Nonlinear Large-Strain Consolidation of Vertical Drains with Coupled Radial–Vertical Flow Considering Hansbo’s Flow and Smearing Effects

Abstract

While early ideal consolidation theories for vertical drains focused primarily on radial flow, numerous coupled radial–vertical seepage models have since been developed to better capture complex flow behavior in practice. To overcome this limitation, a nonlinear large-strain consolidation model for vertical drains with coupled radial-vertical flow is proposed, explicitly incorporating Hansbo’s non-Darcy flow, smear effects, and soil nonlinearity. The finite difference method is then employed to obtain numerical solutions, and the reliability of the proposed numerical scheme is verified by degenerating the model to the radial consolidation case and comparing the results with the corresponding analytical solution. The results indicate that consolidation develops fastest when the permeability coefficient within the smear zone follows a parabolic distribution. Increasing the Hansbo’s flow parameter m and threshold hydraulic gradient parameter I₁ markedly slows down the consolidation process, while the contribution of vertical flow is primarily confined to the early stage. In addition, larger soil nonlinearity parameters I_c and α amplify the influence of radial–vertical coupled flow. Parametric analysis further shows that when the ratio of soil layer thickness to the radius of the influence zone (H/r_e) exceeds 10, the effect of vertical flow becomes negligible, and the consolidation behavior can be reasonably approximated using a radial-flow-only model.

1. Introduction

Vertical drains are widely adopted to accelerate the consolidation of soft ground by shortening drainage paths. In many practical scenarios—particularly when finite drainage boundaries exist, or soil layers are relatively thin—the consolidation process becomes inherently multidimensional, involving coupled radial and vertical flow [1,2]. Recent experimental investigations have further revealed that the reduction ratio in permeability and the radial gradient within the smear zone depend strongly on the intensity of the installation disturbance and soil plasticity [3]. Meanwhile, installation-induced disturbance, nonlinear compressibility, permeability evolution, and large deformation frequently occur simultaneously in soft clays [4,5]. The combined influence of these mechanisms makes the consolidation behavior of vertically drained ground significantly more complex than that described by classical idealized theories.

Substantial progress has been made in modeling different aspects of vertical drain consolidation. Coupled radial–vertical flow formulations have been developed to account for multidimensional seepage under various loading and boundary conditions [4,6,7,8,9]. In parallel, large-strain consolidation theories have been established to capture substantial deformation and evolving soil properties [3,10,11,12,13,14,15]. In addition, nonlinear compression–permeability relationships and non-Darcy flow formulations have been incorporated into consolidation analyses to describe the consolidation behavior of highly compressible soft soils [11,15,16,17,18,19]. In particular, Li et al. [18], based on extensive experimental data, proposed double-logarithmic relationships, lg(1 + e) − lgσ′ and lg(1 + e) − lgk, which demonstrate superior applicability in describing the large-deformation nonlinear consolidation behavior of highly compressible soft soils. Li et al. [19] derived nonlinear consolidation equations and their asymptotic solutions based on this model, effectively reducing discrepancies between theoretical predictions and experimental results. Furthermore, the installation-induced smearing effect and its influence on radial consolidation have been extensively investigated under different permeability distributions and loading modes [8,20,21,22,23,24,25]. These studies have significantly advanced the theoretical understanding of consolidation in vertically drained soft ground.

However, most existing investigations address these mechanisms separately or under simplifying assumptions. Coupled radial–vertical flow models are commonly formulated within small-strain and Darcy flow frameworks [4,6,8]. Large-strain analyses often neglect multidimensional flow or adopt linear permeability relationships [3,14]. Similarly, nonlinear compression and non-Darcy flow effects are typically investigated within one-dimensional or purely radial consolidation settings [19,26]. As a result, a unified theoretical framework that simultaneously integrates large strain, coupled radial–vertical flow, smearing effects, non-Darcy seepage, and nonlinear compressibility–permeability evolution under axisymmetric conditions remains lacking.

To address this gap, the present study develops an axisymmetric large-strain nonlinear consolidation model for vertically drained soft ground. The formulation incorporates coupled radial–vertical flow, installation-induced smearing, Hansbo’s non-Darcy flow rule, and a double-logarithmic nonlinear compression–permeability relationship within a unified governing equation. The model is solved numerically to systematically evaluate the influence of key parameters on consolidation rate and settlement evolution. Comparisons with traditional small-strain radial consolidation solutions are conducted to clarify the significance of the coupled mechanisms. The proposed framework aims to provide a more comprehensive theoretical basis for the analysis and design of vertically drained soft ground under complex engineering conditions.

2. Large-Strain Nonlinear Radial-Vertical Consolidation Model

2.1. Problem Description

Figure 1 presents a simplified consolidation model of vertical-drain-improved soft ground, incorporating the smear effect and radial and vertical flow. In Figure 1, q_u represents the instantaneous uniform load applied on the foundation surface, H is the thickness of the soil layer, and r_w, r_s, and r_e denote the radii of the vertical drain, the smeared zone, and the influence zone, respectively. The radial permeability coefficient of the soil, considering the smear effect, is denoted as k_r(r), while the vertical permeability coefficient is k_v. The reduction in permeability within the smear zone is primarily attributed to remolding-induced destruction of soil fabric, particle reorientation, and clogging of drainage paths during drain installation, as confirmed by laboratory and field observations [1,3,6]. The permeability modification is therefore spatially dependent and may vary from the drain boundary to the influence zone. It is assumed that the outer boundary and bottom of the drain are impermeable, and the ground surface is permeable.

2.2. Basic Assumptions

To solve the consolidation problem of the vertical drain foundation in this study, the following assumptions are made:

(1) The soil is fully saturated, and the compressibility of soil particles and pore water is neglected.

(2) The flow direction is decomposed into radial and vertical components. The flow in the soil follows the Hansbo’s flow model, with the model parameters m and i₁ remaining constant during consolidation [27]. While Teh and Nie [27] considered four cases by distinguishing radial and vertical hydraulic gradients (i_z and i_r) with respect to the threshold gradient i₁, this paper adopts the following unified expression for simplicity and to reduce classification:

$$\overrightarrow{\mathrm{v}}=\left\{\begin{array}{cc}-k{\displaystyle \frac{{\left|i\right|}^{m-1}}{m{i}_1^{m-1}}}\overrightarrow{i}& \left|i\right|\le i_1\\ -k[\left|i\right|-{\displaystyle \frac{i_1(m-1)}{m\left|i\right|}}\overrightarrow{i}]& \left|i\right|\ge i_1\end{array}\right.$$

(1)

where v is the flow velocity in the radial or vertical direction, denoted as $\overrightarrow{{\mathrm{v}}_r}$ and $\overrightarrow{{\mathrm{v}}_z}$, respectively; $\overrightarrow{i}$ is the hydraulic gradient in the radial or vertical direction, denoted as $\overrightarrow{i_r}$ and $\overrightarrow{i_z}$, respectively; k is the permeability coefficient in the radial or vertical direction, denoted as $\overrightarrow{k_r}$ and $\overrightarrow{k_z}$, respectively; i₁ is the threshold hydraulic gradient for the onset of linear flow; and m is an experimentally determined constant, with $\left|i_r\right|={\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}$, $\left|i_z\right|={\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }z}}$.

(3) Large-strain geometric conditions are assumed, with deformation occurring only in the vertical direction.

(4) The nonlinear behavior of the soil is described using the linear relationships lg(1 + e) − lg σ′ and lg(1 + e) − lg k, i.e.,

$$\lg (1+e)-\lg (1+e_0)=I_c(\lg {{\sigma }^{\prime }}_0-\lg {\sigma }^{\prime })$$

(2)

$$\lg (1+e)-\lg (1+e_0)={\displaystyle \frac{1}{\alpha }}(\lg k-\lg k_0)$$

(3)

where e is the void ratio at any time, e₀ is the initial void ratio, σ′ is the effective stress at any time, σ₀′ is the initial effective stress, k is the permeability coefficient (radial or vertical) at any time, k₀ is the initial permeability coefficient, I_c is the slope of the lg(1 + e) − lgσ′ relationship, and α is the reciprocal of the slope of the lg(1 + e) − lgk relationship. For convenience, I_c and α are hereafter referred to as the nonlinear compression parameter and nonlinear permeability parameter, respectively.

(5) The variation in the radial permeability coefficient along the radial direction, i.e., the smear effect, is considered, as expressed in Equation (4). Three smear patterns are assumed [3,4,28], as shown in Figure 2. Experimental studies have shown that permeability reduction in the smear zone may exhibit different spatial characteristics depending on installation method, soil sensitivity, and drainage disturbance [1,6,11]. To capture these potential variations, three representative permeability distribution patterns are considered in this study.

$$k_r(r)=k_{r}f(r)$$

(4)

The functional relationships for patterns 1, 2, and 3 are given by Equations (5), (6), and (7), respectively:

Pattern 1: spatially uniform initial permeability in the smear and influence zones.

$$f(r)=\left\{\begin{array}{ll}\delta ,& r_w\le r\le r_s\\ 1,& r_s<r\le r_e\end{array}\right.$$

(5)

Pattern 2: linear variation across the entire zone;

$$f(r)={\displaystyle \frac{r-r_w}{r_e-r_w}}(1-\delta )+\delta ,r_w\le r\le r_e$$

(6)

Pattern 3: parabolic variation in the smear zone and constant permeability in the influence zone.

$$f(r)=\left\{\begin{array}{cc}(1-\delta )(a_1-b_1+c_1{\displaystyle \frac{r}{r_w}})(a_1+b_1-c_1{\displaystyle \frac{r}{r_w}}),& r_w\le r\le r_s\\ 1,& r_s<r\le r_e\end{array}\right.$$

(7)

where $a_1=1/\sqrt{1-\delta }$; $b_1=s/(s-1)$; $c_1=1/(s-1)$; $s=r_s/r_w$; δ = k_s/k_r, k_s being the permeability coefficient at the drain radius and k_r is the permeability coefficient at the influence zone radius.

(6) It is assumed that the radial permeability coefficient at the influence zone equals the vertical permeability coefficient, i.e., k_z = k_r (where k_r is the permeability coefficient at the radial influence zone) [15,29].

2.3. Governing Equation and Boundary Conditions

The radial hydraulic gradient i is

$$i_r={\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}$$

(8)

where u is the excess pore water pressure at a given radial point, γ_w is the unit weight of water, and r is the radial coordinate of that point.

The vertical hydraulic gradient i is

$$i_{\xi }={\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }\xi }}$$

(9)

where u represents the excess pore water pressure at a certain point in the radial direction; γ_w denotes the unit weight of water; and $\xi$ denotes the vertical coordinate of a certain point in the moving coordinate system.

The relationship between the moving coordinate and the Lagrangian coordinate is as follows:

$${\displaystyle \frac{\mathit{\partial }\mathrm{z}}{\mathit{\partial }\xi }}={\displaystyle \frac{1+e_0}{1+e}}$$

(10)

where e = e (z, t), e represents the void ratio of the soil, e₀ = e (z, 0) denotes the initial void ratio of the soil, z represents the coordinate of a point in the Lagrangian coordinate system, and $\xi$ is the coordinate of a point in the moving coordinate system.

At time t, any unit cell can be selected for analysis, as shown in Figure 3. The distance from the unit cell to the center of the vertical drain is r, the vertical coordinate is z, and the pore water pressure is u. The change in the volume of the unit cell per unit time is

$$\mathrm{d}V=-{\displaystyle \frac{e}{1+e}}{\displaystyle \frac{\mathit{\partial }e}{\mathit{\partial }t}}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}\xi$$

(11)

The pore water inflow from the unit in the radial and vertical directions are, respectively,

$$q_{\xi }={\displaystyle \frac{e}{1+e}}(v_w^{\xi }-v_s^{\xi })r\mathrm{d}\theta \mathrm{d}r$$

(12)

$$q_r={\displaystyle \frac{e}{1+e}}(v_w^r-v_s^r)r\mathrm{d}\theta \mathrm{d}\xi$$

(13)

In the formula, $v_w^{\xi }$, $v_s^{\xi }$, $v_w^r$ and $v_w^r$ represent the actual vertical flow velocity of pore water, the actual vertical flow velocity of soil particles, the actual radial flow velocity of pore water, and the actual radial flow velocity of soil particles, respectively.

The pore water inflow rates from the unit cell in the radial and vertical directions are, respectively,

$$q_{\xi }+\mathrm{d}{q}_{\xi }=\left\{{\displaystyle \frac{e}{1+e}}(v_w^{\xi }-v_s^{\xi })+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\xi }}\left[{\displaystyle \frac{e}{1+e}}(v_w^{\xi }-v_s^{\xi })\right]\mathrm{d}\xi \right\}r\mathrm{d}\theta \mathrm{d}r$$

(14)

$$q_r+\mathrm{d}{q}_r=\left\{{\displaystyle \frac{e}{1+e}}(v_w^r-v_s^r)+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}\left[{\displaystyle \frac{e}{1+e}}(v_w^r-v_s^r)\right]\mathrm{d}r\right\}\left(r+\mathrm{d}r\right)\mathrm{d}\theta \mathrm{d}\xi$$

(15)

Assuming that the radial soil particles are immobile, i.e., $v_s^r=0$, and the higher-order terms of dr can be neglected, the flow changes in the radial and vertical directions of the unit cell are as follows:

$$\mathrm{d}{q}_{\xi }={\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\xi }}\left[{\displaystyle \frac{e}{1+e}}(v_w^{\xi }-v_s^{\xi })\right]r\mathrm{d}\theta \mathrm{d}r\mathrm{d}\xi$$

(16)

$$\mathrm{d}{q}_r=\left\{{\displaystyle \frac{e}{1+e}}{v}_w^r+r{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}\left[{\displaystyle \frac{e}{1+e}}{v}_w^r\right]\right\}\mathrm{d}\theta \mathrm{d}r\mathrm{d}\xi$$

(17)

According to the continuity condition of soil consolidation, the volume change in unit compression equals the net water outflow in both radial and vertical directions of pore water, i.e., $\mathrm{d}V=\mathrm{d}{q}_{\xi }+\mathrm{d}{q}_r$. Substituting Equations (11), (16), and (17) into the equation, we obtain

$${\displaystyle \frac{\mathit{\partial }{v}_r}{\mathit{\partial }r}}+{\displaystyle \frac{v_r}{r}}+{\displaystyle \frac{\mathit{\partial }{v}_z}{\mathit{\partial }\xi }}={\displaystyle \frac{1}{1+e}}{\displaystyle \frac{\mathit{\partial }e}{\mathit{\partial }t}}$$

(18)

where $v_{\xi }=v_w^{\xi }-v_s^{\xi }$, $v_r=v_w^r$ $v_{\xi }$ and $v_r$ represent the vertical and radial flow velocities. Equation (2) can be transformed into

$$e=(1+e_0){({\displaystyle \frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}})}^{I_c}-1$$

(19)

According to the principle of effective stress,

$${\sigma }^{\prime }={{\sigma }^{\prime }}_0+q(t)-u$$

(20)

Taking the partial derivative of Equation (19) with respect to t and substituting Equations (1) and (20) into Equation (18), we obtain

$$\left\{\begin{array}{cc}\begin{array}{l}{\displaystyle \frac{{{\sigma }^{\prime }}_0}{I_c}}({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}})[{\displaystyle \frac{1}{r}}{\displaystyle \frac{k_r}{m{i}_1^{m-1}}}{({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}})}^m+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}[{\displaystyle \frac{k_r}{m{i}_1^{m-1}}}{({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}})}^m]\\ +({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }z}}[{\displaystyle \frac{k_z}{m{i}_1^{m-1}}}{({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }z}})}^m]]={\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }t}}-{\displaystyle \frac{\mathit{\partial }q(t)}{\mathit{\partial }t}}\end{array}& (i\le i_1)\\ \begin{array}{l}{\displaystyle \frac{{{\sigma }^{\prime }}_0}{I_c}}({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}})[{\displaystyle \frac{1}{r}}{k}_r({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}-{\displaystyle \frac{i_1(m-1)}{m}})+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}[k_r({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}-{\displaystyle \frac{i_1(m-1)}{m}})]\\ +({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }z}}[k_z({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }z}}-{\displaystyle \frac{i_1(m-1)}{m}})]={\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }t}}-{\displaystyle \frac{\mathit{\partial }q(t)}{\mathit{\partial }t}}\end{array}& (i\ge i_1)\end{array}\right.$$

(21)

By combining Equations (2) and (3), we obtain

$${\displaystyle \frac{k_r}{k_{r0}}}=({\displaystyle \frac{{{\sigma }^{\prime }}_0}{{\sigma }^{\prime }}}{)}^{\alpha I_c}$$

(22)

Taking into account the nonlinearity of soil compression and smearing effects, combining assumption (6) and substituting Equations (4) and (22) into Equation (21), we can derive the nonlinear large-strain radial consolidation equation considering Hansbo’s flow:

$$\left\{\begin{array}{cc}\begin{array}{l}{\displaystyle \frac{{{\sigma }^{\prime }}_0}{I_c}}[{\displaystyle \frac{1}{r}}({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{1-\alpha I_c}{\displaystyle \frac{k_{0}f(r)}{m{i}_1^{m-1}}}({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}{)}^m+({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}[({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{-\alpha I_c}{\displaystyle \frac{k_{0}f(r)}{m{i}_1^{m-1}}}({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}{)}^m]\\ +({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{1+I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }z}}[({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{-\alpha I_c}{\displaystyle \frac{k_0}{m{i}_1^{m-1}}}({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }\mathrm{z}}}{)}^m]]={\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }t}}-{\displaystyle \frac{\mathit{\partial }q(t)}{\mathit{\partial }t}}\end{array}& (i\le i_1)\\ \begin{array}{l}{\displaystyle \frac{{{\sigma }^{\prime }}_0}{I_c}}[{\displaystyle \frac{1}{r}}({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{1-\alpha I_c}{k}_{0}f(r)({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}-{\displaystyle \frac{i_1(m-1)}{m}})+({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }r}}[({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{-\alpha I_c}{k}_{0}f(r)({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }r}}-{\displaystyle \frac{i_1(m-1)}{m}})]\\ +({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{1+I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }z}}[({\displaystyle \frac{{\sigma }^{\prime }}{{{\sigma }^{\prime }}_0}}{)}^{-\alpha I_c}{k}_0({\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }z}}-{\displaystyle \frac{i_1(m-1)}{m}})]={\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }t}}-{\displaystyle \frac{\mathit{\partial }q(t)}{\mathit{\partial }t}}\end{array}& (i\ge i_1)\end{array}\right.$$

(23)

The boundary conditions are as follows:

$$\left\{\begin{array}{l}u(r_w,z,t)=0\\ {\displaystyle \frac{\mathit{\partial }{u}_r}{\mathit{\partial }r}}{|}_{r=r_e}=0\\ u(r,z,0)=q_u\\ {\displaystyle \frac{\mathit{\partial }{u}_z}{\mathit{\partial }z}}{|}_{z=H}=0\\ u(r,0,t)=0\end{array}\right.$$

(24)

For the case of Pattern 1, the points on both sides of the radius r_s are not continuous. Considering that the discontinuity of the permeability coefficient due to smearing effects only exists in the radial direction, the established continuity condition is

$$\overrightarrow{v_{r1}}=\overrightarrow{v_{r2}}$$

(25)

where $\overrightarrow{v_{r1}}$ represents the flow velocity when the permeability coefficient on the left side at radius r_s is k_s, and $\overrightarrow{v_{r2}}$ represents the flow velocity when the permeability coefficient on the right side at radius rs is k_h.

2.4. Solution of Equations

2.4.1. Non-Dimensionalization

To facilitate numerical solution, the following dimensionless variables are defined: $S={\displaystyle \frac{{{\sigma }^{\prime }}_0}{{{\sigma }^{\prime }}_0}}$, $Q={\displaystyle \frac{q_u}{{{\sigma }^{\prime }}_0}}$, $U={\displaystyle \frac{u}{{{\sigma }^{\prime }}_0}}$, $T_v={\displaystyle \frac{C_{v0}t}{4r_e^2}}$, $C_{v0}={\displaystyle \frac{k_{r0}{{\sigma }_0}^{\prime }}{{\gamma }_w{I}_c}}$, $I={\displaystyle \frac{i{\gamma }_w{r}_e}{{{\sigma }_0}^{\prime }}}$, $I_1={\displaystyle \frac{i_1{\gamma }_w{r}_e}{{{\sigma }_0}^{\prime }}}$, $I_2={\displaystyle \frac{(m-1)}{m}}{I}_1$, $R={\displaystyle \frac{r}{r_e}}$, $R_s={\displaystyle \frac{r_s}{r_e}}$, $R_w={\displaystyle \frac{r_w}{r_e}}$, $Z={\displaystyle \frac{z}{H}}$, $R_h={\displaystyle \frac{r_e^2}{H^2}}$, $T_h={\displaystyle \frac{C_{v0}t}{4r_e^2}}{\displaystyle \frac{r_e^2}{H^2}}=T_v{R}_h$.

By introducing the above dimensionless variables, the control Equation (23) and boundary conditions (24) are transformed into Equations (26) and (27):

$$\left\{\begin{array}{ll}{\displaystyle \frac{1}{m{I}_1^{m-1}}}[{\displaystyle \frac{1}{R}}{(S+Q-U)}^{1-I_c\alpha }f(R)|{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}{|}^{m-1}{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}+& \\ (S+Q-U){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }R}}({(S+Q-U)}^{-I_c\alpha }f(R)|{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}{|}^{m-1}{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}})+& (I\le I_1)\\ {(S+Q-U)}^{1+I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }Z}}({(S+Q-U)}^{-I_c\alpha }|{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}{|}^{m-1}{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}})]={\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }{T}_v}}-{\displaystyle \frac{\mathit{\partial }Q}{\mathit{\partial }{T}_v}}& \\ {\displaystyle \frac{1}{R}}{(S+Q-U)}^{1-I_c\alpha }f(R)({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}-I_2)+& \\ (S+Q-U){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }R}}({(S+Q-U)}^{-I_c\alpha }f(R)({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }R}}-I_2))& (I\ge I_1)\\ +{(S+Q-U)}^{1+I_c}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }Z}}({(S+Q-U)}^{-I_c\alpha }({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }Z}}-I_2))={\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }{T}_v}}-{\displaystyle \frac{\mathit{\partial }Q}{\mathit{\partial }{T}_v}}& \end{array}\right.$$

(26)

The boundary condition is

$$\left\{\begin{array}{l}U(R_w,Z,T)=0\\ {\displaystyle \frac{\mathit{\partial }U(1,Z,T)}{\mathit{\partial }R}}=0\\ U(R,Z,0)=Q_u\\ {\displaystyle \frac{\mathit{\partial }U(R,1,T)}{\mathit{\partial }Z}}=0\\ U(R,0,T)=0\end{array}\right.$$

(27)

2.4.2. Difference Solution of Equation

An explicit finite-difference method is employed in this study to obtain the numerical solution iteratively. Within the radial range of R_w ≤ R ≤ 1, the spatial step size is ΔR, which is evenly divided into n equal parts from the inside out. In the vertical range 0 ≤ Z ≤ 1, the soil layer within the vertical drain is divided into m equal parts from the top to the bottom, with a step size of ΔZ. Time is discretized with a step size of ΔT_v, as shown in Figure 4. Following Teh and Nie [27], the dimensionless hydraulic gradient I at the current time level is evaluated using the excess pore water pressure obtained at the previous time level. Since the adjacent ΔT values are small, the resulting error is not significant.

The previously employed finite-difference formulation is extended here to include the vertical dimension. To obtain the numerical solution, the governing Equation (26) is discretized using the backward difference scheme as follows:

$$\left\{\begin{array}{l}{U}_{i,j}^{k+1}=U_{i,j}^k+{\displaystyle \frac{{\alpha }_{i,j}^k}{m{I}_1^{m-1}}}({\beta }_{i+1/2,j}^k{\varphi }_{i+1/2,j}^j{A}_{i+1/2,j}^j-\\ {\beta }_{i-1/2,j}^j{\varphi }_{i-1/2,j}^j{A}_{i-1/2,j}^j+{\gamma }_{i,j}^k{\psi }_{i,j+1/2}^k{E}_{i,j+1/2}^k-\\ {\gamma }_{i,j}^k{\psi }_{i,j-1/2}^k{E}_{i,j-1/2}^k)+{\displaystyle \frac{{\lambda }_{i,j}^k}{m{I}_1^{m-1}}}{B}_{i,j}^k\\ (I\le I_1)\\ U_i^{j+1}=U_i^j+{\alpha }_i^j({\beta }_{i+1/2}^j{C}_{i+1/2}^j-{\beta }_{i-1/2}^j{C}_{i-1/2}^j+{\gamma }_{i,j}^k{F}_{i,j+1/2}^k-\\ {\gamma }_{i,j}^k{F}_{i,j-1/2}^k)+{\lambda }_i^j{D}_i^j\\ (I\ge I_1)\end{array}\right.$$

(28)

where i represents the number of radial spatial nodes (i = 0, 1, 2, …, n); j represents the number of vertical spatial nodes (j = 0, 1, 2, …, m); k represents the number of time nodes (k = 0, 1, 2, …); $U_{i,j}^k$ is the dimensionless excess pore water pressure at grid point (i, j) at time level k; $R_i=R_w+i\Delta R$; ${\alpha }_{i,j}^k=4(S+Q-U_{i.j}^k){\displaystyle \frac{\Delta T_v}{\Delta R}}$; ${\beta }_{i\pm 1/2,j}^k=(S+Q-{\displaystyle \frac{U_{i\pm 1,j}^k+U_{i,j}^k}{2}}{)}^{-I_c\alpha }f(R)$; $A_{i+1/2,j}^k={\displaystyle \frac{U_{i+1,j}^k-U_{i,j}^k}{\Delta R}}$; $A_{i-1/2,j}={\displaystyle \frac{U_{i,j}^k-U_{i-1,j}^k}{\Delta R}}$; ${\psi }_{i\pm 1/2}^j=|{\displaystyle \frac{U_{i,j\pm 1}^k-U_{i,j}^k}{\Delta Z}}{|}^{m-1}$; ${\gamma }_{i,j}^k=(S+Q-U_{i,j}^k{)}^{-I_c}{R}_h$; ${\varphi }_{i\pm 1/2,j}^k=|{\displaystyle \frac{U_{i\pm 1,j}^k-U_{i,j}^k}{\Delta R}}{|}^{m-1}$; ${\lambda }_{i,j}^k=4{(S+Q-U_{i,j}^k)}^{-I_c\alpha +1}{\displaystyle \frac{\Delta T_v}{R_i}}f(R)$; $B_{i,j}^k ={\displaystyle \frac{U_{i+1,j}^k-U_{i-1,j}^k}{2\Delta R}}$; $C_{i+1/2,j}^k={\displaystyle \frac{U_{i+1,j}^k-U_{i,j}^k}{\Delta R}}-I_2$; $C_{i-1/2,j}^k={\displaystyle \frac{U_{i,j}^k-U_{i-1,j}^k}{\Delta R}}-I_2$; $D_{i,j}^k={\displaystyle \frac{U_{i+1,j}^k-U_{i-1,j}^k}{2\Delta R}}-I_2$; $E_{i,j+1/2}^k={\displaystyle \frac{U_{i,j+1}^k-U_{i,j}^k}{\Delta Z}}$; $E_{i,j-1/2}^k={\displaystyle \frac{U_{i,j}^k-U_{i,j-1}^k}{\Delta Z}}$; $F_{i,j+1/2}^k={\displaystyle \frac{U_{i,j+1}^k-U_{i,j}^k}{\Delta Z}}-I_2$; $F_{i,j-1/2}^k={\displaystyle \frac{U_{i,j}^k-U_{i,j-1}^k}{\Delta Z}}-I_2$.

The initial conditions in Equation (27) can be discretized as follows:

$$U_{i,j}^0=Q,i=0,1,2\dots , n; j=0,1,2\dots ,m$$

(29)

The drainage boundary condition in Equation (27) can be discretized as follows:

$$U_{0,j}^k=0, U_{i,0}^k=0, i=0,1,2\dots , n;j=0,1,2\dots , m, k=1,2\dots$$

(30)

The discretization method for the undrained boundary condition in Equation (27) is the same as that for the boundary condition of the radial flow consolidation equation. A symmetrical virtual node can be assumed at i = n + 1 and j = m + 1, and the discretization can be expressed as

$$U_{n-1,j}^k=U_{n+1,j}^k, U_{i,m-1}^k=U_{i,m+1}^k$$

(31)

The boundary conditions can be obtained as

$$\left\{\begin{array}{c}{U}_{i,0}^k=0\\ U_{0,j}^k=0\\ U_{i,j}^0=Q\\ \begin{array}{l}{U}_{n-1,j}^k=U_{n+1,j}^k\\ U_{i,m-1}^k=U_{i,m+1}^k\end{array}\end{array}\right.$$

(32)

Equations (28) and (32) constitute a closed system of equations, through which the pore water pressure at any given time and location can be obtained through iteration.

For a soil layer with a thickness of H, the formula for the settlement s_t at any given time t is

$$s_t={\displaystyle {\int }_0^H{\displaystyle {\int }_{r_w}^{r_e}\frac{2r}{r_e^2-r_w^2}}\frac{(e_0-e)}{(1+e_0)}\mathrm{d}r}\mathrm{dz}$$

(33)

Substitute Equation (19) into Equation (33), discretize and organize:

$$S_t={\displaystyle \sum _{b=0}^{m-1}\frac{G(b,T_v)+G(b+1,T_v)}{2}}$$

(34)

where $G(b,T_v)={\displaystyle \frac{{\displaystyle \sum _{i=0}^{n-1}{R}_i(1-{(S+Q-U_{i,b})}^{-I_c})+R_{i+1}(1-{(S+Q-U_{i+1,b})}^{-I_c}}}{(1+R_w)n}}$.

The ADC U_p of the vertical-drained improved soft ground, defined by pore pressure, is

$$\begin{array}{ll}{U}_p& =1-{\displaystyle \frac{{\displaystyle {\int }_0^H{\displaystyle {\int }_{r_w}^{r_e}2\pi ru\mathrm{d}r}}\mathrm{d}z}{{\displaystyle {\int }_0^H{\displaystyle {\int }_{r_w}^{r_e}2\pi r{q}_u\mathrm{d}r\mathrm{d}z}}}}=1-{\displaystyle \frac{{\displaystyle {\int }_0^1{\displaystyle {\int }_{R_w}^{1}RU\mathrm{d}R\mathrm{d}Z}}}{{\displaystyle {\int }_0^1{\displaystyle {\int }_{R_w}^{1}R{Q}_u\mathrm{d}R\mathrm{d}Z}}}}\\ & =1-{\displaystyle \sum _{b=0}^{m-1}\frac{\left({\displaystyle \sum _{i=0}^{n-1}{Y}_{i,b}+{\displaystyle \sum _{i=0}^{n-1}{Y}_{i,b+1}}}\right)\Delta Z}{(1-R_w^2)Q_u}}\end{array}$$

(35)

where $Y_{i,b}={\displaystyle \frac{U_{i,b}{R}_{i+1}-U_{i+1,b}{R}_i}{2\Delta R}}(R_{i+1}^2-R_i^2)+{\displaystyle \frac{U_{i+1,b}-U_{i,b}}{3\Delta R}}(R_{i+1}^3-R_i^3)$.

3. Parametric Analysis

3.1. Comparison with Vertical Consolidation with Radial Flow Only

If the vertical flow component is omitted (i.e., the vertical hydraulic gradient is set to zero), the proposed governing equations reduce to the nonlinear large-strain formulation for purely radial consolidation of a vertical-drained improved soft ground. To validate the corresponding finite-difference solution, a comparative analysis was carried out against the radial consolidation model developed by Chen et al. [15]. All model conditions, including fully drained boundaries and a parabolic radial permeability variation in the smear zone, were kept consistent with those used in the reference study. As shown in Figure 5, the results from the two models exhibit excellent agreement, both in the dissipation of excess pore water pressure and in the development of ground settlement, with the numerical curves nearly overlapping. This confirms that the proposed numerical solution accurately reproduces established theoretical results under purely radial flow conditions. The close agreement validates the correctness and reliability of the present model in the degenerated case. It provides a solid basis for subsequent analysis of the coupling effects of radial–vertical flow on the consolidation behavior of vertical drain foundations.

3.2. Comparison with Equivalent Strain Analytical Solutions

To further validate the reliability of the developed finite-difference model solution, a comparative analysis was conducted with the analytical solution for radial–vertical consolidation under the equal-strain assumption, as derived by Zhang et al. [23] for vertical-drained soft ground with continuous drainage boundary. Under the equal-strain condition, the average consolidation degree considering both radial and vertical flows can be expressed as follows:

$$\overline{U}=1-{\displaystyle \sum _{m=0}^{\infty }\frac{2}{M}{e}^{-Yt}}=1-{\displaystyle \sum _{m=0}^{\infty }\frac{2}{M}{e}^{-\frac{A{(\frac{M}{H})}^4-C{(\frac{M}{H})}^2}{1-B{(\frac{M}{H})}^2}t}}$$

(36)

where $M={\displaystyle \frac{2m+1}{2}}\pi ,m=0,1,2,3\cdots$; $A={\displaystyle \frac{r_e^{2}R{k}_w}{2(n^2-1)k_h}}{c}_v$; $B=-{\displaystyle \frac{r_e^{2}R{k}_w}{2(n^2-1)k_h}}$; $C=-(c_v+{\displaystyle \frac{k_w}{k_h(n^2-1)}}{c}_h)$; $R={\displaystyle \frac{n^2}{n^2-1}}[({\displaystyle \frac{k_h}{k_s}}-1)\ln s+\ln n+{\displaystyle \frac{(s^2-3)(s^2-1)}{4n^2}}{\displaystyle \frac{k_h}{k_s}}-{\displaystyle \frac{(s^2-3)(s^2-1)}{4n^2}}]$.

According to assumption 6, assuming c_v = c_h, and that well resistance is neglected (i.e., k_w → ∞), it follows that

$$Y=c_v{({\displaystyle \frac{M}{H}})}^2+{\displaystyle \frac{2c_v}{R{r}_e^2}}$$

(37)

In order to make the parameter values of the numerical solution consistent with those of the analytical solution, dimensionless parameters H = 10, S = 1, Q = 5, R_w = 0.2, I_c = 0.1, α = 0, m = 1 (nonlinear compression and non-Darcy flow were not considered in the analytical solution), and ΔR = 0.01, ΔT_v = 10⁻⁶ were selected to compare the results obtained from finite difference analysis with the results obtained from the equivalent strain analytical solution, as shown in Figure 6. According to the comparison of solutions under the assumptions of equal strain and free strain given by Li et al. [29], the impact of the two assumptions on vertical drain consolidation is mainly reflected in the early stage of consolidation, and the degree of consolidation obtained under the assumption of free strain is greater in the early stage, but gradually approaches over time. The comparison results shown in Figure 6 are basically consistent, further verifying the reliability of the finite difference solution in this paper.

4. Parametric Study

To investigate the nonlinear large-strain consolidation of vertical-drain improved soft ground considering Hansbo’s flow and smear effects under coupled radial–vertical flow, a parametric study is conducted based on the present solution. The analyzed factors include the pattern of radial permeability variation, the nonlinear compression parameter (I_c) and nonlinear permeability parameter (α) obtained from the double-logarithmic relationship, and the parameters m and the dimensionless threshold hydraulic gradient I₁ in the Hansbo’s flow model. For computational convenience, the following baseline parameters are adopted: r_w = 0.2 m, r_e = 1 m, r_s = 0.5 m, σ₀′ =10 kPa, q_u = 50 kPa, ΔR = 0.01, ΔT_v = 10⁻⁶, m = 2, I₁ = 1, I_c = 0.1, α = 10, δ = 2/3, H = 5 m, with Pattern 1 for radial permeability variation.

4.1. Influence of Smear Zone Permeability Pattern

Figure 7 shows the influence of different radial permeability variation patterns on the ADC (U_p) and settlement (S_t). It can be observed that Pattern 3 (parabolic variation within the smear zone) yields the fastest consolidation rate, followed by Pattern 2 (linear variation across the entire zone), and Pattern 1 (constant permeability in the smear zone) results in the slowest rate. Regarding settlement, Pattern 3 requires the shortest time to reach a given settlement, while Pattern 1 requires the longest. All three patterns, however, converge to the same final settlement. Compared to Pattern 1, Pattern 2 exhibits maximum relative deviations of 16.2% in the average degree of consolidation (ADC) and 11.2% in settlement. For Pattern 3, the corresponding deviations reach 25.7% and 19.2%, respectively. These results demonstrate that the spatial distribution of permeability, rather than merely its average value, plays a decisive role in governing consolidation behavior. The faster consolidation observed under Pattern 3 is primarily attributed to the relatively higher permeability in the vicinity of the drain. Since excess pore water must ultimately flow toward the drain, the hydraulic resistance near the drain exerts a dominant control on the dissipation rate. A permeability distribution that reduces resistance in this critical drainage region significantly enhances pore pressure dissipation and accelerates settlement development. In contrast, Pattern 1 maintains a uniformly reduced permeability within the smear zone, leading to greater overall hydraulic resistance and slower consolidation. Although these trends are consistent with those observed in purely radial consolidation, the influence of the permeability pattern on both ADC and settlement is more pronounced when vertical flow is incorporated. This can be attributed to the fact that variations in the overall permeability field simultaneously alter the vertical flow velocity, thereby exerting a stronger impact on the consolidation response and settlement development.

4.2. Influence of Nonlinear Compression Parameter (I_c) and Nonlinear Permeability Parameter (α)

Based on the study by Li et al. [19] regarding the nonlinear compression parameter (I_c) and nonlinear permeability parameter (α) in double-logarithmic coordinates, the values of I_c are predominantly concentrated in the range of 0.08–0.12, while α typically varies between 6 and 14. Accordingly, analyses in this study are conducted using I_c values of 0.08, 0.1, and 0.12, and α values of 6, 10, and 14, respectively. Figure 8 presents the influence of I_c and α on ADC and settlement over time (T_v). As I_c increases, U_p at a given T_v decreases, whereas the settlement increases, with both the rate of settlement and the final settlement also rising. An increase in α slows the progress of consolidation and settlement, though the final settlement remains unchanged. Relative to the baseline case (I_c = 0.1), the scenario with I_c = 0.08 shows a maximum deviation of 19.3% in U_p and a final settlement difference of 0.151 m. For I_c = 0.12, the corresponding deviations are 15.7% in U_p and 0.143 m in settlement. When compared to α = 10, the case with α = 6 yields maximum deviations of 43.5% in U_p and 29.2% in settlement, while for α = 14, the deviations are 28.7% in U_p and 21.8% in settlement. That is, the nonlinear compression parameter I_c primarily controls the final settlement magnitude. In contrast, the nonlinear permeability parameter α governs consolidation speed, and both factors significantly affect consolidation and settlement.

Figure 9 compares the average degree of consolidation (ADC, U_p) obtained from the coupled radial–vertical flow model with that from the radial-flow-only model for different Hansbo’s flow parameters m and I₁. At any given time factor Tv, the coupled model consistently predicts a higher U_p than the radial-only model, because vertical seepage near the drainage boundary is non-negligible and accelerates excess pore-pressure dissipation. For all investigated values of m and I₁, the coupled solution exhibits a higher consolidation rate than the radial-only solution. The relative difference between the two solutions is more pronounced at early times and decreases as consolidation proceeds. For instance, in Figure 9a, the relative difference is approximately 20% at Tv = 0.01 and reduces to about 4% at Tv = 0.5, indicating that vertical seepage mainly influences the early stage of consolidation, with a limited effect during the intermediate and late stages.

4.3. Influence of Hansbo’s Flow Parameters m and I₁

Figure 10 presents the influence of Hansbo’s flow parameter m (with values 1, 1.5, 2, 2.5) on ADC U_p and settlement, respectively. As shown in Figure 10, an increase in the flow parameter m or I₁ slows the overall consolidation rate; however, this effect is mainly apparent in the later stages, while the early-stage consolidation progress remains largely unaffected. Furthermore, higher values of m and I₁ result in greater settlement at a given time without affecting the final settlement magnitude. Specifically, relative to Darcy’s flow (m = 1), the maximum relative deviations in the ADC for m = 1.5, 2.0, and 2.5 are 6.6%, 10.7%, and 13.6%, respectively, while the corresponding deviations in settlement are 4.9%, 8.3%, and 11.1%. Likewise, as the dimensionless threshold hydraulic gradient parameter I₁ increases from 0.5 to 1, 2, and 3, the maximum relative deviations in the ADC reach 12.7%, 22.6%, and 27.6%, respectively, while the corresponding deviations in settlement are 11.2%, 20.3%, and 25.1%. Therefore, if the influence of Hansbo’s flow on the consolidation settlement rate is neglected, the consolidation settlement process of the vertically drained grounds will be significantly overestimated, leading to an unsafe prediction of the actual consolidation time in engineering.

Figure 11 presents the excess pore pressure distributions along radial and vertical directions, respectively, for time factor T_v = 0.1 under different m and I₁ values. As seen, the results for radial flow closely resemble those derived from the purely radial consolidation model. It can be observed that beyond a certain depth at this time instant, the variation in excess pore pressure becomes negligible, corresponding to an essentially zero vertical hydraulic gradient. Figure 11 indicates that flow in deeper regions is dominated by radial flow, whereas both radial and vertical flow components contribute significantly in shallower zones.

4.4. Influence of Soil Layer Thickness

In the classical vertical drain consolidation model that considers only radial flow, the consolidation process is independent of depth H, and thus H does not affect consolidation or settlement rate. To examine the effect of incorporating vertical flow, Figure 12 illustrates the influence of the soil layer thickness H (normalized by r_e) on U_p and settlement of vertical drains with coupled radial–vertical flow. For comparison, the responses of the radial-flow-only model have also been illustrated in Figure 12. The results indicate that when the ratio of soil layer depth to influence zone radius H/r_e = 5, the maximum relative deviation in ADC between the coupled model and the radial-only model is 20.2%; when H/r_e = 10, this deviation decreases to 3.8%. As seen, the influence of increasing H/r_e on the consolidation rate gradually decreases and is primarily concentrated in the intermediate stage of consolidation. Regarding settlement, the magnitude at any given time is approximately proportional to H/r_e times the settlement from the radial-flow-only model. Therefore, for H/r_e > 10, the purely radial consolidation model can provide a reasonable approximation for U_p, and the settlement can be estimated by scaling the radial-flow-only settlement result.

5. Conclusions

Building on the previous study on radial consolidation using Hansbo’s flow model, this paper further incorporates vertical consolidation. A new governing equation for vertical drain foundation consolidation is derived, discretized using the finite difference method, and solved iteratively. The effects of large strain, Hansbo’s flow, layer depth, and vertical flow are analyzed. The main conclusions are as follows:

• The spatial distribution pattern of permeability within the smear zone significantly affects consolidation behavior under coupled radial–vertical flow. Different distribution forms lead to distinct consolidation rates, indicating that not only the average permeability but also its radial variation influences the consolidation process within a multidimensional flow framework.
• Larger values of Hansbo’s flow parameters m and I₁ reduce the consolidation rate due to increased hydraulic resistance associated with non-Darcy flow. The discrepancy between coupled and purely radial consolidation is most pronounced in the early stage, with the maximum deviation in ADC exceeding 20% under certain conditions, indicating that vertical flow mainly accelerates the initial dissipation of excess pore pressure. With increasing depth and time, radial flow gradually becomes dominant and the difference between the two models diminishes.
• Increasing the nonlinear compression and permeability parameters enhances the relative deviation between the coupled and purely radial models, indicating that soil nonlinearity amplifies the interaction between radial and vertical flow. Nevertheless, radial flow remains the primary consolidation mechanism, while vertical flow plays a supplementary yet non-negligible role under strongly nonlinear conditions.
• At any given time, the average degree of consolidation predicted by the coupled radial–vertical model exceeds that of the purely radial model. However, as the normalized depth H/r_e increases, the contribution of vertical drainage progressively diminishes. When H/r_e > 10, the consolidation response converges to that of the purely radial model, and simplified radial solutions can provide reasonable engineering estimates. In contrast, for relatively thin soil layers, pronounced smear effects, or highly nonlinear soil behavior, neglecting vertical flow and large-strain coupling may lead to noticeable deviations in predicted consolidation rate and settlement evolution.

Overall, the present study demonstrates that the interaction among the large strain, non-Darcy flow, nonlinear compressibility––permeability relationship, and multidimensional seepage can significantly influence consolidation behavior under certain geometric and material conditions. The proposed integrated framework therefore provides a more reliable theoretical basis for analyzing vertically drained soft ground under complex engineering scenarios.

Future research may further extend the model in several directions:

(i)

Incorporating anisotropic permeability conditions to better represent natural clay deposits;

(ii)

Considering non-instantaneous or cyclic loading scenarios, particularly relevant to traffic or staged construction;

(iii)

Validating the model against field case histories or centrifuge tests to quantify predictive accuracy;

(iv)

Coupling the formulation with constitutive models that account for creep or secondary compression, enabling long-term settlement prediction.

Such extensions would enhance the applicability of the model to more complex and realistic engineering conditions.