Penetration Grouting Mechanism of Bingham Fluid in Porous Media Based on Fractal Theory

Abstract

Penetration grouting is a significant grouting technique. The pore structure has important impacts on the infiltration mechanism of slurry in porous media. In this study, based on fractal theory, a theoretical penetration grouting model for Bingham fluid is established. An experimental apparatus for simulating the penetration process of Bingham fluid with a constant flow rate is developed. A series of penetration-grouting experiments are conducted to validate the theoretical model established in this study and analyze the impacts of the water–cement ratio and flow rate on the slurry injection pressure. The results show that the theoretical values of the slurry pressure along the penetration direction obtained from the penetration grouting model match the experimental values well. This indicates that the proposed model can better describe the process of slurry infiltration and provide valuable support for related grouting projects.

1. Introduction

Permeation grouting is one of the most important techniques used in grouting reinforcement engineering. It is widely applied in various fields, such as hydraulic engineering and highway construction, due to its limited disturbance to the grouted medium [1]. Extensive studies on the permeation-grouting theory have been conducted, including the slurry permeation and diffusion models for Newtonian fluids, Bingham fluids, and power-law fluids. Maag formulated the classical theoretical model for Newtonian fluid permeation grouting [1]. Yang et al. [2,3] investigated the permeation and diffusion mechanisms of Bingham fluids and power-law fluids in sand layers.

On the basis of these foundational studies, scholars have further extended the permeation grouting theory by considering varies factors such as time-dependent and spatial variations in slurry viscosity, seepage effects during the diffusion process, and the tortuosity effects in porous media. With respect to the time-dependent and spatial variability effects of slurry viscosity, Ruan [4] established time-dependent viscosity diffusion equations for cement-based and solution-based grouting materials. Yang et al. [5,6] investigated the permeation and diffusion models of Bingham fluids, considering time-dependent viscosity for spherical, cylindrical, and column-semi-spherical geometries. The cylindrical permeation grouting mechanisms of power-law cement slurries with time-varying rheological parameters was also explored [7]. Building on Maag’s classical theory, Zhou [8] optimized Maag’s formula by substituting the average viscosity with time-dependent viscosity. Additionally, Zhang et al. [9] analyzed the permeation and diffusion mechanisms of rapidly setting grouts, by considering the time–space variations in slurry viscosity. With respect to the seepage effect on the slurry diffusion process, Saada et al. [10,11,12,13] carried out laboratory experiments to examine the impact of density, consolidation stress, cement concentration, and grouting flow rate on the seepage effects of cement slurries. Feng et al. [14] proposed a theoretical model for calculating the diffusion distance of particulate slurries under laminar flow conditions. Wang et al. [15] discussed the impact of seepage and proposed an expression for the spatial distribution of viscosity-decay rate. Li et al. [16] analyzed the impact of slurry seepage on the porosity and permeability of sandy media during the cement slurry permeation process. With respect to tortuosity effects, Zhang et al. [17], Lu et al. [18], and Yang et al. [19] investigated the permeation and diffusion mechanisms for Newtonian fluids, Bingham fluids, and power-law fluids, through considering the diffusion pathways. Wang et al. [20] investigated the permeation grouting mechanism of viscous time-varying fluids, considering the diffusion path.

Fractal theory is a crucial tool for describing the pore characteristics of porous media, which makes it powerful in representing the influence of microscopic structural parameters of pores on slurry diffusion during the permeation grouting process. Yu et al. [21] provided a comprehensive discussion on the application of fractal theory in describing transport properties in porous media. Yun et al. [22] developed a model to calculate the initiation pressure gradient for Bingham fluid flow, considering the fractal characteristics of porous media and capillary pressure. Zhang et al. [23] derived a permeability calculation formula for power-law fluids, incorporating the fractal features of pore size distribution and the tortuous flow paths within the porous medium.

In summary, significant research on the permeation and diffusion mechanisms of Bingham fluids has been conducted. Fractal theory has also been utilized to characterize the tortuous flow paths of slurries. However, existing theoretical models for the permeation and diffusion of Bingham fluids still can be further enhanced to reflect the complex pore characteristics of porous media and their impact on slurry diffusion. Additionally, the parameters involved in fractal-based slurry permeation models are complex and difficult to determine from conventional geotechnical tests. This restricts the experimental validation of the theoretical models, thereby constraining their application.

This study aims to propose a theoretical model of permeation grouting for Bingham fluids based on fractal theory and validate the model by experiments. Pure cement slurry, which is widely used in grouting engineering practices, is adopted as the Bingham fluid [24] in the experiments. This work is organized as follows: Firstly, a theoretical model for the one-dimensional steady-flow permeation grouting of Bingham fluids is established based on fractal theory. Subsequently, slurry permeation and diffusion experiments are conducted with varying water–cement ratios and slurry flow rates. Finally, the theoretical and experimental results are compared to validate the applicability of the theoretical model. The spatial and temporal variations in grouting pressure under different water–cement ratios and grouting rates are also analyzed.

2. Theoretical Model of Permeation Grouting for Bingham Fluid Based on Fractal Theory

2.1. Assumptions

The assumptions of the permeation grouting theory model include the following:

• The slurry is a homogeneous, incompressible Bingham fluid;
• The viscosity variation of slurry is not considered;
• The flow regime of slurry during diffusion is laminar, ignoring the effects of gravity and filtration;
• The pore radius in the porous medium satisfies $r_{min}/r_{max}<{10}^{-2}$.

2.2. The Diffusion Model of Bingham Fluid in a Circular Pipe

The rheological equation for Bingham fluid is as follows [25]:

$$\begin{array}{c}\tau ={\tau }_0+{\mu }_p\gamma \end{array}$$

(1)

where $\tau$ is the shear stress, ${\tau }_0$ is the yield stress, ${\mu }_p$ is the plastic viscosity, and $\gamma$ is the shear rate ( $\gamma =-dv/dr$, where $v$ is the fluid velocity and $r$ is the radius of the circular pipe).

In a complex morphology of pore structures in porous media, if the tortuous path of the slurry flow within the porous medium is not considered, it is often assumed that the flow channel can be represented as a long straight circular pipe. The flow of slurry in the porous medium is therefore considered as the superposition of the flow through all these channels [25]. The flow of the Bingham fluid in a single circular pipe is illustrated in Figure 1. The radius of the circular pipe is $r_0$. The radius of a small slurry element along the axis of the pipe is ${r<r}_0$, while the length is $dl$. The pressure at the two ends of the element is $p$ and $p+dp$, respectively, with a shear stress $\tau$ acting opposite to the direction of the flow velocity.

The force balance equation for the slurry element can be written as follows:

$$\pi r^{2}dp+2\pi r\tau dl=0$$

(2)

The shear stress distribution is as follows:

$$\tau =-{\displaystyle \frac{r}{2}}{\displaystyle \frac{dp}{dl}}$$

(3)

Around the axis of the pipe, the fluid experiences a lower shear stress. When $\tau \le {\tau }_0$, there is no relative motion between fluid particles, leading to the existence of a central stagnant core region $0\le r\le r_p$. In this region, fluid velocities are uniform and equal to $v_p$. For $r_p\le r\le r_0$, fluid particles are in a state of shear motion.

By substituting $r=r_p$ and $\tau ={\tau }_0$ into Equation (3), the following is obtained:

$$r_p=2{\tau }_0/\left(-{\displaystyle \frac{dp}{dl}}\right)$$

(4)

By substituting Equation (3) into Equation (1) and simplifying, the following is obtained:

$${\displaystyle \frac{dv}{dr}}={\displaystyle \frac{1}{{\mu }_p}}\left({\displaystyle \frac{r}{2}}{\displaystyle \frac{dp}{dl}}+{\tau }_0\right)$$

(5)

By integrating Equation (5) and applying the boundary condition $r=r_0$, $v=0$, the velocity distribution of the Bingham fluid in a circular pipe can be described as follows:

$$v={\displaystyle \frac{1}{{\mu }_p}}\left[-\left({\displaystyle \frac{dp}{4dl}}\right)\left(r_0^2-r^2\right)-{\tau }_0\left(r_0-r\right)\right] \left(r_p\le r\le r_0\right)$$

(6)

and

$$v_p={\displaystyle \frac{1}{{\mu }_p}}\left[-\left({\displaystyle \frac{dp}{4dl}}\right)\left(r_0^2-r_p^2\right)-{\tau }_0\left(r_0-r_p\right)\right] \left(0\le r\le r_p\right)$$

(7)

In a single circular pipe, the flow rate $q$ is the sum of the flow rates in the shear zone $\left(r_p\le r\le r_0\right)$ and the core region $\left(0\le r\le r_p\right)$:

$$q=\pi r_p^2{v}_p+{\int }_{r_p}^{r_0}2\pi rvdr$$

(8)

Equations (4)–(7) are then incorporated into Equation (8) to obtain the total flow rate $q$ for the Bingham fluid in a circular pipe.

$$q={\displaystyle \frac{\pi r_0^4}{8{\mu }_p}}\left(-{\displaystyle \frac{dp}{dl}}\right)\left[1-{\displaystyle \frac{4}{3}}\left({\displaystyle \frac{\frac{2{\tau }_0}{r_0}}{-\frac{dp}{dl}}}\right)+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\frac{2{\tau }_0}{r_0}}{-\frac{dp}{dl}}}\right)}^4\right]$$

(9)

In order to determine the condition that the flow rate $q$ in the pipe is zero, Equation (9) is solved:

$$-{\displaystyle \frac{dp}{dl}}={\displaystyle \frac{2{\tau }_0}{r_0}}=\delta$$

(10)

where $\delta$ represents the yield pressure gradient of the Bingham fluid.

Since the pressure gradient $-dp/dl$ is significantly greater than the yield pressure gradient $\delta$ during the grouting process, Equation (9) can be simplified as follows:

$$q={\displaystyle \frac{\pi r_0^4}{8{\mu }_p}}\left(-{\displaystyle \frac{dp}{dl}}\right)\left[1-{\displaystyle \frac{4}{3}}\left({\displaystyle \frac{\frac{2{\tau }_0}{r_0}}{-\frac{dp}{dl}}}\right)\right]$$

(11)

The above equation describes the flow rate of the Bingham fluid in a single conduit, assuming that the fluid flow path in the porous medium can be approximated as a long and straight circular pipe.

2.3. Fractal Theory-Based Diffusion Model of Bingham Fluid Penetration Grouting

However, due to the complex pore structure of the porous medium, the actual seepage pathway of the Bingham fluid is tortuous, as illustrated in Figure 2.

Porous media pores exhibit fractal characteristics [22], and the pore size distribution conforms to fractal scaling relationships:

$$N\left(\ge r_0\right)={\left({\displaystyle \frac{r_{max}}{r_0}}\right)}^{D_f}$$

(12)

where $N$ is the number of pores; $r_0$ is the pore radius; $r_{max}$ is the maximum pore radius; and $D_f$ is the pore fractal dimension, $1<D_f<2$.

The length of curved pore channels $l_t$ can be expressed in fractal theory [22]:

$$l_t=l_0^{D_T}{\left(2r_0\right)}^{1-D_T}$$

(13)

where $l_t$ is the curved length of the channel; $l_0$ is the straight length of the channel; and $D_T$ is the tortuosity fractal dimension, $1<D_T<2$. When $D_T=1$, it indicates that the flow channel is straight.

By substituting $l_t$, which represents the straight pore channel in Equation (11), with the actual curved pore channel $l$ represented by the pore fractal dimension, the following is obtained:

$$q={\displaystyle \frac{\pi r_0^4}{8{\mu }_p}}\left(-{\displaystyle \frac{dp}{d{l}_0}}\right)\left[{\displaystyle \frac{l_0^{1-D_T}{\left(2r_0\right)}^{D_T-1}}{D_T}}-{\displaystyle \frac{4}{3}}{\displaystyle \frac{\frac{2{\tau }_0}{r_0}}{\left(-\frac{dp}{d{l}_0}\right)}}\right]$$

(14)

The above equation represents the flow rate of the Bingham fluid flowing through a pore with radius $r_0$. Therefore, the flow rate $Q$ through the entire cross-section of the porous medium can be expressed as follows:

$$Q=-{\int }_{r_{min}}^{r_{max}}qdN$$

(15)

where $Q$ is the total flow rate through the porous medium; $r_{min}$ is the minimum pore radius; and $r_{max}$ is the maximum pore radius.

Substituting Equation (14) into Equation (15) and integrating yields the following:

$$Q={\displaystyle \frac{\pi D_f}{8{\mu }_p}}\left(-{\displaystyle \frac{dp}{d{l}_0}}\right)\left\{{\displaystyle \frac{l_0^{1-D_T}{2}^{D_T-1}{r}_{max}^{D_T+3}}{D_T\left(D_T-D_f+3\right)}}\left[1-{\left({\displaystyle \frac{r_{min}}{r_{max}}}\right)}^{D_T-D_f+3}\right]-{\displaystyle \frac{4r_{max}^3}{3\left(3-D_f\right)}}{\displaystyle \frac{2{\tau }_0}{\left(-\frac{dp}{d{l}_0}\right)}}\left[1-{\left({\displaystyle \frac{r_{min}}{r_{max}}}\right)}^{3-D_f}\right]\right\}$$

(16)

Generally, in porous media, $r_{min}/r_{max}<{10}^{-2}$. It is considered that ${1<D}_T<2$ and ${1<D}_f<2$ [22]; therefore, ${\left(r_{min}/r_{max}\right)}^{D_T-D_f+3}\ll 1$ and ${\left(r_{min}/r_{max}\right)}^{3-D_f}\ll 1$.

Thus, neglecting higher-order terms, the expression for the flow rate of the Bingham fluid in porous media can be simplified as follows:

$$Q={\displaystyle \frac{\pi D_f{r}_{max}^{3+D_T}}{{\mu }_p{l}_0^{D_T-1}{2}^{{4-D}_T}{D}_T\left(D_T-D_f+3\right)}}\left[\left(-{\displaystyle \frac{dp}{d{l}_0}}\right)-{\displaystyle \frac{{l_0^{D_T-1}\tau }_0{D}_T\left(D_T-D_f+3\right)}{3\left(3-D_f\right)2^{D_T-4}{r}_{max}^{D_T}}}\right]$$

(17)

According to fractal theory, the cross-sectional area of a unit cell can be represented as follows [21]:

$$S={\displaystyle \frac{\pi D_f{r_{max}}^2\left(1-\phi \right)}{\left(2-D_f\right)\phi }}$$

(18)

where $S$ is the cross-sectional area of the unit cell and $\phi$ is the areal porosity.

The average flow velocity of Bingham fluid can then be expressed as follows:

$$\overline{v}={\displaystyle \frac{Q}{S}}={\displaystyle \frac{r_{max}^{1+D_T}\left(2-D_f\right)\phi }{{\mu }_p{l}_0^{D_T-1}{2}^{{4-D}_T}{D}_T\left(D_T-D_f+3\right)\left(1-\phi \right)}}\left[\left(-{\displaystyle \frac{dp}{d{l}_0}}\right)-{\displaystyle \frac{{l_0^{D_T-1}\tau }_0{D}_T\left(D_T-D_f+3\right)}{3\left(3-D_f\right)2^{D_T-4}{r}_{max}^{D_T}}}\right]$$

(19)

where $\overline{v}$ is the average flow velocity on the cross-section of the unit cell.

In porous media, the Darcy seepage velocity $V$ at any point in the unit cell satisfies the following relationship with the average flow velocity $\overline{v}$ [25]:

$$V=\varphi \overline{v}$$

(20)

where $V$ is the Darcy seepage velocity and $\varphi$ is the volumetric porosity. The relationship between volumetric porosity $\varphi$ and areal porosity $\phi$ can be expressed as follows [26]:

$$\varphi =\Gamma \phi$$

(21)

where $\Gamma$ is the tortuosity, $\Gamma =1$ indicates that the flow channel is straight, and the areal porosity and volumetric porosity are equal.

According to the law of mass conservation, the relationship between the grouting flow rate $q$, seepage velocity $V$, and the cross-sectional area $A$ for fluid diffusion can be expressed as follows:

$$q=VA$$

(22)

where $q$ is the grouting flow rate and $A$ is the cross-sectional area for fluid diffusion.

By combining Equations (19)–(22) and simplifying, the following is obtained:

$$dp=-{\displaystyle \frac{D_T\left(D_T-D_f+3\right)}{l_0^{1-D_T}{2}^{D_T-4}{r}_{max}^{D_T}}}\left[{\displaystyle \frac{{\mu }_p\left(\mathsf{\Gamma }-\varphi \right)q}{{Ar}_{max}\left(2-D_f\right){\varphi }^2}}+{\displaystyle \frac{{\tau }_0}{3\left(3-D_f\right)}}\right]d{l}_0$$

(23)

Therefore, the relationship between the grouting pressure $p_{\mathrm{c}}$ and the grout diffusion distance $l_{\mathrm{m}}$ can be expressed as follows:

$$p_c={\int }_0^{l_{\mathrm{m}}}\left\{-{\displaystyle \frac{D_T\left(D_T-D_f+3\right)}{l_0^{1-D_T}{2}^{D_T-4}{r}_{max}^{D_T}}}\left[{\displaystyle \frac{{\mu }_p\left(\mathsf{\Gamma }-\varphi \right)q}{{Ar}_{max}\left(2-D_f\right){\varphi }^2}}+{\displaystyle \frac{{\tau }_0}{3\left(3-D_f\right)}}\right]\right\}d{l}_0+p_0$$

(24)

In the case of $l_{\mathrm{m}}=0$, the grout diffusion distance is zero and the grout pressure $p_c=p_0$, where $p_0$ is the initial grouting pressure.

Under condition of a steady-state Bingham fluid (slurry) flow, the relationship between the grout pressure and fluid diffusion distance can be expressed as follows:

$$p_c=-{\displaystyle \frac{\left(D_T-D_f+3\right)}{2^{D_T-4}{r}_{max}^{D_T}}}\left[{\displaystyle \frac{{\mu }_p\left(\mathsf{\Gamma }-\varphi \right)q}{{Ar}_{max}\left(2-D_f\right){\varphi }^2}}+{\displaystyle \frac{{\tau }_0}{3\left(3-D_f\right)}}\right]l_{\mathrm{m}}^{D_T}+p_0$$

(25)

2.4. Parameter Values

In the relationship between the grout pressure and diffusion distance represented by Equation (25), the slurry viscosity ${\mu }_p$ and yield stress ${\tau }_0$ can be measured using a viscometer. The slurry flow rate $q$, fluid diffusion cross-sectional area $A$, grout diffusion distance $l_{\mathrm{m}}$, and the volumetric porosity $\varphi$ of the porous medium can be obtained through experiments.

The tortuosity $\Gamma$ can be obtained from the volumetric porosity $\varphi$ through the following expression [26]:

$$\mathsf{\Gamma }=1-{\displaystyle \frac{\varphi }{2}}+{\displaystyle \frac{\sqrt{1-\varphi }}{4}}+{\displaystyle \frac{\left(\varphi +1+\sqrt{1-\varphi }\right)\sqrt{9-5\varphi -8\sqrt{1-\varphi }}}{8\varphi }}$$

(26)

The fractal dimension of porosity $D_f$ can be represented by the maximum and minimum pore radii of the porous medium and the areal porosity $\phi$ according to the following relationship [21]:

$$D_f=2-{\displaystyle \frac{\mathit{ln}\phi }{\mathit{ln}\left(\frac{r_{min}}{r_{max}}\right)}}$$

(27)

In Equation (27), the areal porosity $\phi$ can be derived from the measured volumetric porosity $\varphi$ using Equation (21). The ratio of the minimum to maximum pore radii is related to the areal porosity $\phi$ according to the following relationship [21]:

$${\displaystyle \frac{r_{min}}{r_{max}}}={\displaystyle \frac{\sqrt{2}}{d^{+}}}\sqrt{{\displaystyle \frac{1-\phi }{1-0.342\phi }}}$$

(28)

where $d^{+}$ is typically set to 24. Therefore, the fractal dimension of porosity $D_f$ can be calculated from the volumetric porosity $\varphi$.

The maximum pore radius $r_{max}$ is related to tortuosity $\mathsf{\Gamma }$, the fractal dimension of porosity $D_f$, the areal porosity $\phi$, and the permeability $k$ according to the following relationship [26]:

$$r_{max}=\sqrt{{\displaystyle \frac{8k\mathsf{\Gamma }\left(4-D_f\right)\left(1-\phi \right)}{\left(2-D_f\right)\phi }}}$$

(29)

where $k$ is the permeability of the porous medium, which can be measured experimentally. Therefore, the maximum pore radius $r_{max}$ can be calculated from the volumetric porosity $\varphi$ and the permeability $k$.

The tortuosity fractal dimension $D_T$ can be expressed in terms of the porosity fractal dimension $D_f$, the maximum and minimum pore radii, and the areal porosity $\phi$ as follows [21]:

$$D_T=1+ln{\displaystyle \frac{1}{2}}\left[1+{\displaystyle \frac{1}{2}}\sqrt{1-\phi }+\sqrt{1-\phi }{\displaystyle \frac{\sqrt{{\left(\frac{1}{\sqrt{1-\phi }}-1\right)}^2+\frac{1}{4}}}{1-\sqrt{1-\phi }}}\right]/\mathit{ln}\left({\displaystyle \frac{\frac{D_f-1}{{D_f}^{\frac{1}{2}}}{\left[\frac{1-\phi }{\phi }\frac{\pi }{4\left(2-D_f\right)}\right]}^{\frac{1}{2}}}{\frac{r_{min}}{r_{max}}}}\right)$$

(30)

The tortuosity fractal dimension $D_T$ can ultimately be transformed into an expression involving the volumetric porosity $\varphi$.

To summarize, the theoretical model for the permeation and grouting diffusion of the Bingham fluid based on fractal theory, developed in this study, relies on the following parameters: slurry viscosity ${\mu }_p$ and yield stress ${\tau }_0$; volumetric porosity $\varphi$ and permeability $k$ of the porous medium; slurry flow rate $q$; fluid diffusion cross-sectional area $A$; and grout diffusion distance $l_{\mathrm{m}}$. All these parameters can be determined from conventional geotechnical tests.

2.5. Scope of Application

Equation (25) is proposed based on the assumption of laminar flow; hence, it is not applicable for turbulent flow. The permeation and diffusion of the Bingham fluid in porous media can be determined based on its generalized Reynolds number $Re$. When $Re\le 2000$, the slurry diffuses in a laminar flow regime. When $Re>2000$, the slurry diffuses in a turbulent flow regime.

The Reynolds number $Re$ can be determined using the following formula:

$$Re={\displaystyle \frac{\rho \overline{v}D}{{\mu }_p}}$$

(31)

where $\rho$ is the density of the Bingham fluid and $D$ is the diameter of the pore channel in the porous medium.

3. Experiments of Permeation Grouting in Porous Media

To analyze the applicability and accuracy of the fractal-based theoretical model for Bingham fluid permeation grouting, a series of permeation grouting experiments are conducted. These experiments facilitate a comparative analysis between theoretical predictions and experimental results.

3.1. Experimental Apparatus

In order to study the diffusion patterns of the Bingham fluid with varying injection pressures and different fluid properties, a one-dimensional visual permeation grouting simulation apparatus is designed (Figure 3).

The grouting column model employs Polymethyl Methacrylate (PMMA), composed of four sections of identical dimensions, which enables the slurry flow to be visualized. Each cylindrical tube section measures $\mathsf{\Phi }$100 mm $\times$ 200 mm, with the total length of the apparatus reaching 80 cm. The bottom is equipped with adjustable supports. The tubes are filled with sand (the injected medium). The slurry is injected from the bottom and disperses upwards along the Polymethyl Methacrylate (PMMA). The inlet and outlet openings are covered with filter screens to prevent structural damage to the samples.

The slurry pressure measurements are conducted using a DM-YB1820 (Nanjing Danmo Electronic Technology Co., Ltd, Nanjing, China) static resistance strain gauge connected to infiltration pressure sensors. The pressure sensors are positioned at distances of 0, 10, 20, 30, 40, 50, 62.5, and 77.5 cm from the slurry inlet. The sensors located from 0 to 50 cm have a measurement range of 200 kPa, while those at 62.5 cm and 77.5 cm have a measurement range of 100 kPa, all with an accuracy of 0.3%.

3.2. Experimental Design

P.O 42.5 Portland cement is employed as the grouting material. During the experiment, the properties of the cement slurry are altered by adjusting the water–cement ratio (w/c ratio). The w/c ratios tested are 0.80, 1.00, and 1.25. The rheological equations for the cement slurry at different w/c ratios are shown in

Table 1. Rheological equations of three cement grout.

Water–Cement Ratio	Rheological Equation
0.80	$\tau =2.2078+0.0201\gamma$
1.00	$\tau =0.8593+0.0169\gamma$
1.25	$\tau =0.1136+0.0159\gamma$

.

The grouting medium used in the experiment is Chinese ISO standard sand with particle sizes ranging from 1 to 2 mm. The sand samples are initially in a dry state. Before each test, the sand samples are cleaned and dried. Based on the specified porosity, the required mass of sand samples is calculated and filled accordingly. The sand is filled in increments, with each filling step involving leveling the sand and assessing the filled height against the scale values on the model wall to ensure compliance. The permeability coefficient of the grouting medium is determined through the constant-head permeability tests [27]. The main parameters of the grouting medium are listed in

Table 2. Main parameters of the injected medium.

Parameter	Bulk Density (-)	Particle Size Range (mm)	Volumetric Porosity (-)	Permeability (m2)
value	2.65	1~2	0.394	2.54 × 10−8

.

In this study, the permeation grouting experiments are conducted with the constant flow rate grouting method. The slurry flow rates are set at 1 L/min, 2 L/min, and 3 L/min, respectively. The experiments involve two primary variables: the water–cement ratio (W/C ratio) of the slurry and the slurry flow rate. To systematically investigate the effects of these variables on slurry dispersion, five distinct permeation grouting tests were designed (refer to

Table 3. Design of test conditions.

Experimental Condition Number	Grouting Flow Rate (L/min)	Water–Cement Ratio (-)
1	1	1.00
2	2	1.00
3	3	1.00
4	2	0.80
5	2	1.25

).

In test conditions 1, 2, and 3, the water–cement ratio is held constant, in order to examine the impact of varying slurry flow rates on the dispersion characteristics of the slurry. Conversely, in test conditions 2, 4, and 5, the slurry flow rate is held constant, to predominantly analyze how variations in the water–cement ratio influence slurry dispersion. Throughout the experimental process, the slurry flow rate is meticulously monitored using an electromagnetic flowmeter. Simultaneously, the injection pressure and the pressure distribution along the grout diffusion path within the injected medium are continuously measured and recorded in real-time via pressure sensors. The duration of each test is precisely timed with a stopwatch.

3.3. Experimental Results and Analysis

3.3.1. Validation of Experimental Model Effectiveness

In the experimental setup, fluctuations in flow rate during slurry injection are observed. It is important to validate that the slurry flow rate can remain roughly constant, not strongly affected by the fluctuation in the slurry injection pump. As illustrated in Figure 4, the injection time and the volume of injected slurry exhibit a linear relationship, suggesting a constant flow rate of slurry. The slurry flow rate is approximately 3.18 L/min with an error within 10% (condition 3 in Table 3), indicating that the experimental conditions are achieved as designed.

3.3.2. Impact of Grouting Parameters on Grouting Pressure

Figure 5 and Figure 6 present the grouting pressure (slurry injection pressure). The results show that the grouting pressure increases nonlinearly over time. As the grouting process continues, both the grouting pressure and its rate of increase rise progressively. This increase in grouting pressure is primarily due to the flow resistance of slurry in the injected medium. Over time, the resistance builds up, leading to a corresponding increase in grouting pressure.

Figure 5 illustrates the effect of the water–cement ratio on the grouting pressure associated with slurry permeation and diffusion. Across different water–cement ratios, the grouting pressure consistently increases with grouting time (or the volume of injected slurry). This means that as the duration of grouting increases, the slurry injection pressure gradually rises. However, at the same grouting time (or volume), an increase in the water–cement ratio results in a decrease in the slurry injection pressure. The primary reason for this trend is that a higher water–cement ratio reduces the plastic viscosity and yield stress of the slurry. Consequently, the resistance that the slurry encounters during its diffusion process is lower with a higher water–cement ratio, in turn, leading to a decrease in the grouting pressure.

Figure 6 shows the impact of the slurry injection rate on the grouting pressure associated with slurry permeation and diffusion. Across different slurry injection rates, the grouting pressure consistently increases with the duration of the grouting process (or the volume of injected slurry). This indicates that as the grouting time extends, the slurry injection pressure progressively rises. As the slurry injection rate increases, the distance that the slurry permeates and diffuses within the same period is greater. This results in a higher flow resistance that the slurry must overcome, leading to an increase in the slurry injection pressure.

4. Comparison of Theoretical and Experimental Results

In this section, we conduct a comparative analysis between the results of permeation grouting experiments under different conditions and the theoretical calculations from the fractal theory-based diffusion model. This comparison aims to validate the fractal theory-based model of Bingham fluid permeation and diffusion proposed in this study. In addition to the model proposed in this study, the analysis also includes other two theoretical models established in earlier studies: the model for Bingham fluid permeation and diffusion without considering the permeation and diffusion paths, and the model for Bingham fluid permeation and diffusion considering the permeation and diffusion paths. A brief overview of these models is provided as follows.

4.1. The Theoretical Model of Permeation and Diffusion for Bingham Fluid

In the theoretical model of permeation and diffusion for Bingham fluid without considering the permeation and diffusion paths, the pressure spatiotemporal distribution is expressed as follows [28]:

$$p_c=-\left({\displaystyle \frac{{\mu }_{p}q}{Ak}}+{\displaystyle \frac{2{\tau }_0}{3}}\sqrt{{\displaystyle \frac{2\varphi }{k}}}\right)l_m+p_0$$

(32)

In the theoretical model considering the permeation and diffusion paths, the pressure spatiotemporal distribution is expressed as follows [18]:

$$p_c=-\left({\displaystyle \frac{{\mu }_{p}q}{Ak}}{l}_m+{\displaystyle \frac{{\tau }_0\varphi }{6k\eta }}{l}_m^2\right)+p_0$$

(33)

where $\eta$ represents the length ratio of the porous channels, which is related to the areal porosity $\phi$ by the following equation:

$$\eta =6ln\left({\displaystyle \frac{\sqrt{2}}{24\phi }}\sqrt{{\displaystyle \frac{1-\phi }{1-0.342\phi }}}\right)\sqrt{{\displaystyle \frac{2\left(1-0.342\phi \right)\pi }{\left[2ln\left(\frac{\sqrt{2}}{24}\sqrt{\frac{1-\phi }{1-0.342\phi }}\right)-ln\phi \right]\phi ln\phi }}}$$

(34)

4.2. Comparative Analysis of Experimental Results and Theoretical Calculations

The calculation parameters for theoretical models corresponding to the permeation grouting experiments are listed in

Table 4. Calculation parameters.

Experimental Condition Number	$\mathit{q}$ (L/min)	${\mathit{p}}_0$ (kPa)	$\mathit{t}$ (s)	$\mathit{A}$ (m2)
1	1	23.84	130	0.00785
2	2	28.73	80
3	3	31.04	46
4	2	32.60	80
5	2	24.60	80

. These parameters are substituted into Equations (25), (32), and (33), respectively, for each experimental condition, yielding the relationship curve of slurry pressure with diffusion distance under grouting pressure $p_0$ (Figure 7).

As shown in Figure 7, taking experimental conditions 3 and 4 as examples, the theoretical model that neglecting permeation diffusion pathways exhibit a linear decay with an increasing slurry diffusion distance. This significantly overestimates the slurry pressure obtained from experiments. On the other hand, the model that considers both the slurry permeation path and the fractal theory-based model can capture the general changing trend over the diffusion distance. The fractal theory-based model can better predict the slurry diffusion distance under certain grouting pressure, and better describe the slurry pressure over distance.

In the experiments conducted in this section, the slurry injection stops at the point at which the slurry is just about to flow out of the overflow pipe. Therefore, the slurry diffusion distance in all tests is set to 80 cm. Under different experimental conditions, the calculated slurry diffusion distances from the model neglecting permeation pathways are 3.5 to 6.3 times larger than the experimental values, showing significant discrepancies. The reason is that the approach assumes slurry diffusion occurs uniformly in a circular pipe, disregarding the influence of pore structures, thereby causing notable deviations from the experimental results.

In contrast, considering permeation diffusion pathways reduces the calculated slurry diffusion distances to 0.4 to 1.4 times of the experimental values, but it overlooks the effects of porous medium structural parameters on slurry diffusion.

On the other hand, the slurry diffusion distances calculated based on fractal theory range from 0.8 to 1.1 times of the experimental values (Figure 8), showing the greatest agreement. Overall, the proposed Bingham fluid permeation diffusion model based on fractal theory comprehensively characterizes slurry diffusion paths and pore-structure features. Its theoretical pressure calculations closely match experimental values.

4.3. Error Analysis of Theoretical Results

Figure 9 and Figure 10 show the deviation in slurry pressure between theoretical (fractal theory-based diffusion model) and experiment results under various conditions. As presented in the figures, the deviation is positively correlated with the slurry diffusion distance. As the slurry diffusion distance increases, more pores become clogged by the slurry, resulting in faster pressure attenuation observed in experiments, thereby increasing the deviation from theoretical values.

As presented in Figure 9, the higher the grouting rate (slurry injection rate), the smaller the deviation between the theoretical and experimental values. One possible reason is that at a lower grouting rate, the slurry flows for a longer time through the porous medium, causing a greater difference in the viscosity of the slurry along the diffusion distance. There is a significant difference in slurry viscosity between the vicinity of the injection point and the distant areas. However, the theoretical model based on fractal theory proposed in this study does not account for the temporal variability in slurry viscosity.

Figure 10 reveals that under experimental condition 4 (slurry water–cement ratio of 0.8), the theoretical values of pressure based on the fractal-theory model closely match the experimental results. As the slurry water–cement ratio increases, however, the deviation between theoretical and experimental values becomes larger. One possible reason is the filtration effect when slurry flows through a porous medium. In this process, some particles in the slurry are retained in pores, which reduces flow resistance at greater diffusion distances, resulting in lower slurry pressure. Moreover, higher water–cement ratios enhance the filtration effect [14]. The experimental observations in this study also confirm significant filtration effects (Figure 11).

Figure 11 depicts a comparison between the bottom and top sections after the experiment at condition 5 (slurry water–cement ratio of 1.25). The bottom of the sample (inlet) appears darker and denser, indicating a higher slurry concentration, whereas the top of the sample (outlet) appears lighter and more diluted, illustrating noticeable slurry filtration effects. However, the fractal theory-based Bingham fluid permeation diffusion theoretical model does not consider the filtration effects of the slurry. The theoretical calculations of the slurry pressure at higher water–cement ratios tend to be larger than the experimental values, with discrepancies increasing as the water–cement ratio rises.

5. Conclusions

A theoretical penetration grouting model for Bingham fluid based on fractal theory was proposed in this study and validated by pure-cement slurry diffusion experiments. The primary conclusions are listed as follows:

(1)

In addressing the influence of pore structure on the diffusion of Bingham fluid slurry, this study establishes a theoretical model based on fractal theory. The parameters involved in the proposed model can be determined from conventional geotechnical tests.

(2)

A one-dimensional permeation grouting simulation apparatus was developed. The experimental setup allows for the comprehensive analysis of the spatiotemporal pressure variations of the pure-cement slurry under different experimental conditions.

(3)

The theoretical model for Bingham fluid permeation grouting based on fractal theory proposed in this study aligns closely with the experimental results. It comprehensively considers the pore structure and the tortuosity effect on the pure-cement slurry permeation and diffusion. This model provides a significant reference for the design of grouting operations and can contribute valuable insights to related construction practices.