotech J 1998;35(3):491–9.[27] Xu W, Liu SH, Wang LJ, Wang JB. Analytical theory of soft ground consolidation

under vacuum preloading combined with electro-osmosis. J Hohai Univ (Nat Sci)2011;30(2):169–75. [in Chinese].

[28] Su J, Wang Z. The two-dimensional consolidation theory of electro-osmosis.Géotechnique 2003;53(8):759–63.

[29] Li Y, Gong XN, Lu MM, Guo B. Coupling consolidation theory under combinedaction of load and electro-osmosis. Chin J Geotech Eng 2010;32:77–81. [inChinese].

[30] Wang LJ, Liu SH, Wang ZJ, Zhang K. A consolidation theory for one-dimensionallarge deformation problems under combined action of load and electroosmosis. EngMech 2013;30(12):91–8. [in Chinese].

[31] Wu H, Qi WG, Hu LM, Wen QB. Electro-osmotic consolidation of soil with variablecompressibility, hydraulic conductivity and electro-osmosis conductivity. ComputGeotech 2017;85:126–38.

[32] de Wet M. Electro-kinetics, infiltration and unsaturated flow. In: Alonso E, Delage P(Eds.), Unsaturated soil; 1995. p. 283–291.

[33] Gabrieli L, Jommi C, Musso G, Romero E. In: Shao JF, Burlion N (Eds.), Thermo-hydro-mechanical and chemical coupling in geomaterials and applications: pro-ceedings of the 3rd international symposium GeoProc; 2008. Wiley, ISTE. p.203–10.

[34] Tamagnini C, Jommi C, Cattaneo F. A model for coupled electro-hydro-mechanicalprocesses in fine grained soil accounting for gas generation and transport. An AcadBras Cienc 2010;82(1):1–25.

[35] Wang LJ, Liu SH, Wang JB, Zhu H. Numerical analysis of electroosmostic con-solidation based on coupled electrical field-seepage field-stress field. Rock SoilMech 2012;33(6):1094–911. [in Chinese].

[36] Yuan J, Hicks MA. Influence of gas generation in electro-osmosis consolidation. IntJ Numer Anal Meth Geomech 2016;40(11):1570–93.

[37] Zhou YD, Deng A, Liu ZX, Yang AW, Zhang H. One-dimensional electroosmosisconsolidation model considering variable saturation. Chin J Geotech Eng2017;39(8):1524–9. [in Chinese].

[38] Gardner WR. Some steady-state solutions of the unsaturated moisture flow equationwith application to evaporation from a water table. Soil Sci 1958;85:228–32.

[39] Bjerrum L, Moum J, Eide O. Application of electro-osmosis to a foundation problemin a Norwegian quick clay. Géotechnique 1967;17(3):214–35.

[40] Li Y, Gong X, Lu M, Tao Y. Non-mechanical behaviors of soft clay in two-dimen-sional electro-osmotic consolidation. J Rock Mech Geotech Eng 2012;4(3):282–8.

[41] Zhuang YF, Huang Y, Liu F, Zou W, Li Z. Case study on hydraulic reclaimed sludgeconsolidation using Electrokinetic Geosynthetics. In: 10th ICG, Berlin, Germany;September 2014. p. 21–25.

[42] Micic S, Shang JQ, Lo KY, Lee YN, Lee SW. Electro-kinetic strengthening of a marinesediment using intermittent current. Can Geotech J 2001;38(22):287–302.

[43] Li Y, Gong XN, Zhang XC. Experimental research on effect of applied voltage onone-dimensional electroosmotic drainage. Rock Soil Mech 2011;32(3).709–714+721.

L. Wang et al. Computers and Geotechnics 105 (2019) 27–36

36