Multi-dimensional electro-omosis consolidation of clays

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Yuan - Multi-dimensional electro-osmosis consolidation of clays

Multi-dimensional electro-osmosis consolidation of clays

J.Yuan, Delft University of Technology, Delft, The Netherlands, j.yuan-1@tudelft.nl

M.A.Hicks, Delft University of Technology, Delft, The Netherlands, m.a.hicks@tudelft.nl

J.Dijkstra, Delft University of Technology, Delft, The Netherlands, j.dijkstra@tudelft.nl

ABSTRACT

Electro-osmosis consolidation is an innovative and effective ground improvement method for soft clays. But electro-osmosis is also a very complicated process, as the mechanical behaviour, and hydraulic and electrical properties of the soil are changing rapidly during the treatment process; this makes electro-osmosis hard to describe and simulate. Traditional electro-electro-osmosis consolidation theory cannot provide a satisfactory solution, because it does not directly consider the mechanical behaviour of the soil and the coupling between the soil deformation, electro-osmosis flow and pore pressure. A numerical model for the electro-osmosis consolidation of clay in multi-dimensional domains is presented, with the coupling of the soil mechanical behaviour, pore water transport and electrical fields being considered. Three fully coupled governing equations considering force equilibrium, pore water transport and electrical distribution are presented and solved using COMSOL. The model is verified against the well-known classical analytical solution for electro-osmosis consolidation. A two-dimensional numerical model is then simulated to investigate the settlement and excess pore pressure profile during the electro-osmosis consolidation process. It is found that the peak excess pore pressure is developed near the bottom of the anode and that the maximum settlement is developed near the top of the anode. Moreover, excess pore pressures and settlements develop very rapidly at the beginning of the electro-osmosis treatment, but then become slower with time.

1. INTRODUCTION

A wide variety of ground improvement methods have been developed for economic foundation solutions on soft soils (for a state-of-the-art, see Chu et al. [1] and cited references). An alternative method, however, is the use of electro-osmosis. Electro-osmosis is important for many geo-engineering applications, such as improving friction pile capacity (Milligan [2]), the strengthening and stabilization of soft clays (Casagrande [3]; Lo et al. [4]; Burnotte et al. [5]), controlling the pore water at excavation sites (Bjerrum et al. [6]) and dewatering of mine tailings (Sprute and Kelsh [7]; Sprute et al. [8]; Lockhart [9]; Fourie et al. [10]). Consolidation in clays due to applying an electric current may occur as the result of two distinct mechanisms; the osmosis under electric gradients will cause fluid flow from anodes to cathodes resulting in pore pressure changes and a consequential increase in effective stress in the clay; a less important effect is the hardening of the soil due to the generation of heat and electro-chemical reactions during the process. Electro-osmosis consolidation is a coupling process involving mechanical behaviour, hydraulic flow and electrical flow.

The theory of one dimensional electro-osmosis consolidation was first developed by Esrig [11]. Based on Esrig’s equation, Wan and Mitchell [12] presented an analytical solution for one dimensional preloading and electro-osmosis consolidation. Feldkamp and Belhomme [13] later provided a large-deformation one dimensional electro-osmosis consolidation model. A two dimensional finite-element solution was given by Lewis and Garner [14], who considered the coupling effect of the electric and hydraulic gradients. Shang [15] developed a two dimensional analytical model combining preloading and electro-osmosis consolidation of clay soils. Iwata and Jami [16] presented a numerical model to simulate the combined electro-osmosis dewatering and mechanical response using the Terzaghi-Voigt combined model to consider creep deformation. However, none of the models cited above directly consider the mechanical behaviour of soil or the full multi-physical coupling that occurs during the electro-osmosis consolidation process.

In this study, a numerical model for the electro-osmosis consolidation of clay in multi-dimensional domains is developed. Three fully coupled governing equations considering force equilibrium, pore water transport and electrical current flow are presented and solved using COMSOL. The proposed approach is verified via comparison with the analytical solution developed by Esrig [11]. The results of pore pressure changes for the two cases are compared. Finally, a two-dimensional numerical model is simulated to investigate the settlements and excess pore pressure profile during the electro-osmosis consolidation process.

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2. THEORETICAL AND NUMERICAL FORMULATION

In this paper, an isotropic saturated soil with incompressible pore liquid and soil particles is considered. The governing equations for the electric potential and hydraulic head are derived based on the following assumptions: the current due to electrophoresis of the fine grained particles is negligible [11]; the flow of fluid due to the electrical and a hydraulic gradients may be superimposed to obtain the total flow [11]; Ohm’s law is valid; Darcy’s lay is valid; the electrical gradient caused by movement of ions is negligible compared to applied electrical field.

2.1. Mechanical equilibrium and stress-strain constitutive relationship

The stress equilibrium equation, for small deformations, can be expressed by

T

 

A σ f 0

(1)

where A contains the spatial derivatives:

0

T

y

z

x

y

x

z

y

x

z





_





_













 

_















_







A

(2)

and

σ

represents the total stress vector: T y z x xy yz zx









 



σ

(3)

and f represents the vector of body forces: T x y z

F





 



f

(4)

where F ,_x F and _y F are the body forces in the x, y and z directions, respectively. According to the _z principle of effective stress, the total stress can be written as:

'



p



σ σ m

(5) where

σ

'

and p are the effective stress vector and pore pressure, repetitively, and



1 1 1 0

0

0 

T



m

(6)

The strain-displacement relationship can be written as:



ε Au

(7)

whereεis the total strain vector: T x y y xy yz zx









 



ε

(8)

and is the displacement vector:





T

u

v

w



u

(9)

where u, v and w are the displacements in the x, y and z directions, respectively. The elastic constitutive relationship for soil is:

'



σ

Dε

(10)

(3)





1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 2 0 0 0 0 0 2(1 ) 1 2 0 0 0 0 0 2(1 ) 1 2 0 0 0 0 0 2(1 )

E(1

)

1 (1 2 )

                 



           





































_

_





_

_





























D

(11)

where E and



are the elastic modulus and Poisson’s ratio, respectively.

Hence, by substituting eq.(2)-eq.(11) into eq.(1) the mechanical equilibrium governing equation can be expressed by

p



 

T T

A DAu A m

f 0

(12)

2.2. Pore water transport

The mass conservation of the pore water can be expressed by:

0 n

t



   



v

(13)

where n is the porosity of the soil, t is time and is the velocity of the pore water in the soil, which comprises two components. One is the hydraulic flow caused by the gradients of pore water pressure and the other is the electro-osmosis flow caused by electrical potential gradients. From Darcy’s law the hydraulic flow can be expressed as:





w w w w

k

p



z



 





v

(14)

where k ,_w



_w and z are the coefficient of hydraulic conductivity, the unit weight of water and the elevation, respectively.

The fluid flux due to electro-osmosis is [17] eo

  

k

eo

V

v

(15)

where k is the coefficient of electro-osmosis conductivity and V is the electrical potential. _eo

According to Esrig’s assumption, these two independent flows can be combined to give the total flow:





w w eo w

k

p



z

k

V



 





 

v

(16)

Because the soil is saturated and incompressible, the change of porosity can be expressed in terms of the deformation as T T

n

t





_



_



u

m

m A

(17)

Consequently, the equation of pore water mass conservation can be written in the following form by substituting eq.(16) and eq.(17) into eq.(13):









0

T w w eo w

k

p

z

k

V

t









_{   }

_

_

_{    }

_







_

_

u

m A

(18)

2.3. Electrical transport

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Yuan - Multi-dimensional electro-osmosis consolidation of clays e

I

  



V

(19)

where I is the electrical current and



_eis the electrical conductivity of the soil. By applying the conservation of charge and assuming the current is steady state:

0

e

I

R

 



(20)

where R is the current source. _e Substituting eq.(19) into eq.(20) gives;





e

V



R

e

0  







(21)

2.4. Final governing equations

The primary variables, that is, displacements, pore pressure and electrical potential, are coupled through the governing equations for mechanical equilibrium, pore water transport and electrical current transport, i.e. eqs. (12), (18) and (21) respectively. These governing equations are solved using the COMSOL Multiphysics PDE interface.

3. VERIFICATION AND INVESTIGATION

The proposed approach is here verified against the well-known classical 1D analytical solution of electro-osmosis consolidation. Then two dimensional electro-electro-osmosis consolidation is investigated using the proposed approach.

3.1. 1D Verification

Esrig [11] developed a one dimensional electro-osmosis consolidation theory, in which the governing equation can be expressed as

2 2 2 2 w eo v w

k

p

V

p

k

m

x

t





_



_



(22)

where is the coefficient of compressibility.

The analytical solution of above equation can be obtained through the method of variable separation and Laplace transform [18]:

 

2 2 2 0

1

2 ,

sin sin

exp(

)

n eo eo w w n w w

k

V

m x

p x t

V x

m

T

k

m

L









 









 







_

_











(23)

where m=n+1/2, ∙ / ∙ is the time factor, and L is the distance between the anode and cathode.

Furthermore, the maximum negative excess pore pressure developed at the anode is given by [19] max eo w w

k

p

V

k



 



(24)

The one dimensional model is shown in Fig. 1 and is considered to be originally 1.0m thick. The material parameters are listed in Table 1 and are typical for a soft clay. These have been used for Esrig’s solution and for the finite element analysis using COMSOL. For the latter, a two dimensional plane strain FEM model has been used, but with suitable boundary conditions to impose the one dimensional condition. 3-node triangular elements and a linear solver are used in this study.

Table 1: Material parameters

Variable Physical meaning Unit Value

Hydraulic conductivity m/s 2.0E-09

Electro-osmosis conductivity m /V ∙ s 2.0E-09 Coefficient of compressibility 1/Pa 1.0E-06

Unit weight of pore water kN/m 9.8E+03

E Young’s modulus Pa 7.4E+05

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Figure 1: 1D electro-osmosis consolidation Figure 2: Excess pore pressure profile at anode The initial excess pore pressure is set to zero throughout the problem domain. The cathode is the top boundary and is free draining, whereas the anode is the bottom boundary and is fixed and impermeable, as seen in Fig.1. The electrical potential is maintained at 10 volts at the anode. The left and right boundaries are impermeable with respect to both pore pressure and electrical potential, to ensure one-dimensionality. The simulation time is 100 days.

According to eq.(24), the maximum negative excess pore pressure developed at the anode will be -98.1 kPa. Fig. 2 shows the evolution of pore pressure with time at the anode due to electro-osmosis. The negative pore pressure develops rapidly at the beginning and reaches the maximum value at around day 100. An excellent correlation is found between the results of the proposed approach and Esrig’s theoretical solution. This demonstrates that the proposed approach is able to correctly consider the coupling behaviour in 1D electro-osmosis consolidation.

3.2. 2D Investigation

A two dimensional electro-osmosis consolidation model has also been investigated using the proposed approach. The impact of the electro-osmosis consolidation is assessed by analysing the excess pore pressures and settlements.

A square domain of side length 1m is presented in Fig.3. In order to investigate the coupling behaviour in the soil, a surcharge load is applied on the top surface. The boundary conditions employed are as follows: the anode is along the left edge, which is set as impermeable and on rollers; the right edge is the cathode, and is free draining and on rollers; the bottom boundary is impermeable and fixed; the top surface is free draining and a uniform surcharge pressure, q=100 kPa, is applied. An electrical potential of 10 V is applied at the anode, and the initial conditions and material parameters are the same as for the previous one dimensional analysis, besides the use of a Yong’s modules of E=1000 kPa.

The excess pore pressure profiles at day 0, day 10 and day 100 are shown in Fig. 4, in which automatic scaling has been used for the contours. At the start of the analysis, at day 0, the postive pore pressure developed is around 98.8 kPa because of the surcharge loading. As time progresses there are two evolution processes: the positive pore pressure reduces due to the free draining boundaries; at the same

Figure 3: 2D Electro-osmosis consolidation model

‐100 ‐90 ‐80 ‐70 ‐60 ‐50 ‐40 ‐30 ‐20 ‐10 0 0.01 0.1 1 10 100 Excess por e pressure (kPa) Time (Day) Analytical solution FEM Solution

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Figure 4: Excess pore pressure profiles at day 0, 10 and 100

Figure 5: Excess pore pressure-time relationship Figure 6: Excess pore pressure profiles along

bottom boundary

time negative pore pressures develop at the anode caused by the electro-osmosis. At day 10, the maximum negative pore pressure of -12.2 kPa has developed near the top of the anode, whereas the largest postive pore pressure of 18.9 kPa is near the middle of bottom boundary. At day 100 the time dependent system reaches the steady state, with a maximum negative pore pressure of -61.7 kPa having been developed near the bottom of anode.

Fig. 5 shows the excess pore pressure–time relationship at the bottom of the anode (x=0, y=0 m). In the early stages of the analysis, the electro-osmosis dominates the pore water flow and the negative excess pore pressures grow rapidly near the anode. As the negative excess pore pressures increase, the hydraulic gradients becoming larger and the entire system reaches the steady state after around 40 days.

The excess pore pressure distrubitions along the bottom of the model at different times are shown in Fig. 6. The dissipation of the excess presures is related to both the distance to the cathode and anode, since the electro-osmosis causes the development of negative pore pressure changes along anode and the cathode is free draining. So, at early stages of the analysis the middle part has the highest pore pressure, but if the time is long enough the pore pressure profile stabilises. Note that the initial conditions and surcharge load do not influence the final profiles of this time dependent problem.

The settlement profiles at day 1, 10 and 100 are shown in Fig. 7. At day 1, the settlement is mainly developed near the cathode and is about 32 mm; this because there is free drainage at the cathode. Later, negative pore pressures developed by electro-osmosis become a dominating factor; at day 10, the settlement is almost uniform over the whole domain, and the maxmium value is around 68 mm near the anode. At day 100, the maxmium settlement is 92 mm near the anode.

Figure 7: Displacement profiles at day 1, 10 and 100

‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 120 0.01 0.1 1 10 100 Excess por e pressure (kPa) Time (Day) Pore pressure (x=0, y=0) ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 Excess por e pressure (Kpa) Distance to anode (m) 1 Days 5 Days 10 Days 20 Days

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Figure 8: Settlement-time relationship Figure 9: Settlement profiles at top surface

To investigate the surface settlement during electro-osmosis consolidation, the evolution of the surface settlement at the top of anode (x=0, y=1 m) with time is shown in Fig. 8. The settlement increases quickly between day 1 and day 10, and becomes stable by day 100 when the maxmium settlement has reached around 95mm. It can also be observed from Fig. 8 that most of the surface settlement takes place within the first 10 days. As mentioned before, the excess pore pressure plays an important role in the mechanical equilibrium, which controls the soil deformation as is evident by comparing Fig. 5 and Fig. 8.

Furthermore, the settlement profiles of the top surface at different times are shown in Fig. 9. At day 1 the settlement at the cathode is larger than at the anode; at this stage the settlement is mainly controlled by the surcharge load. As the negative pore pressure develops near the anode, the settlement at the anode surface increases quicker than at the cathode and, between days 5 and 10, the settlements are almost uniform along the top surface. As the excess pore pressures continue to grow near the anode, the settlement at the anode becomes larger than at the cathode at the steady state. Fig. 9 also shows that, by applying the combined surcharge loading and electro-osmosis consolidation, a more unifrom consolidation rate can be obtained.

4. CONCLUSIONS

A formulation for the consideration of multi-dimensional electro-osmosis consolidation has been presented in this paper. Three coupled governing equations for force equilibrium, pore water transport and electrical transport are presented and solved using finite elements. The proposed approach is verified against a well-known analytical 1D solution of electro-osmosis consolidation.

A hypothetical two dimensional electro-osmosis consolidation problem considering surcharge load is then investigated. The settlements and excess pore pressure profiles during the electro-osmosis consolidation process are studied. It is found that the peak excess pore pressure is developed near the bottom of the anode and that the maximum settlement is developed near the top of the anode. Moreover, excess pore pressure and settlements develop very rapidly at the beginning of the electro-osmosis treatment, but then became slower with time.

In this new developed model, the mechanical behaviour of soil is considered and the soil settlements are obtained directly; hence the real coupling effect in the electro-osmosis process is modelled. Although unsaturated flow and material and geometric nonlinearity are not considered in this paper, the formulation is amenable to further development to include these factors.

REFERENCES

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[3] Casagrande, L., Stabilization of soils by means of electro-osmosis-State of the art. Journal of the Boston Society of Civil Engineers Section, American Society of Civil Engineers, 1983. 69(2): p. 255-302. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.01 0.1 1 10 100 Selttment (m) Time (Day) Displaceme nt(x=0,y=1) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0 0.2 0.4 0.6 0.8 1 Settlement (m) Distance to anode (m) 1 Days 5 Days 10 Days 20 Days 100 Days

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[19] Mitchell, J.K., In-place treatment of foundation soils. Journal of the Soil Mechanics and Foundations Division, 1970. 96(SM1): p. 73-110.