Multi-dimensional electro-osmosis consolidation of clays

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1 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 Multi-dimensional electro-osmosis consolidation of clays J.Yuan, Delft University of echnology, Delft, he Netherlands, j.yuan-1@tudelft.nl M.A.Hicks, Delft University of echnology, Delft, he Netherlands, m.a.hicks@tudelft.nl J.Dijkstra, Delft University of echnology, Delft, he Netherlands, j.dijkstra@tudelft.nl ABSRAC 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 electroosmosis hard to describe and simulate. raditional electro-osmosis consolidation thry cannot provide a satisfactory solution, because it does not directly consider the mechanical behaviour of the soil and the coupling beteen the soil deformation, electro-osmosis flo and pore pressure. A numerical model for the electro-osmosis consolidation of clay in multi-dimensional domains is presented, ith the coupling of the soil mechanical behaviour, pore ater transport and electrical fields being considered. hree fully coupled governing equations considering force equilibrium, pore ater transport and electrical distribution are presented and solved using COMSOL. he model is verified against the ell-knon classical analytical solution for electro-osmosis consolidation. A to-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. Morver, excess pore pressures and settlements develop very rapidly at the beginning of the electro-osmosis treatment, but then become sloer ith time. 1. INRODUCION A ide 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, hoever, is the use of electro-osmosis. Electro-osmosis is important for many g-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 ater at excavation sites (Bjerrum et al. [6]) and deatering 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 to distinct mechanisms; the osmosis under electric gradients ill cause fluid flo 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 flo and electrical flo. he thry of one dimensional electro-osmosis consolidation as 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 to dimensional finite-element solution as given by Leis and Garner [14], ho considered the coupling effect of the electric and hydraulic gradients. Shang [15] developed a to dimensional analytical model combining preloading and electro-osmosis consolidation of clay soils. Iata and Jami [16] presented a numerical model to simulate the combined electro-osmosis deatering and mechanical response using the erzaghi-voigt combined model to consider creep deformation. Hoever, 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. hree fully coupled governing equations considering force equilibrium, pore ater transport and electrical current flo are presented and solved using COMSOL. he proposed approach is verified via comparison ith the analytical solution developed by Esrig [11]. he results of pore pressure changes for the to cases are compared. Finally, a to-dimensional numerical model is simulated to investigate the settlements and excess pore pressure profile during the electro-osmosis consolidation process.

2 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June HEOREICAL AND NUMERICAL FORMULAION In this paper, an isotropic saturated soil ith incompressible pore liquid and soil particles is considered. he governing equations for the electric potential and hydraulic head are derived based on the folloing assumptions: the current due to electrophoresis of the fine grained particles is negligible [11]; the flo of fluid due to the electrical and a hydraulic gradients may be superimposed to obtain the total flo [11]; Ohm s la is valid; Darcy s lay is valid; the electrical gradient caused by movement of ions is negligible compared to applied electrical field Mechanical equilibrium and stress-strain constitutive relationship he stress equilibrium equation, for small deformations, can be expressed by A σ f 0 (1) here A contains the spatial derivatives: x y z A y x z (2) z y x and σ represents the total stress vector: σ x y z xy yz zx (3) and f represents the vector of body forces: f Fx Fy F z (4) here F x, Fy and Fz are the body forces in the x, y and z directions, respectively. According to the principle of effective stress, the total stress can be ritten as: σ σ' mp (5) here σ ' and p are the effective stress vector and pore pressure, repetitively, and m (6) he strain-displacement relationship can be ritten as: ε Au (7) hereε is the total strain vector: ε x y y xy yz zx (8) and is the displacement vector: u u v (9) here u, v and are the displacements in the x, y and z directions, respectively. he elastic constitutive relationship for soil is: σ ' Dε (10) here the elastic stress-strain matrix D is given by:

3 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June E(1 ) 1 1 D 1 (1 2 ) 1 2 (11) (1 ) (1 ) (1 ) here 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 A DAu A mp f 0 (12) 2.2. Pore ater transport he mass conservation of the pore ater can be expressed by: n v 0 (13) t here n is the porosity of the soil, t is time and is the velocity of the pore ater in the soil, hich comprises to components. One is the hydraulic flo caused by the gradients of pore ater pressure and the other is the electro-osmosis flo caused by electrical potential gradients. From Darcy s la the hydraulic flo can be expressed as: v k p z here k, and z are the coefficient of hydraulic conductivity, the unit eight of ater and the elevation, respectively. he fluid flux due to electro-osmosis is [17] v kv (15) here k is the coefficient of electro-osmosis conductivity and V is the electrical potential. According to Esrig s assumption, these to independent flos can be combined to give the total flo: k v p z k V (16) Because the soil is saturated and incompressible, the change of porosity can be expressed in terms of the deformation as n u t t t m m A (17) Consequently, the equation of pore ater mass conservation can be ritten in the folloing form by substituting eq.(16) and eq.(17) into eq.(13): u k p z k V 0 ma (18) t 2.3. Electrical transport According to Ohm s la, the electrical current flo can be expressed by: (14)

4 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 I V (19) e here I is the electrical current and e is the electrical conductivity of the soil. By applying the conservation of charge and assuming the current is steady state: I R e 0 here Re is the current source. Substituting eq.(19) into eq.(20) gives; V R (21) 0 e e 2.4. Final governing equations he primary variables, that is, displacements, pore pressure and electrical potential, are coupled through the governing equations for mechanical equilibrium, pore ater transport and electrical current transport, i.e. eqs. (12), (18) and (21) respectively. hese governing equations are solved using the COMSOL Multiphysics PDE interface. 3. VERIFICAION AND INVESIGAION he proposed approach is here verified against the ell-knon classical 1D analytical solution of electroosmosis consolidation. hen to dimensional electro-osmosis consolidation is investigated using the proposed approach D Verification Esrig [11] developed a one dimensional electro-osmosis consolidation thry, in hich the governing equation can be expressed as 2 2 k p V p k 2 m 2 v (22) x x t here is the coefficient of compressibility. he analytical solution of above equation can be obtained through the method of variable separation and Laplace transform [18]: k 2 1 n k V m x 2 2 p xt, Vx sin sin exp( m ) 2 k k n0 m L (23) here m=n+1/2, / is the time factor, and L is the distance beteen the anode and cathode. Furthermore, the maximum negative excess pore pressure developed at the anode is given by [19] k pmax V (24) k he one dimensional model is shon in Fig. 1 and is considered to be originally 1.0m thick. he material parameters are listed in able 1 and are typical for a soft clay. hese have been used for Esrig s solution and for the finite element analysis using COMSOL. For the latter, a to dimensional plane strain FEM model has been used, but ith suitable boundary conditions to impose the one dimensional condition. 3- node triangular elements and a linear solver are used in this study. able 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 eight of pore ater kn/m 9.8E+03 E Young s modulus Pa 7.4E+05 Poisson s ratio (20)

5 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 Excess pore pressure (kpa) Analytical solution FEM Solution ime (Day) Figure 1: 1D electro-osmosis consolidation Figure 2: Excess pore pressure profile at anode he initial excess pore pressure is set to zero throughout the problem domain. he cathode is the top boundary and is free draining, hereas the anode is the bottom boundary and is fixed and impermeable, as seen in Fig.1. he electrical potential is maintained at 10 volts at the anode. he left and right boundaries are impermeable ith respect to both pore pressure and electrical potential, to ensure onedimensionality. he simulation time is 100 days. According to eq.(24), the maximum negative excess pore pressure developed at the anode ill be kpa. Fig. 2 shos the evolution of pore pressure ith time at the anode due to electro-osmosis. he negative pore pressure develops rapidly at the beginning and reaches the maximum value at around day 100. An excellent correlation is found beteen the results of the proposed approach and Esrig s thretical solution. his demonstrates that the proposed approach is able to correctly consider the coupling behaviour in 1D electro-osmosis consolidation D Investigation A to dimensional electro-osmosis consolidation model has also been investigated using the proposed approach. he 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. he boundary conditions employed are as follos: the anode is along the left edge, hich 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. he excess pore pressure profiles at day 0, day 10 and day 100 are shon in Fig. 4, in hich 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 to evolution processes: the positive pore pressure reduces due to the free draining boundaries; at the same Figure 3: 2D Electro-osmosis consolidation model

ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 Figure 4: Excess pore pressure profiles at day 0, 10 and 100 Excess pore pressure (kpa) 120 100 80 60

1 1 10 100 ime (Day) Figure 5: Excess pore pressure-time relationship Excess pore pressure (Kpa) 120 100 80 60 40 20 0 20 40 60 80 1 Days 5 Days 10 Days 20 Days 0 0.2 0.4 0.6 0.

At day 10, the maximum negative pore pressure of -12.2 kpa has developed near the top of the anode, hereas the largest postive pore pressure of 18.9 kpa is near the middle of bottom boundary.

5 shos the excess pore pressure time relationship at the bottom of the anode (x=0, y=0 m).

As the negative excess pore pressures increase, the hydraulic gradients becoming larger and the entire system reaches the steady state after around 40 days.

he 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

6 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 Figure 4: Excess pore pressure profiles at day 0, 10 and 100 Excess pore pressure (kpa) Pore pressure (x=0, y=0) ime (Day) Figure 5: Excess pore pressure-time relationship Excess pore pressure (Kpa) Days 5 Days 10 Days 20 Days Distance to anode (m) 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 kpa has developed near the top of the anode, hereas 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, ith a maximum negative pore pressure of kpa having been developed near the bottom of anode. Fig. 5 shos 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 ater flo and the negative excess pore pressures gro 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. he excess pore pressure distrubitions along the bottom of the model at different times are shon in Fig. 6. he 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. he settlement profiles at day 1, 10 and 100 are shon 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 hole 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

7 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 Selttment (m) Displaceme nt(x=0,y=1) Settlement (m) Days 5 Days 10 Days 20 Days 100 Days ime (Day) Distance to anode (m) Figure 8: Settlement-time relationship Figure 9: Settlement profiles at top surface o investigate the surface settlement during electro-osmosis consolidation, the evolution of the surface settlement at the top of anode (x=0, y=1 m) ith time is shon in Fig. 8. he settlement increases quickly beteen day 1 and day 10, and becomes stable by day 100 hen the maxmium settlement has reached around 95mm. It can also be observed from Fig. 8 that most of the surface settlement takes place ithin the first 10 days. As mentioned before, the excess pore pressure plays an important role in the mechanical equilibrium, hich 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 shon 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, beteen days 5 and 10, the settlements are almost uniform along the top surface. As the excess pore pressures continue to gro near the anode, the settlement at the anode becomes larger than at the cathode at the steady state. Fig. 9 also shos 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. hree coupled governing equations for force equilibrium, pore ater transport and electrical transport are presented and solved using finite elements. he proposed approach is verified against a ell-knon analytical 1D solution of electro-osmosis consolidation. A hypothetical to dimensional electro-osmosis consolidation problem considering surcharge load is then investigated. he 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. Morver, excess pore pressure and settlements develop very rapidly at the beginning of the electro-osmosis treatment, but then became sloer ith time. In this ne 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 flo and material and gmetric nonlinearity are not considered in this paper, the formulation is amenable to further development to include these factors. REFERENCES [1] Chu, J., S. Varaksin, U. Klotz and P. Menge, Construction Processes. "State of the art Report". 17th Int. Conf. Soil Mech. Gtech. Eng., Alexandria, Egypt, Vol. 4: p [2] Milligan, V., First application of electro-osmosis to improve friction pile capacity - three decades later. Proceedings - ICE: Gtechnical Engineering, (2): p [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, (2): p

8 ISSMGE - C 211 International Symposium on Ground Improvement IS-GI Brussels 31 May & 1 June 2012 [4] Lo, K.Y., I.I. Inculet and K.S. Ho, Electroosmotic strengthening of soft sensitive clays. Can Gtech J, : p [5] Burnotte, F., G. Lefebvre and G. Grondin, A case record of electroosmotic consolidation of soft clay ith improved soil-electrode contact. Can Gtech J, (6): p [6] Bjerrum, L., J. Moum and O. Eide, Application of electro-osmosis to a foundation problem in a Noregian quick clay. Géotechnique, (3): p [7] Sprute, R.H. and D.J. Kelsh, Deatering fine-particle suspensions ith direct current. In: Proc. Int. Symp. Fine Particle Processes,vol La Vegas, Nevada, p [8] Sprute, R.H., D.J. Kelsh and S.L. hompson, Electrokinetic densification of solids in a coal mine sediment pond : a feasibility study. Report of Investigations. 1982, Avondale, Md.: U.S. Dept. of the Interior, Bureau of Mines. [9] Lockhart, N.C., Electro-deatering of fine suspensions. Advances in Solid-liquid Separation, 1986: p [10] Fourie, A.B., D.G. Johns and C.F. Jones, Deatering of mine tailings using electrokinetic gsynthetics. Can Gtech J, (2): p [11] Esrig, M.I., Pore pressures, consolidation and electrokinetics. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng, (4 SM): p [12] Wan,. and J.K. Mitchell, Electroosmotic consolidation of soils. Journal of Gtechnical Engineering Division, ASCE, (5): p [13] Feldkamp, J.R. and G.M. Belhomme, Large-strain electrokinetic consolidation: thry and experiment in one dimension. Gtechnique, (4): p [14] Leis, R., A finite element solution of coupled electrokinetic an hydodynamic flo in porous media. International Journal for Numerical Methodes in Enginerring : p [15] Shang, J.Q., Electroosmosis-enhanced perloasing consolidation via vertical drains. Can Gtech J, : p [16] Iata, M. and M.S. Jami, Analysis of combined electroosmotic deatering and mechanical expression operation for enhancement of deatering. Drying echnology, (7): p [17] Casagrande, L., Electro-osmotic stabilization of soils. Journal of the Boston Society of Civil Engineers, (1): p [18] Mitchell, J.K., Fundamentals of Soil Behavior. 2nd ed ed. 1993, Ne York: John and Sons. [19] Mitchell, J.K., In-place treatment of foundation soils. Journal of the Soil Mechanics and Foundations Division, (SM1): p

Numerical simulation of elasto-plastic electro-osmosis consolidation at large strain

Acta Geotechnica (216) 11:127 143 DOI 1.17/s1144-15-366-z RESEARCH PAPER Numerical simulation of elasto-plastic electro-osmosis consolidation at large strain Jiao Yuan Michael A. Hicks Received: 1 November