Modelling solute transport and deformation of clay soil under the chemo-hydro-mechanical coupling actions based on fractal theory

Abstract

The interactions between chemical, hydraulic, and mechanical processes in clay soils has garnered significant attention in various fields. It is important to study the transport of chemical substances and the deformation of the clay layer under the combined action of chemical solution and mechanical loading. Clay is a typical porous medium, primarily composed of soil skeleton and pores, with the shape, size, and distribution of clay pores being random, making it difficult to accurately describe using traditional geometry. A fractal model can be used to simulate clay soils. This study leverages the fractal theory of clay, incorporating the generalized effective stress principle in chemical solutions and a fractal model for the diffusion of chemical substances. A chemo-hydro-mechanical coupling model based on fractal theory is developed. Simulation results show that an increase in fractal dimension corresponds to a heightened roughness of the pore surfaces and an increased tortuosity of the pore channels, which significantly amplify the resistance to solute diffusion, thereby retarding the rate of solute transport. As a result, the magnitude of the maximum negative pore pressure within the clay layer, leading to a longer time required for complete pore pressure dissipation. Additionally, the deformation of the clay layer is larger.

Introduction

The interactions between chemical, hydraulic, and mechanical processes in clay soils has garnered significant attention in various fields, including environmental protection, waste management, and geotechnical engineering. Understanding the chemo-hydro-mechanical (CHM) coupling in clay soils is essential for evaluating the long-term safety and stability of critical infrastructures such as deep geological repositories, municipal solid waste landfills, and coastal foundations. Previous research has demonstrated that clay soils, characterized by low porosity and permeability, exhibit complex responses when subjected to external loads and chemical environments. The migration of solutes and the resulting chemical precipitation can lead to changes in soil porosity and strength, a phenomenon known as chemical enhancement^1,2,3. Since the pioneering work of Greenberg et al. in the 1970s⁴, numerous studies have focused on the CHM coupling in soils, with particular emphasis on the one-dimensional consolidation process^5,6,7,8,9. These studies have highlighted the significance of chemical reactions, pore-fluid flow, and soil deformation in the overall behavior of clay soils.

All previous research on the effects of chemical actions on clay has been conducted within the framework of traditional geometry. Clay is a typical porous medium, primarily composed of soil skeleton and pores, with the shape, size, and distribution of clay pores being random, making it difficult to accurately describe using traditional geometry¹⁰. Fractal theory has been employed by many scholars to study the widespread irregular phenomena in nature. Numerous researchers have introduced fractal theory into soil science^11,12,13, geosciences^14,15, and other fields. Xu et al.^16,17,18,19 have applied fractal theory to soil mechanics, particularly in the study of unsaturated clay mechanics, conducting a series of exploratory studies that have yielded pioneering results, demonstrating the applicability and accuracy of fractal theory in soil mechanics research. Figure 1 is the Menger sponge model, which is commonly used fractal model for simulating porous media²⁰. The Menger sponge is created by dividing a unit cube into 27 smaller cubes with edge lengths of 1/3, removing all the smaller cubes that do not touch the outer boundary of the larger cube, and then repeating this operation on the remaining cubes. By the i-th step, the number of small cubes is 20ⁱ and the edge length of each small cube is (1/3)⁻ⁱ. The fractal dimension of the Menger sponge is 2.737.

Fig. 1

Menger sponge.

Research in fractal soil mechanics has yielded many achievements both domestically and internationally. Liu et al.²¹ used nitrogen adsorption and Pb²⁺ adsorption tests to determine the surface fractal dimension of GMZ-Na bentonite. Although the surface fractal dimensions of the bentonite measured by different methods showed slight differences, the results indicated that the surface of GMZ-Na bentonite possesses typical fractal characteristics. Sun et al.²² utilized mercury intrusion porosimetry, combined with image analysis, to study the surface fractal dimension of compacted bentonite, and the results showed that the surface fractal dimension of bentonite decreases with the reduction of matric suction. Huang et al.²³ investigated the specific surface area and adsorption properties of GMZ-Ca bentonite, finding a correlation between surface fractal dimension and adsorption performance: the smaller the surface fractal dimension of the bentonite, the smoother its surface, and the poorer its adsorption capacity. In the meantime, the transport of chemical solutes in clay is related to surface adsorption capacity, which in turn is associated with surface geometric characteristics. Therefore, the transport of chemical solutes in clay and the deformation of clay under the coupled effects of chemical, hydraulic, and mechanical actions are related to surface fractal dimensions²⁴. The resistance experienced by clay particles during transport is related to the tortuosity of the pore structure, which can also be reflected by the grain fractal dimension. From the perspective of fractal theory, the use of fractal dimensions can more accurately quantify the impact of geometric characteristics on the coupled consolidation properties of clay under chemical, hydraulic, and mechanical actions.

This paper, based on the fractal theory of clay and the principle of generalized effective stress in chemical solution environments, establishes a coupled consolidation model for saturated clay considering the fractal characteristics of soil. Using the COMSOL Multiphysics 6.0 finite element simulation software, numerical simulations of consolidation and solute migration of clay in chemical solutions under two sets of conditions and two boundary conditions are conducted. The impact of clay surface fractal dimensions on the coupled consolidation under chemical, hydraulic, and mechanical actions is analyzed.

Theoretical model

Generalized effective stress of clay in chemical solutions based on fractal theory

The concept of generalized effective stress in the context of clay behavior in chemical solutions is an extension of the traditional Terzaghi’s effective stress principle. Traditionally, the effective stress p_e is defined as the difference between the total stress p and the pore water pressure p_w, which can be expressed as:

$$p_{e} = p - p_{w}$$

(1)

However, when considering the impact of chemical solutions on clay, the generalized effective stress principle takes into account additional factors that influence the behavior of the soil skeleton and the interaction with the chemical environment²⁵. The surface fractal dimension of clay provides a measure of the complexity and roughness of the soil surface, which can significantly affect the adsorption capacity, permeability, and mechanical behavior of the clay in the presence of chemical solutions. In the presence of chemical solutions, the transport of chemical solutes and the adsorption capacity of clay are closely related to the surface geometric characteristics, which can be quantified using the surface fractal dimension. The generalized effective stress principle, therefore, incorporates the effects of surface fractal dimensions on the mechanical behavior of clay, particularly in the context of chemical, hydraulic, and mechanical coupling effects.

Xu et al.²⁶ based on the fractal model of clay pores, derived a fractal model that describes how the ratio of the volume of pore water to the volume of the clay body in pure water changes with effective stress.

$$e_{m} = \frac{{V_{w} }}{{V_{m} }} = C_{m} p^{{D_{s} - 3}}$$

(2)

where e_m represents the ratio of pore water volume to the volume of the clay body; V_w is the volume of pore water; V_m is the volume of the clay body; C_m denotes the volume change constant of the clay; and D_s is the surface fractal dimension of the clay.

Barbour and Fredlund²⁷ based on the stress–strain relationship of clay bodies in chemical solutions, revised the effective stress and proposed the concept of modified effective stress for clay bodies in salt solutions. They linked the effective stress to the osmotic suction of the chemical solution, and the modified effective stress can be expressed as:

$$p_{e} = p + p_{\pi }$$

(3)

where ${p}_{\pi }=\alpha \pi$, 0 < α < 1.

As shown in Fig. 2, the arrangement of clay particles has a hierarchical structure. Microscopically, external loads and the osmotic suction of salt solutions act together on the crystal grains of clay, causing changes in the volume of aggregates. However, the effect of osmotic suction is different from that of external loads, it does not cause changes in macrospores. Therefore, it is necessary to establish a relationship that expresses the osmotic suction at the micro level as osmotic stress at the macro level. Xu et al.²⁷ provided the following relationship to connect the two:

$$p_{\pi } = \pi \left( {\frac{p}{\pi }} \right)^{{D_{s} - 2}}$$

(4)

Fig. 2

Osmotic suction and stress applied on clay.

By substituting Eq. (4) into Eq. (3), the expression for the generalized effective stress p_e experienced by clay in salt solutions can be obtained:

$$p_{e} = p + p_{\pi } = p + \pi \left( {\frac{p}{\pi }} \right)^{{D_{s} - 2}}$$

(5)

The generalized effective stress takes into account the osmotic suction that clay experiences in chemical solutions, which can more accurately reflect the actual forces exerted on the clay in chemical solutions. This principle is fundamental in understanding the behavior of clay in chemical solutions and is crucial for the development of models that predict the mechanical response of clay under complex chemical, hydraulic, and mechanical conditions.

Static equilibrium

The change in volume of the CBL may occur due to mechanical and chemical loading and it can be expressed as²⁸

$$\Delta {\varepsilon }_{v}=-{m}_{v}\Delta {p}_{e}-{m}_{c}\Delta c$$

(6)

where ε_v is the strain of the soil layer; c indicates the chemical concentration.

Therefore, the partial derivative of the volumetric strain concerning time can be expressed as

$$\frac{\partial {\varepsilon }_{v}}{\partial t}=-{m}_{v}\frac{\partial {p}_{e}}{\partial t}-{m}_{c}\frac{\partial c}{\partial t}$$

(7)

Conservation of pore water

The osmotic and hydraulic water flows follow the principle of superposition producing the coupled pore flow equation

$$v={v}_{h}+{v}_{c}=-\frac{{k}_{h}}{{\gamma }_{w}}\frac{\partial u}{\partial z}+{k}_{c}\frac{\partial c}{\partial z}$$

(8)

where v is the discharge velocity vector of the pore water; v_h denotes the hydraulic flow velocity; v_c indicates the chemo-osmotic flow velocity; k_c is the coefficient of chemo-osmosis permeability, k_c = ωRTk_h, where ω denotes the osmotic efficiency. Osmotic efficiency is a characterization parameter of the semi-permeable membrane characteristics of clay, which depends on the type of clay, porosity, type and concentration of chemical substances et al.^29,30. ω equal to 0 means that the clay has no semi-permeable membrane properties, and ω equal to 1 means that the clay is an ideal semi-permeable membrane, and the osmotic efficiency of clay usually ranges from 0 to 0.7³¹.

The mass conservation equation of pore fluid in saturated soil can be expressed by the following equation²⁸

$$\frac{\partial (n\rho )}{\partial t}+\nabla \cdot \left(n\rho {{\varvec{v}}}_{f}\right)=0$$

(9)

where ρ is the density of the pore fluid; n is the porosity; v_f denotes the absolute velocity of the pore fluid, which can be expressed as

$${{\varvec{v}}}_{f}=\frac{{\varvec{v}}}{n}-{{\varvec{v}}}_{s}$$

(10)

where v_s indicates the velocity of solid soil. Neglecting the convective components of the time derivative of density and porosity, and ignoring the change of total density of pore liquid due to a change in the amount of absorbed water induced by a unit increment of the concentration of the chemical, Eq. (9) can be rearranged as

$$\frac{\partial n}{\partial t}+n\frac{\partial {v}_{s}}{\partial z}+\frac{\partial v}{\partial z}=0$$

(11)

Since it is assumed that the solid soil skeleton is incompressible, the mass conservation equation of the solid phase can be expressed as

$$\frac{\partial n}{\partial t}=\left(1-n\right)\frac{\partial {v}_{s}}{\partial z}$$

(12)

The relationship between porosity and the volumetric strain of the soil layer can be expressed as

$$\frac{\partial {\varepsilon }_{v}}{\partial t}=\frac{1}{1-n}\frac{\partial n}{\partial t}$$

(13)

Substituting Eqs. (7), (8), (12), and (13) into Eq. (11), one obtains that

$$-{m}_{v}\frac{\partial {p}_{e}}{\partial t}-\frac{{k}_{h}}{{\gamma }_{w}}\frac{{\partial }^{2}u}{\partial {z}^{2}}-{m}_{c}\frac{\partial c}{\partial t}+{k}_{c}\frac{{\partial }^{2}c}{\partial {z}^{2}}=0$$

(14)

Conservation of chemical species

The transport of chemical substances can be expressed by the following equation³²

$$J={J}_{a}+{J}_{D}+{J}_{{D}_{u}}$$

(15)

where J represents the coupling flux of chemicals; J_a is the convection flux; J_D is the nonadvective flux caused by the chemical concentration gradient, which conforms to Fick’s law; J_Du is the ultrafiltration term caused by the hydraulic gradient.

$${J}_{a}=cv=c\left(-\frac{{k}_{h}}{{\gamma }_{w}}\frac{\partial u}{\partial z}+{k}_{c}\frac{\partial c}{\partial z}\right)$$

(16)

$${J}_{D}=-n{D}^{e}\frac{\partial c}{\partial z}$$

(17)

$${J}_{{D}_{u}}={D}_{u}\frac{\partial u}{\partial z}=\frac{\omega c{k}_{h}}{{\gamma }_{w}}\frac{\partial u}{\partial z}$$

(18)

where D^e is the modified effective diffusion coefficient, which can be calculated by³³

$$D_{e} = D^{*} n^{{1 + 2(D_{s} - 2)}}$$

(19)

D_u denotes the ultrafiltration coefficient. Therefore, the coupled flux of chemicals can be expressed as

$$J=-\left(1-\omega \right)\frac{c{k}_{h}}{{\gamma }_{w}}\frac{\partial u}{\partial z}-\left(n{D}^{e}-c{k}_{c}\right)\frac{\partial c}{\partial z}$$

(20)

In saturated porous media, the conservation equation for the transport of a single solute in the pore solution can be expressed by the following equation²⁸

$$\frac{\partial \left(nc\right)}{\partial t}+\nabla \cdot {{\varvec{v}}}_{c}=0$$

(21)

where v_c is the average velocity of the solute per unit area in the clay, which can be expressed as

$${{\varvec{v}}}_{c}={\varvec{J}}+cn{{\varvec{v}}}_{s}$$

(22)

Since the velocity of the soil skeleton and the pore fluid are very small, the convective component of the time derivative of the concentration and porosity can be neglected. Substituting Eq. (7), (12), (13), (20), and (22) into Eq. (21), one obtains that

$$-{c}_{0}{m}_{v}\frac{\partial {p}_{e}}{\partial t}-\left(1-\omega \right)\frac{{c}_{0}{k}_{h}}{{\gamma }_{w}}\frac{{\partial }^{2}u}{\partial {z}^{2}}+\left({n}_{0}-{c}_{0}{m}_{c}\right)\frac{\partial c}{\partial t}-\left({n}_{0}{D}^{e}-{c}_{0}{k}_{c}\right)\frac{{\partial }^{2}c}{\partial {z}^{2}}=0$$

(23)

Equations (14) and (23) delineate the interrelation between variations in excess pore pressure and fluctuations in chemical concentration, both under consistent mechanical and chemical loadings, thereby embodying the comprehensive coupling of chemico-hydro-mechanical (CHM) interactions. Specifically, the term ${m}_{v}\frac{\partial u}{\partial t}$ encapsulates the mass alteration attributable to volumetric changes of the soil matrix when subjected to mechanical forces. Conversely, ${m}_{c}\frac{\partial c}{\partial t}$ signifies the mass variation stemming from chemical osmotic processes. The expression ${c}_{0}{m}_{v}\frac{\partial u}{\partial t}$ illustrates the modification in chemical constituents within the soil due to shifts in the stress field. Lastly, $\left({n}_{0}-{c}_{0}{m}_{c}\right)\frac{\partial c}{\partial t}$ represents the alteration in soil chemical substances influenced by chemical gradients, with the latter component accounting for the convective effects.

Validation

To verify the accuracy of the fractal model and the generalized effective stress in the consolidation of saturated clay under chemo-hydro-mechanical coupling, the calculated data when D_s equals 2 were compared with the analytical solution data of Kaczmarek and Hueckel²⁸. The model simulations have been carried out using the COMSOL Multiphysics 6.0 (https://www.comsol.com) software³⁴. The results are shown in Fig. 3. It can be observed that when D_s equals 2, the results from the finite element simulation using the fractal model match well with the analytical calculations from the conventional model, demonstrating the feasibility and correctness of applying the fractal model and the generalized effective stress to the chemo-hydro-mechanical coupling consolidation of saturated clay.

Fig. 3

The comparison between the fractal model (D_s = 2) and the analytical solutions of conventional model: (a) Excess pore-water pressure, and (b) Chemical concentration.

Model application

Problem description

The model presented in this section are aimed at studying the chemical transport processes in semi-permeable clay liners in municipal solid waste landfills and the deformation behavior of the clay liner. The weight of the waste, waste degradation, leachate formation, and seepage create a complex and challenging scenario, which requires ensuring and maintaining the optimal performance of the clay liner to restrict the movement or spread of pollutants. The model is used to simulate multiple coupled landfill processes in a 1D clay liner domain. Simulated the chemo-hydro-mechanical coupling consolidation under two types of boundary conditions, which are shown in Fig. 4.

Fig. 4

Model schematic diagram.

Boundary condition 1: The upper boundary of the soil layer is permeable, and the lower boundary is impermeable.

$$\begin{gathered} {\text{Upper }}\;{\text{boundary}}:\;u(0,t) = 0,c(0,t) = c_{0} \hfill \\ {\text{Lower }}\;{\text{boundary}}:\;\frac{\partial u}{{\partial z}}\left( {H,t} \right) = 0,\;\frac{\partial c}{{\partial z}}\left( {H,t} \right) = 0 \hfill \\ \end{gathered}$$

Boundary condition 2: The upper and lower boundaries of the soil layer are both permeable.

$$\begin{gathered} {\text{Upper}}\;{\text{ boundary}}:\;u(0,t) = 0,c(0,t) = c_{0} \hfill \\ {\text{Lower boundary}}:\;u(H,t) = 0,\;c(H,t) = 0 \hfill \\ \end{gathered}$$

According to some typical literatures^28,32, the material properties used in the analysis are given in Table 1. The mechanical load and chemical concentration on the upper surface of the soil layer is uniformly distributed and constant.

Table 1 Physical parameters of the saturated soil layer^28,32chem.

Results

Figures 5 and 6 represent the spatiotemporal distribution of pore solution concentration under two boundary conditions. For the first type of boundary condition, as the salt solution enters the soil mass through advection or diffusion, it takes approximately 3 years for the chemical substances to be transported to the bottom of the soil layer, whereas the conventional model predicts this to occur in about 0. 3 years. It is evident that considering the fractal characteristics of clay, the diffusion of chemical substances is much slower. This is because, compared to the conventional straight pipe-type diffusion, the diffusion based on the fractal model takes into account the irregularity and roughness of the pore structure, which hinders the diffusion of chemical substances. It takes about 40 years for the pore solution concentration at the bottom of the soil layer to reach the same concentration as the external environment, at which point consolidation is complete, and the soil layer remains stable, the conventional model only requires about 15 years for this process. For the second type of boundary condition, due to the permeable boundary at the bottom accelerating the seepage velocity of pore water, the speed at which chemical substances enter the soil layer through advection is faster, with chemical substances reaching the bottom of the soil layer in about 0. 5 years. The time to reach stability is about 15 years. For the second type of boundary condition, at the time of stability, the pore solution concentration in the soil layer decreases with increasing depth.

Fig. 5

Distribution of chemical concentration under the first type of boundary conditions.

Fig. 6

Distribution of chemical concentration under the second type of boundary conditions.

Figures 7 and 8 represent the spatiotemporal distribution of excess pore water pressure under two boundary conditions. For the first type of boundary condition, the maximum negative pore pressure in the soil layer calculated by the chemo-hydro-mechanical coupling consolidation model considering fractal theory is greater than that calculated by the conventional model. This is because an increase in fractal dimension slows down the diffusion of chemical substances, which in turn slows down the rate at which the osmotic pressure difference between the inside and outside decreases. Consequently, the time for complete dissipation of pore pressure is prolonged, and the entire consolidation process takes longer. For the second type of boundary condition, the maximum negative pore pressure occurs in the middle of the soil layer and is much smaller than the maximum negative pore pressure under the first type of boundary condition. Additionally, due to the permeable boundary condition at the bottom accelerating the drainage of water within the soil, the dissipation of pore pressure is much faster.

Fig. 7

Distribution of excess pore water pressure under the first type of boundary conditions.

Fig. 8

Distribution of excess pore pressure under the second type of boundary conditions.

Discussion

Effect of surface fractal dimension on chemical concentration

Figures 9 and 10 illustrate the variation of pore solution concentration over time under the first and second boundary conditions, respectively, for different fractal dimensions. It can be observed that the higher the fractal dimension, the slower the development of pore solution concentration. This is because a larger fractal dimension indicates a rougher pore surface and more tortuous pore channels, which consequently provides a stronger resistance to solute diffusion, leading to slower diffusion rates. Conversely, a smaller fractal dimension suggests a smoother pore surface and straighter pore channels, resulting in faster solute diffusion. When the fractal dimension equals 2, the medium’s pore channels become straight cylindrical bundles, and the solute diffusion model reverts to the corresponding form in the conventional straight capillary bundle model of porous media. The model is the traditional CHM model. Additionally, by comparing Figs. 9 and 10, it can be seen that under the first type of boundary condition, all pore solution concentrations within the soil layer eventually increase to match the external solution concentration. The deeper the depth, the longer the time required for diffusion to be completed. Under the second type of boundary condition, after stabilization, the pore solution concentrations at different depths within the soil layer vary, decreasing with increasing depth. As previously analyzed, the presence of external mechanical loads has a negligible impact on the transport of chemical solutes, and thus, the discussion of pore solution concentration under the sole influence of chemical loads is not further elaborated.

Fig. 9

Effect of surface fractal dimension on chemical concentration of CHM coupling consolidation under the first type of boundary conditions.

Fig. 10

Effect of surface fractal dimension on chemical concentration of CHM coupling consolidation under the second type of boundary conditions.

Figures 11 and 12 depict the changes in excess pore pressure over time under the first and second boundary conditions, respectively, when both chemical and mechanical loads are acting, for different fractal dimensions. It can be observed that, in general, the larger the fractal dimension, the greater the maximum negative pore pressure during consolidation, and the longer the time required for complete dissipation of pore pressure. However, the early consolidation stage shows irregularity in the decrease of pore pressure. This is because, although an increase in fractal dimension slows down the migration of solutes and the reduction of osmotic pressure difference between the soil layer and its surroundings, leading to a faster decrease in soil pore pressure, on the other hand, a larger fractal dimension indicates a rougher pore surface and more tortuous pore channels, which strongly hinders the seepage of water in the soil. Therefore, the drainage rate of pore water under external loads decreases, and the dissipation of pore pressure slows down. These two effects occur simultaneously, when the fractal dimension’s impact on the migration of chemical solutes is greater than its impact on the seepage of pore water, the dissipation of pore pressure will be faster, otherwise, it will be slower. Thus, in the early stages of consolidation, the fractal dimension’s impact on pore pressure dissipation is irregular under different conditions.

Fig. 11

Effect of surface fractal dimension on excess pore pressure of CHM coupling consolidation under the first type of boundary conditions.

Fig. 12

Effect of surface fractal dimension on excess pore pressure of CHM coupling consolidation under the second type of boundary conditions.

Comparing Figs. 11 and 12, it can be found that under the first type of boundary condition, the fractal dimension has a greater impact on pore pressure, especially at the bottom of the soil layer, where the sensitivity to fractal dimension is more distinct between the two boundary conditions. This is because the bottom of the first type of boundary condition is closed, while the bottom of the second type of boundary condition is open, hence the roughness of the pores reflected by the size of the fractal dimension has a different impact on the seepage of pore water under the two boundary conditions. Clearly, its impact on the seepage of water in the soil under the first type of boundary condition is greater, especially at the bottom of the soil layer. Under the first type of boundary condition, the pore water at the bottom can only be drained upwards, while under the second type of boundary condition, the pore water at the bottom can be drained from the bottom, and the total length of the seepage path through the pores is much smaller, thus the roughness of the pores has a much smaller impact.

Effect of surface fractal dimension on deformation

Figures 13 illustrates the changes in deformation over time under the first and second boundary conditions with different fractal dimensions. It can be observed that, due to the permeable boundary conditions of the second type accelerating the drainage of pore water, settlement progresses more rapidly. Under the first type of boundary condition, the maximum deformation of the soil during consolidation and the final settlement after consolidation stabilization are both greater. As the fractal dimension increases, the rate at which chemical substances enter the soil layer decreases, the decay rate of the osmotic pressure difference slows down, and the rate of soil layer deformation increases. The larger the fractal dimension, the greater the maximum deformation of the soil layer during consolidation and the final settlement after consolidation stabilization. Additionally, the larger the fractal dimension, the longer the time required for the soil layer deformation to stabilize. It can also be seen from the figures that under the first type of boundary condition, the fractal dimension has a greater impact on soil deformation, compared to the simultaneous action of chemical and mechanical loads, the impact of the fractal dimension on soil deformation is greater when there is no mechanical load.

Fig. 13

Effect of surface fractal dimension on deformation of CHM coupling consolidation: (a) Boundary condition 1, (b) Boundary condition 2.

Conclusion

This study is based on the fractal theory of clay and introduces the generalized effective stress principle and the fractal model of chemical substance diffusion in the chemical solution environment of clay. A chemo-hydro-mechanical coupling control model considering the fractal characteristics of soil is established, and numerical simulations of consolidation and solute migration of clay in chemical solutions under two boundary conditions are conducted to analyze the impact of clay surface fractal dimension on chemo-hydro-mechanical coupling consolidation. The following conclusions can be drawn:

• (1) The fractal dimension takes into account the irregularity and roughness of pores. Compared to conventional models, the migration of chemical solutes in clay based on the fractal model is much slower. The larger the fractal dimension, the rougher the pore surface and the more tortuous the pore channels, thus providing a stronger resistance to solute diffusion, resulting in slower diffusion, conversely, the smaller the fractal dimension, the smoother the pore surface and the straighter the pore channels, leading to faster diffusion. When the fractal dimension equals 2, the medium’s pore channels become straight cylindrical bundles, and the solute diffusion model degenerates into the corresponding form in the conventional straight capillary bundle model of porous media. Under the first type of boundary condition, all pore solution concentrations within the soil layer will eventually increase to match the external solution concentration. The deeper the depth, the longer the time required for diffusion to be completed. Under the second type of boundary condition, after stabilization, the pore solution concentrations at different depths within the soil layer vary, decreasing with increasing depth.

• (2) The larger the fractal dimension, the greater the maximum negative pore pressure during consolidation, and the longer the time required for complete dissipation of pore pressure. However, the decrease in pore pressure during the early stages of consolidation is irregular. This is because, although an increase in fractal dimension slows down the migration of solutes and the reduction of osmotic pressure difference between the soil layer and its surroundings, leading to a faster decrease in soil pore pressure, on the other hand, a larger fractal dimension indicates a rougher pore surface and more tortuous pore channels, which strongly hinders the seepage of water in the soil. Therefore, the drainage rate of pore water under external loads decreases, and the dissipation of pore pressure slows down. These two effects occur simultaneously, when the fractal dimension’s impact on the migration of chemical solutes is greater than its impact on the seepage of pore water, the dissipation of pore pressure will be faster, otherwise, it will be slower. Thus, in the early stages of consolidation, the fractal dimension’s impact on pore pressure dissipation is irregular under different conditions.

• (3) Under the first type of boundary condition, the fractal dimension has a greater impact on pore pressure, especially at the bottom of the soil layer, where the sensitivity to the fractal dimension is more distinct between the two boundary conditions. This is because the bottom of the first type of boundary condition is closed, while the bottom of the second type of boundary condition is open, hence the roughness of the pores reflected by the size of the fractal dimension has a different impact on the seepage of pore water under the two boundary conditions. Obviously, its impact on the seepage of water in the soil under the first type of boundary condition is greater, especially at the bottom of the soil layer. Under the first type of boundary condition, the pore water at the bottom can only be drained upwards, while under the second type of boundary condition, the pore water at the bottom can be drained from the bottom, and the total length of the seepage path through the pores is much smaller, thus the roughness of the pores has a much smaller impact.

• (4) The larger the fractal dimension, the greater the maximum deformation of the soil layer during consolidation and the final settlement after consolidation stabilization, and the longer the time required for the soil layer deformation to stabilize. Additionally, under the first type of boundary condition, the fractal dimension has a greater impact on soil deformation, compared to the simultaneous action of chemical and mechanical loads, the impact of the fractal dimension on soil deformation is greater when there is no mechanical load.