Keyword

1 Introduction

The problem of dynamic response of unsaturated soils under dynamic loading has been widely studied in many fields, such as earthquake engineering, oilfield exploration, and geophysics.

Currently, in studying the dynamic properties of unsaturated soils, most are based on the equivalent fluid model [1] and the mixture theory [2] as the theoretical basis. Yang [3] studied the propagation of elastic waves in unsaturated porous media using the equivalent fluid model. Yuanqiang Cai [4] investigated the significant influence of saturation change on elastic wave propagation in saturated and unsaturated porous media based on the equivalent fluid model, indicating that the change of saturation should be emphasized in the study of soil dynamics. However, the equivalent fluid model considers the liquid and the gas as the same fluid, which shows that this kind of model does not belong to the three-phase medium model, and the equivalent fluid only applies to the highly saturated soil. Therefore, the application range of the equivalent fluid model is limited.

Mixture theory can effectively study complex porous media's deformation and overall motion and helpfully describe unsaturated soil's mechanical properties. Therefore, Hu Yayuan [5] introduced the isotropic linear elastic equation of unsaturated double medium based on the mixture theory to analyze the complex multi-physical field coupling effect of geotechnical soil; Lu [6] established the phase field model of porous medium in the freezing situation based on the mixture theory.

Porosity directly reflects the degree of compactness of the soil and is an important parameter affecting the fluid transport and permeability properties within unsaturated soils. Zhou Fengxi [7] used the theory of mixtures to give the dynamic control equations of unsaturated porous elastic media, studied the effect of unsaturated soil porosity, shear modulus, and other factors on the fluctuation response of the soil and carried out parametric analysis, Zhou Song et al. [8] studied the nature of heat conduction in unsaturated bentonite soil and analyzed the effect of porosity and other factors on the effective heat conduction characteristics of the soil.

The soil skeleton is bound to be deformed in unsaturated soil under the action of dynamic load, which in turn leads to a corresponding change in soil porosity [9]. However, the above studies regarded the porosity as a constant, ignoring the dynamic shift in porosity under dynamic loading and the effect on the dynamic properties of unsaturated soil, and the change of porosity will lead to a series of changes in the mechanical properties of soil permeability. Failure to consider dynamic porosity can lead to a decrease in model prediction accuracy, limiting the understanding of soil deformation mechanisms and affecting long-term performance assessment. However, some studies have concluded that soil permeability does not change with the change in porosity [10]. Therefore, this paper does not consider the influence of the change of porosity on permeability.

In summary, the mixture theory can characterize the complex interactions between multiphase substances and deal with complex media's ontological relationships. Therefore, this paper takes the mixture theory as the theoretical basis, establishes the dynamic response equation of unsaturated soil considering the dynamic porosity, and numerically solves it through the Comsol Multiphysics PDE module to analyze the dynamic characteristics of two-dimensional unsaturated soil under the action of vertically concentrated sinusoidal loading. Studies using dynamic modelling of unsaturated soils incorporating dynamic porosity can provide more accurate predictions of vertical displacements and pore water pressures. This approach takes into account the changes in the pore structure of the soil body under dynamic loading, which enables a more realistic simulation of the response behaviour of the soil body and provides critical guidance for engineering design.

2 Equations Governing the Dynamics of Unsaturated Soils Considering Dynamic Porosity

The correctness of the mixture theory that can be used to solve the skeleton deformation problem of unsaturated soils has been proved theoretically [13], and it has been proved by a large number of experiments that the motion of liquids and gases in the pores of unsaturated soils can be described by Darcy's law. [19].

In order to effectively establish the dynamic control equation of unsaturated soil considering dynamic porosity, the following assumptions are made:

  1. (1)

    The three phases of unsaturated soil have the same temperature and thermal effects are not taken into account;

  2. (2)

    There is a clear demarcation line between the three phases of unsaturated soil;

  3. (3)

    The unsaturated soil is a homogeneous isotropic material;

  4. (4)

    The transformation of mass, internal energy and momentum between the phases is not taken into account.

2.1 Ontological Relationships for Soil Skeleton Deformation

The effective stress obtained based on the deformation work is [11]

$$\bar{T}_{ij} = T_{ij} - [S_{\rm{r}} p_l + (1 - S_{\text{r}} )p_{\rm{g}} ]l$$
(1)

where T̅ij and Tij are the practical and total stresses of the unsaturated soil, respectively; Sr is the degree of saturation of the unsaturated soil; pl and pg are the liquid-phase and gas-phase pressures, respectively; and I is the unit tensor.

Relationship between each of the corresponding variables and their displacements in unsaturated soils [12]:

$$\varepsilon_{\rm{a}} = - \frac{{[\nabla X_{\text{a}} + (\nabla X_{\text{a}} )^T ]}}{{2}}$$
(2)

where: when a = s, l, g, Xa is the displacement of the solid, liquid, and gas phases, respectively.

The ontological relationship for the unsaturated soil skeleton is [12]

$$\bar{T}_{ij} = \lambda eI + {2}\mu \varepsilon_{\rm{s}}$$
(3)

where e = trɛs. λ and μ are Lame constants for the unsaturated soil skeleton.

2.2 Equations of Control for Unsaturated Soil Dynamics

The volume fractions of each phase of the unsaturated soil are:

$$\begin{gathered} \Phi_{\text{s}} = 1 - n\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(a) \hfill \\ \Phi_l = nS_{\rm{r}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(b) \hfill \\ \Phi_{\text{g}} = n(1 - S_{\text{r}} )\;\;\;\;\;\;\;\;\;\;(c) \hfill \\ \end{gathered}$$
(4)

where Φs, Φl, and Φg are the volume fractions of the solid, liquid, and gas phases, respectively, n is the porosity of the unsaturated soil.

The liquid-phase and vapor-phase pressures of unsaturated soils are [13]:

$$\begin{gathered} p_l = - \lambda_{{\text{c}}l} \nabla \cdot X_{\rm{s}} + M_{ll} \zeta_l + M_{lg} \zeta_{\rm{g}} \;\;\;\;\;\;(a) \hfill \\ p_{\rm{g}} = - \lambda_{{\text{c}}{\rm{g}}} \nabla \cdot X_{\text{s}} + M_{{\rm{g}}l} \zeta_l + M_{{\rm{gg}}} \zeta_{\text{g}} \;\;\;\;(b) \hfill \\ \end{gathered}$$
(5)

Where: λcv = (Λsv + Λlv + Λgv)/αv, Λ is the equilibrium interaction force, \(\Phi_{\text{a}}^{ + }\) is the volume fraction of the phases in the unsaturated soil at static equilibrium, ζ is the volume increment of the phases in the unsaturated soil, Uv are the displacements of the liquid phase and the gaseous phase concerning the solid phase, respectively. a, b = s, l, g. v = l, g.

Equation (5a) with Eq. (5a) can be simplified to:

$$\begin{gathered} - \nabla \cdot U_l = c_{l1} p_l + c_{l2} p_{\rm{g}} + c_{l3} \nabla \cdot X_{\rm{s}} \;\;\;\;\;\;\;\;(a) \hfill \\ - \nabla \cdot U_{\rm{g}} = c_{{\rm{g}}1} p_l + c_{{\rm{g}}2} p_{\rm{g}} + c_{{\rm{g}}3} \nabla \cdot X_{\rm{s}} \;\;\;\;\;\;(b) \hfill \\ \end{gathered}$$
(6)

Where: Δc = MllMgg − MglMlg, cl1 = Mgg/Δc, cl2 = cg1 = −Mlg/Δc, cg2 = Mll/Δc, cl3 = (Mggλcl – Mlgλcg)/Δc, cg3 = (Mllλcg − Mglλcl)/Δc.

The equations governing the dynamics of unsaturated soils are [13]

(7)

Where: γl and γg are the actual densities of the liquid phase and gas phase in unsaturated soil, respectively. ml = γl/Φl, mg = γg/Φg, ρ is the relative density of unsaturated soil, ρ = Φsγs + Φgγg + Φlγl; \(\eta_l\) and \(\eta_g\) are the viscosity coefficients of the liquid phase and the gas phase, respectively. Kll, Klg, Kgl and Kgg are the permeability of the liquid phase to the solid phase, the permeability of the gas phase to the liquid phase, the permeability of the liquid phase to the gas phase, and the permeability of the gas phase to the solid phase, respectively; and bl, bg and b are the external body forces of the liquid phase, the gas phase, and the solid phase, respectively.

Since the liquid phase and gas phase in unsaturated soil can flow freely and stably, the effect of gas-phase pressure gradient on the liquid-phase flow is negligible compared to the liquid-phase pressure gradient on the liquid-phase flow; similarly, the effect of liquid-phase pressure gradient on the gas-phase flow is negligible compared to the gas-phase pressure gradient on the gas-phase flow of the deterministic role of the gas-phase flow, which can be obtained from the Klg → ∞, Kg/ → ∞.. Equations (4b) and (4c) are obtained by bringing them into Eqs. (7a) and (7b), respectively:

$$\begin{gathered} \frac{\gamma_l }{{nS_{\rm{r}} }}\frac{\partial^2 U_l }{{\partial t^2 }} + \gamma_l \frac{{\partial^2 X_{\rm{s}} }}{\partial t^2 } = - \nabla p_l - B_{ll} \frac{\partial U_l }{{\partial t}} + \gamma_l b_l \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(a) \hfill \\ \frac{{\gamma_{\rm{g}} }}{{n(1 - S_{\rm{r}} )}}\frac{{\partial^{2} U_{\rm{g}} }}{{\partial t^{2} }} + \gamma_{\rm{g}} \frac{{\partial^{2} X_{\rm{s}} }}{{\partial t^{2} }} = - \nabla p_{\rm{g}} - B_{{\rm{gg}}} \frac{{\partial U_{\rm{g}} }}{\partial t} + \gamma_{\rm{g}} b_{\rm{g}} \;\;\;\;\;\;\;\;(b) \hfill \\ \end{gathered}$$
(8)

where Bll =  \(\eta_l\)/Kll, Bgg = \(\eta_g\)/Kgg.

Substituting Eqs. (1), (2), (3), (4a), (4b) & (4c) into Eq. (7c) gives:

(8c)

Equations (8a), (8b) and (8c) are the equations governing the dynamics of unsaturated soils expressed in terms of solid-phase displacement Xs, liquid-phase pressure pl, and gas-phase pressure pg.

2.3 Dynamic Porosity Modeling of Unsaturated Soils

The mass conservation equation for solid phase media, i.e., the equation for porosity with time, is [14]:

$$\frac{{{\text{d(1 - }}n{\rm{)}}}}{{{\text{d}}t}} + \frac{{{1} - n}}{{\rho_{\text{s}} }}\frac{{{\text{d}}\rho_{\text{s}} }}{{{\text{d}}t}} + (1 - n)\nabla \cdot v_{\text{s}} = 0$$
(9)

Where ρs is the relative density of the solid phase, ρs is the absolute velocity of the solid phase.

The material derivative of the solid phase in unsaturated soils can be simplified as:

$$\frac{{{\text{d(}} \cdot {)}}}{{{\text{d}}t}} \approx \frac{{\partial {(} \cdot {)}}}{\partial t}$$
(10)

The relationship between the absolute velocity of the solid phase and the bulk strain in unsaturated soils is [15]:

$$\nabla \cdot v_{\text{s}} { = }\frac{{{\text{d}}e}}{{{\text{d}}t}} \approx \frac{\partial e}{{\partial t}}$$
(11)

The solid phase density in unsaturated soils can be expressed as [16]:

$$\frac{{{1 - }n}}{{\rho_{\text{s}} }}\frac{{{\text{d}}\rho_{\text{s}} }}{{{\text{d}}t}} = - (1 - \alpha^{1} )\frac{{{\text{d}}e}}{{{\text{d}}t}} + \frac{{\alpha^{1} - n}}{{K_{\text{s}} }}(\chi_l \frac{{{\text{d}}p_l }}{{{\text{d}}t}}{ + }\chi_{\text{g}} \frac{{{\text{d}}p_{\text{g}} }}{{{\text{d}}t}}) - (\alpha^{1} - n)\beta_{\rm{T}} \frac{{{\rm{d}}T}}{{{\rm{d}}t}}$$
(12)

Where χl and χg are Bishop's effective stress coefficients, βT is the coefficient of thermal expansion, T is the temperature, α1 ≡ 1 – K/Ks. K is the bulk compression modulus of the solid phase under drained conditions, Ks is the bulk compression modulus of the solid phase.

From Eqs. (9)–(12), the equation for the change in porosity with time can be derived as:

$$\frac{\partial n}{{\partial t}} = (\alpha^1 - n)(\frac{\partial e}{{\partial t}}{ + }\frac{\chi_l }{{K_{\text{s}} }}\frac{\partial p_l }{{\partial t}} + \frac{{\chi_{\text{g}} }}{{K_{\text{s}} }}\frac{{\partial p_{\text{g}} }}{\partial t} - \beta_{\rm{T}} \frac{\partial T}{{\partial t}})$$
(13)

Since thermal effects are not taken into account in this paper and Eq. (1) is used as the effective stress for unsaturated soils, Eq. (13) can be simplified as:

$$\frac{\partial n}{{\partial t}} = (\alpha^1 - n)(\frac{\partial e}{{\partial t}} + \frac{{S_{\text{r}} }}{{K_{\text{s}} }}\frac{\partial p_l }{{\partial t}} + \frac{{1 - S_{\text{r}} }}{{K_{\text{s}} }}\frac{{\partial p_{\text{g}} }}{\partial t})$$
(14)

When the change in porosity is not significant, Eq. (14) can be obtained by integrating over time:

$$n{ = }n_{0} { + }(\alpha^{1} - n_{0} )(e + \frac{{S_{\text{r}} }}{{K_{\text{s}} }}p_l + \frac{{1 - S_{\text{r}} }}{{K_{\text{s}} }}p_{\text{g}} )$$
(15)

where n0 is the initial porosity of the unsaturated soil.

Equations (8a), (8b), (8c), and (15) are the equations governing the dynamics of unsaturated soils considering dynamic porosity.

3 Based on Comsol Multiphysics PDE Model Construction

3.1 Generalised Transformations of the Equations Governing the Dynamics of Unsaturated Soils Considering Dynamic Porosity

The general form of the system of partial differential equations of generalized type is:

$$e_{\rm{a}} \frac{{\partial^{2} u}}{{\partial t^{2} }} + d_{\rm{a}} \frac{\partial u}{{\partial t}} + \nabla \cdot \Gamma = f$$
(16)

where a is the mass factor, da is the damping factor, Γ is the conserved flux, and f is the source term.

Equation (8a), Eq. (8b), and Eq. (8c) are generalized and transformed in a two-dimensional cartesian coordinate system according to the solution form provided by the Comsol Multiphysics PDE module:

According to the form of solving the system of partial differential equations of generalized type in Comsol Multiphysics PDE, it is necessary to rewrite Eq. (6) and Eq. (15) into the form of matrices that satisfy Eq. (16) and the corresponding matrices are thus obtained as:

$$d_{\rm{a}} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {B_{ll} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {B_{ll} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {B_{{\rm{gg}}} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {B_{{\rm{gg}}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)$$
(17)
$$u = \left[ {\begin{array}{*{20}c} {U_{\rm{x}} } & {U_{\text{y}} } & {U_{l{\text{x}}} } & {U_{l{\text{y}}} } & {U_{{\text{gx}}} } & {U_{{\text{gy}}} } & {p_l } & {p_{\text{g}} } & n \\ \end{array} } \right]^T$$
(18)
$$\Gamma = \left[ {\begin{array}{*{20}c} { - \nabla (\mu U_{\text{x}} )} & { - \nabla (\mu U_{\text{y}} )} & 0 & 0 & 0 & 0 & { - U_l } & { - U_{\text{g}} } & 0 \\ \end{array} } \right]^T$$
(19)
$$e_{\rm{a}} = \left( {\begin{array}{*{20}c} {\gamma_{\text{s}} + n\left( {\gamma_{\text{g}} - \gamma_{\text{s}} } \right) \, + nS_{\text{r}} \left( {\gamma_l - \gamma_{\text{g}} } \right)} & 0 & {\gamma_l } & 0 & {\gamma_{\text{g}} } & 0 & 0 & 0 & 0 \\ 0 & \rho & 0 & {\gamma_l } & 0 & {\gamma_{\text{g}} } & 0 & 0 & 0 \\ {\gamma_l } & 0 & {\frac{\gamma_l }{{nS_{\text{r}} }}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\gamma_l } & 0 & {\frac{\gamma_l }{{nS_{\text{r}} }}} & 0 & 0 & 0 & 0 & 0 \\ {\gamma_{\text{g}} } & 0 & 0 & 0 & {\frac{{\gamma_{\text{g}} }}{{n(1 - S_{\text{r}} )}}} & 0 & 0 & 0 & 0 \\ 0 & {\gamma_{\text{g}} } & 0 & 0 & 0 & {\frac{{\gamma_{\text{g}} }}{{n(1 - S_{\text{r}} )}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)$$
(20)
$$f = \left( {\begin{array}{*{20}c} { - S_{\text{r}} \frac{\partial p_l }{{\partial x}} - {(}1 - S_{\text{r}} )\frac{{\partial p_{\text{g}} }}{\partial x}{ + }\gamma_{\text{s}} + [n(\gamma_{\text{g}} - \gamma_{\text{s}} ) + nS_{\text{r}} (\gamma_l - \gamma_{\text{g}} )b_{\text{x}} {] + (}\lambda + \mu )(\frac{{\partial^{2} U_{\text{x}} }}{{\partial x^{2} }} + \frac{{\partial^{2} U_{\text{y}} }}{\partial x\partial y})} \\ { - S_{\text{r}} \frac{\partial p_l }{{\partial y}} - {(}1 - S_{\text{r}} )\frac{{\partial p_{\text{g}} }}{\partial y}{ + }\gamma_{\text{s}} + [n(\gamma_{\text{g}} - \gamma_{\text{s}} ) + nS_{\text{r}} (\gamma_l - \gamma_{\text{g}} )b_{\text{y}} {] + (}\lambda + \mu )(\frac{{\partial^{2} U_{\text{x}} }}{\partial x\partial y} + \frac{{\partial^{2} U_{\text{y}} }}{\partial y^2 })} \\ { - \frac{\partial p_l }{{\partial x}} + \gamma_l b_{l{\text{x}}} } \\ { - \frac{\partial p_l }{{\partial y}} + \gamma_l b_{l{\text{y}}} } \\ { - \frac{{\partial p_{\text{g}} }}{\partial x} + \gamma_{\text{g}} b_{{\text{gx}}} } \\ { - \frac{{\partial p_{\text{g}} }}{\partial y} + \gamma_{\text{g}} b_{{\text{gy}}} } \\ {c_{l1} p_l + c_{l2} p_{\text{g}} + c_{l3} \nabla \cdot X_{\text{s}} } \\ {c_{{\text{g}}1} p_l + c_{{\text{g}}2} p_{\text{g}} + c_{{\text{g}}3} \nabla \cdot X_{\text{s}} } \\ {n_0 + (\alpha^1 - n_0 )(e + \frac{{S_{\text{r}} }}{{K_{\text{s}} }}p_l + \frac{{1 - S_{\text{r}} }}{{K_{\text{s}} }}p_{\text{g}} ) - n} \\ \end{array} } \right)$$
(21)

Where Ux and Uy are the displacements of the solid phase of unsaturated soil in the horizontal and vertical directions, respectively. Ulx and Uly are the displacements of the liquid phase compared to the solid phase in the horizontal and vertical directions, respectively. Ugx and Ugy are the displacements of the gaseous phase compared to the solid phase in the horizontal and vertical directions, respectively. bx, blx, and bgx are the external body forces of the solid, liquid, and gaseous phases of unsaturated soil in the horizontal direction, respectively, and by, by and bgy are the external body forces of the solid, liquid and gas phases in the vertical direction, respectively.

3.2 Geometrical Modeling of Two-Dimensional Unsaturated Soils

To study the dynamic response characteristics of two-dimensional unsaturated soil considering dynamic porosity under vertical concentrated harmonic loading, a two-dimensional unsaturated soil model is established in Comsol Multiphysics PDE module, and the geometrical model is set to have a height and a width of 2 × 104 m, with the center of the upper top surface of the model as the origin of the co-ordinates, the positive half-axis of the y-axis vertically downward, and the positive half-axis of the x-axis horizontally to the right. The axis is vertically downward, the x-axis is horizontal to the right, and the top surface of the soil body is permeable, air permeable, and impermeable; impermeable two different boundary conditions: the bottom surface boundary is fixed and impermeable, impermeable boundary conditions, the left and right boundaries of the two along the x-direction is zero, and both are impermeable, impermeable boundary conditions (Fig. 1).

Fig. 1.
figure 1

Comparison of the results of this paper with those of He Wenhai [17] considering dynamic porosity

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4 Simulation and Analysis

4.1 Model Verification

To demonstrate the reliability of the research methodology in this paper, The dynamic equations of unsaturated soil considering dynamic porosity were degraded to the dynamic governing equations of saturated soil considering dynamic porosity, and the results were compared with the calculations in the literature [17]. The degraded soil contains only a liquid phase, saturation degree Sr = 1, and the mechanical parameter about the gas phase in the expression is 0. The geometric model is set to have impermeable boundary conditions at the top surface, bottom surface, and left and proper boundaries, and the dimensionless mechanical parameter, the geometric model, the magnitude of the excitation and its imposition are the same as that in the literature [17], and Comsol Multiphysics PDE solves the equation. The patterns of dimensionless vertical displacement and dimensionless liquid phase pressure with time are shown in Figs. 2a and b.

As can be seen from Fig. 2, when the constructed model degraded saturated soil dynamic control equations, the results of this paper are basically in agreement with the calculation results of literature [17] and literature [18]. Therefore, the reliability of the research method in this paper can be illustrated.

In this paper, considering the actual situation of the project, a group of artificial fill as unsaturated soil is selected for research, and according to the relevant engineering specifications as well as engineering experience, to determine its specific mechanical parameters as shown in Table 1 [20], and the harmonic load f(t) acted vertically on the soil at the center of the upper top surface:

$$f\left( t \right) = A_0 \sin (2\pi \omega t{\rm{)}}$$
(22)

where A0 is the load amplitude, A0 = 1 × 106 Pa; ω is the harmonic load frequency, in this paper ω = 10 Hz.

Table 1. Mechanical parameters of unsaturated soils
Full size table

4.2 Changing Law of Unsaturated Soil Porosity Under Different Boundary Conditions

Unsaturated soil in the lower bottom surface and the left and right boundaries are impermeable, impermeable premise, when the upper top surface is permeable, permeable and impermeable, impermeable two different boundary conditions, under the action of harmonic load f(t), the rule of change of the change of porosity at the origin of the coordinates of the law of change with time is shown in Fig. 3

Fig. 2.
figure 2

Variation of porosity of unsaturated soils with time

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Fig. 3.
figure 3

Variation of vertical displacement of unsaturated soil with time

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As can be seen in Fig. 3, in the process of harmonic loading, with the change of time, unsaturated soil in the upper top surface permeable, permeable and impermeable, impermeable with two different boundary conditions, the porosity is fluctuating decreasing trend. The change rule of porosity in the first cycle (0 < t < 0.75 s) during the harmonic loading is thus analyzed. When 0 < t < 0.375 s, the soil skeleton is continuously compressed by the vertical downward load, resulting in a gradual decrease in porosity; when 0.375 s < t < 0.5625 s, the load is vertically upward along the y-axis, but due to the existence of inertia, the soil skeleton is still continuously compressed, and porosity continues to decrease; when t = 0.5625 s, the inertia disappears, and the porosity reaches the minimum value, when 0.5625 s < t < 0.75 s, the load is still vertically upward along the y-axis, and the soil skeleton is compressed to a gradually decreasing degree, and the porosity shows an increasing trend. During the subsequent action of harmonic load, the rule of change of porosity is the same as that of the first cycle. Therefore, the porosity of unsaturated soil shows a fluctuating decrease.

It can also be seen from Fig. 3 that the porosity of unsaturated soil changes more under the boundary conditions of water permeability and air permeability at the upper top surface. This is because when the upper top surface is impermeable, the liquid and gas phases in the pores of the soil can not be freely discharged from the upper top surface, and the pressure of the liquid and gas phases in the pores resists the deformation of the soil body under the action of load. Under the boundary condition of water and air permeability on the upper top surface, the liquid and gas phases in the soil pore space can be discharged freely. The load borne by the liquid and gas phases will be borne by its skeleton, and the soil skeleton is continuously compressed. Therefore, the porosity of unsaturated soil changes more.

4.3 Effect of Dynamic Porosity on the Dynamic Response of Unsaturated Soils

Effect of Dynamic Porosity on Vertical Displacement of Unsaturated Soils

Unsaturated soil in the lower bottom surface and the left and right boundaries are impermeable, impermeable premise, when the upper top surface is permeable, permeable and impermeable, impermeable two different boundary conditions, under the action of harmonic load f(t), the change rule of the vertical displacement of the soil body at the origin of the coordinates to time is shown in Fig. 4.

As shown in Fig. 4, under the effect of harmonic loading, with time, the vertical displacement of unsaturated soil exhibits a fluctuating increase under two different boundary conditions at the top surface: permeable to water and air and impermeable to both, regardless of whether the dynamic porosity is considered or not. Taking the vertical displacement of soil during the first cycle of harmonic loading as an example for analysis, the load moves vertically downward along the y-axis in the 1st half of the cycle, continuously compressing the soil skeleton, resulting in an increase in vertical soil displacement. During the 1/2 to 3/4 period of the cycle, the load moves vertically upward along the y-axis. Still, due to inertia, the soil skeleton continues to be compressed, increasing vertical soil displacement. In the final 1/4 of the cycle, as the inertia dissipates and the load continues to move vertically upward along the y-axis, the degree of compression on the soil skeleton decreases, causing a decrease in vertical soil displacement. In subsequent cycles of harmonic load application, the pattern of change in the soil's vertical displacement is the same as in the first cycle. Therefore, regardless of whether dynamic porosity is considered, the vertical displacement of unsaturated soil shows a fluctuating increase.

As seen in Fig. 4, under the boundary conditions allowing for permeability and aeration at the top surface, the vertical displacement of unsaturated soil is more significant than that under non-permeable and non-aerated conditions at the same surface, regardless of the consideration of dynamic porosity. This occurs because, under the non-permeable and non-aerated conditions at the top surface, the liquid and gas phases within the soil pores cannot freely escape, being retained inside the soil body. When the soil is subjected to external loads, the pressure from the liquid and gas phases in the soil pores can resist the deformation of the soil. In contrast, under the conditions allowing for permeability and aeration at the top surface, the liquid and gas phases within the soil pores can freely escape, leading to a zero pressure increment under the load. Therefore, the vertical displacement of the unsaturated soil is more significant.

Furthermore, as also observed in Fig. 4, dynamic porosity significantly affects the vertical displacement of unsaturated soil, primarily manifesting as smaller vertical displacement under the same top surface boundary conditions when dynamic porosity is considered. This is because, under harmonic load, the porosity dynamically decreases over time (as shown in Fig. 3), enhancing the interaction between the soil skeleton and the liquid and gas phases in the pores. The resistance increases as the soil skeleton is compressed under load, resulting in a smaller vertical displacement of the unsaturated soil.

Effect of Dynamic Porosity on Pore Liquid Phase Pressure in Unsaturated Soils. Unsaturated Soil in the Lower

Unsaturated soil in the lower bottom surface and the left and right boundaries are impermeable, impermeable premise, when the upper top surface is permeable, permeable and impermeable, impermeable two different boundary conditions, respectively, in the harmonic load f(t) under the action of the liquid-phase pressure in the pore space of the unsaturated soil at the origin of the coordinates of the law of change over time is shown in Fig. 5.

Fig. 4.
figure 4

Variation of liquid phase pressure of unsaturated soil with time

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Fig. 5.
figure 5

Variation of gas phase pressure of unsaturated soil with time

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As can be seen from Fig. 5, under the action of harmonic loading, with the growth of time, when the upper top surface of the soil body is impermeable and impermeable boundary condition, the pressure of the liquid phase in the pores of the soil body shows a fluctuating growth, no matter whether the dynamic porosity is considered or not. This is because, under the boundary conditions of the impermeable and impermeable upper top surface, the liquid phase in the soil pore space cannot be discharged freely from the upper top surface and is retained inside the soil body, and under the action of sinusoidal load, the soil body is fluctuating compression, which leads to fluctuating compression of the liquid phase in the pore space, and the liquid-phase pressure shows a fluctuating growth tendency. Under the boundary condition of water permeability and air permeability on the upper top surface, the pore liquid phase pressure increment is zero, regardless of whether dynamic porosity is considered. The main reason is that when the upper top surface is water-permeable and air-permeable, the liquid phase in the pore space can be discharged freely, and the liquid phase pressure change is not affected by the vertical deformation of the soil. Hence, the liquid phase pressure increment in the pore space of the unsaturated soil is zero.

Consideration of dynamic porosity has a specific effect on the liquid phase pressure in the pores of unsaturated soils, as can be seen from Fig. 5; under the boundary conditions of the impermeable and impermeable upper top surface, the liquid phase pressure in the pores of the soil is relatively tiny when dynamic porosity is considered. This is because compared to not considering dynamic porosity, in the presence of dynamic porosity, as previously mentioned, the porosity decreases, and the soil skeleton is compressed when the resistance increases. Hence, the vertical displacement of the soil body is smaller (as shown in Fig. 4), and the degree of compression becomes smaller, resulting in the degree of compression of the pore liquid phase being smaller; therefore, the unsaturated soil pore liquid phase pressure is more minor.

Effect of Dynamic Porosity on Pore Gas Phase Pressure in Unsaturated Soils

Unsaturated soil in the lower bottom surface and the left and right boundaries are impermeable, impermeable premise, when the upper top surface of the soil body is permeable, permeable and impermeable, impermeable two different boundary conditions, in the harmonic load f(t), in the origin of the coordinates of the soil body pore space in the law of change of the law of the gas-phase pressure with time is shown in Fig. 6.

As can be seen from Fig. 6, under the action of harmonic loading, with the change of time, when the boundary of the upper top surface of the soil body is impermeable, the pressure of the gas phase in the pore space of the soil body shows a fluctuating growth, regardless of whether the dynamic porosity is considered or not. As mentioned above, under the boundary condition that the upper top surface is impermeable and impermeable, the gas phase in the pores of the soil body cannot be discharged freely and is retained inside the soil body, and under the action of harmonic load, the soil body is compressed by fluctuation, which leads to the fluctuation compression of the gas phase in the pores as well, and therefore, the pore gas-phase pressure shows a fluctuating growth. Under the boundary conditions of water permeability and air permeability at the upper top surface of the unsaturated soil, the increment of gas-phase pressure in the pores of the soil body is zero, regardless of whether dynamic porosity is considered. This is because the gas-phase in the pores of the soil body can be discharged freely, and the vertical deformation of the soil body doesn't affect the change of the gas-phase pressure in the pores.

It can also be seen from Fig. 6 that under the boundary conditions of the impermeable and impermeable upper top surface, the soil pore gas-phase pressure is smaller when dynamic porosity is considered. As mentioned above, when dynamic porosity is considered, under harmonic loading, the porosity decreases, and the soil body movement is affected by resistance. The soil body is compressed to a smaller extent, which causes the degree of compression of the gas phase in the soil pores to be smaller, so the soil pore gas-phase pressure is smaller when dynamic porosity is considered.

From Figs. 5 and 6, it can be seen that when the unsaturated soil is impermeable and impermeable on the upper top surface, the fluctuation amplitude of the liquid phase pressure in the soil pore space is much larger than the fluctuation amplitude of the gas phase pressure. This is because the liquid-phase pressure is much larger than the gas-phase pressure under harmonic loading, which indicates that the load borne by the liquid-phase is much larger than the gas-phase. The degree of compression is also more significant, so the amplitude of liquid-phase pressure fluctuation is much larger than that of gas-phase pressure fluctuation.

From Fig. 4, 5 and 6, it can be seen that the dynamic porosity does not affect the change rule of unsaturated soil dynamic response. Still, it affects the intensity size of its dynamic response. In the study of the dynamic response of unsaturated soil, full consideration of dynamic porosity can improve its accuracy.

5 Conclusion

This paper takes mixture theory as the theoretical basis, establishes the expression of the dynamic response of unsaturated soil considering dynamic porosity, and numerically solves and analyses the two-dimensional unsaturated soil based on Comsol Multiphysics PDE module, and the results show that:

  1. (1)

    The porosity, vertical displacement, liquid-phase pressure, and gas-phase pressure of unsaturated soil are directly affected by the boundary conditions of the upper top surface. Compared with the boundary conditions of the impermeable and impermeable upper top surface, when the upper top surface of the soil is permeable and permeable, the liquid phase and gas phase in the pore space can be discharged freely from the upper top surface. The pressure of the liquid and gas phases in the pore space is zero, and the soil porosity and vertical displacement change are more significant.

  2. (2)

    In the study of the dynamics of unsaturated soil, whether to consider the dynamic porosity only on the soil body response strength size does not affect the change rule of its dynamic response. When considering dynamic porosity, harmonic loading, the soil porosity decreases, the soil skeleton and pore liquid phase, gas phase interaction between the enhancement of the soil skeleton is compressed resistance becomes larger; the soil vertical displacement becomes smaller, the liquid phase and gas phase is compressed to a lesser extent, the liquid phase and the gas phase pressure is also reduced accordingly. This shows that considering dynamic porosity helps to reduce research errors.

The dynamic model of unsaturated soils considering dynamic porosity is closer to the behaviour of real soils, which also leads to more accurate mechanical models and analytical methods for better prediction of the nonlinear characteristics of soils under complex loading conditions.

In view of the lack of directly comparable experimental results, this paper validates the established mechanical model and numerical simulation method by using the results of related literature, based on which numerical simulations of the dynamic response of two-dimensional unsaturated soils excited by harmonic loading under different conditions are carried out. It provides a basis for the next work to be carried out by scholars who study the dynamic response characteristics of unsaturated porous media considering dynamic porosity.