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The electro-osmotic consolidation included electrochemical processes (electrolysis and the transport of ion), temperature changes, and mechanical processes. The following sections explain the interrelationship between these three processes based on the experimental data and relative theories.

### Electrochemical processes

During electro-osmotic consolidation, ions are transported in the soil under the direct current. The transport of ions contains diffusion, electromigration, and electro-osmotic flux advection, which are shown in Eq. (1):

$$begin{aligned} & J_{j}^{d} = D_{j} tau nnabla ( – c_{j} ) \ & J_{j}^{m} = u_{j} tau nc_{j} nabla ( – E) \ & J_{j}^{e} = c_{j} k_{e} nabla ( – E) \ end{aligned}$$

(1)

where *J*_{j}^{d} is the diffusive mass flux, *J*_{j}^{m} is the migration mass flux, *J*_{j}^{e} is the electro-osmotic mass flux, *D*_{j} is the diffusion coefficient, (tau) is the tortuosity factor, *n* is the porosity of the clay samples, *c*_{j} is the molar concentration, *u*_{j} is the ionic mobility, *E* is the electrical potential, and *k*_{e} is the electro-osmotic permeability coefficient. From previous literature^{26,27}, the electromigration of *J*_{j}^{m} is a major contributing component to the total mass flux. Therefore, the mass flux of H^{+} and OH^{−} increased with a rise in the duration and voltage [as per Eq. (1)].

H^{+} and O_{2} are generated on the anode surface and OH^{−} and H_{2} are generated on the cathode surface during an electrolysis reaction. The electric current is expended in electrolysis reaction^{26}. Therefore, the amount of H^{+} at the anode and of OH^{−} at the cathode has a positive correlation with the duration and voltage of the experiment (the current increases with a rise in the voltage, which will be discussed in “Temperature change process” section). The specific reaction equation is shown in Eq. (2):

$$begin{aligned} & 2H_{2} O – 4e^{ – } = O_{2} uparrow , + , 4H^{ + } quad left( {{text{anode}}} right) \ & 4H_{2} O + 4e^{ – } = , 2H_{2} uparrow , + , 4OH^{ – } quad left( {{text{cathode}}} right) \ end{aligned}$$

(2)

Under the electromigration process, OH^{−} is transported from the surface of the cathode to the anode. The transport of H^{+} (transported from the surface of the anode to the cathode) can neutralize OH^{−} and can obstruct the transport of OH^{−}. In addition, H^{+} is adsorbed by the surface of clay particles (negative charge) due to the buffering capacity of the clay (clay minerology: illite, kaolinite, and smectite)^{27}. Therefore, the pH value increased from the anode to the cathode, as shown in Fig. 3. In addition, the pH value at a normalized distance of 0.1 to the anode decreased during electro-osmotic consolidation, and increased with a rise in voltage, which was mainly because the amount of H^{+} increased with a rise in the duration and voltage. In addition, existing Cl^{−} transported from the cathode to the anode under the electromigration process, and a part of Cl^{−} gathered near the anode^{27}. Therefore, the molar concentration of Cl^{−} of the nearby anode was much higher than that of the nearby cathode, which was showed in the insert graph of Fig. 3. The insert graph of Fig. 3 shows that the molar concentration of Cl^{−} decreased with electro-osmotic going on. It is mainly because that the Cl^{−} lost electrons on the anode surface and generated chlorine, which reduced the molar concentration of Cl^{−} of the nearby anode.

### Temperature change process

The saturated clay sample consisted of clay particles and pore water. As per Eq. (3), the current value is determined by the external voltage and the apparent electrical resistance of the clay^{28}. The apparent electrical resistance of the clay was determined by the apparent electrical resistivity of clay, as well as the clay sample’s geometrical shape.

$$I = frac{{U_{m} }}{{R_{app} }} = frac{{U_{m} }}{{rho_{app} frac{L}{S}}}$$

(3)

where *U*_{m} is the voltage, *R*_{app} is the apparent electrical resistance of the clay, (rho_{app}) is the apparent electrical resistivity of the clay. The apparent electrical resistivity of the saturated clay can be expressed by Eq. (4):

$$rho_{app} = f(n,rho_{w} ,T)_{{}}$$

(4)

The apparent electrical resistivity of the clay (rho_{app}) has a positive correlation with the electrical resistivity of the pore water (rho_{w}), and a negative correlation with porosity *n* and temperature *T*^{28,29}.

The curves of the apparent electrical resistance of the clay and the temperature change are shown in Fig. 4a, b. The apparent electrical resistance of the clay decreased in the initial 120 min of 12 V. Although the porosity *n* decreased during electro-osmotic consolidation^{13}, which increased the apparent electrical resistivity of the clay. However, there were two main contributions that led to a greater reduction in the apparent electrical resistance of the clay than that of the increase in apparent electrical resistance (which came from the decrease in porosity *n*). The two main contributions were: On the one hand, pore water contained an amount of Na^{+} [molar conductivity 5.01 × 10^{3}/(S m^{2} mol^{−1})] and Cl^{−} [molar conductivity 7.63 × 10^{3}/(S m^{2} mol^{−1})] in the initial stage. The electrolysis reaction (discussed in “Electrochemical processes” section) produced H^{+} [molar conductivity 34.98 × 10^{3}/(S m^{2} mol^{−1})] and OH^{–} [molar conductivity 19.8 × 10^{3}/(S m^{2} mol^{−1})] during electro-osmotic consolidation, which decreased the apparent electrical resistance in this period. On the other hand, the Joule heating generated in the marine clay samples due to conveyance of the current increased the clay samples’ temperature (as shown in Fig. 4b). As per Eq. (4), the apparent electrical resistivity of the clay decreased with a rise in temperature.

As the pore water discharged from the cathode, the average water content decreased (which will be discussed in “Mechanical process” section), which increased the apparent electrical resistivity value of the clay. At the same time, the temperature decreased with a rise in the apparent electrical resistivity of the clay and reached a stable value at a later period of the electro-osmotic consolidation experiments, as shown in Fig. 4a, b.

As per Eq. (3), a high-voltage current is higher than that of a low-voltage current when the clay samples’ electrical resistance is the same. Based on Joule’s law, a high current corresponds to a high Joule heating. Therefore, the maximum temperature rise of 12 V was higher than that of 6 V, which is shown in Fig. 4b.

### Mechanical process

#### Electro-osmotic permeability coefficient and pore water pressure

The electro-osmotic permeability coefficient *k*_{e} can be calculated by the drainage volume *Q* in Fig. 5a, the electrical potential gradient *i*_{e}, the clay sample cross-section *A*, and the experiment time (Delta t).

$$k_{e} = frac{Q}{{i_{e} ADelta t}}$$

(5)

The *k*_{e} values (which are shown in the inset graph of Fig. 5a) increased with a rise in voltage in the initial 300 min, which was mainly because the temperature rise promoted a rise in the electro-osmotic permeability coefficient *k*_{e}^{30}.

Equation (6) is the expression of the consolidation coefficient, and Eq. (7) is the expression of negative pore water pressure *u*(*x*,*t*)^{31}.

$$C_{v} = frac{{k_{h} }}{{m_{v}gamma_{w}}}$$

(6)

$$uleft( {x,t} right) = – frac{{k_{e} gamma_{w} }}{{k_{h} }}frac{{U_{m} x}}{L} + frac{{2k_{e} gamma_{w} U_{m} }}{{k_{h}pi^{2} }}sumlimits_{n = 0}^{infty } {frac{{left( { – 1} right)^{n} }}{{left( {n + frac{1}{2}} right)^{2} }}} cdot sinfrac{{left( {n + frac{1}{2}} right)pi x}}{L}left[ {exp – left( {n + frac{1}{2}} right)^{2} pi^{2} T_{v} } right]{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt}$$

(7)

where *k*_{h} is the hydraulic permeability coefficient in the vertical direction, *x* is the distance to the cathode, *C*_{v} is the consolidation coefficient, and *m*_{v} is the coefficient of volume compressibility.

As per Eq. (7), the pore water pressure *u*(*x*, *t*) is a negative value and the absolute value of the pore water pressure *u*(*x*, *t*) increased during electro-osmotic consolidation. Based on the effective stress principle, the effective stress (sigma^{^{prime}}) increases with the development of negative pore water pressure. The settlement of consolidation is caused by a rise in the effective stress, and was observed during electro-osmotic consolidation, which is shown in Fig. 5b. A high temperature increased the hydraulic permeability coefficient *k*_{h}^{32}. Therefore, a rise in voltage increased the consolidation coefficient *C*_{v}^{13}. *C*_{v} had a negative correlation with the 50% degree of consolidation time^{4}, increasing with the rise in voltage (inset graph of Fig. 5b).

Correspondingly, the average water content reduced, as shown in the inset graph of Fig. 6. The absolute value of the negative pore water pressure and the effective stress (sigma^{^{prime}}) increased from the cathode to the anode, which had a positive correlation with voltage. Therefore, the water content decreased from the cathode to the anode and decreased with the rise in voltage during electro-osmotic consolidation, which is shown in Fig. 6. Furthermore, the settlement of 12 V was higher than that of 9 V and 6 V after 48 h electro-osmotic consolidation.

The variable coefficient *cov* was used to measure the water content discreteness among the clay samples:

$${text{cov}} = frac{{sigma_{water_content} }}{{mu_{water_content} }}$$

(8)

where (mu_{water_content}) is the mean value of the water content, and (sigma_{water_content}) is the mean square error of the water content. The water content otherness in different positions increased with a rise in the variable coefficient. In the inset graph of Fig. 6, the water content variable coefficient *cov* increased during electro-osmotic consolidation. As can be seen from Eq. (7) and the previous literature^{14}, the absolute value of the negative pore water pressure *u*(*x,t*) otherness increased during electro-osmotic consolidation. Based on the effective stress principle, the effective stress otherness and water content otherness increased, correspondingly.

#### Horizontal shrinkage

During electro-osmotic consolidation, the negative pore water pressure *u*(*x*, *t*) induced the vertical and horizontal stress increment (Delta sigma_{eo} (x,t)) in the clay samples. The critical stress condition of horizontal shrinkage can be expressed by Eq. (9), which was modified from^{33}:

$$Delta sigma_{eo} (x,t) = left| {u(x,t)} right| > frac{{k_{0} cdot sigma_{v0}^{,} }}{{1 – k_{0} }}$$

(9)

$$k_{0} = left( {{1 – }sin phi^{^{prime}} } right) cdot left( {{text{OCR}}} right)^{{sin phi^{^{prime}} }}$$

(10)

where *k*_{0} is the earth pressure coefficient, and (sigma_{v0}^{^{prime}}) is the initial effective stress, which is equal to the initial consolidation pressure on the clay samples. *k*_{0} can be described by the stress history *OCR* and the effective internal friction angle (varphi^{^{prime}}). When the horizontal stress increment (Delta sigma_{e0} {text{(x,t) > 1}}{.8}sigma_{v0}^{^{prime}}) [by putting the parameter (varphi^{^{prime}}) = 20.9° into Eqs. (9) and (10)], horizontal shrinkage of the clay samples occurred. The horizontal shrinkage value can be calculated by Eq. (11):

$$D_{shrinkage} = D_{initial} {-} , D_{treatment}$$

(11)

where *D*_{shrinkage} is the horizontal shrinkage value, *D*_{initial} is the initial diameter of the clay, and *D*_{treatment} is the diameter of a clay sample cross-section after electro-osmotic consolidation, which was measured by a caliper.

Figure 7a shows that the horizontal shrinkage increased during the experiment, which was mainly because the horizontal stress increment (Delta sigma_{eo} (x,t)) increased. As per Eqs. (9) and (7), the horizontal stress increment (Delta sigma_{eo} (x,t)) increased from the cathode to the anode. In theory, the horizontal shrinkage value should increase from the cathode to the anode. However, anode–clay contact created a frictional force, which reduced the horizontal shrinkage value at the anode–clay contact site. Thus, maximum horizontal shrinkage did not appear at the anode–clay contact site. In addition, the horizontal shrinkage value was almost zero near the cathode (horizontal shrinkage schematic photo is shown in Fig. 7b). This was because the horizontal stress increment (Delta sigma_{eo} (x,t)) in this area was lower than that of the critical stress value. As per Eqs. (7) and (9), the horizontal stress increment (Delta sigma_{eo} (x,t)) was proportional to the external voltage value *U*_{m}. Furthermore, horizontal shrinkage increased with a rise in the external voltage *U*_{m}.

#### Volume differences

In the preloading consolidation experiment using an oedometer, no horizontal shrinkage occurred and bubbles were generated in the clay samples. Karunaratne and Chew et al.^{3,34} observed that the settlement of the soil foundation was not obvious after electro-osmosis consolidation; however, an increase in the drainage volume of the soil foundation was obvious during electro-osmotic consolidation, which was mainly because of the existence of a “volume difference” . During the electro-osmotic consolidation of marine clay: (1) Bubbles were produced by the electrolysis reaction^{35} and moved into the marine clay, resulting in an “unsaturated” status of the marine clay^{36}; (2) horizontal shrinkage accompanied settlement during electro-osmotic consolidation. Due to these two reasons, the drainage volume was much higher than the change in vertical settlement volume *V*_{vertical}. The volume difference of electro-osmotic consolidation can be expressed by Eq. (12):

$$Delta V = V_{all} – V_{vertical} = V_{shrinkage} + V_{air}$$

(12)

where *V*_{all} is the drainage volume of electro-osmotic consolidation; *V*_{vertical} is the change in vertical settlement volume; *V*_{shrinkage} is the change in horizontal shrinkage volume; and *V*_{air} is the bubble volume in the clay samples.

Horizontal shrinkage and bubbles in the marine clay increased during electro-osmotic consolidation, which was discussed in “Electrochemical processes” and “Horizontal shrinkage” sections. As a result, the volume difference (Delta V) increased with time, as shown in Fig. 8. In addition, the horizontal shrinkage and the amount of bubbles (i.e., the electrolysis reaction) increased with a rise in voltage *U*_{m}, as shown in Fig. 7a and in the discussion of “Horizontal shrinkage” section. Therefore, the higher the voltage, the higher the volume difference.

### Coupling analysis of the electro-osmotic consolidation process

In Fig. 9a, we assumed that the initial clay properties were uniform. As shown in Fig. 9b, when direct current was applied to the marine clay samples, the pore water moved from the anode to the cathode, and then discharged from the cathode. As a result, the average water content reduced during the electro-osmotic consolidation. At the same time, an absolute value of the negative pore water pressure *u*(*x*, *t*) developed and increased from the cathode to the anode, which induced vertical effective stress and increased the horizontal stress increment (Delta sigma_{eo} (x,t)). Firstly, vertical settlement occurred in the marine clay samples, but horizontal shrinkage was not observed in the initial stage. Secondly, the horizontal stress increment (Delta sigma_{eo} (x,t)) was higher than the critical stress condition, and horizontal shrinkage and volume difference were observed, which is shown in Fig. 9d. Vertical settlement and horizontal shrinkage reduced the porosity of the clay samples.

During electro-osmotic consolidation, an electrolysis reaction occurred on the surface of the electrodes, which is shown in Fig. 9b, c. H^{+} (anode product) was transported from the anode to the cathode, while OH^{–} had the opposite transport direction. During the transport of ions, H^{+} and OH^{–} met in the middle section of the marine clay samples, which initiated a neutralization reaction. Due to the high water content and the conductivity in the clay near the cathode, the electrical potential was much lower than the ideal value, which reduced the effect of the reduction in water content near the cathode. In addition to H^{+} and OH^{−}, the surface of the electrode produced bubbles, and the bubbles moved into the marine clay samples, which is another reason for the volume difference.

A temperature rise was observed in this research, which was caused by Joule heating. The temperature increased with a rise in the current, but decreased with a reduction in the porosity of the clay samples. It is beneficial for the mass flux of ions to increase, because effective ionic mobility has a positive relationship with temperature. Moreover, a rise in temperature promoted the electro-osmotic consolidation effect and the consolidation coefficient, which increased the effect of the reduction in water content and the consolidation coefficient. However, the temperature rise induced an increase in the energy coefficient.

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