Modified (Spherical and Cylindrical) Permeation Diffusion Model Considering Deep Bed Filtration Effect

Abstract

The study of the slurry reinforcement mechanism is mainly focused on the interaction between slurry and soil. The seepage effect of the slurry always exists no matter what way the slurry interacts with the soil around the pile. In the process of slurry diffusion, the porosity of the soil, the permeability of the slurry, and the slurry pressure vary due to some cement particles being blocked by the soil particle skeleton. Therefore, the study of the slurry filtration effect is of great significance for predicting the permeation and diffusion law of slurry. In this paper, a macroscopic linear filtration model was introduced and the changes of slurry properties in the permeation diffusion process were considered. Firstly, a spherical (cylindrical) permeation diffusion model, which takes the linear filtration effect and the variation of slurry viscosity into account, was derived based on the conservation of mass. Furthermore, in order to more accurately reflect the influence of the filtration effect on the slurry permeation diffusion model, a polynomial nonlinear filtration model was proposed, and the numerical solution for the permeation diffusion model was derived using finite difference and finite element methods. Finally, the numerically simulated values, the measured values, and the values from the spherical permeation diffusion model that does not consider slurry viscosity variations were compared. The results indicate that the grout pressure is inconsistent with the measured value without considering the effect of the filtration. The initial grouting pressure calculated by the model in this paper is slightly larger, and the required grouting pressure over time is greater than that without considering the filtration effect, regardless of whether the grout diffuses in a spherical or cylindrical manner. The results of this study can contribute to a better understanding of grouting engineering and provide some theoretical guidance for actual grouting.

1. Introduction

Grouting technology has become an essential approach for enhancing soil strength and reducing soil permeability. The injected slurry can effectively fill soil voids and cement the soil particles together, thereby reinforcing the foundation. The filtration effect during slurry permeation is an important factor affecting slurry diffusion [1,2]. when the slurry is injected into the soil, some cement particles will be blocked by the soil particle skeleton and gradually filtered out, decreasing the slurry concentration. This leads to soil pore clogging and adds to the difficulty of grouting, referred to as the filtration effect. The filtration effect is prevalent in the permeation grouting process of granular soil and plays a crucial role. Therefore, it is necessary to study the influence of the filtration effect on the permeation diffusion law of slurry.

Currently, scholars have conducted a great deal of research on the diffusion theory of penetration grouting, and the representative permeation grouting theories include the spherical diffusion theory, the cylindrical diffusion theory, Carroll’s theory, Baker’s formula, Lombad G formula, etc., and the corresponding calculation formulas are established [3,4,5]. Bouchelaghem [6] introduced a one-dimensional macroscopic permeation model considering the filtration effect of cement slurry in saturated porous media. The model took into account the diffusion and dispersion of cement slurry and the coupled interaction between the slurry and soil skeleton. The results indicated that the soil skeleton would generate strains on the order of 10⁻⁶. On this basis, Maghous [7] and Saada [8] developed a macroscopic model considering the linear filtration effect and derived the analytical solution for the cylindrical seepage diffusion model of slurry in porous media; in order to account for more intricate situations, a numerical solution for the seepage diffusion model was established using the finite difference method. Kim [9] conducted network analysis on porous media and conducted in-depth research on the diffusion model of sand column infiltration grouting. Wang [10] established a cylindrical diffusion model of sand layer seepage grouting by using a tortuous circular pipe as the internal seepage channel of sand layer and analyzed the attenuation law of slurry pressure and the influencing factors of diffusion radius. Zhao [11] deduced the diffusion mechanism equation of columnar seepage grouting of time-varying viscosity Newtonian fluid and analyzed the application scope of the equation. Yang [12] established a cylindrical-hemispherical penetration diffusion model and analyzed the diffusion parameters of slurry. Huang [13] revised the Maag’s spherical diffusion model by using the law of vacuum pressure distribution and predicted the law of slurry diffusion in vacuum grouting. Fang [14] analyzed the injectability of grout and established a spherical seepage diffusion model considering the seepage effect of grout based on the linear filtration effect.

Meanwhile, many scholars have also studied the influence of grouting parameters on permeation diffusion. Under the assumption that the slurry diffuses along the hemisphere, Ye [15] established a hemisphere permeation diffusion model and analyzed the effects of grouting pressure, grouting time, initial viscosity of the slurry, and soil permeability coefficient on the diffusion radius and pressure of the slurry. Wu [16] established a spherical permeation diffusion model considering the influence of soil unloading effect, analyzed the parameters of the model, and studied the permeation and diffusion characteristics of power-law fluid in unloading soil. Ding [17] established the rheological equation and seepage motion equation of Newtonian fluid with rheological parameters changing with time and analyzed the diffusion mechanism for Newtonian fluids in spherical penetration grouting.

In summary, it has become an extensive and beneficial research topic to analyze and study the diffusion mode of grout and the optimization of grouting parameters by using the permeation diffusion model. The above research has promoted the development of infiltration diffusion grouting theory. However, the model considering both spherical diffusion theory and cylindrical diffusion theory is rarely involved, especially the model considering both the filtration effect and nonlinearity. In the present paper, a macroscopic linear filtration model was introduced and the mass balance of water, cement, and soil during slurry filtration were considered. Based on the conservation of mass, a spherical (cylindrical) permeation diffusion model, which takes the linear filtration effect and the variation of slurry viscosity into account, was derived. On this basis, the impact of the filtration effect and the time-dependent changes in slurry viscosity on the diffusion of grouting slurry were studied, Additionally, a polynomial nonlinear filtration model was proposed, and numerical solutions for the permeation diffusion model were derived using finite difference and finite element methods. The rationality of the model was verified by comparing the calculation results considering the effect of slurry filtration, the calculation results without considering the effect of slurry filtration, and the test results.

2. Macroscopic Filtration Model

2.1. Macroscopic Filtration Model

From a macroscopic perspective, in the grouting process, the soil at the pile toe is regarded as a porous medium composed of the soil skeleton (soil particles) and the slurry (water and cement). The porosity of the soil per unit volume $\varphi$ is given by Equation (1):

$$\varphi ={\varphi }_w+{\varphi }_c=1-{\varphi }_s$$

(1)

where ${\varphi }_w,{\varphi }_c,{\varphi }_s$ are the volume fractions of water, cement, and soil particles per unit volume of soil, respectively.

In the context of the filtration effect, the cement filtered out is treated as soil particles, and it is assumed that there is a mass exchange between cement particles and soil particles. The mass of unfiltered cement particles in the unit volume of slurry is represented by the parameter $-\mu$, and it is directly proportional to the volume fraction function $h\left(\delta \right)$ of cement particles in the flowing slurry within the soil. The filtration phenomenon can be expressed using the Equations (2) and (3) [18]:

$$\mu ={\rho }_{c}h\left(\delta \right)$$

(2)

$$\delta ={\varphi }_c/\varphi$$

(3)

where ${\rho }_c$ represents the density of cement (kg/m³), which is a constant. $\delta$ denotes the volume fraction of cement in the flowing slurry within the soil, and its initial value can be obtained through the conversion of the water–cement ratio of the slurry.

In the equations expressing the filtration effect, $h\left(\delta \right)$ is a function of the volume fraction of cement $\delta$ in the flowing slurry within the soil, which can be determined through experiments [19]. According to the initial condition $h\left(\delta =0\right)=0$, Equation (4) can be obtained by the Taylor series expansion of $h\left(\delta \right)$ at $\delta =0$ yields.

$$h\left(\delta \right)={\displaystyle \sum _{n=0}^{\infty }\frac{h^n\left(0\right)}{n!}{\delta }^n}=h^{\prime }\left(0\right)\delta +h^{″}\left(0\right){\delta }^2+h^{‴}\left(0\right){\delta }^3\dots +h^n\left(0\right){\delta }^n$$

(4)

Since ${\delta }^3$ is much less than 1, the higher term of ${\delta }^n\left(n>3\right)$ can be ignored and the above equation simplifies to Equation (5):

$$h\left(\delta \right)=-\left(a\delta +b{\delta }^2+c{\delta }^3\right)$$

(5)

where $a,b,c$ represent the filtration coefficient (all greater than zero), and $a=-h^{\prime }\left(0\right), b=-h^{″}\left(0\right), c=-h^{‴}\left(0\right)$ can be obtained by experimental fitting. If $b=c=0$, the filtration law reduces to a linear Equation: $\frac{\mu }{{\rho }_c}=-a\delta$.

Tarafdar [20] established the linear model by Equation (6):

$$\frac{\partial \varphi }{\partial t}=-\left[v_0^{-}{\rho }_{c}F\left({\varphi }_0-\varphi \right)\right]\delta$$

(6)

The linear filtration equation assumes that the change in porosity of the soil during slurry diffusion ${\varphi }_0-\varphi$ is negligible; therefore, the influencing factor of the filtration effect on porosity $F\left({\varphi }_0-\varphi \right)$ is approximated as 1 and $a\approx v_0^{-}{\rho }_c$. In the actual grouting process, the change in the porosity of the soil cannot be ignored. The slurry undergoes a noticeable filtration phenomenon and pore clogging may even occur. Therefore, the linear filtration equation is no longer applicable.

2.2. Influence of Filtration Effect

2.2.1. Slurry Density and Viscosity

During the grouting process, after the cement particles are intercepted by the soil skeleton, the density of the slurry that can pass through the pores ${\rho }_g$ decreases, and the density of the slurry ${\rho }_g$ can be expressed as Equation (7):

$${\rho }_g={\rho }_w+\left({\rho }_c-{\rho }_w\right)\delta \left(r,t\right)$$

(7)

where ${\rho }_w$ is the density of water (kg/m³), which is a constant, and ${\eta }_g$ represents the plastic viscosity of the slurry $\left(kPa\cdot s\right)$.

Fang [14] assumed that the slurry viscosity does not vary with time, and it can be estimated by Equation (8):

$${\eta }_g={\eta }_{g0}=0.00426e^{\frac{{\delta }_0}{0.04453}}+7.006$$

(8)

In order to account for the influence of the filtration effect and time on slurry viscosity, the present paper introduces the filtration coefficient $a$ and time $t$ for exponential function correction to the above equation, and the plastic viscosity of the slurry can be expressed by Equation (9):

$${\eta }_g={\eta }_{g_0}{e}^{-at}=\left(0.00426e^{\frac{{\delta }_0}{0.04453}}+7.006\right)e^{-at}$$

(9)

2.2.2. Permeability

When the filtration effect of the slurry occurs, the volume fraction of cement in the flowing slurry within the soil decreases, and the soil porosity $\varphi$ reduces, resulting in a decrease in the soil permeability $k$; as hydraulic flow into the pores encounters resistance, even minor changes in porosity have a significant impact on permeability. A hyperbolic function was used to approximate the impact of filtration-effect-induced porosity changes on permeability [8], and the permeability coefficient of soil can be expressed by Equation (10):

$$k=\frac{k_0}{1+\lambda \left[\varphi \left(r,t\right)-{\varphi }_0\right]}$$

(10)

where ${\varphi }_0$ and $k_0$ are the initial porosity of the soil and the initial permeability of the slurry (m²), respectively, and $\lambda$ is the test parameter and takes negative values.

3. Permeation Diffusion Model Considering Filtration Effect

3.1. Modeling

The basic assumptions of the model in this paper are as follows. (1) The soil at the pile toe is assumed to be a uniform isotropic porous medium. (2) Both soil particles and cement particles are considered rigid and incompressible. (3) It is assumed that the cement slurry within the soil flows in a continuous permeation diffusion pattern, following Darcy’s law. (4) Only the slurry flow velocity in the radial direction needs to be considered. (5) During the grouting process, the dynamic dispersion effect of the slurry is neglected, assuming that the macroscopic velocity of water and cement particles is consistent, i.e., $v_w=v_c=v$.

In the grouting process, the cement slurry undergoes a filtration effect, the unfiltered cement particles can be regarded as part of the soil particles, and the new soil skeleton gradually forms, consisting of the original soil particles and the filtered cement particles. Since only the radial flow of slurry is considered, the flow rate $v$ is only related to the diffusion radius $r$ and time $t$ after entering the soil pores. According to the principle of mass conservation, differential continuity equations for water, cement particles, and soil particles can be expressed as Equations (11)–(13), respectively:

$$\frac{\partial \left({\rho }_w{\varphi }_w\right)}{\partial t}+\nabla \left({\rho }_w{\varphi }_w\overrightarrow{v}\right)=0$$

(11)

$$\frac{\partial \left({\rho }_c{\varphi }_c\right)}{\partial t}+\nabla \left({\rho }_c{\varphi }_c\overrightarrow{v}\right)=\mu$$

(12)

$$\frac{\partial \left({\rho }_s{\varphi }_s\right)}{\partial t}\approx -u$$

(13)

Assuming that the density of unfiltered soil particles is approximately equal to the density of cement, i.e., ${\rho }_s$ is approximately equal to ${\rho }_c$, the density of water ${\rho }_w$, the density of cement ${\rho }_c$ and the density of soil ${\rho }_s$ are constants independent of time. The mass balance Equations (11)–(13) can be simplified as Equations (14)–(16):

$$\frac{\partial {\varphi }_w}{\partial t}+\mathrm{div}\left({\varphi }_w\overrightarrow{v}\right)=0$$

(14)

$$\frac{\partial {\varphi }_c}{\partial t}+\mathrm{div}\left({\varphi }_c\overrightarrow{v}\right)=\frac{\mu }{{\rho }_c}=h\left(\delta \right)$$

(15)

$$\frac{\partial {\varphi }_s}{\partial t}=-h\left(\delta \right)$$

(16)

By combining Equations (14)–(16), and incorporating Equation (1) about the porosity and Equation (2) about the filtration law, the differential expression for soil porosity and the fluid differential continuity equation can be derived as Equations (17) and (18):

$$\frac{\partial \varphi }{\partial t}=\frac{\mu }{{\rho }_c}=h\left(\delta \right)$$

(17)

$$\mathrm{div}\left(\varphi \overrightarrow{v}\right)=0$$

(18)

Before reaching the boundary $r=r_0$ (the interface between the grouting pipe and pile toe soil), i.e., $r=r_0^{-}$, it is assumed that the flow rate of the slurry is $v_0^{-}$ (which can be obtained from the grouting test), the slurry flow rate at the $r=r_0^{+}$ interface is $v_0^{+}\left(r_0,t\right)$, and the volume fraction of unfiltered cement particles is ${\delta }_0$; the mass balance conditions of water and cement particles at the boundary $r=r_0$ can be expressed as Equations (19a) and (19b), and Equation (20) can be obtained by summing the two equations up.

$$\left(1-{\delta }_0\right)v_0^{-}={\varphi }_w\left(r_0,t\right)v_0^{+}\left(r_0,t\right)$$

(19a)

$${\delta }_0{v}_0^{-}={\varphi }_c\left(r_0,t\right)v_0^{+}\left(r_0,t\right)$$

(19b)

$$v_0^{+}\left(r_0,t\right)=\frac{v_0^{-}}{\varphi \left(r_0,t\right)}$$

(20)

where $r_0$ is the radius of the grouting pipe (m), and $r$ is the slurry diffusion radius (m).

Equation (21) represents the boundary condition at $r=r_0$ of the volume fraction of cement in the flowing slurry within the soil.

$$\delta \left(r=r_0,t\right)={\delta }_0$$

(21)

The initial conditions at $t=0$ in the slurry diffusion region ($r_0<r\le r_0+R$) can be expressed as Equations (22) and (23):

$$\delta \left(r_0<r\le r_0+R,t=0\right)=0$$

(22)

$$\varphi \left(r_0<r\le r_0+R,t=0\right)={\varphi }_0$$

(23)

According to Gauss’s law, by combining Equations (20)–(23), the basic equation for slurry diffusion can be presents as Equation (24), ($r_0<r\le r_0+R, t\ge 0$):

$$\varphi \left(r,t\right)v\left(r,t\right)=\frac{r_0^m}{r^m}{v}_0^{+}\left(r_0,t\right)\varphi \left(r_0,t\right)=\frac{r_0^m}{r^m}{v}_0^{-}$$

(24)

where $m$ is the slurry diffusion shape. $m=1$ indicates cylindrical diffusion and $m=2$ denotes spherical diffusion.

Assuming that the interface between the region where the diffusion of the liquid extends up to its maximum radius and the unaffected soil region is impermeable, the boundary conditions of the volume fraction of cement $\delta$ in the flowing slurry within the soil, the porosity $\varphi$ of the soil, and the slurry flow rate $v$ at $r\ge r_0+R$ can be calculated via Equations (25)–(27), respectively.

$$\delta \left(r\ge r_0+R,t\right)=0$$

(25)

$$\varphi \left(r\ge r_0+R,t\right)={\varphi }_0$$

(26)

$$v\left(r=r_0+R,t\right)=\frac{r_0^m}{r^m}\frac{v_0^{-}}{{\varphi }_0}$$

(27)

Equation (28) is the slurry diffusion model considering the filtration effect which can be obtained by combining Equations (15), (17) and (24).

$$\varphi \frac{\partial \delta }{\partial t}+\frac{r_0^m}{r^m}{v}_0^{-}\frac{\partial \delta }{\partial r}=h\left(\delta \right)\left(1-\delta \right)$$

(28)

3.2. The Relation between Slurry Diffusion Radius and Time

The grouting takes $t=0$ and $r=r_0$ as staring point, and it is assumed that the slurry diffuses outward radially with a spherical (cylindrical) surface, and the diffusion surface is an impermeable boundary. The geometric boundary on the diffusion surface can be expressed by Equation (29):

$$r=f\left(t\right),t=f^{-1}\left(r\right)$$

(29)

The diffusion radius is a function of time. Beyond the diffusion front of the slurry $r>f\left(t\right)$, the soil porosity remains at its initial value ${\varphi }_0$; the cement volume fraction is 0, and it is unaffected by the flow of the slurry. The flow rate on the diffusion front of the slurry can be represented by Equation (30):

$$\overline{v}\left(r=f\left(t\right)\right)=\dot{r}=\dot{f}\left(t\right)=\frac{1}{{\left[f^{-1}\left(r\right)\right]}^{\prime }}$$

(30)

As $v\left(r\ge f\left(t\right),t\right)=\frac{r_0^m}{r^m}\frac{v_0^{-}}{{\varphi }_0}$, and on the diffusion front, $\overline{v}\left(r=f\left(t\right)\right)=\frac{r_0^m}{r^m}\frac{v_0^{-}}{{\varphi }_0}$, the relations between $\forall t$ and $\forall r$ during the grouting process on the diffusion front can be expressed by Equations (31) and (32):

$$t=f^{-1}\left(r\right)=\frac{{\varphi }_0}{r_0^m{v}_0^{-}}{\displaystyle {\int }_{r_0}^r{r}^m}dr=\frac{{\varphi }_0{r}_0^{}}{\left(m+1\right)v_0^{-}}\left(\frac{r^{m+1}}{r_0^{m+1}}-1\right)$$

(31)

$$r=f\left(t\right)=r_0^{}\sqrt[m+1]{\frac{\left(m+1\right)v_0^{-}}{{\varphi }_0{r}_0^{}}t+1}$$

(32)

3.3. Analytical Solution for a Spherical (Cylindrical) Permeation Model Based on the Linear Filtration Law

3.3.1. Analytical Solution for Cement Volume Fraction and Porosity

To simplify the calculations, the porosity in Equation (28) is assumed to be independent of time and diffusion distance, i.e., $\varphi \left(r,t\right)={\varphi }_0$ is a constant. From Equations (20) and (24), we obtain $v_0^{+}\left(r_0,t\right)=\frac{v_0^{-}}{{\varphi }_0}$ and $v\left(r\right)=\frac{r_0^m}{r^m}\frac{v_0^{-}}{{\varphi }_0}=\frac{r_0^m}{r^m}{v}_0^{+}$. By introducing a linear filtration effect model $h\left(\delta \right)=-a\delta$, the slurry diffusion model (28) is simplified as Equation (33):

$${\varphi }_0\frac{\partial \delta }{\partial t}+v_0^{-}\frac{r_0^m}{r^m}\frac{\partial \delta }{\partial r}=-a\delta \left(1-\delta \right)$$

(33)

According to the boundary condition (21) and the initial condition (22), as well as slurry diffusion parameters (31) and (32), the analytical expression for the volume fraction $\delta$ of cement in the flowing slurry within the soil is expressed as Equation (34):

$$\delta \left(r,t\right)=\{\begin{array}{ll}\frac{1}{1+\left(\frac{1}{{\delta }_0}-1\right)\exp \left[\frac{a{r}_0^{}}{\left(m+1\right)v_0^{-}}\left(\frac{r^{m+1}}{r_0^{m+1}}-1\right)\right]}& r_0<r\le f\left(t\right)\\ 0& r>f\left(t\right)\end{array}$$

(34)

By substituting Equation (34) into Equation (17), the porosity $\varphi$ can be calculated by Equation (35):

$$\varphi \left(r,t\right)=\{\begin{array}{ll}{\varphi }_0-a\delta \left(r,t\right)\left[T-\frac{{\varphi }_0{r}_0^{}}{\left(m+1\right)v_0^{-}}\left(\frac{r^{m+1}}{r_0^{m+1}}-1\right)\right]& r_0<r\le f\left(t\right)\\ 0& r>f\left(t\right)\end{array}$$

(35)

where $T$ is the total grouting time.

By substituting Equations (34) and (35) into Equation (3), we can obtain ${\varphi }_c=\delta \varphi$. Injected cement mass per unit volume $q_c$ can be calculated by Equation (36):

$$q_c={\rho }_c{\varphi }_c+{\displaystyle \int -\mu \left(r,t\right)}dt$$

(36)

3.3.2. Analytical Solution for Pressure Gradient

Whether the slurry diffuses in a spherical or cylindrical shape, under the grouting pressure, it will overcome the resistance from the weak soil layer on the pile side and its own gravity. It will flow upwards along the pile side, displacing and reinforcing the mud layer on the pile side, thereby enhancing the strength of the soil adjacent to the pile. Due to the significantly lower cohesion of the mud skin (weak layer) on the pile side compared to the surrounding soil, the influence of the weak layer on the grouting pressure was neglected to simplify the calculations. Equation (37) can be obtained according to Darcy’s Law:

$$-\frac{\partial p}{\partial r}=\varphi \overrightarrow{v}\frac{{\eta }_g}{\kappa }+{\rho }_{g}g$$

(37)

where $p$ is the slurry flowing pressure (kPa).

Substituting the Equations (7)–(10), (34) and (35) into the above Equation (37), the analytical solution for the pressure gradient $\nabla p$ can be obtained. Through integration, the pressure can be calculated by Equation (38):

$$-\frac{\partial p}{\partial r}=\frac{{\eta }_{g_0}{e}^{-a{f}^{-1}\left(r\right)}}{\frac{{\kappa }_0}{1-\lambda a\delta \left(r,t\right)\left[T-f^{-1}\left(r\right)\right]}}\frac{r_0^m}{r^m}{v}_0^{-}+\left[{\rho }_w+\left({\rho }_c-{\rho }_w\right)\delta \left(r,t\right)\right]g$$

(38)

3.4. Numerical Solution for Infiltration Model Based on Nonlinear Filtration Law

In the previous section, to facilitate the derivation of analytical solutions, the filtration model was simplified to a linear one under the assumption of a very large water–cement ratio (very small initial cement volume fraction), which does not align with practical engineering conditions. In order to more accurately reveal the influence of the filtration effect on slurry diffusion, a general functional form Equation (5) of the filtration equation is introduced, and numerical solutions of the permeation diffusion model are derived by using semi-implicit finite difference discretization in the time domain and finite element discretization in the spatial domain.

Assuming that the porosity ${\varphi }^n$ and the cement volume fraction ${\delta }^n$ at time $t^n$ are known, the finite difference discretization in the time domain is applied to Equations (17) and (28), resulting in Equations (39) and (40) in the form of a semi-implicit time discretized for $\varphi$ and $\delta$.

$$\frac{{\varphi }^{n+1}-{\varphi }^n}{\Delta t}=h\left({\delta }^n\right)$$

(39)

$${\varphi }^{n+1}\frac{{\delta }^{n+1}-{\delta }^n}{\Delta t}+\frac{r_0^m}{r^m}{v}_0^{-}\frac{\partial {\delta }^{n+1}}{\partial r}=h\left({\delta }^n\right)\left(1-{\delta }^n\right)$$

(40)

where $\Delta t$ is time increment, and ${\varphi }^n$ and ${\delta }^n$ are the porosity and cement volume fraction at time $t^n$ in the region, respectively; $h\left({\delta }^n\right)=-\left[a{\delta }^n+b{\left({\delta }^n\right)}^2+c{\left({\delta }^n\right)}^3\right]$.

Combining Equations (38) and (39) yields the spatial differential Equation (41):

$$\begin{array}{l}\Delta t\frac{r_0^m}{r^m}{v}_0^{-}\frac{\partial {\delta }^{n+1}}{\partial r}+{\delta }^{n+1}\left[{\varphi }^n+h\left({\delta }^n\right)\Delta t\right]\\ =h\left({\delta }^n\right)\Delta t\left(1-{\delta }^n\right)+{\delta }^n\left[{\varphi }^n+h\left({\delta }^n\right)\Delta t\right]\end{array}$$

(41)

Using the finite element method, ${\delta }^{n+1}$ is obtained. The diffusion region of the slurry is divided into $N_e$ three-node elements with $N_n$ nodes. The interval of the $i$ element is ${\mathsf{\Omega }}_i=\left[r_0+\frac{R}{N_e}\left(i-1\right),r_0+\frac{R}{N_e}i\right]$, $\left(i=1,2,\cdots ,N_e\right)$, as shown in Figure 1.

At a certain spatial domain, ${\mathsf{\Omega }}_i$, $U_j$ is the vector matrix of the cement volume fraction ${\delta }^{n+1}$ of a node, and the relationship between them can be expressed by Equations (42) and (43).

$${\delta }^{n+1}=N{\left(r\right)}^{\mathrm{T}}\cdot U_j$$

(42)

$$\frac{\partial {\delta }^{n+1}}{\partial r}\left(r\right)=B{\left(r\right)}^{\mathrm{T}}\cdot U_j$$

(43)

where $N$ and $B$ are vectors of difference functions with respect to the diffusion radius.

The overall nodal force matrix can be represented by Equation (44):

$$F=K\cdot U$$

(44)

where $U={\left({\delta }_j^{n+1}\right)}_{1\le j\le N_n}$ is the overall cement volume fraction at $t^{n+1}$, $K={\displaystyle \sum _{i=1}^{N_e}{K}_i}$ is the overall stiffness matrix, and $F={\displaystyle \sum _{i=1}^{N_e}{F}_i}$ is the overall nodal force matrix at $t^{n+1}$.

The single node stiffness matrix can be calculated by Equation (45), and substituting it into Equation (46), the global nodal force matrix is obtained.

$$K_i=\Delta t{r}_0^m{v}_0^{-}{\displaystyle {\int }_{{\mathsf{\Omega }}_{{}_i}}N\cdot B^{\mathrm{T}}dr+{\displaystyle {\int }_{{\mathsf{\Omega }}_{{}_i}}\left[{\varphi }^n+h\left({\delta }^n\right)\Delta t\right]N\cdot N^{\mathrm{T}}{r}^{m}dr}}$$

(45)

$$F={\displaystyle {\int }_{{\mathsf{\Omega }}_{{}_i}}\left\{h\left({\delta }^n\right)\Delta t\left(1-{\delta }^n\right)+{\delta }^n\left[{\varphi }^n+h\left({\delta }^n\right)\Delta t\right]\right\}N{r}_{}^{m}dr}$$

(46)

Based on the boundary condition Equation (21) at $r=r_0$ and initial conditions Equations (22) and (23) at $t=0$, we obtain ${\varphi }^0={\varphi }_0$ and ${\delta }^0=0$. The numerical process can be realized through iterative time-stepping.

4. Engineering Application

4.1. Project Overview

To verify the validity and feasibility of the derived spherical (cylindrical) permeation diffusion model in this paper, with the engineering case in the previous study [14] as an example, the variation of grouting pressure with grouting time, as well as the diffusion and distribution of slurry are analyzed. A comparison is made between the results obtained from the proposed model and traditional calculations. The project overview is as follows: The test site is located in Hangzhou. The pile toe is situated in a cobblestone layer (including some sandy soil). According to the ratio $N=15.4$ between the particle size $D_{15}=0.6 \mathrm{m}\mathrm{m}$ accounting for 15% of the grading curve of this soil layer and the particle size $d_{85}=0.039 \mathrm{m}\mathrm{m}$ accounting for 85% of the cement particle distribution curve, it can be inferred that this soil layer has poor injectability and significant filtration effect [21]. The values of the initial grouting parameters and related parameters in the slurry diffusion model are shown in

Table 1. Values of initial grouting parameters and diffusion model parameters.

Parameter	Initial Water–Cement Ratio	Initial Cement Volume Fraction ${\mathit{\delta }}_0$	Initial Grouting Radius ${\mathit{r}}_0/\mathit{m}$	Initial Grouting Rate ${\mathit{v}}_0/\mathit{m}\cdot {\mathit{s}}^{-1}$	Initial Porosity ${\mathit{\varphi }}_0$	Initial Permeability $\mathbf{Coefficient}{\mathit{k}}_0/\mathit{m}\cdot {\mathit{s}}^{-1}$	Filtration Coefficient $\mathit{a}/{\mathit{s}}^{-1}$	Permeability Coefficient Test Parameter $\mathit{\lambda }$
Value	0.7	0.32	0.15	0.015	0.385	0.0005	0.00029	−34

. The filtration coefficient and permeability test parameters of the soil were derived from the indoor unidirectional grouting test on the in-situ soil samples.

4.2. Result Analysis

The grouting pressures distributed along the diffusion radius at different grouting time obtained from the computational model in this paper were compared with the calculated and measured results from the previous study, as shown in Figure 2. Firstly, from the measured results, it can be observed that the pressure of the slurry at the pile toe has a tendency to decay along the diffusion radius, which indicates the necessity of considering the filtration effect in the slurry permeation process. Secondly, the grouting pressure gradually decreases along the diffusion radius, and when the slurry diffuses to a certain position, the grouting pressure no longer decreases; under the same grouting rate and grouting pressure, the pressure decay rate of the spherical permeation diffusion model is faster than that of the cylindrical model, while the cylindrical model has a larger slurry diffusion radius.

The results of numerical simulation without considering the effect of the filtration and the measured data are shown in

Table 2. Comparison of calculated and measured data without considering the effect of the filtration (Unit: KPa).

Diffusion Radius (m)	t = 60 s			t = 300 s			t = 600 s			t = 1200 s
	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data
0.2	932	932	1195	932	932	1428	932	932	1826	932	932	2000
0.5	735	556	—	735	556	—	735	556	—	735	556	—
1.2	531	422	417	531	422	417	531	422	417	531	422	417
2.0	462	—	—	462	—	—	462	—	—	462	—	—
2.5	443	—	—	443	—	—	443	—	—	443	—	—

. As can been observed, when the slurry filtration effect is not considered, the slurry pressure is independent of the grouting time and is not consistent with the measured value.

Table 3. Comparison of grouting pressure distribution of two models (Unit: KPa).

Diffusion Radius (m)	t = 60 s				t = 300 s				t = 600 s				t = 1200 s
	No Filtering Effect		Consider Filtering Effect		No Filtering Effect		Consider Filtering Effect		No Filtering Effect		Consider Filtering Effect		No Filtering Effect		Consider Filtering Effect
	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2	m = 1	m = 2
0.2	932	932	1123	939	932	932	1425	1358	932	932	2080	1843	932	932	2860	2786
0.5	735	556	596	439	735	556	750	542	735	556	935	682	735	556	1187	765
1.2	531	422	438	419	531	422	441	420	531	422	486	422	531	422	549	425
2.0	462	—	412	—	462	—	425	—	462	—	443	—	462	—	468	—
2.5	443	—	1123	939	443	—	—	—	443	—	—	—	443	—	—	—

shows the comparison of grouting pressure distribution calculated by two models with and without considering the filtration effect. It can be observed that when the filtration effect is considered, the initial grouting pressure required is greater than that without considering the filtration effect over time, regardless of whether the grouting infiltration diffusion is in a spherical or cylindrical manner. However, the attenuation rate of grout pressure along the diffusion radius when considering the filtration effect is much greater than that without considering the filtration effect, and the attenuation amplitude is more and more obvious with the extension of grouting time. This is because the soil pores at the pile tip are continuously filled with slurry during the grouting process, and its diffusion radius gradually increases. The slurry filtration effect becomes more and more obvious at a certain diffusion radius.

Table 4. Comparison between the calculation data of the proposed model and the literature model. (Unit: KPa).

Diffusion Radius (m)	t = 60 s			t = 300 s			t = 600 s			t = 1200 s
	m = 1	m = 2	Literature (m = 2)	m = 1	m = 2	Literature (m = 2)	m = 1	m = 2	Literature (m = 2)	m = 1	m = 2	Literature (m = 2)
0.2	1123	939	938	1425	1358	1302	2080	1843	1819	2860	2786	2636
0.5	596	439	430	750	542	500	935	682	609	1187	765	750
1.2	438	419	419	441	420	420	486	422	422	549	425	426
2.0	412	—	412	425	—	—	443	—	—	468	—	—
2.5	1123	939	—	—	—	—	—	—	—	—	—	—

shows that the initial grouting pressure calculated by the spherical permeation diffusion model in this paper is slightly larger than the calculation results in the previous study. The reason is that the present research considers the grouting pressure required to overcome the self-weight of the slurry.

As listed in

Table 5. Comparison of calculated and measured data by proposed model (Unit: KPa).

Diffusion Radius (m)	t = 60 s			t = 300 s			t = 600 s			t = 1200 s
	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data	m = 1	m = 2	Measured Data
0.2	1123	939	1195	1425	1358	1428	2080	1843	1826	2860	2786	2000
0.5	596	439	—	750	542	—	935	682	—	1187	765	—
1.2	438	419	417	441	420	417	486	422	417	549	425	417
2.0	412	—	—	425	—	—	443	—	—	468	—	—
2.5	1123	939	—	—	—	—	—	—	—	—	—	—

, when the grouting time is not more than 600 s, the grouting pressure at the grouting hole calculated by the cylindrical seepage diffusion model is closer to the measured value. When the grouting time t is greater than 600 s, the deviation between the theoretical and measured grouting pressure values calculated by the two models increases with the increase of grouting time. On one hand, the reason is that the filtration coefficient and permeability test parameters in the calculation model were determined based on the unidirectional grouting test of the undisturbed soil indoors. Soil disturbance, test equipment, and test conditions may all lead to deviation of parameter values. On the other hand, in order to facilitate the measurement of grouting pressure at the pile end, the pressure gauge is buried at the center of the pile end section rather than at the grouting hole. Compared with the actual grouting pressure at the grouting hole, the measured grouting pressure is smaller, and this deviation becomes more and more obvious with the increase of grouting time. However, the model in this article can already consider the influence of the filtration effect well and predict the trend of grouting pressure in the soil.

5. Conclusions

This study investigates the impact of the filtration effect and the time-dependent changes in slurry viscosity on the diffusion of grouting slurry. Based on the slurry injectability standards, the research considers the mass balance of water, cement, and soil particles during the slurry filtration process. The study introduces a macroscopic linear filtration model, accounting for variations in slurry properties during the permeation diffusion process. A spherical (cylindrical) permeation diffusion model that incorporates linear filtration effects and the self-weight of the slurry is derived, and the analytical solutions of cement volume fraction, porosity and pressure gradient are given. Additionally, a polynomial nonlinear filtration model is proposed. The study also derives numerical solutions for the permeation diffusion models using finite difference and finite element methods. With the engineering case in a previous study as an example, this study examines variations in grouting pressure and the diffusion and distribution of slurry under different grouting durations. The effectiveness of the model proposed in the present paper is confirmed through a comparison between the theoretical model results and experimental data. The results demonstrate that the grout pressure is inconsistent with the measured value without considering the effect of the filtration. The attenuation rate of slurry pressure along the diffusion radius when considering the filtration effect is much greater than that without considering the filtration effect, and the attenuation amplitude becomes more and more obvious with the extension of grouting time. The required grouting pressure over time is greater than that without considering the filtration effect regardless of whether the grout diffuses in a spherical or cylindrical manner. The pressure decay rate of the spherical seepage diffusion model is faster than that of the cylindrical model under the same grouting rate and grouting pressure, while the corresponding cylindrical model has a larger diffusion radius of the slurry. The initial grouting pressure calculated by the model in this paper is slightly larger than the calculation results of the linear spherical permeation diffusion model; the reason is that the present research considers the grouting pressure required to overcome the self-weight of the slurry, which is also more in line with the actual situation. However, the applicability of this model to the gravel layer is proved by an engineering example, and the applicability to other formations will be the focus of our research work.