1 Introduction

Electroosmosis is a phenomenon that describes the process of pore water driving from anode to cathode when the electric gradient is applied, forming an electroosmotic flow and leading to the dewatering and consolidation of porous media. Since Casagrande [4] discovered the favorable effect of electroosmosis on the permanent stabilization of soil, the practical application has been extensively adopted in soil engineering [1, 2, 13, 18], especially when electro-kinetic geosynthetics (EKG) that could effectively avoid anodic corrosion was invented [8, 11, 17, 36, 39]. Electro-osmotic treatment areas are sometimes stacked on original soils with high water content [14, 39]. Since the horizontal EKG electrodes are usually arranged at intervals, pore water in these original soils can flow through the gaps and readily infiltrate into the treatment layer. Previous studies assumed that the anode arranged at intervals is impermeable and only focused on the consolidation effect of the treatment layer itself, neglecting the influence of the original soil.

Theoretical predictions of the consolidation process and consolidation completion time are necessary to facilitate engineering design. In comparison to numerical simulation, the analytical approach saves time and is more convenient for engineers to apply in actual designs. Based on the assumption that the pore water flow caused by the potential gradient and hydraulic gradient can be linearly superimposed, Esrig [6] developed the first one-dimensional (1D) model of electroosmotic consolidation and derived analytical solutions for excess pore water pressure and average degree of consolidation. Subsequently, analytical two-dimensional (2D) models [20, 24] and axisymmetric models [16, 34] were proposed for electroosmotic dewatering. Recently, more complicated analytical models of electroosmotic consolidation have been proposed from different aspects, including the combined use with surcharge and vacuum preloading [10, 22, 27, 30], non-linear properties of soil [31, 33, 35], and the decrease of effective voltage [7, 32]. The above models have contributed significantly to the knowledge of electroosmotic behavior. However, it should be noted that these models rely on the assumption of impermeable anode.

In recent years, EKG has been proposed and widely used in electroosmosis [9, 15]. The arrangement of strip-shaped EKG electrodes, similar to prefabricated vertical drains (PVD) [3, 21, 25] or prefabricated horizontal drains (PHD) [5, 19, 23], are usually intermittent instead of continuous. Similar to PHD, horizontal EKG arrangement way enables simultaneous dredging and consolidation, shortening the construction period, and this way allows the EKG to settle in conjunction with the soil, avoiding bending of the plates caused by settlement deformation [26, 29]. Wang et al. [28] used horizontal electrodes in sludge consolidation tests and compared them with vertical electrodes, demonstrating significant advantages of the former in terms of drainage efficiency and economy. In a recent study by Wan et al. [26], model tests were conducted to evaluate the effectiveness of horizontal electroosmotic consolidation utilizing a conductive plastic drainage plate and its drainage performance was demonstrated to be exceptional. Therefore, the present study focuses on the horizontal strip-shaped EKG in an intermittent arrangement way, which means that the pore water would penetrate through the gaps. On the other hand, dredging and filling engineering is commonly employed in coastal areas where the original soil is characterized by soft soil layers [8, 23]. During the consolidation process, there is a possibility that pore water from the original soil may seep into the treatment layer [14, 37], potentially impacting the electroosmotic consolidation efficiency of the treatment layer. However, to the authors’ knowledge, the previous models are almost based on the assumption that the anode is impermeable, which would be limited in their ability to investigate the influence of the permeable anode on the consolidation of the electroosmotic layer.

This study aims to propose a simplified 1D electroosmotic consolidation model of the layered soil considering the effect of original soil under permeable anode on the consolidation process. Firstly, the rationality of the 1D model is justified by comparing the results of a 2D model. The analytical solutions for excess pore water pressure and average degree of consolidation are then derived. Furthermore, the accuracies of the solutions are verified by existing analytical solutions and numerical results. Finally, the influence of original soil under permeable anode on the consolidation behavior is thoroughly investigated through parametric studies.

2 Theoretical analysis

2.1 Model development

A typical schematic diagram of electroosmotic consolidation of soft soil using horizontal EKG is displayed in Fig. 1a. The horizontal strip-shaped EKG is arranged at intervals on the original soil, similar to the deployment of prefabricated horizontal drains, allowing the pore water to flow through the gaps. This provides a permeable electrode boundary. The treatment layer, where the electrodes are placed at the two ends of the soil mass, rests on the original soft soil, creating the scenario of a double-layer electroosmosis system. To facilitate drainage, the cathode is positioned on the top side of the electroosmotic layer, while the anode is placed on the bottom side. This allows the pore water in the original soil to flow through the gaps between the anodes, creating a permeable anode boundary.

Fig. 1
figure 1

Schematic diagram of electroosmotic drainage of double-layered system incorporating original soil: a arrangement of horizontal strip EKG at intervals; b simplified 1D model for the electroosmotic drainage

Full size image

To analyze the consolidation problem in the layered system involving the original soil, a mathematical model needs to be developed. A 2D model is appropriate for simulating the strip EKG due to its significant length compared to its width and spacing. However, the strip EKG arranged at intervals results in partial permeable (pore water can flow through the gaps) and partial impermeable (pore water cannot flow through the EKG) at the boundary between the electroosmotic layer and the original soil. This makes it difficult to solve the 2D model analytically. Therefore, the pore water is free to flow in most regions at the interface between original soil and electroosmotic layer. Therefore, we propose a simplified 1D model (as shown in Fig. 1b), which assumes that the anode is permeable throughout the entire section.

The rationalization of the simplified 1D model is further illustrated by comparing its results with those of the 2D model. A 2D model, which can consider the variations of the width and spacing of the electrodes, is developed and solved by the finite element method. The detailed model information and results are implemented in the supplementary material. The results indicate that the 1D model accurately calculates the average degree of consolidation throughout the region. Although there are some differences in the electric field and the average excess pore water pressure near the electrodes, it can still reflect the overall pattern of aviation of the average pore pressure at the horizontal direction along the depth. Thus, the simplified 1D model can be considered as appropriate.

In Fig. 1b, the Eulerian coordinate, z, is defined as a positive number downward from a fixed datum plane that is coincident with the cathode, and the anode is fixed at z = H, where both cathode and anode boundaries are permeable. The thickness of original soil is set as L. The material parameters of the electroosmotic layer are the hydraulic conductivity khe, the electrical conductivity kee, and the compression coefficient mve. The material parameters of the original soil include the hydraulic conductivity khu and the compression coefficient mvu.

The major assumptions used are as follows:

  1. (1)

    Both the electrical potential and excess pore water pressure exhibit positional dependence; the electrical potential is assumed to remain constant over time, whereas the excess pore water pressure varies. The electrical potential Ve(z) is characterized by a linear depth function [6, 20].

    $$ V_{{\text{e}}} (z) = V_{\max } \frac{z}{H} $$
    (1)

    where Vmax denotes the applied maximum effective voltage. H is the thickness of electroosmotic layer.

  2. (2)

    Darcy’s law and Ohm’s law are valid for the electroosmotic layer and the original soil, and the velocity of pore water flow induced by hydraulic and electrical gradients for the electroosmotic layer can be linearly superimposed [6]. The velocities of the water flow for the two layers can be expressed as

    $$ v_{{\text{e}}} = v_{{{\text{he}}}} + v_{{{\text{ee}}}} = - \frac{{k_{{{\text{he}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial u_{{\text{e}}} (z,t)}}{\partial z} - k_{{{\text{ee}}}} \frac{{\partial V_{{\text{e}}} (z)}}{\partial z} $$
    (2)
    $$ v_{{\text{u}}} = - \frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial u_{{\text{u}}} (z,t)}}{\partial z} $$
    (3)

    where ve and vu are the velocities of the water flow in electroosmotic and unelectroosmotic layers (original soil). vhe and vee are the velocities induced by the hydraulic gradient and the voltage gradient in electroosmotic layer. ue and uu are the excess pore water pressure in the electroosmotic and unelectroosmotic layers, respectively. The weight of water γw = 9.8 kN/m3.

  3. (3)

    The pore water is able to flow freely through the interface at anode as horizontal EKG is arranged at intervals and the width of EKG is usually much smaller than the gaps. The excess pore water pressure and water flow velocity at the interface change continuously.

  4. (4)

    The hydraulic conductivity, electroosmosis conductivity, and coefficient of volume compressibility are assumed to be constant in each layer.

  5. (5)

    The thermal-osmotic and chemical-osmotic effects are disregarded.

On the basis of assumptions (2) and (5), the one-dimensional governing equations that describe water flow in the double-layered system can be expressed as [6, 12, 38]

$$ m_{{{\text{ve}}}} \frac{{\partial u_{{\text{e}}} (z,t)}}{\partial t} = \frac{{k_{{{\text{he}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial^{{2}} u_{{\text{e}}} (z,t)}}{{\partial z^{{2}} }} + k_{{{\text{ee}}}} \frac{{\partial^{{2}} V_{{\text{e}}} (z)}}{{\partial z^{{2}} }} $$
(4)
$$ m_{{{\text{vu}}}} \frac{{\partial u_{{\text{u}}} (z,t)}}{\partial t} = \frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial^{{2}} u_{{\text{u}}} (z,t)}}{{\partial z^{{2}} }} $$
(6)

The excess pore water pressure at the initial time is given by

$$ u_{{\text{e}}} (z,0) = u_{{\text{u}}} (z,0) = 0 $$
(6)

The top boundary (cathode) can be expressed as

$$ \left. {u_{{\text{e}}} (z,t)} \right|_{{z = {0}}} { = 0} $$
(7)

There is usually no water flow at the underlying bedrock, wherein the bottom boundary is set as impermeable.

$$ \left. {\frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial u_{{\text{u}}} (z,t)}}{\partial z}} \right|_{z = H + L} { = 0} $$
(8)

According to assumption (4), the interfacial continuity conditions can be given by

$$ \left. {u_{{\text{e}}} (z,t)} \right|_{z = H} { = }\left. {u_{{\text{u}}} (z,t)} \right|_{z = H} $$
(9)
$$ \left. {\left[ {\frac{{k_{{{\text{he}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial u_{{\text{e}}} (z,t)}}{\partial z} + k_{{{\text{ee}}}} \frac{{\partial V_{{\text{e}}} (z)}}{\partial z}} \right]} \right|_{z = H} { = }\left. {\frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial u_{{\text{u}}} (z,t)}}{\partial z}} \right|_{z = H} $$
(10)

2.2 Analytical solution

The analytical solution of the governing Eqs. (4) and (5) with the above boundary conditions is difficult to solve since that continuous condition Eq. (10) is inhomogeneous. In order to homogenize Eq. (10), the classical variables ξe and ξu were usually introduced according to the previous works [6, 20, 24]

$$ \xi_{{\text{e}}} (z,t) = u_{{\text{e}}} (z,t) + \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}V_{{\text{e}}} (z) $$
(11)
$$ \xi_{{\text{u}}} (z,t) = u_{{\text{u}}} (z,t) + \frac{{k_{{{\text{eu}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{hu}}}} }}V_{{\text{u}}} (z) $$
(12)

where keu is electrical conductivity of original soil; the voltage applied in original soil, Vu(z), equals zero. Once these variables are introduced, Eq. (10) is converted to be homogeneous, while the other continuous condition Eq. (9) turns to be inhomogeneous. Visibly, such dummy variables are invalid. Therefore, the new variables ζe and ζu are introduced to achieve the homogenization of all the continuous conditions.

$$ \zeta_{{\text{e}}} (z,t) = u_{{\text{e}}} (z,t) + f_{{\text{e}}} \left( z \right) $$
(13)
$$ \zeta_{{\text{u}}} (z,t) = u_{{\text{u}}} (z,t) + f_{{\text{u}}} \left( z \right) $$
(14)

where fe(z) and fu(z) are the functions of z. Then, rewriting Eqs. (4) and (5) yields the following

$$ m_{{{\text{ve}}}} \frac{{\partial \zeta_{{\text{e}}} (z,t)}}{\partial t} = \frac{{k_{{{\text{he}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial^{{2}} \zeta_{{\text{e}}} (z,t)}}{{\partial z^{{2}} }} + f^{\prime\prime}_{{\text{e}}} \left( z \right) $$
(15)
$$ m_{{{\text{vu}}}} \frac{{\partial \zeta_{{\text{u}}} (z,t)}}{\partial t} = \frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\frac{{\partial^{{2}} \zeta_{{\text{u}}} (z,t)}}{{\partial z^{{2}} }} + f^{\prime\prime}_{{\text{u}}} \left( z \right) $$
(16)

and Eqs. (6)–(10) are transformed into Eqs. (17)–(22).

$$ \zeta_{{\text{e}}} (z,0) = f_{{\text{e}}} \left( z \right) $$
(17)
$$ \zeta_{{\text{u}}} (z,0) = f_{{\text{u}}} \left( z \right) $$
(18)
$$ \left. {\zeta_{{\text{e}}} (z,t)} \right|_{z = 0} { = }f_{{\text{e}}} \left( 0 \right) $$
(19)
$$ \left. {\frac{{\partial \zeta_{{\text{u}}} (z,t)}}{\partial z}} \right|_{z = H + L} { = }f^{\prime}_{{\text{u}}} \left( z \right) $$
(20)
$$ \left. {\left[ {\zeta_{{\text{e}}} (z,t) - f_{{\text{e}}} \left( z \right)} \right]} \right|_{z = H} { = }\left. {\left[ {\zeta_{{\text{u}}} (z,t) - f_{{\text{u}}} \left( z \right)} \right]} \right|_{z = H} $$
(21)
$$ \frac{{k_{{{\text{he}}}} }}{{\gamma_{{\text{w}}} }}\left. {\left[ {\frac{{\partial \zeta_{{\text{e}}} (z,t)}}{\partial z} - f^{\prime}_{{\text{e}}} \left( z \right) + \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}\frac{{\partial V_{{\text{e}}} (z)}}{\partial z}} \right]} \right|_{z = H} { = }\frac{{k_{{{\text{hu}}}} }}{{\gamma_{{\text{w}}} }}\left. {\left[ {\frac{{\partial \zeta_{{\text{u}}} (z,t)}}{\partial z} - f^{\prime}_{{\text{u}}} \left( z \right)} \right]} \right|_{z = H} $$
(22)

To homogenize the two continuous conditions Eqs. (21) and (22), Eqs. (23) and (24) need to be satisfied, respectively.

$$ f_{{\text{e}}} \left( H \right){ = }f_{{\text{u}}} \left( H \right) $$
(23)
$$ f^{\prime}_{{\text{u}}} \left( H \right){ = }\frac{{k_{{{\text{he}}}} }}{{k_{{{\text{hu}}}} }}\left( {f^{\prime}_{{\text{e}}} \left( H \right) - \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}\frac{{\partial V_{{\text{e}}} (H)}}{\partial z}} \right) $$
(24)

The next mission is to find suitable fe(z) and fu(z). According to Eqs. (15) and (16), let

$$ \frac{{d^{2} f_{{\text{e}}} \left( z \right)}}{{dz^{2} }} = g_{{\text{e}}} \left( z \right) $$
(25)
$$ \frac{{d^{2} f_{u} \left( z \right)}}{{dz^{2} }} = g_{{\text{u}}} \left( z \right) $$
(26)

where ge(z) and gu(z) denote functions with respect to z. Integration of fe(z) and fu(z) yield

$$ f_{{\text{e}}} \left( z \right) = \int_{0}^{z} {\int_{0}^{y} {g_{{\text{e}}} \left( x \right)} } dxdy + A_{{\text{e}}} z + B_{{\text{e}}} $$
(27)
$$ f_{{\text{u}}} \left( z \right) = \int_{H}^{z} {\int_{H}^{y} {g_{{\text{u}}} \left( x \right)} } dxdy + A_{{\text{u}}} z + B_{{\text{u}}} $$
(28)

where Ae, Be, Au, Bu, are undetermined coefficients. Substituting Eqs. (27) and (28) into Eqs. (23) and (24), we have

$$ \int_{0}^{H} {\int_{0}^{y} {g_{{\text{e}}} \left( x \right)} } dxdy + A_{{\text{e}}} H + B_{{\text{e}}} { = }\int_{H}^{H} {\int_{H}^{y} {g_{{\text{u}}} \left( x \right)} } dxdy + A_{{\text{u}}} H + B_{{\text{u}}} $$
(29)
$$ A_{{\text{u}}} = \int_{H}^{H} {g_{{\text{u}}} \left( x \right)} dx + \frac{{k_{{{\text{he}}}} }}{{k_{{{\text{hu}}}} }}\left[ {\int_{0}^{H} {g_{{\text{e}}} \left( x \right)} dx + A_{{\text{e}}} - \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}\frac{{\partial V_{{\text{e}}} (H)}}{\partial z}} \right] $$
(30)

From the two equations, Eqs. (29) and (30), there are six unknowns (or functions) in the equation, allowing for countless combinations of solutions. Thus, we can set ge(x) and gu(x) to zero. Additionally, substitute Eqs. (27) and (28) into the boundary conditions Eqs. (19) and (20). We find that setting Be and Au to zero can homogenize the boundary conditions. Hence, Ae and Bu can be obtained as

$$ A_{{\text{e}}} = \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}\frac{{\partial V_{{\text{e}}} (H)}}{\partial z} $$
(31)
$$ B_{{\text{u}}} = \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}\frac{{\partial V_{{\text{e}}} (H)}}{\partial z}H $$
(32)

Therefore, the original problem for electroosmotic consolidation of layered soil with inhomogeneous continuity conditions is transformed into a new problem with homogeneous ones. A detailed description of the new problem is shown in Appendix A.

The general solutions to Eqs. (A1) and (A2) can be obtained as

$$ \zeta_{{\text{e}}} (z,t) = \sum\limits_{m = 1}^{\infty } {g_{{m{\text{e}}}} (z)f_{m} } (t) $$
(33)
$$ \zeta_{{\text{u}}} (z,t) = \sum\limits_{m = 1}^{\infty } {g_{{m{\text{u}}}} (z)f_{m} } (t) $$
(34)

Substituting Eqs. (33) and (34) into Eqs. (A1) and (A2), we have

$$ \frac{{k_{{{\text{he}}}} }}{{m_{{{\text{ve}}}} \gamma_{{\text{w}}} }}\frac{{g^{\prime\prime}_{{m{\text{e}}}} (z)}}{{g_{{m{\text{e}}}} (z)}} = \frac{{f^{\prime}_{m} (t)}}{{f_{m} (t)}} = - \beta_{{m{\text{e}}}} $$
(35)
$$ \frac{{k_{{h{\text{u}}}} }}{{m_{{{\text{vu}}}} \gamma_{{\text{w}}} }}\frac{{g^{\prime\prime}_{{m{\text{u}}}} (z)}}{{g_{{m{\text{u}}}} (z)}} = \frac{{f^{\prime}_{m} (t)}}{{f_{m} (t)}} = - \beta_{{m{\text{u}}}} $$
(36)

where βme and βmu are the positive unknowns for the electroosmotic layer and unelectroosmotic layer, respectively. According to the continuity condition, it can be found that βme = βmu = βm.

Solving Eqs. (35) and (36), gme(z), gmu(z) and fm(t) can be obtained as

$$ g_{{m{\text{e}}}} (z) = A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z) + B_{{m{\text{e}}}} \cos (\lambda_{{m{\text{e}}}} z) $$
(37)
$$ g_{{m{\text{u}}}} (z) = C_{{m{\text{u}}}} \sin (\lambda_{{m{\text{u}}}} z) + D_{{m{\text{u}}}} \cos (\lambda_{{m{\text{u}}}} z) $$
(38)
$$ f_{m} (t) = c_{m} e^{{ - \beta_{m} t}} $$
(39)

where \(\lambda_{{m{\text{e}}}} = \sqrt {m_{{{\text{ve}}}} \gamma_{{\text{w}}} \beta_{m} /k_{{{\text{he}}}} }\), \(\lambda_{{m{\text{u}}}} = \sqrt {m_{{{\text{vu}}}} \gamma_{{\text{w}}} \beta_{m} /k_{{{\text{hu}}}} }\). There are four undetermined coefficients, Ame, Bme, Cmu and Dmu, in Eqs. (37) and (38), which is relatively tedious. To facilitate the solution, more concise general solutions of Eqs. (35) and (36) are adopted by considering the boundary conditions, gme(z) and gmu(z) can be obtained as

$$ g_{{m{\text{e}}}} (z) = A^{\prime}_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z) $$
(40)
$$ g_{{m{\text{u}}}} (z) = C^{\prime}_{{m{\text{u}}}} \cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right] $$
(41)

in which A’me and C’mu are undetermined coefficients.

According to Eqs. (33) and (34), the general solution can be further expressed as

$$ \zeta_{{\text{e}}} (z,t) = \sum\limits_{m = 1}^{\infty } {c_{m} A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)e^{{ - \beta_{m} t}} } $$
(42)
$$ \zeta_{{\text{u}}} (z,t) = \sum\limits_{m = 1}^{\infty } {c_{m} \cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]e^{{ - \beta_{m} t}} } $$
(43)

Combining the continuity conditions Eqs. (A7) and (A8), the following equations can be obtained.

$$ A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} H){ = }\cos \left( {\lambda_{{m{\text{u}}}} L} \right) $$
(44)
$$ A_{{m{\text{e}}}} k_{{{\text{he}}}} \lambda_{{m{\text{e}}}} \cos (\lambda_{{m{\text{e}}}} H){ = }k_{{{\text{hu}}}} \lambda_{{m{\text{u}}}} \sin \left( {\lambda_{{m{\text{u}}}} L} \right) $$
(45)

Accordingly, Ame and βm can be obtained from the two Eqs. (44) and (45).

Substituting Eqs. (42) and (43) into the initial condition Eqs. (A3) and (A4), we can obtain the following equations

$$ \zeta_{{\text{e}}} (z,0) = \sum\limits_{m = 1}^{\infty } {c_{m} A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)} = \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} V_{\max } }}{{k_{{{\text{he}}}} }}\frac{z}{H} $$
(46)
$$ \zeta_{{\text{u}}} (z,0) = \sum\limits_{m = 1}^{\infty } {c_{m} \cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]} = \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} V_{\max } }}{{k_{{{\text{he}}}} }} $$
(47)

Furthermore, using the orthogonal relationship

$$ \begin{gathered} m_{{{\text{ve}}}} \int_{0}^{H} {A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)A_{{n{\text{e}}}} \sin (\lambda_{{n{\text{e}}}} z)dz} \hfill \\ + m_{{{\text{vu}}}} \int_{H}^{H + L} {\cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]\cos \left[ {\lambda_{{n{\text{u}}}} (z - H - L)} \right]dz} = 0\begin{array}{*{20}c} , & {\left( {m \ne n} \right)} \\ \end{array} \hfill \\ \end{gathered} $$
(48)

the parameter cm can be obtained as

$$ c_{m} = \frac{{m_{{{\text{ve}}}} \int_{0}^{H} {\zeta_{{\text{e}}} (z,0)A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)dz} + m_{{{\text{vu}}}} \int_{H}^{H + L} {\zeta_{u} (z,0)\cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]dz} }}{{m_{{{\text{ve}}}} \int_{0}^{H} {\left[ {A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)} \right]^{2} dz + m_{{{\text{vu}}}} \int_{H}^{H + L} {\cos^{2} \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]dz} } }} $$
(49)

As a result, the analytical solution to excess pore water pressure can be obtained

$$ u_{{\text{e}}} (z,t) = \sum\limits_{m = 1}^{\infty } {c_{m} A_{{m{\text{e}}}} \sin (\lambda_{{m{\text{e}}}} z)e^{{ - \beta_{m} t}} } - \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}V_{{\text{e}}} (z) $$
(50)
$$ u_{{\text{u}}} (z,t) = \sum\limits_{m = 1}^{\infty } {c_{m} \cos \left[ {\lambda_{{m{\text{u}}}} (z - H - L)} \right]e^{{ - \beta_{m} t}} } - \frac{{k_{{{\text{ee}}}} \gamma_{{\text{w}}} }}{{k_{{{\text{he}}}} }}V_{\max } $$
(51)

Here, the average excess pore water pressure of electroosmotic layer, which considers the effect of original soil, is calculated by

$$ u_{{{\text{avg}}}} (t) = \frac{1}{H}\int_{{0}}^{H} {u_{{\text{e}}} (z,t)dz} $$
(52)

and the corresponding average degree of consolidation is given by

$$ U_{{}}^{*} (t) = 1 - \frac{{\int_{0}^{H} {[u_{{\text{e}}} (z,t) - u_{{\text{e}}} (z,\infty )]dz} }}{{\int_{0}^{H} {[u_{{\text{e}}} (z,0) - u_{{\text{e}}} (z,\infty )]dz} }} $$
(53)

3 Verification

As there is no known exact solution for the electroosmotic consolidation of a double-layered system incorporating original soil. In this section, we aim to address this gap in knowledge by verifying our proposed solution using simplified analytical solution. Subsequently, a numerical approach is adopted to further confirm its accuracy.

3.1 Verification against analytical solution for single layer

The first verification example is conducted to compare the present solution with the existing analytical solution for electroosmotic consolidation of a single layer [6]. The soil properties for the electroosmosis layer with a thickness of 1 m are as follows: khe = 2 × 10−9 ms−1, kee = 5 × 10−9m2s−1 V−1, mve = 5 × 10−3 kPa−1. The voltage is fixed at V = 10 V. The thickness of original soil is set to a sufficiently small value (L = 0.01 m), and thus the present model can be taken as a single electroosmosis layer. The comparison results with respect to excess pore water pressure along depth at different Tv (Tv = Cvet/H2) is the time factor, where Cve = khe/(mveγw) denotes consolidation coefficient) are illustrated in Fig. 2, which suggests excellent agreement.

Fig. 2
figure 2

Comparison with the existing analytical solution for electroosmotic consolidation of single layer

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3.2 Verification against numerical results for double-layered system with original soil

In order to validate the proposed solution in a general setting, a finite element method (FEM) based on COMSOL Multiphysics is utilized to solve the present equations, allowing us to further confirm the effectiveness of the proposed solution. The soil parameters of both the electroosmotic layer and the original soil remain the same as in the previous validation example, while the voltage is not applied to the original soil. The original soil layer is set to a thickness of 1 m, and the voltage of the electroosmosis layer is set to 10 V. The excess pore water pressure profiles at different Tv are compared in Fig. 3, revealing that the values obtained from the present solution are in agreement with those obtained from the FEM solution. Thus, these verification examples provide strong evidence that the proposed analytical solution are robust and reliable for the electroosmotic dewatering of double-layered system incorporating original soil.

Fig. 3
figure 3

Comparison with the numerical results for electroosmotic consolidation of double-layered system incorporating original soil

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4 Parametric study

In practice, the variation of hydraulic permeability coefficient and compression coefficient of the original soil depending on different soil types has a wide range, which has a significant impact on the drainage process and is the focus of parameter analysis in this paper. In order to make the results more universal, the properties of the electroosmotic layer are kept unchanged and the parameters of the original soil are varied (as shown in Table 1). Let a = khu/khe and b = mvu/mve. Thus, the main parameters in the present study include a, b, L, ke and V. From Eqs. (4)-(10), it is obvious that the electroosmotic drainage capability is proportional to the electroosmotic coefficient (or voltage), and thus the electroosmotic coefficient and the applied voltage are kept constant and are not discussed.

Table 1 Soil properties for parametric study
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4.1 Effect of original soil under permeable anode on excess pore water pressure distribution

In order to quantify the influence of the original soil under permeable anode on the distribution of the excess pore water pressure ue, the critical thickness of original soil, which means that the ue value is unchanged when the thickness of original soil exceeds this value, is defined. It is observed from Fig. 4a that the excess pore water pressure at anode (uz=H) rapidly increases from − 228 to − 185 as H increases from 0 to 0.4 m when a = 0.1, and then the variation of uz=H is negligible when H increases again, which means that the critical thickness under this condition is 0.4 m. This indicates that the critical thickness of the layer is smaller when the permeability coefficient of original soil is relatively small due to the fact that the velocity of the pore water in original soil penetrating into the electroosmotic layer is slower. When a = 1 and 10, the critical depth of original soil greatly increases (it is not shown in the figures as the critical depth is much larger than 1 m), since the pore water in original soil can flow quickly to the electroosmotic layer.

Fig. 4
figure 4

Excess pore water pressure profiles for different ratios a = khu/khe at Tv = 1.0: a a = 0.1; b a = 1; c a = 10

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The effects of different compression coefficient ratios b = mvu/mve on the distribution of excess pore water pressure at Tv = 1.0 are depicted in Fig. 5. The larger b is, the more significant is the effect of original soils under the permeable anode. For b = 0.1 and 1, the critical thickness of original soil is much greater than 1 m, whereas for b = 10, the critical thickness is reduced to only no more than 0.6 m. This is because larger compressibility means that the soil is readily compressed and the pore water is more susceptible to drainage from the original soil and passes through the permeable anode, following which it infiltrates into the electroosmotic layer.

Fig. 5
figure 5

Excess pore water pressure profiles for different ratios b = mvu/mve at Tv = 1.0

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To more comprehensively investigate the spatial and temporal variation rules of a and b on the excess pore water pressure profile during the entire process of electroosmosis, the cloud maps of the excess pore water pressure distribution for the case with L = 0.6 m are displayed in Fig. 6. As can be observed, there are obvious sharp corners at the interfaces when the permeability coefficient of the original soil layer is low (a = 0.1) (Fig. 6a and b), and the larger b is, the more remarkable the sharp corner is. The obvious sharp corner indicates that the difference between the excess pore water pressure at the top and bottom of original soil (termed as ∆uu) is large, which also means that the pore water in the deeper soil layer is discharged slowly. The ∆uu value at Tv = 1.0 for a = 0.1 and b = 10 reaches up to 110 kPa (Fig. 6b). Because the pore water in the original soil with a lower permeability coefficient (a = 0.1) is difficult to flow. On the contrary, the pore water in the original soil with a higher permeability coefficient (a = 10) can flow rapidly and would be sucked into the electroosmotic layer readily by the negative pressure at the permeable anode, and thus the excess pore water pressure in the original soil layer is always approaching the anode (Fig. 6c and d). The ∆uu value at Tv = 1.0 for a = 10 and b = 10 is only 5 kPa (Fig. 6d).

Fig. 6
figure 6

Cloud maps of the distribution of excess pore water pressure along the depth with time: a a = 0.1, b = 0.1; b a = 0.1, b = 10; c a = 10, b = 0.1; d a = 10, b = 10

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When the compression coefficient of the original soil is relatively small (b = 0.1), the excess pore water pressure can quickly tend to stabilize and it is basically stabilized when Tv ≥ 2 (Fig. 6a and c). While the required time to reach stabilization for the excess pore water pressure increases up to Tv > 20 when the compression coefficient of the original soil is relatively large (b = 10).

Notably, the ultimate excess pore water pressure profiles for all cases are identical and the ultimate uu in original soil is equal to that of the permeable anode (Fig. 6), indicating that they are not influenced by the properties of original soil. This phenomenon can be explained by Eq. (51), which shows that the ultimate uu is dependent on the permeability coefficient of the electroosmotic layer and the applied voltage, but independent of the characteristics of the original soil. This inspires us to utilize non-inserted electrodes for consolidating the original soil in blowfill engineering.

4.2 Effect of original soil under permeable anode on average excess pore water pressure

This section investigates the effect of the original soil on the average excess pore water pressure uavg of the electroosmotic layer. The variation curves of the uavg values with time for different values of a or b are presented in Figs. 7 and 8, respectively. As can be observed, the original soil layer only delays the drainage process of the electroosmotic layer (termed as delayed effect of the original soil on electroosmosis) without affecting the ultimate uavg value. Because the ultimate uavg value only depends on the ratio of kee/khe and the electrical potential distribution as can be seen from Eq. (50). When a = 0.1, the effect of the original soil on uavg is small at the beginning of electroosmosis, and then gradually increases with time. Furthermore, the larger L is, the more significant is the delayed effect at the later stage of electroosmosis (Fig. 7a). While for a = 10, the delayed effect is more significant at the overall consolidation process of the electroosmotic layer, and this effect is more significant for larger L (Fig. 7b). When the compression coefficient of the original soil is smaller than electroosmotic layer (b = 0.1), the delayed effect is slight (Fig. 8a). While significant delay effects occur for even small values of L when b = 10. For example, the time corresponding to uavg = − 100 kPa at L = 0.2 m (Tv = 4.1) is much larger than that for L = 0 (Tv = 0.7) (Fig. 8b).

Fig. 7
figure 7

Effect of original soil on the average excess pore water pressure for different ratios a = khu/khe: a a = 0.1; b a = 10

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Fig. 8
figure 8

Effect of original soil on the average excess pore water pressure for different ratios b = mvu/ mve: a b = 0.1; b b = 10

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In order to visually reflect the extent of the influence of original soil on the average excess pore water pressure uavg, the difference in uavg with and without consideration of the original soil at the same consolidation time, ∆uavg, is defined:

$$ \Delta u_{avg} = \, \Delta u_{avg} \left( {L > 0} \right) - \, \Delta u_{avg} \left( {L = 0} \right) $$
(54)

Obviously, the larger ∆uavg means more significant delayed effect of original soil on electroosmotic consolidation. The curves of ∆uavg with respect to time for different a and b are plotted in Figs. 9 and 10. As shown in the figures, ∆uavg first increases rapidly to a certain peak ∆uavg_max with increasing Tv, and then decreases immediately to zero. For a = 0.1 (Fig. 9a) or b = 10 (Fig. 10b), the ∆uavg values are almost equivalent for various L in the preliminary stage of electroosmosis, while as Tv increases, generally the larger L results in the larger ∆uavg and the peak values ∆uavg_max increase gradually. However, the variation of ∆uavg_max is modest with increasing L again for L > 0.6 m. On the other hand, a more pronounced delay effect occurs in the thicker original soil in the later stage of electroosmosis. The general tendencies of ∆uavg with Tv for a = 10 (Fig. 9b) or b = 0.1 (Fig. 10a) are similar to those for a = 0.1 or b = 10, but the peak values ∆uavg_max always increase significantly with increasing L. Moreover, at the early stage of electroosmosis, the larger L exhibits the more unfavorable effect of original soil on consolidation process of electroosmotic layer.

Fig. 9
figure 9

Variation of ∆uavg with Tv for different ratios a = khu/khe: a a = 0.1; b a = 10

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Fig. 10
figure 10

Variation of ∆uavg with Tv for different ratios b = mvu/mve: a b = 0.1; b b = 10

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Comparing Fig. 9a and b, the value ∆uavg_max for a = 10 is considerably larger than that for a = 0.1, while the impact time of the former terminates faster. For example, the ∆uavg_max value for a = 10 (53.4 kPa) is over twice as large as that for a = 0.1 (23.2 kPa) when L = 1 m, while the impact time of the former (Tv < 10) is much shorter than that of the latter (Tv > 20). When the compression of the original soil is comparatively large (b = 10), not only is the maximum peak (e.g., ∆uavg_max = 82.8 kPa for L = 1 m) much larger than that for the b = 0.1 (e.g., ∆uavg_max = 10.2 kPa for L = 1 m), but also the impact time of the former (Tv≈ 100) is significantly larger than that of the latter (Tv ≈ 5).

4.3 Effect of original soil under permeable anode on average degree of consolidation

The effects of original soil under permeable anode on the average degree of consolidation of the electroosmotic layer, U*, are discussed in this section. In experience, the electric field may not be removed until the average degree of consolidation reaches 90% (U*90). Therefore, to better illustrate the effect of original soil on the drainage process, Tv90 is defined as the required time to achieve U*90. For a = 0.1, Tv90 for L = 0.2, 0.4, 0.6, 0.8, and 1.0 m are 1.5, 1.8, 2.4, 3.3, and 3.7, respectively. The values of Tv90 increase by 67, 100, 167, 267, and 311%, respectively, in comparison to those without considering the original soil (Tv90 = 0.9). On the other hand, Tv90 equals to 0.9 when the original soil is not considered (L = 0), and at this time, the values of U*90 corresponding to L = 0.2, 0.4, 0.6, 0.8, 1.0 m are 0.88, 0.77, 0.71, 0.70, 0.70, respectively (Fig. 11a). Similar regularities are observed for a = 10, the corresponding Tv90 increases more than threefold as L increases from 0 to 1.0 m (Fig. 11b). While one notable contrast is that the influence of the original soil on the entire consolidation process is significant, leading to a dramatic decrease in U*90 from 0.9 to 0.46 at Tv = 0.9 as L increases from 0 to 1.0 m.

Fig. 11
figure 11

Variation of the average degree of consolidation with time factor for different ratios a = khu/khe: a a = 0.1; b a = 10

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The ratio of compression coefficient between original soil and electroosmotic layer (b = mvu/mve) has a significant impact on the average degree of consolidation of electroosmotic layer and the effect increases rapidly with increasing b. For b = 0.1, The time required to reach U*90 corresponding to L = 1.0 m is Tv90 = 1.17, which increases by 23% compared to the case without considering the original soil (Fig. 12a). While for b = 10, the effect of the original layer is quite remarkable. The Tv90 value reaches up to 5.7 for L = 0.2 m, which is about 500% higher than that without considering the original soil. On the other hand, when Tv = 0.9, U* is only 0.32 for L = 0.2 m (Fig. 12b).

Fig. 12
figure 12

Variation of the average degree of consolidation with time factor for different ratios b = mvu/mve: a b = 0.1; b b = 10

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For the sake of a more comprehensive description of the effect of original soil on Tv90, the relationships of Tv90 with the ratios of a = khu/khe or b = mvu/mve are plotted in Fig. 13. As can be observed from the figures, the ratio of permeability coefficient a has a relatively insignificant effect, while the ratios of thickness or compression coefficient produce the much more significant delayed effect on the consolidation process of the electroosmotic layer. The Tv90 curves almost appear as approximately horizontal lines for different a, especially when L = 0.2 (Fig. 13a), indicating that the effect of a is minimal. As L increases to 1 m, Tv90 shows a tendency of first slightly increasing and then decreasing with an increase of a. The difference of Tv90, i.e., ∆Tv90 = Tv90 (L > 0) − Tv90 (L = 0), increases remarkably with increasing L, illustrating that the thickness of original soil plays an important role in the delayed effect. On the contrary, the Tv90 values dramatically increase with an increase of b. When b = 0.1, the ∆Tv90 values approach zero for different L, while Tv90 increases up to 6.4 times for b = 2 and L = 1 m.

Fig. 13
figure 13

Variations of Tv90 with a a = khu/khe; b b = mvu/ mve

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5 Conclusions

Traditional analytical models usually assume that the anode arranged at intervals is an impermeable boundary, which cannot analyze the influence of the original soil. In this paper, a 1D analytical model considering the effect of original soil under the permeable anode, simulating the horizontal EKG in spaced arrangement, on the consolidation effect of electroosmotic layer is proposed. The rationality of the 1D model is justified by comparing the results of a 2D numerical model. The validation examples show that the analytical solution possesses excellent accuracy. Parametric analyses are carried out utilizing this model, and results show that the original soil under the permeable anode may have a significantly delayed effect on the consolidation process of the electroosmotic layer for the reason that the pore water in the original soil will be penetrated into the electroosmotic layer. The main conclusions are as follows:

  1. 1.

    The critical thickness of original soil decreases with a decrease of a since the pore water in original soil with a lower permeability coefficient penetrating into the electroosmotic layer is slower. The critical thickness is only 0.4m for a=0.1 at Tv=1.0.

  2. 2.

    The larger b causes a more significant effect of original soil on the distribution of the excess pore water pressure and a smaller critical thickness of original soil since the original soil with larger compressibility is readily compressed and the pore water is more susceptible to infiltrating into the electroosmotic layer. The critical thickness is reduced to only 0.6m or less for b=10.

  3. 3.

    The ultimate excess pore water pressure is independent of the properties of original soil, while the excess pore water pressure distribution in the original soil during consolidation is significantly affected by the value of a. In the present case, the difference between the excess pore water pressure of the top and bottom in original soil at Tv=1.0 for a=0.1 and b=10 reaches 110 kPa, while the one for a=10 is only 5 kPa.

  4. 4.

    The more significant delayed effect on the consolidation of the electroosmotic layer for thicker original soil with a larger compression coefficient, while the hydraulic permeability coefficient of original soil has an insignificant effect. As L increase from 0 to 1.0m, the required time corresponding to 90% average degree of consolidation (Tv90) undergoes an increase of over threefold regardless of the value of a, while Tv90 increases up to more than 6 times for b=2.

The present analytical model facilitates precise and straightforward prediction and assessment of the impact of original soil on the electroosmotic consolidation of electroosmotic layer. Additionally, we can take advantage of the fact that the anode is permeable to directly consolidate shallower original soil by laying the electrodes horizontally without the need to insert them into the original soil.