Introduction

Soft clays in the coastal areas of southeastern China, and in some inland areas deposited by lakes and rivers, is prevalent and is typically characterized by high moisture content1, elevated compressibility, diminished shear strength2, and low permeability3. Notably, the high compressibility often results in substantial or uneven ground settlement. The low permeability extends the duration of settlement, potentially compromising the structural integrity of the building. Moreover, the reduced shear strength frequently leads to inadequate ground bearing capacity and stability, failing to meet engineering specifications. Consequently, appropriate treatment of soft clay grounds is imperative prior to construction activities. Traditional reinforcement methods like surcharge preloading4 or vacuum preloading5 often exhibit low efficiency, extended construction durations, and elevated treatment costs6. Furthermore, laboratory studies have indicated that the electrical conductivity of fine-grained soils, across various grain sizes, generally stays around 10–9·m2/(V·s) under commonly applied voltage gradients7,8. Consequently, electro-osmosis is being increasingly utilized for consolidating low-permeability soils.

Currently, studies on the consolidation of saturated soil through the combination of electro-osmosis and surcharge preloading have made some advancements in analytic theory9,10,11,12,13, laboratory and field experiments14,15,16,17, and numerical simulations18. Notably, studies have employed electrokinetic prefabricated vertical drains (EVDs) to integrate these two techniques in field tests for the consolidation of soft soils19, demonstrating their effectiveness. These studies show that whereas surcharge preloading speeds up the discharge of water accumulated on the electrodes and hinders the formation of cracks; at the same time electro-osmosis increases the rate of drainage. As a result, the combination of electro-osmosis and surcharge preloading exhibits a synergistic effect, reinforcing each other’s efficacy. Notably, in the project of soil consolidation using electro-osmosis-enhanced preloading, Li et al.20 initially developed an axisymmetric consolidation model based on the assumption of equal strain and briefly discussed the related consolidation properties. Additionally, Wu and Hu10 introduced an analytical solution for axisymmetric free-strain consolidation focusing solely on electro-osmosis and posited that the free-strain assumption more accurately represents the influence of the radial electric fields. Consequently, there is a pressing need to delve into the theory of axisymmetric free-strain consolidation for saturated soil under electro-osmosis-enhanced surcharge preloading.

In traditional consolidation theory for saturated soils, soil boundaries are typically assumed to be either fully permeable or completely impermeable, yielding straightforward solutions to complex geomechanical problems. With the advancement of mathematical methods and computer technology and considering that soil boundaries in practical engineering are often between permeable and impermeable, researchers have sought to enhance traditional consolidation theory by introducing semi-permeable boundaries21. However, the definition of a semi-permeable boundary condition remains ambiguous, and the drainage capacity of the interface cannot be quantitatively expressed22. Additionally, semi-permeable boundaries cannot satisfy the initial conditions, and the resulting analytical solutions lack explicit expressions. To address this, researchers have proposed a continuous drainage boundary that, by adjusting interface parameters, can approximate the varying drainage capacities of actual soil layer boundaries23. This boundary condition not only strictly meets the initial conditions but also enables the effective derivation of solutions with explicit expressions24. These interface parameters can be determined through the inversion of measured pore pressure values or through experimental simulations25,26,27.

This study aims to formulate an axisymmetric free-strain consolidation solution for saturated soils incorporating a continuous drainage boundary under electro-osmotic-enhanced preloading. To achieve this, a novel solution method, the Laplace–Hankel transform, coupled with boundary homogenization, is employed to yield semi-analytical solutions. The validity of the proposed solutions is confirmed through comparison with existing analytical solutions in the literature. Subsequently, these solutions are utilized to explore the consolidation behavior of soil under the combined electro-osmotic and surcharge preloading. This investigation will serve as a reference and guide for projects that use combined electro-osmotic and surcharge loading for soil consolidation.

Theoretical analysis

Mathematical modeling

In the case of electric vertical wells (electric plastic drainage boards or perforated iron pipes) for drainage consolidation, the type of vertical well arrangement generally consists of horizontal and axisymmetric types, and the corresponding consolidation model can be simplified to a plane-strain model and an axisymmetric model. A schematic diagram of the consolidation of saturated soil under electroosmosis–surcharge preloading is shown in Fig. 1a. When the vertical wells are arranged in the hexagonal shape shown in Fig. 1b, the corresponding consolidation model can be simplified to the axisymmetric simplified model shown in Fig. 1c. The model parameters include the radius (re), representing the distance between the cathode and anode, the equivalent radius of the cathodic EVD (rw), and the thickness (h) of the saturated soil layer. The boundaries of the influence zone are characterized as follows: the bottom (z = h) and outer (r = re) boundaries are impermeable, while the inner boundary (r = rw) is permeable. The top boundary (z = 0) accommodates the variation of boundary permeability over time during consolidation and is defined as a continuous drainage boundary25. \(q_{{0}}\) denotes the external loading. \(k_{e}\) refers to the radial electrical permeability coefficient, and \(k_{w * }\) (where the subscripts * = r or z denote its corresponding direction, respectively) represents the hydraulic permeability coefficient.

Fig. 1
figure 1

Schematic sketches of electroosmosis–surcharge preloading consolidation of saturated soil: (a) consolidation system; (b) plan layout; (c) computational sketch.

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Governing equations and solution conditions

The governing equations for excess pore water pressure (EPWP) in saturated soils under coupled electrical and mechanical fields are formulated based on the following assumptions: (1) A single layer of soil is saturated and horizontally isotropic; with negligible compression of soil particles and water. Additionally, the volume of drainage from the soil unit per unit time equals the volume change of the soil. (2) The hydraulic and electrical permeability coefficients of the soil are considered constant during consolidation10. (3) The radial potential distribution within the soil remains unchanged with respect to time and depth. (4) Electrode voltage losses and thermal and concentration differences are not considered.

With reference to Wu and Hu10, the radial and vertical flow velocity under electric and hydraulic gradient fields can be expressed as follows:

$$v_{wr} = - \frac{{k_{wr} }}{{\gamma_{w} }}\frac{{\partial u_{w} }}{\partial r} - k_{e} \frac{\partial \Phi }{{\partial r}}$$
(1)
$$v_{wz} = - \frac{{k_{wz} }}{{\gamma_{w} }}\frac{{\partial u_{w} }}{\partial z}$$
(2)

where \(\gamma_{w}\) represents the unit weight of water; \(u_{w}\) denotes the EPWP; \(\Phi\) is the electrical potential.

According to assumption (1) and conservation of water volume:

$$\frac{{\partial \left( {\Delta V_{w} /V_{0} } \right)}}{\partial t} = - \frac{1}{r}\left( {v_{wr} + r\frac{{\partial v_{wr} }}{\partial r}} \right) - \frac{{\partial v_{wz} }}{\partial z}$$
(3)

here, \(V_{0}\) represents the initial volume of the soil cell, and \(\Delta V_{w}\) denotes the volume change of water within the unit.

In accordance with the assumption of linear elasticity, the relationship between the volumetric strain and EPWP in the soil is given by:

$$\frac{{\partial \left( {\Delta V_{w} /V_{0} } \right)}}{\partial t} = \frac{{\partial \varepsilon_{v} }}{\partial t} = m_{v} \frac{{\partial u_{w} }}{\partial t}$$
(4)

where \(\varepsilon_{v}\) is the volumetric strain, and \(m_{v}\) indicates the coefficient of volume compressibility.

Combining Eqs. (1–4), the controlling equation20 is given by

$$\frac{{\partial u_{w} }}{\partial t} - \frac{{k_{wr} }}{{\gamma_{w} m_{v} }}\left( {\frac{1}{r}\frac{{\partial u_{w} }}{\partial r} + \frac{{\partial^{2} u_{w} }}{{\partial r^{2} }}} \right) - \frac{{k_{e} }}{r}\frac{\partial \Phi }{{\partial r}} - k_{e} \frac{{\partial^{2} \Phi }}{{\partial r^{2} }} - \frac{{k_{wz} }}{{\gamma_{w} m_{v} }}\frac{{\partial^{2} u_{w} }}{{\partial z^{2} }} = 0$$
(5)

In this study, it is assumed that the initial EPWP remains constant along depth.

$$u_{w} \left( {r,z,t} \right)\left| {_{t = 0} } \right. = p_{0}$$
(6)

where \(p_{0}\) represents the initial EPWP generated in the soil under the external loading.

The boundary conditions, as depicted in Fig. 1 and described in the corresponding text, are mathematically represented by Eqs. (7–9).

$$u_{w} \left( {r,z,t} \right)\left| {_{{r = r_{w} }} } \right. = 0$$
(7)
$$\frac{{\partial u_{w} \left( {r,z,t} \right)}}{\partial z}\left| {_{z = h} } \right. = 0$$
(8)
$$u_{w} \left( {r,z,t} \right)\left| {_{z = 0} } \right. = p_{0} e^{ - bt}$$
(9)

here, b represents the interface parameter, which assumes values greater than zero. Furthermore, as b approaches zero, the top boundary tends towards a fully impermeable condition, whereas it approaches a fully permeable boundary as b tends to infinity25.

It is essential to emphasize that the outer boundary at r = re is impermeable. In line with the law of charge conservation, excluding alterations in soil parameters and potential losses at the anode10, the voltage is described by:

$$\frac{{\partial \Phi^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \Phi }{{\partial r}} = 0$$
(10)

Combining Eq. (1) and Eq. (10), the outer boundary is shown below

$$\frac{{\partial u_{w} \left( {r,z,t} \right)}}{\partial r}\left| {_{{r = r_{e} }} } \right. = \frac{{ - k_{e} \gamma_{w} }}{{k_{wr} }}\frac{{\Phi_{a} }}{\ln N}\frac{1}{{r_{e} }}$$
(11)

Derivation of semi-analytical solutions

Considering Eq. (10), the governing Eq. (5) can be reformulated as follows:

$$\frac{{\partial u_{w} }}{\partial t} - \frac{{k_{wr} }}{{\gamma_{w} m_{v} }}\left( {\frac{1}{r}\frac{{\partial u_{w} }}{\partial r} + \frac{{\partial^{2} u_{w} }}{{\partial r^{2} }}} \right) - \frac{{k_{wz} }}{{\gamma_{w} m_{v} }}\frac{{\partial^{2} u_{w} }}{{\partial z^{2} }} = 0$$
(12)

An intermediate variable, denoted as presented in Eq. (13), is introduced to homogenize the radial boundary condition.

$$v_{w} (r,z,t) = u_{w} (r,z,t) + \Gamma \ln \frac{r}{{r_{w} }}$$
(13)

where \(\Gamma\) represents a constant, \(\Gamma = {{k_{e} \gamma_{w} \Phi_{a} } \mathord{\left/ {\vphantom {{k_{e} \gamma_{w} \Phi_{a} } {\left[ {k_{wr} \ln \left( {{{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}} \right)} \right]}}} \right. \kern-0pt} {\left[ {k_{wr} \ln \left( {{{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}} \right)} \right]}}\).

Substituting Eq. (13) into Eq. (12), the governing equation, initial condition, and boundary conditions are expressed as:

$$\left. \begin{gathered} \hfill \frac{{\partial v_{w} }}{\partial t} - \frac{{k_{wr} }}{{\gamma_{w} m_{v} }}\left( {\frac{1}{r}\frac{{\partial v_{w} }}{\partial r} + \frac{{\partial^{2} v_{w} }}{{\partial r^{2} }}} \right) - \frac{{k_{wz} }}{{\gamma_{w} m_{v} }}\frac{{\partial^{2} v_{w} }}{{\partial z^{2} }} = 0, \\ \hfill v_{w} \left( {r,z,t} \right)\left| {_{{t = t_{0} }} } \right. = p_{0} + \Omega \ln \frac{r}{{r_{w} }}, \\ \hfill v_{w} \left( {r,z,t} \right)\left| {_{{r = r_{w} }} } \right. = 0, \, \frac{{\partial v_{w} \left( {r,z,t} \right)}}{\partial r}\left| {_{{r = r_{e} }} } \right. = 0, \\ \hfill v_{w} \left( {r,z,t} \right)\left| {_{z = 0} } \right. = p_{0} e^{ - bt} + \Gamma \ln \frac{r}{{r_{w} }}, \, \frac{{\partial v_{w} \left( {r,z,t} \right)}}{\partial z}\left| {_{z = h} } \right. = 0 \\ \end{gathered} \right\}$$
(14)

The Laplace transform22 of Eq. (14) results in:

$$\left. \begin{gathered} \hfill \frac{{\partial^{2} v_{w,L} }}{{\partial z^{2} }} + \frac{{k_{wr} }}{{k_{wz} }}\left( {\frac{1}{r}\frac{{\partial v_{w,L} }}{\partial r} + \frac{{\partial^{2} v_{w,L} }}{{\partial r^{2} }}} \right) - \frac{{\gamma_{w} m_{v} }}{{k_{wz} }}sv_{w,L} = \frac{{ - \gamma_{w} m_{v} }}{{k_{wz} }}\left( {p_{0} + \Gamma \ln \frac{r}{{r_{w} }}} \right), \\ \hfill v_{w,L} \left( {r,z,s} \right)\left| {_{{r = r_{w} }} } \right. = 0, \, \frac{{\partial v_{w,L} \left( {r,z,s} \right)}}{\partial r}\left| {_{{r = r_{e} }} } \right. = 0, \\ \hfill v_{w,L} \left( {r,z,s} \right)\left| {_{z = 0} } \right. = \frac{{p_{0} }}{b + s} + \frac{\Gamma }{s}\ln \frac{r}{{r_{w} }}, \, \frac{{\partial v_{w,L} \left( {r,z,s} \right)}}{\partial z}\left| {_{z = h} } \right. = 0 \\ \end{gathered} \right\}$$
(15)

where \(v_{w,L}\) represents the Laplace transform functions of \(v_{w}\), \(v_{w,L} = \int_{0}^{\infty } {v_{w} } {\text{e}}^{ - st} {\text{d}}t\). The letter s is a complex number frequency parameter.

The finite Hankel transform26,27 of Eq. (15) in the domain (rw, re) leads to:

$$\left. \begin{gathered} \hfill \frac{{\partial^{2} v_{w,LH} }}{{\partial z^{2} }} - \alpha_{n} v_{w,LH} = \beta_{n} , \\ \hfill v_{w,LH} \left( {\lambda_{n} ,z,s} \right)\left| {_{z = 0} } \right. = \chi_{n} , \\ \hfill \frac{{\partial v_{w,LH} \left( {\lambda_{n} ,z,s} \right)}}{\partial z}\left| {_{z = h} } \right. = 0 \\ \end{gathered} \right\}$$
(16)

in which

$$\alpha_{n} = \frac{{\gamma_{w} m_{v} }}{{k_{wz} }}s + \frac{{k_{wr} }}{{k_{wz} }}\lambda_{n}^{2}$$
(17)
$$\beta_{n} = \frac{{ - \gamma_{w} m_{v} }}{{k_{wz} }}\int_{{r_{w} }}^{{r_{e} }} {\left( {p_{0} + \Gamma \ln \frac{r}{{r_{w} }}} \right)D_{0} \left( {\lambda_{n} r} \right)r} dr$$
(18)
$$\chi_{n} = \int_{{r_{w} }}^{{r_{e} }} {\left( {\frac{{p_{0} }}{b + s} + \frac{\Gamma }{s}\ln \frac{r}{{r_{w} }}} \right)D_{0} \left( {\lambda_{n} r} \right)r} dr$$
(19)
$$D_{0} \left( {\lambda_{n} r} \right) = J_{0} \left( {\lambda_{n} r} \right)Y_{0} \left( {\lambda_{n} r_{w} } \right) - J_{0} \left( {\lambda_{n} r_{w} } \right)Y_{0} \left( {\lambda_{n} r} \right)$$
(20)

where \(v_{w,LH}\) (\(= \int_{{r_{w} }}^{{r_{e} }} {v_{w,L} \cdot D_{0} \left( {\lambda_{n} r} \right)r{\text{d}}r}\)) represents the finite Hankel transform functions of \(v_{w,L}\). \(J_{k} \left( {\lambda_{n} r} \right)\), \(Y_{k} \left( {\lambda_{n} r} \right)\) (subscript k = 0 or 1) denotes the Bessel functions of the first and second kind of k-order at r. \(\alpha_{n}\), \(\beta_{n}\), \(\chi_{n}\), \(D_{0} \left( {\lambda_{n} r} \right)\) are all intermediate variables. \(\lambda_{n}\) indicates the relevant eigenvalues and is the positive root of the following transcendental Eq. (21).

$$J_{0} \left( {\lambda_{n} r_{w} } \right)Y_{1} \left( {\lambda_{n} r_{e} } \right) - J_{1} \left( {\lambda_{n} r_{e} } \right)Y_{0} \left( {\lambda_{n} r_{w} } \right) = 0$$
(21)

The solution of Eq. (16) gives:

$$v_{w,LH} \left( {\lambda_{n} ,z,s} \right) = \frac{{\left( {\beta_{n} + \alpha_{n} \chi_{n} } \right)\cosh \left[ {\left( {h - z} \right)\sqrt {\alpha_{n} } } \right]sech\left( {h\sqrt {\alpha_{n} } } \right) - \beta_{n} }}{{\alpha_{n} }}$$
(22)

Thus

$$u_{w,LH} \left( {\lambda_{n} ,z,s} \right) = v_{w,LH} \left( {\lambda_{n} ,z,s} \right) - \frac{\Gamma }{s}\int_{{r_{w} }}^{{r_{e} }} {\ln \frac{r}{{r_{w} }}D_{0} \left( {\lambda_{n} r} \right)r} dr$$
(23)

The Laplace-domain solution of uw can be derived from the definition of the finite Hankel inversion29,

$$u_{w,L} (r,z,s) = \frac{{{\uppi }^{2} }}{2}\sum\limits_{n = 1}^{\infty } {\frac{{\lambda_{n}^{2} \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{{\varvec{e}}} } \right)} \right]^{2} u_{w,LH} \left( {\lambda_{n} ,z,s} \right)}}{{J_{0}^{2} \left( {\lambda_{n} r_{w} } \right) - \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{e} } \right)} \right]^{2} }}} D_{0} \left( {\lambda_{n} r} \right)$$
(24)

The formula of the radial average EPWP is calculated based on Eq. (24). That is:

$$\overline{u}_{w,L} (z,s) = \frac{{\pi^{2} }}{{r_{e}^{2} - r_{w}^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\lambda_{n}^{2} \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{e} } \right)} \right]^{2} u_{w,LH} \left( {\lambda_{n} ,z,s} \right)}}{{J_{0}^{2} \left( {\lambda_{n} r_{w} } \right) - \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{e} } \right)} \right]^{2} }} \cdot } \int_{{r_{w} }}^{{r_{e} }} {D_{0} \left( {\lambda_{n} r} \right)r} dr$$
(25)

Similarly, the vertical average EPWP is expressed as follows:

$$\overline{\overline{u}}_{w,L} (s) = \frac{{{\uppi }^{2} }}{{h\left( {r_{e}^{2} - r_{w}^{2} } \right)}}\sum\limits_{n = 1}^{\infty } {\frac{{\lambda_{n}^{2} \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{e} } \right)} \right]^{2} \int_{0}^{h} {u_{w,LH} \left( {\lambda_{n} ,z,s} \right)} dz}}{{J_{0}^{2} \left( {\lambda_{n} r_{w} } \right) - \left[ {J_{0}^{\prime } \left( {\lambda_{n} r_{e} } \right)} \right]^{2} }} \cdot } \int_{{r_{w} }}^{{r_{e} }} {D_{0} \left( {\lambda_{n} r} \right)r} dr$$
(26)

With reference to related research31,32, the average degree of consolidation in the electroosmotic consolidation is defined as follows.

$$\overline{U} = 1 - \frac{{\overline{\overline{u}}_{w} (t) - \overline{\overline{u}}_{wf} }}{{u_{w}^{0} - \overline{\overline{u}}_{wf} }}$$
(27)

Therefore, the average degree of consolidation in the Laplace domain can be formulated as:

$$\overline{U}_{,L} = \frac{1}{s} - \frac{{\overline{\overline{u}}_{w,L} (s) - {{\overline{\overline{u}}_{wf} } \mathord{\left/ {\vphantom {{\overline{\overline{u}}_{wf} } s}} \right. \kern-0pt} s}}}{{u_{w}^{0} - \overline{\overline{u}}_{wf} }}$$
(28)

The EPWP and the average degree of consolidation in the Laplace domain, which are derived based on the free strain assumption, cannot be directly derived from the analytical expressions in the time domain due to the complexity of their mathematical expressions. For this reason, the Stehfest’s method33 was used for the numerical Laplace inversion.

Experimental study

Soil preparation and analysis

Soil samples used in the experiment were collected from a construction site in Shanghai, representing the silty clay, characterized by its gray color. The collected soil samples were air-dried, impurities removed, crushed, and sieved through a 2 mm nylon sieve. Basic physical properties of the clay used in the experiments were tested according to the “Standard for geotechnical testing method”34, with results shown in Table 1. The hydraulic permeability coefficient (kh) was determined by a variable head permeability test. The electroosmotic coefficient (ke) was calculated according to the method described in the literature35. In this method, the hydraulic and electroosmotic flows are set in opposite directions by means of a soil column test. In this case, the hydraulic flow is driven from bottom to top by the head difference (ih) and the electro-osmotic flow is driven from top to bottom by the potential difference (ie), When the flows reach equilibrium, they satisfy the condition where kh·ih = ke·ie. From this equilibrium the electroosmotic coefficient can be calculated from the hydraulic permeability coefficient. Further details can be found in detail in reference35.

Table 1 Physical parameters of the silty clay.
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Experimental apparatus and procedure

Figure 2 illustrates the schematic diagram of the experimental apparatus for the consolidation of soils with electroosmosis–surcharge preloading. The model box consists of a circular acrylic barrel and a rigid acrylic cover with a height of 50 cm, an outer diameter of 50 cm and an inner diameter of 47.6 cm. The experimental procedures were summarised as follows: (1) Before the test started, the soil sample was placed in a model box and the pre-consolidation height of the soil was measured to be 36 cm. An EVD (5 cm wide, 0.4 cm thick, 30 cm high) connected to the cathode line was inserted in the centre and connected to the PU tube through a sealed pipe. (2) Four custom-made anode EVDs were inserted evenly into the soil along a circle with a radius of 18 cm centred on the cathode EVD. (3) Top with a sealing cap and connect the PU tubing. (4) After connecting the cathode and anode wires to a DC power supply, an equivalent load of 20 kPa was applied within 5 min and the power supply was switched on simultaneously to carry out the electroosmosis–surcharge preloading consolidation. (5) During the test, soil settlement was monitored in real time by a deformation instrument; drainage was read in real time by an electronic scale and recorded by an electronic camera. (6) At the end of the test, the instruments were dismantled sequentially and the test data were collated.

Fig. 2
figure 2

Experimental apparatus and settlement monitoring.

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Verification

Comparison of the proposed solution with the existing solutions

To assess the reliability of the proposed solutions in this study, they were juxtaposed with existing analytical solutions across three scenarios: only considering electroosmosis10, only considering surcharge preloading36 within the conventional drainage boundary, and axisymmetric consolidation with continuous drainage boundary under surcharge preloading conditions37. The validation employed calculation parameters sourced from the literature10, detailed in Table 2. As depicted in Fig. 3, the solutions presented exhibit substantial alignment with those in extant literature across various scenarios, where the time factor Tv is equivalent to \({{k_{wr} t} \mathord{\left/ {\vphantom {{k_{wr} t} {\left( {m_{v} \gamma_{w} h^{2} } \right)}}} \right. \kern-0pt} {\left( {m_{v} \gamma_{w} h^{2} } \right)}}\). This attests to the effectiveness of the solution methodology and the resultant findings, underscoring the broad applicability of the proposed solutions in this manuscript.

Table 2 Physical and material parameters of saturated soft soil.
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Fig. 3
figure 3

Comparison of EPWP of the proposed solution with the existing solution.

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Comparison of obtained solutions with experimental results

The consolidation coefficients of soils under different pressures were calculated and analyzed by compression consolidation tests34 for the experimental validation. Table 1 lists some of the parameters required for the consolidation theory. In addition, the EVD equivalent radius can be calculated by referring to32 (\(r_{w} = 0.25w + 0.35t = 1.39\)\(cm\), re = 18 cm). During the experiment, a constant voltage of 18 V was applied during the test (the voltage gradient was always 1 V/cm). Figure 4 presents a comparison between the test-monitored degree of consolidation and the results calculated by the semi-analytical solution (SAS) in this paper. The comparison between the two shows that the solution obtained in this paper is capable of better predicting the degree of consolidation of soil using electro-osmosis combined with surcharge preloading when the upper boundary permeability parameter b takes the value of 10–4 s−1. Conversely, if the upper boundary is regarded as an ideal boundary, the predictions obtained using the semi-analytical solution will differ significantly from the experimental results.

Fig. 4
figure 4

Comparison of the settlement obtained from the proposed solution with that obtained from the experiment.

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Parameter sensitivity analysis

In this section, the effect of pertinent parameters, including the hydraulic-to-electrical permeability coefficient ratio, applied voltage, and interface parameter, on the consolidation process of saturated soils under electroosmotic improved surcharge preloading is examined, utilizing the parameters outlined in Table 2. Before performing parametric sensitivity analyses, Fig. 5 shows the distribution of the EPWP at different time factors within an ideal vertical well ground with a fully permeable top boundary (where the interface parameter b takes a value of 10–4 s−1). This distribution reveal spatial differences in the degree and rate of soil consolidation. Larger EPWPs usually indicate that consolidation has not yet proceeded sufficiently in the area and that there is still a large potential for drainage and consolidation of the soil, while smaller EPWPs imply that consolidation is closer to completion at that location. Two significant observations are evident: firstly, during free strain consolidation, EPWP predominantly dissipates along the diagonal direction; however, near the anode, dissipation is accelerated due to the coupled effect of the radial electric field (i.e., the EPWP value appears to increase and then decrease along the radial direction). Secondly, it can be seen from the EPWP values at different time factors that when the EPWP value is less than zero, , at which time it is only affected by electroosmosis, the EPWP value at any location within the soil body is dissipated along the diagonal. This indicates that the EPWP values in the pre-dissipation and post-dissipation phases (with zero value as the critical point) is mainly affected by surcharge preloading and electroosmosis, respectively.

Fig. 5
figure 5

Spatial–temporal distribution of EPWP at different time factor.

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Figure 6 presents the effect of varying radial hydraulic-to-electrical permeability coefficient ratios on the dissipation of EPWP. As the ratio increases, corresponding to an increase in the hydraulic permeability coefficient, the rate of EPWP dissipation accelerates during the initial stages. However, in the later stages, the absolute value of the negative EPWP diminishes due to the influence of electro-osmosis. This trend, in conjunction with the boundary conditions at r = re, suggests an inverse relationship between the absolute value of the negative EPWP and the radial hydraulic-to-electrical permeability coefficient ratio for a given electric potential gradient. Furthermore, a reduced hydraulic permeability coefficient leads to slower EPWP dissipation in the initial stage and results in a more negative EPWP generated by electro-osmosis. This implies that electro-osmosis effectively facilitates consolidation in soils with low hydraulic permeability.

Fig. 6
figure 6

The dissipation of EPWP under different radial hydraulic-to-electrical permeability ratios.

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Figure 7 demonstrates the dissipation of EPWP and the variation of the average degree of consolidation when considering electroosmosis, surcharge preloading and their combined application. As shown in Fig. 7a, when electroosmotic consolidation is applied alone, a negative EPWP is generated, and the region eventually stabilizes at a final value. When instantaneous uniform surcharge preloading is used for consolidation alone, the initial excess pore water pressure gradually dissipates until it approaches zero. Moreover, when both techniques are used simultaneously, the EPWP at any time factor is the sum of the EPWP generated by each technique when used individually. This indicates that when electroosmosis and surcharge preloading are applied for soil consolidation, their effects on EPWP are linearly additive. In addition, the final negative EPWP induced by electroosmosis is positively correlated with the applied voltage. As can be clearly seen in Fig. 7b, electroosmosis-enhanced surcharge preloading significantly improves consolidation efficiency, with the consolidation rate increasing as the applied voltage is increased.

Fig. 7
figure 7

Effect of surcharge preloading and applied voltage on (a) the dissipation of EPWP and (b) the development of average degree of consolidation.

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Figure 8 depicts the distribution of EPWP along depth for different values of the top boundary interface parameter (which corresponds to soft ground with different permeability top boundary) at a time factor of 0.1. As the interface parameter increases from 0, the permeability of the top boundary of the soft soil ground transitions from completely impermeable to fully permeable. This suggests that the axisymmetric free-strain consolidation solution under electro-osmotic-assisted preloading can effectively predict the consolidation behavior of an soft soil ground enhanced with vertical drain with different permeability top boundary by adjusting the interface parameters appropriately. This is another indication of the generality of the solution obtained in this paper, which is applicable to a variety of soft ground with different permeability top boundaries.

Fig. 8
figure 8

The distribution of EPWP under different interface parameters.

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Conclusions

According to the assumptions of continuous drainage boundary and free strain, this paper adopts the Laplace–Hankel transform method, aiming at finding the theoretical solution of electroosmosis–surcharge preloading consolidation closer to the actual condition, which can provide guidance for the engineering. The conclusions obtained after analytical modeling and parametric analysis include:

  1. (1)

    The spatial distribution of the consolidation solutions obtained based on the free-strain assumption shows the EPWP at different times, which is convenient to reveal the coupling effect generated by the electric field, thereby offering a framework to explore the influence of the radial electric field on consolidation properties.

  2. (2)

    Electro-osmosis, as a technique particularly suitable for consolidation of low-permeability soils, combined with surcharge preloading, effectively improves the consolidation efficiency of soft soil, and the two are linearly superimposable.

  3. (3)

    The absolute value of the final negative EPWP induced by electro-osmosis grows with an increase in the hydraulic-to-electrical permeability coefficient ratio or applied voltage, indicating a direct proportional relationship between them.

  4. (4)

    The proposed solutions for axisymmetric consolidation of saturated soft soils under electro-osmotic-enhanced surcharge preloading are universally applicable; they can be employed to replicate real-world scenarios with varying permeability top boundaries by establishing appropriate interface parameters. Additionally, the test results indicate that assuming that the permeable boundary is a continuous permeable boundary will facilitate the modelling of real boundary permeability.