Experimental and Numerical Studies of Self-Expansible Polyurethane Slurry Diffusion Behavior in a Fracture Considering the Slurry Temperature

Abstract

Polyurethane grouting material (polymer) has been widely used in rock mass fracture grouting. The previous test of polymer slurry grouting in planar fracture did not consider the influence of temperature; therefore, in this paper, a test of polymer slurry grouting in planar fracture was firstly conducted in order to explore the diffusion characteristics of polymer slurry. The results show that the preheating temperature of polymer slurry has a great influence on the diffusion of slurry in fractures, which should be considered in the research of polymer fracture grouting. In addition, the slurry temperature is related to the chemical reaction of the polymer itself. However, the existing polymer fracture grouting models ignore the influence of temperature and the chemical reaction of the polymer slurry itself, which lack the rationality to reveal the diffusion behavior of the polymer slurry in fracture. Therefore, in this paper, a chemical reaction-fluid dynamic (CF) model was established. In addition, the Particle Swarm Optimization (PSO) algorithm was used to obtain the chemical reaction kinetic parameters of the polymer slurry. Based on the CF model, the diffusion characteristics of the polymer slurry during the fracture diffusion process were calculated. The applicability of the CF model was verified by comparing the experimental data with the calculated results. Finally, the influence of polymer preheating temperature and ambient temperature on slurry fracture grouting behavior was explored by the CF model. The research in this article provides some theoretical reference for the design of grouting parameters in fracture grouting engineering.

1. Introduction

Grouting is one of the main methods for reinforcing and preventing seepage in rock mass fractures and has been widely used in fields such as water conservancy, transportation, and mining. It is of great guiding significance to deeply understand the migration law of slurry in rock fractures for grouting design and construction [1,2,3,4,5,6,7,8,9,10,11,12].

Many scholars have conducted extensive research on the mechanism of slurry diffusion through experiments. Draganović et al. [13] developed a slot with different fracture openings and tested the permeability law of cement slurry in the fracture. Mohammed et al. [14] used a plexiglass disk with a diameter of 0.5 m to simulate microfractures and studied the seepage characteristics of slurry in fractures with different openings. Sui et al. [15] carried out a single fracture grouting experiment with chemical grouting material under dynamic water conditions in order to analyze the influence of fracture width, initial water flow velocity, slurry gelling time, and grouting volume on grouting efficiency. Yang et al. [16] made different roughness fracture models with two serrated concrete slabs and carried out a hydrodynamic grouting experiment. The effects of viscosity, water flow velocity, grouting pressure, and roughness coefficient of carbon fiber composite cement on the slurry diffusion behavior in fractures were analyzed. Funehag et al. [17] made a microfracture model with water pressure and studied the viscosity, yield stress, and diffusion behavior of Bingham fluid. Li et al. [18] developed a grouting device to conduct shear tests and confined tests for the single-fracture chlorite schist.

Compared with experiment, the numerical simulation method can visually reflect the continuous dynamic variation process of slurry diffusion in fractures. Wallner [19] firstly proposed that the slurry flow in the fracture channel can be regarded as a Bingham fluid. Based on this, Hassler et al. [20] established a one-dimensional channel fracture model and simulated the slurry diffusion mechanism in fracture. After that, Eriksson et al. [21] studied the penetration and filtration abilities of slurry by using one-dimensional channels. Yang et al. [22] predicted the slurry penetration in a fracture according to a numerical simulation based on the Monte-Carlo method. Luo et al. [23] developed the seepage model of Bingham grouts flowing in fracture networks based on the simulation of fracture networks of rock masses. Saeidi et al. [24] used UDEC software to analyze the influence of the roughness and strength of fracture surfaces on grout penetration. Zhou et al. [25] developed a numerical model with cohesive finite elements to simulate the grouting of rock mass fractures, which couples the stress-seepage-damage field. Xu et al. [26] considered the seepage stress coupling of rock mass fracture and used UDEC software to simulate the multiple hole grouting problem. Mohajerani et al. [27] proposed a fully explicit algorithm to describe the slurry propagation in the rock joints. Oge [28] predicted the cementitious slurry quantity using the neuro-fuzzy inference system and multiple regression. Dehghanipoodeh et al. [29] used the discrete fracture network-discrete element method (DFN-DEM) to ensure the reality of the fracture pattern. The mechanical properties of both intact and rock joints and their interactions were taken into account. Wang et al. [12] developed the two-phase rough rock fracture model to determine the grout behavior under water washing. Zhu et al. [30] established an analytical grouting diffusion model of a single rough fracture under constant-pressure control. Li et al. [31] proposed a Sequential Diffusion and Solidification (SDS) method to reveal the diffusion behavior of quick-setting slurry in dynamic water.

The above research on grouting mainly focuses on constant-density slurries such as cement and water glass. In recent years, the polymer grouting material and its high-pressure injection technology have been widely used in engineering repair and reinforcement due to its fast reaction speed, high expansibility, and good durability [32,33]. Unlike the non-expansive slurry, whose diffusion mechanism is mainly driven by grouting pressure, the expansion and movement of polymer slurry in rack mass fracture mainly relies on the pressure generated by its own chemical reaction. The expansion performance of polymer materials is related to factors such as temperature, environmental pressure, and the confinement of surrounding media; thus, its grouting mechanism is more complex than the non-expansive slurry. In order to explain the diffusion mechanism of polymer slurry in fracture, Li et al. [34] used a quasi-three-dimensional numerical model to analyze the polymer slurry diffusion mechanism in the parallel fracture. Hao et al. [35] studied the polymer slurry diffusion mechanism in a single water-free fracture by simulation. Li et al. [36] used Youngs method to solve the VOF function so as to track the polymer slurry’s moving interface during its diffusion process. Liang et al. [37] established a polymer fracture grouting diffusion model based on the rheological properties of the slurry and the theory of viscous hydrodynamics.

Some scholars have noticed the high exothermic phenomenon in the reaction process of polymer slurry. Shi et al. [38] studied the temperature changes of polyurethane materials during the curing process. Hao [39] found that under the same ambient temperature, the expansion rate of polymer increases with the increase in the preheating temperature of the slurry. The reason is that when the preheating temperature increases, the chemical reaction rate of the slurry is faster, which accelerates the heating system and the gasification of physical blowing agents, resulting in an acceleration of the slurry expansion rate.

It can be deduced that during the flow process of slurry, thermal conduction will occur between slurry and the surrounding medium due to their temperature difference. This phenomenon will change the temperature field distribution of the slurry and affect the slurry’s reaction rate and diffusion process. Especially for the rock mass fracture, with the large extension range and small opening, the contact area between the slurry and the fracture wall is large, and the impact of heat conduction on the temperature field and diffusion process of the slurry is significant. However, the impact of temperature on the polymer diffusion behavior in fracture is limited to qualitative understanding and a lack of systematic and in-depth research.

Therefore, in this paper, based on the mechanism of the polymer polymerization reaction and computational fluid dynamics, a chemical reaction-fluid dynamic simulation model was established, which contains the polymer slurry and the fracture wall. The quasi-three-dimensional coordinate grid finite volume method was used to discretize the slurry flow control equation, and the three-dimensional coordinate grid finite volume method was used to discretize the heat conduction equation of the fracture wall. The influence of the slurry preheating temperature and ambient temperature on the slurry diffusion behavior is analyzed, aiming to deeply understand the diffusion mechanism of polymer fracture grouting, which can provide some guidance for the design of polymer grouting technology and the formulation of construction plans.

2. Test Studies

2.1. Test Materials

The expansive polyurethane polymer studied in this article is composed of a mixture of components A and B, with a mixing ratio of 1.1:1. The composition of the material’s mass fraction ratio is shown in

Table 1. Composition of the polymer’s mass fraction ratio.

Component	Ingredient	Mass Fraction Ration
Component A	Polymethylene polyphenyl polyisocyanate	100%
Component B	Hard foam polyether polyol system	50%
	Additive flame retardant	10%
	Catalyst system	2%
	Surface active agent	0.5%
	Physical blowing agent	5%
	Chemical blowing agent	0.5%
	Others	32%

.

As shown in Figure 1, the reaction of polymers can be divided into chemical reaction and physical reaction, in which the chemical reaction mainly includes gelling reaction and blowing reaction. The gelling reaction generates polyurethane from isocyanate and polyhydric alcohols. The blowing reaction generates urea and carbon dioxide from isocyanate and water (chemical blowing agent). The chemical reactions are high-exothermic; under this influence, the physical blowing agent changes from liquid a state to a gas state. The expansion and diffusion of polymers are driven by gaseous physical blowing agents and carbon dioxide [40,41,42,43].

2.2. Test Device and Procedure

The polymer slurry fracture grouting test device shown in Figure 2 mainly consists of two PMMA boards with 120 mm × 120 mm × 15 mm and a fracture with a width of 7 mm. The upper and lower boards are fixed together with steel bars and screws. The grouting hole is drilled on the center of upper board. The polymer slurry is injected into the grouting hole through the grouting gun. The whole test was conducted indoors with an ambient temperature of 30 °C. The grouting volume was 750 g each time. The grouting preheating temperature was set at 30 °C, 40 °C, and 50 °C, respectively.

2.3. Test Results and Analysis

The real-time diffusion range of polymer slurry was recorded by an HD camera, which is shown in Figure 3. It can be seen that after the polymer slurry is injected into the fracture, a chemical reaction occurs. The slurry gradually expands and spreads from the grouting hole in a circular shape and transforms from the initial transparent liquid to a light yellow solid.

Figure 4 shows the variation of polymer diffusion radius under three preheating temperature conditions. The diffusion radius of polymer after consolidation under three working conditions was measured to be 0.25 m, 0.461 m, and 0.462 m, respectively. In addition, it was found from Figure 3 and Figure 4 that the diffusion range of the slurry under preheating temperatures of 40 °C and 50 °C is very close, larger than the diffusion radius of the slurry under a preheating temperature of 30 °C. The higher the preheating temperature, the faster the diffusion rate of the slurry. The test result further proves the influence of temperature on the diffusion process of polymer slurry in fracture.

3. Method of Chemical Reaction-Fluid Dynamic Model

3.1. Chemical Reaction Kinetic Equation

The kinetic equation of the gelling reaction is as follows:

$$\frac{\mathrm{d}{X}_{\mathrm{OH}}}{\mathrm{d}t}=A_{\mathrm{OH}}\exp (-\frac{E_{\mathrm{OH}}}{RT})c_{\mathrm{OH},0}(1-X_{\mathrm{OH}})(\frac{c_{\mathrm{NCO},0}}{c_{\mathrm{OH},0}}-2\frac{c_{\mathrm{W},0}}{c_{\mathrm{OH},0}}{X}_{\mathrm{W}}-X_{\mathrm{OH}}){\left[1+r_{\mathrm{BL}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{BL}}}+r_{\mathrm{W}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{W}}}\right]}^{-1}$$

(1)

The kinetic equation of the blowing reaction is as follows:

$$\frac{\mathrm{d}{X}_{\mathrm{W}}}{\mathrm{d}t}=A_{\mathrm{W}}\exp (-\frac{E_{\mathrm{W}}}{RT})c_{\mathrm{OH},0}(1-X_{\mathrm{W}})(\frac{c_{\mathrm{NCO},0}}{c_{\mathrm{OH},0}}-2\frac{c_{\mathrm{W},0}}{c_{\mathrm{OH},0}}{X}_{\mathrm{W}}-X_{\mathrm{OH}}){\left[1+r_{\mathrm{BL}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{BL}}}+r_{\mathrm{W}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{W}}}\right]}^{-1}$$

(2)

The subscript P is the polyurethane mixture, W is water (chemical blowing agent), BL is the liquid physical blowing agent, NCO is isocyanate, and OH is polyhydric alcohol. A_OH and A_W are the pre-exponential factors, E_OH and E_W are the activation energies; these four parameters are collectively referred to as chemical reaction kinetic parameters (CRKP). c_i,0 is the initial concentration of each component, and c_i is the current concentration of each component. X_i is their fractional conversion. R is the universal gas constant. T is the polymer temperature. r_i is the mass fraction of each component, and ρ_i is their density [44,45].

3.2. Chemical Reaction Kinetic Parameter Identification

Different slurry proportions have different CRKP, which directly determines the chemical reaction degree of polymer slurry. The traditional way to determine these parameters was through the Stephen method, which can only be implemented for a single chemical reaction. Considering that the expansion and diffusion processes of polymer slurry contain two chemical reactions, a better method is required to avoid the coupling effect of two chemical reactions. Therefore, in this paper, in order to obtain the CRKP of polymer slurry, a PSO inversion method for CRKP identification was proposed. The CRKP was obtained through the PSO algorithm according to the variation of polymer temperature and density measured by the polymer-free expansion test. The polymer slurry-free expansion test process is shown in Figure 5.

For the polymer CRKP identification PSO method, assuming there is a single particle i in the swarm n, its searching velocity is v_i = (v_i1, v_i2, v_i3, v_i4). Its current position is x_i = (A_OH, E_OH, A_W, E_W). Its corresponding fitness value is f_i. The optimal position of particle i is p_i = (A_OH, E_OH, A_W, E_W). The optimal position of the swarm is p_g = (A_OH, E_OH, A_W, E_W). The fitness values of p_i and p_g are expressed as f_pi and f_g, respectively.

In each iteration, particles move according to the following equation:

$$v_i^t=w{v}_i^{t-1}+c_1{r}_1(p_i^{t-1}-x_i^{t-1})+c_2{r}_2(p_g^{t-1}-x_g^{t-1})$$

(3)

where t and t − 1 represent iteration times, r₁ and r₂ are the random factors, and c₁ and c₂ are the learning factors. w is the inertia weight, which is used to control search speed and balance the search capability.

The new position of each particle is updated as follows:

$$x_i^t(A_{\mathrm{OH}},E_{\mathrm{OH}},A_{\mathrm{W}},E_{\mathrm{W}})=x_i^{t-1}(A_{\mathrm{OH}},E_{\mathrm{OH}},A_{\mathrm{W}},E_{\mathrm{W}})+v_i^t(v_{i1},v_{i2},v_{i3},v_{i4})$$

(4)

The polymer temperature and density variation with time are calculated when particle i reaches a new position x_i = (A_OH, E_OH, A_W, E_W), and its fitness value f_it is calculated as follows:

$$f_i^t={\displaystyle \sum _{k=1}^m\left|T_k-{T_k}^{\prime }\right|}+{\displaystyle \sum _{j=1}^l\left|D_j-{D_j}^{\prime }\right|}$$

(5)

where T_k is the test polymer temperature, T_k′ is the calculated polymer temperature at the same time, D_j is the test polymer density, D_j′ is the calculated polymer density, k is the serial number of temperature recording points, m is the total number of temperature recording points, j is the serial number of density recording points, and l is the total number of density recording points.

The polymer CRKP identification PSO procedure is shown in Figure 6.

The inversion CRKP results corresponding to p_g = (A_OH, E_OH, A_W, E_W) are shown in

Table 2. Chemical reaction kinetic parameters of the polymer slurry.

CRKP	Searching Space		Inversion Value
	Minimum Value	Maximum Value
AOH (m3·s−1·mol−1)	2.3	3	2.628
EW (J·mol−1·k−1)	30,000	34,000	32,450.557
AOH (m3·s−1·mol−1)	25.676	29.676	27.457
EW (J·mol−1·k−1)	37,410	41,000	39,375.422

. The temperature and density variation calculated from p_g = (A_OH, E_OH, A_W, E_W) are compared with test results, which are shown in Figure 7.

It can be seen from Figure 7 that both the calculated temperature curve and density curve of polymer slurry are in good agreement with the test results, which proves that the obtained CRKP is effective and can be used for numerical analysis of polymer fracture grouting.

3.3. Density Equation

The polymer slurry can be regarded as a homogeneous fluid composed of three kinds of liquids and two kinds of gases. The liquid part consists of water, a polyurethane mixture, and a liquid physical blowing agent. The gas part consists of a gaseous physical blowing agent and carbon dioxide. The polymer density can be calculated using the following equation:

$${\rho }_{\mathrm{F}}=\frac{1+r_{\mathrm{BL},0}+r_{\mathrm{W},0}}{\frac{r_{{\mathrm{CO}}_2}1000RT}{p{M}_{{\mathrm{CO}}_2}}+\frac{r_{\mathrm{BG}}1000RT}{p{M}_{\mathrm{B}}}+\frac{r_{\mathrm{BL}}}{{\rho }_{\mathrm{BL}}}+\frac{r_{\mathrm{W}}}{{\rho }_{\mathrm{W}}}+\frac{1}{{\rho }_{\mathrm{P}}}}$$

(6)

where subscript BG is the gaseous physical blowing agent and CO₂ is carbon dioxide. r_i,0 is the initial mass fraction.

At the beginning, the physical blowing agent exists in the slurry as a liquid (r_BL,0), and gradually vaporizes when heated. The relationship between r_BG and r_BL is as follows:

$$r_{\mathrm{BG}}=r_{\mathrm{BL},0}-r_{\mathrm{BL}}$$

(7)

Figure 8 shows the change of r_BL with temperature during the polymer chemical reaction process, which refers to the method in [46].

r_W and r_CO2 are calculated as follows:

$$r_{\mathrm{W}}=\frac{c_{\mathrm{W}}{M}_{\mathrm{W}}}{1000{\rho }_{\mathrm{P}}}$$

(8)

$$r_{{\mathrm{CO}}_2}=\frac{c_{\mathrm{W},0}{X}_{\mathrm{W}}{M}_{{\mathrm{CO}}_2}}{(1000{\rho }_{\mathrm{P}})-r_{{\mathrm{CO}}_2,\mathrm{D}}}$$

(9)

where M_i is the molecular weight and r_CO2,D is the initial mass fraction of carbon dioxide.

3.4. Polymer Flowing Control Equation

The polymer slurry satisfies the following mass conservation equation during the process of reaction:

$$\frac{\partial \rho }{\partial t}+\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}+\frac{\partial (\rho w)}{\partial z}=0$$

(10)

where t represents time and u, v, and w represent the velocity components of the velocity vector u in the x, y and z directions.

The polymer slurry follows the momentum conservation equation in the process of expansion and diffusion, which can be expressed as follows:

$$\{\begin{array}{l}\frac{\partial (\rho u)}{\partial t}+\nabla \cdot (\rho uu)=\nabla \cdot (\mu \nabla u)-\frac{\partial p}{\partial x}+S_u\\ \frac{\partial (\rho v)}{\partial t}+\nabla \cdot (\rho vu)=\nabla \cdot (\mu \nabla v)-\frac{\partial p}{\partial y}+S_v\\ \frac{\partial (\rho w)}{\partial t}+\nabla \cdot (\rho wu)=\nabla \cdot (\mu \nabla w)-\frac{\partial p}{\partial z}+S_w\end{array}$$

(11)

where ρ is density, p is the pressure of the fluid microelement, μ is the dynamic viscosity, and S_u, S_v, and S_w are the source items in x, y and z directions.

The polymer slurry in fractures also satisfies the law of energy conservation, and the energy conservation equation is as follows:

$$\frac{\partial \left({\rho }_{\mathrm{F}}{C}_{\mathrm{F}}T\right)}{\partial t}+\nabla \cdot \left({\rho }_{\mathrm{F}}{C}_{\mathrm{F}}uT\right)=\nabla \cdot \left(k\nabla T\right)+S_T$$

(12)

where T is the temperature. For the polymer slurry, the source term S_T can be expressed as the following:

$$S_{\mathrm{T}}=\frac{{\rho }_{\mathrm{F}}{\alpha }_{\mathrm{F}}}{{\rho }_{\mathrm{F},0}}\left(\Delta H_{\mathrm{OH}}{c}_{\mathrm{OH},0}\frac{\mathrm{d}{X}_{\mathrm{OH}}}{\mathrm{d}t}+\Delta H_{\mathrm{W}}{c}_{\mathrm{W},0}\frac{\mathrm{d}{X}_{\mathrm{W}}}{\mathrm{d}t}-\Delta H_{\mathrm{BG}}{\rho }_{\mathrm{F},0}\frac{\mathrm{d}{r}_{\mathrm{BG}}}{\mathrm{d}t}\right)$$

(13)

where α is the volume fraction of polymer slurry in solution domain, ρ_F,0 is the initial density of polymer slurry, and ΔH_i is the reaction heat.

C_F represents the specific heat capacity of the slurry, and its calculation method is as follows:

$$C_{\mathrm{F}}=C_{\mathrm{P}}+r_{{\mathrm{CO}}_2}{C}_{{\mathrm{CO}}_2}+r_{\mathrm{W}}{C}_{\mathrm{W}}+r_{\mathrm{BG}}{C}_{\mathrm{BG}}+r_{\mathrm{BL}}{C}_{\mathrm{BL}}$$

(14)

where C_i is the heat capacity of each component [47,48].

k_F is the thermal conductivity coefficient of the fluid, and its calculation method is as follows:

$$k_{\mathrm{F}}=\{\begin{array}{l}8.70\times {10}^{-8}{\rho }_{\mathrm{F}}^2+8.46\times {10}^{-5}{\rho }_{\mathrm{F}}+1.16\times {10}^{-2}{\rho }_{\mathrm{F}}^{-2} {\rho }_{\mathrm{F}} \ge 48 {\mathrm{kg}/\mathrm{m}}^3\\ 9.37\times {10}^{-6}{\rho }_{\mathrm{F}}^2-7.35\times {10}^{-5}{\rho }_{\mathrm{F}}+2.956\times {10}^{-2}{\rho }_{\mathrm{F}}^{-2} {\rho }_{\mathrm{F}} <48 {\mathrm{kg}/\mathrm{m}}^3\end{array}$$

(15)

The mass conservation equation and the momentum conservation equation can be expressed as the flowing control equation:

$$\frac{\partial (\rho \phi )}{\partial t}+\nabla \cdot (\rho u\phi )=\nabla \cdot (\Gamma \nabla \phi )+S$$

(16)

where φ is the general variable, Γ is the generalized diffusion coefficient. The four terms from left to right represent the transient term, convective term, diffusion term, and source term, respectively.

The chemical reaction process of the polymer slurry in the fracture is described by a mass conservation equation that ignores the diffusion term. Therefore, the component mass conservation equation of the gelling reaction is as follows:

$$\frac{\partial }{\partial t}\left(\rho {\alpha }_{\mathrm{F}}{X}_{\mathrm{OH}}\right)+\nabla \cdot \left(\rho {\alpha }_{\mathrm{F}}{X}_{\mathrm{OH}}u\right)=\rho {\alpha }_{\mathrm{F}}{Q}_{\mathrm{Kin},\mathrm{OH}}$$

(17)

where the source term describes the rate of change in the conversion rate of polyhydric alcohol. According to Formula (1), its expression is as follows:

$$Q_{\mathrm{Kin},\mathrm{OH}}=A_{\mathrm{OH}}\exp (-\frac{E_{\mathrm{OH}}}{RT})c_{\mathrm{OH},0}(1-X_{\mathrm{OH}})(\frac{c_{\mathrm{NCO},0}}{c_{\mathrm{OH},0}}-2\frac{c_{\mathrm{W},0}}{c_{\mathrm{OH},0}}{X}_{\mathrm{W}}-X_{\mathrm{OH}}){\left[1+r_{\mathrm{BL}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{BL}}}+r_{\mathrm{W}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{W}}}\right]}^{-1}$$

(18)

The component mass conservation equation of the blowing reaction is as follows:

$$\frac{\partial }{\partial t}\left(\rho {\alpha }_{\mathrm{F}}{X}_{\mathrm{W}}\right)+\nabla \cdot \left(\rho {\alpha }_{\mathrm{F}}{X}_{\mathrm{W}}u\right)=\rho {\alpha }_{\mathrm{F}}{Q}_{\mathrm{Kin},\mathrm{W}}$$

(19)

where the source term describes the rate of change in the conversion rate of water. According to Formula (2), its expression is as follows:

$$Q_{\mathrm{Kin},\mathrm{W}}=A_{\mathrm{W}}\exp (-\frac{E_{\mathrm{W}}}{RT})c_{\mathrm{OH},0}(1-X_{\mathrm{W}})(\frac{c_{\mathrm{NCO},0}}{c_{\mathrm{OH},0}}-2\frac{c_{\mathrm{W},0}}{c_{\mathrm{OH},0}}{X}_{\mathrm{W}}-X_{\mathrm{OH}}){\left[1+r_{\mathrm{BL}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{BL}}}+r_{\mathrm{W}}\frac{{\rho }_{\mathrm{P}}}{{\rho }_{\mathrm{W}}}\right]}^{-1}$$

(20)

3.5. Heat Conduction Equation of the Fracture Wall

After calculating the temperature field of the polymer slurry using the energy conservation equation, the temperature field of the fracture wall under thermal conduction is obtained by solving the heat conduction equation. The heat conduction equation is described by the energy conservation equation. As there is only heat exchange and no flow phenomenon between the polymer slurry and the fracture wall, without considering the convection and source terms of the energy conservation equation, it can be expressed as follows:

$$\frac{\partial \left(\rho CT\right)}{\partial t}=k{\nabla }^{2}T$$

(21)

where ρ represents the density of fracture wall, C is the specific heat capacity of fracture wall, and k is the thermal conductivity coefficient of fracture wall.

4. Solution of CF Model

4.1. Discretization of the Momentum and Energy Conservation Equation

Since the length of the z-axis is much smaller than the length of the x-axis and y-axis when calculating the planar fracture grouting problem, the slurry flows in the fracture can be regarded as a Hele-Shaw flow, which is shown as follows:

$$\frac{\partial \phi }{\partial z}=-\frac{6}{h}\phi$$

(22)

where h is the thickness of the fracture. In this way, a 3D problem can be simplified to a 2D problem.

Discrete the momentum conservation equation and energy conservation equation by using the quasi-three-dimensional coordinate grid FVM, and they can be obtained as follows:

$${\displaystyle {\int }_t^{t+\Delta t}{\displaystyle {\int }_{\Delta v}\frac{\partial (\rho \phi )}{\partial t}}}\mathrm{d}V\mathrm{d}t+{\displaystyle {\int }_t^{t+\Delta t}{\displaystyle {\int }_{\Delta v}\mathrm{div}(\rho u\phi )}}\mathrm{d}V\mathrm{d}t={\displaystyle {\int }_t^{t+\Delta t}{\displaystyle {\int }_{\Delta v}\mathrm{div}(\mu \mathrm{grad}\phi )}}\mathrm{d}V\mathrm{d}t+{\displaystyle {\int }_t^{t+\Delta t}{\displaystyle {\int }_{\Delta v}S}}\mathrm{d}V\mathrm{d}t$$

(23)

The following symbols are introduced to sort out the control integral equation for the momentum discrete equation:

$$\{\begin{array}{l}{F}_e={\left(\rho u\right)}_e{A}_e F_w={\left(\rho u\right)}_w{A}_w F_n={\left(\rho v\right)}_n{A}_n F_s={\left(\rho v\right)}_s{A}_s\\ D_e={\mu }_e{A}_e/{\left(\delta x\right)}_e D_w={\mu }_w{A}_w/{\left(\delta x\right)}_w D_n={\mu }_n{A}_n/{\left(\delta y\right)}_n D_s={\mu }_s{A}_s/{\left(\delta y\right)}_s\\ a_P=a_W+a_E+a_S+a_N+12\mu A/h +(F_e-F_w)+(F_n-F_s)+a_P^0\end{array}$$

(24)

For the energy discrete equation, the symbols are as follows:

$$\{\begin{array}{l}{F}_e={\left(\rho uCA\right)}_e F_w={\left(\rho uCA\right)}_w F_n={\left(\rho vCA\right)}_n F_s={\left(\rho vCA\right)}_s\\ D_e=k_e{A}_e/{\left(\delta x\right)}_e D_w=k_w{A}_w/{\left(\delta x\right)}_w D_n=k_n{A}_n/{\left(\delta y\right)}_n D_s=k_s{A}_s/{\left(\delta y\right)}_s\\ a_P=a_W+a_E+a_S+a_N +(F_e-F_w)+(F_n-F_s)+a_P^0\end{array}$$

(25)

where Fi is the flow rate through the interface, Di is the diffusion conductivity of the interface, Ai is the interface area, and $\delta$ is the width of the control volume.

Therefore, Equation (23) becomes the following:

$$a_P{\phi }_P=a_W{\phi }_W+a_E{\phi }_E+a_S{\phi }_S+a_N{\phi }_N+b$$

(26)

where a_P, a_W, a_E, a_S, a_N, and b are coefficients. They are expressed as follows:

$$\{\begin{array}{l}{a}_W=D_w+\max (0,F_w) a_E=D_e+\max (0,-F_e)\\ a_S=D_s+\max (0,F_s) a_N=D_n+\max (0,-F_n)\\ b=S_c\Delta V+a_P^0{\phi }_P^0 \\ a_P^0=\frac{{\rho }_P^0\Delta V}{\Delta t}\end{array}$$

(27)

The quasi-three-dimensional FVM model can greatly improve the model calculation efficiency compared with the traditional FVM model.

4.2. The SIMPLE Algorithm

The SIMPLE algorithm is used to solve the discrete control Equation (26). Set the initial guessing pressure to p*. Set the pressure correction p′ as the error between the correct pressure p and the guessing pressure p*. Set the guessing velocity as (u*,v*) and the correct velocity as (u,v). Substitute p*and p into the discrete control Equation (26) to obtain (u*,v*) and (u,v). Substitute the corrected velocity value into the discrete equation and organize it to obtain the following pressure correction equation:

$$a_P{p^{\prime }}_P=a_W{p^{\prime }}_W+a_E{p^{\prime }}_E+a_S{p^{\prime }}_S+a_N{p^{\prime }}_N+b^{\prime }$$

(28)

4.3. Discretization of the Heat Conduction Equation for Fracture Walls

Discrete the heat conduction equation by using the three-dimensional coordinate grid FVM, and it can be obtained as follows:

$$a_{P}T=a_E{T}_E^0+a_W{T}_W^0+a_N{T}_N^0+a_S{T}_S^0+a_B{T}_B^0+a_T{T}_T^0+a_P^0{T}^0$$

(29)

where

$$\begin{array}{l}{a}_E=k_e{A}_e/{\left(\delta x\right)}_e, a_W=k_w{A}_w/{\left(\delta x\right)}_w\\ a_N=k_n{A}_n/{\left(\delta y\right)}_n, a_S=k_s{A}_s/{\left(\delta y\right)}_s\\ a_B=k_b{A}_b/{\left(\delta z\right)}_b, a_T=k_t{A}_t/{\left(\delta z\right)}_t\\ a_P^0=\frac{\rho C\mathrm{d}V}{\mathrm{d}t}-a_E-a_W-a_N-a_S-a_B-a_T\\ a_P=\frac{\rho C\mathrm{d}V}{\mathrm{d}t}\end{array}$$

(30)

5. Numerical Simulation

5.1. Model Description

Figure 9 is the planar fracture grouting simulation model, with a length of 120 cm and a thickness of 7 mm. The boundaries of front, back, left and right are free surface boundaries, which are the same as the experiment conditions. The upper and lower boundaries are heat conduction boundaries, as there exists heat conduction between the slurry and the fracture wall. The grouting hole is located in the center of the model. Considering the symmetry of the fracture geometry structure, a quarter part along the center point is taken as the analysis region in order to improve computing efficiency.

The material parameters used in this model are shown in

Table 3. Material parameters.

Parameter	Parameter Value	Parameter	Parameter Value
AOH	2.7029 m3·s−1·mol−1	(ΔH)W	8.6 × 104 kJ/kg
AW	26.54 m3·s−1·mol−1	(ΔH)BG	20.68 × 104 kJ/kg
EOH	35,195.707 m3·s−1·mol−1	CP	1800 J/kg·K
EW	42,046.74 m3·s−1·mol−1	CCO2	836.6 J/kg·K
cOH,0	4079.6 mol/m3	CW	294.7 J/kg·K
cNCO,0	4557.7 mol/m3	CBG	1000 J/kg·K
cW,0	294.7 mol/m3	CBL	1159 J/kg·K
ρS	1890 kg/m3	CS	1680 J/kg·K
ρP	1100 kg/m3	R	8.314 J/(mol·K)
ρBL	1228 kg/m3	μ	1 Pa·s
ρW	1000 kg/m3	kS	0.2 W·(m·K)−1
(ΔH)OH	7.705 × 104 kJ/kg	r0	17.6 cm

.

5.2. Basic Assumptions

This model satisfies the following assumptions:

• The slurry is an isotropic, homogeneous, continuous fluid;
• The slurry is in a laminar flow state during the reaction process;
• The fracture planar surface is rigid. It will not deform due to the grout pressure during the diffusion process.

5.3. Numerical Simulation Result and Verification

5.3.1. Diffusion Radius

Figure 10 shows the diffusion range and velocity vector field distribution of the slurry at different times under an injection volume of 750 g, an ambient temperature of 30 °C, and a preheating temperature of 40 °C. It can be seen that with the chemical reaction process of the slurry, the polymer slurry diffuses around in a circular shape in the fracture. Along the diffusion radius direction, the slurry flow rate reaches its maximum at the edge of the filling area, gradually decreases towards the center point, and approaches 0 at the slurry flow center. The velocity vector direction of the slurry is vertical to the interface at the edge of the filling area.

Figure 11 shows the comparison between the calculated result and tested data of the polymer slurry diffusion radius with time under injection volume of 750 g, ambient temperature of 30 °C, and preheating temperature of 40 °C. It can be seen that the diffusion radius variation process obtained from the experimental research is slightly faster than the simulation result, while the overall trend of the slurry diffusion radius obtained by the two methods is basically consistent. The average relative error between the calculated result and the tested data is −2.23%, which is in good agreement.

5.3.2. Polymer Temperature

Figure 12 shows the comparison between the calculated polymer temperature and the tested polymer temperature at the distance of 0.1 m and 0.2 m from the center point under injection volume of 750 g, ambient temperature of 30 °C, and preheating temperature of 40 °C. It can be seen that the polymer temperature of the slurry at the measurement point firstly increases, gradually decreasing after reaching to the peak point. In the stage of temperature rise, the experimental results are faster than the numerical results, and the temperature peak is larger. In the stage of temperature decrease, the experimental results and numerical results are almost the same. The temperature changes of the slurry obtained by the two methods are relatively consistent. The average relative errors between them are 1.63% at a distance of 0.1 m from the center point and 3.05% at a distance of 0.2 m from the center point.

5.3.3. Polymer Density

Figure 13a is the polymer density variation with time at the center point, and Figure 13b is the density distribution from the center point along the radius direction. It can be seen from Figure 13a that the density of the slurry gradually decreases over time. For Figure 13b, the density of the polymer slurry in the filling area is basically the same at the same time, with a slight decrease from the center point along the diffusion radius direction.

In order to verify the accuracy of the polymer density, the tested density obtained from the final polymer consolidated body was compared with the calculated density at the corresponding position under grouting amount 750 g, ambient temperature 30 °C, and preheating temperature of 40 °C. The polymer consolidated body was cut along the radius direction from the center point with a slice size of 10 cm × 10 cm × 7 mm, which is shown in Figure 14.

The density of cut slices was calculated based on their weight and volume size. After that, their density was compared with the average calculated slurry density at the corresponding positions; the comparison result is shown in Figure 15. It can be seen that the density of the polymer consolidation obtained from the experiment is in good agreement with the calculated density values at the corresponding positions. The average relative error between the calculated density and the tested density is 1.44%.

The above comparative analysis of the calculated results and tested values about slurry diffusion radius, slurry temperature changes, and slurry final density indicates that the established simulation model can reflect the polymer diffusion characteristics in fractures well, which can be used to simulate and analyze the slurry diffusion process in fractures.

6. Discussion of the Influence of Temperature

Based on the test results in Section 2, in order to further analyze the influence of preheating temperature on the polymer diffusion process in fractures, 10 slurry preheating temperatures were set up from 30 °C to 39 °C, and a simulation analysis was conducted on the polymer diffusion process under each condition.

Figure 16 shows the variation of the polymer diffusion range in the fracture at 0 s, 6 s, 14 s, 18 s, 26 s, and 30 s under different preheating temperature conditions. It can be seen that the polymer slurry gradually diffuses in a circular shape from the grouting hole. When the preheating temperature increases, the final diffusion radius of the slurry gradually increases. The final diffusion radii of the slurry under the preheating temperature of 30–34 °C are 0.25 m, 0.28 m, 0.31 m, 0.34 m, and 0.38 m, respectively. The final diffusion radii of the slurry under the preheating temperature of 35–39 °C are 0.448 m, 0.452 m, 0.456 m, 0.459 m, and 0.462 m, respectively.

Figure 17 shows the variation of the slurry diffusion radius with time under different preheating temperature conditions. From 0 s to 6 s, the diffusion rate of slurry under ten conditions is basically the same. After 6 s, the diffusion rate of the slurry corresponding to different preheating temperatures gradually shows significant differences. The diffusion rate of the slurry in the preheating temperature range of 30–34 °C was much lower than the corresponding value in the preheating temperature range of 35–39 °C.

Figure 18a shows the polymer temperature variation at a distance of 0.1 m from the center point under different preheating temperature conditions. It can be seen that under the ambient temperature of 30 °C, the polymer temperature firstly increases rapidly and gradually decreases after reaching the peak value. As the preheating temperature increases, the time for the slurry temperature to reach its peak gradually decreases, which is 51 s, 50 s, 48 s, 47 s, 45 s, 44 s, 43 s, 41 s, 40 s, and 39 s, respectively. Figure 18b shows the corresponding peak temperature of the slurry under different preheating temperature conditions. It can be seen that the peak temperature increases with the increase in preheating temperature, and the relationship between the two is approximately linear.

Considering the variability of ambient temperature during actual grouting construction, the influence of different ambient temperatures on the slurry diffusion process in fractures was calculated and analyzed. Figure 19 shows the variation of the slurry diffusion radius with time under ambient temperatures of 15 °C, 20 °C, 25 °C, and 30 °C, with slurry preheating temperatures of 34 °C and 36 °C, respectively. It can be seen that under the same preheating temperature conditions, the higher the environmental temperature, the faster the slurry diffusion rate, and the larger the diffusion radius. When the ambient temperature rises to a certain value, the final diffusion radius of the slurry no longer changes significantly. At a preheating temperature of 34 °C, the slurry diffusion radius reaches 0.281 m, 0.314 m, 0.34 m, and 0.382 m under ambient temperatures of 15 °C, 20 °C, 25 °C, and 30 °C, respectively. At a preheating temperature of 36 °C, the slurry diffusion radius under ambient temperatures of 25 °C and 30 °C is very close, reaching 0.448 m and 0.452 m.

7. Conclusions

The main conclusions obtained through this study are the following:

(1) Based on the slurry chemical reaction mechanism and computational fluid dynamics theory, a chemical reaction-fluid dynamic simulation model was established, which contains the polymer slurry and the fracture wall. The chemical reaction kinetic parameters of the polymers were obtained through the PSO inversion method. The quasi-three-dimensional coordinate grid FVM was used to discretize the slurry flow control equation, and the three-dimensional coordinate grid FVM was used to discretize the fracture wall thermal conductivity equation. The coupling solution of the chemical reaction process, flow field, and temperature field of the slurry-fracture system has been achieved;

(2) A polymer fracture grouting model test was conducted. The applicability of the CF model was verified by comparing the tested data with the calculated results. The comparison results show that the average relative errors of the slurry diffusion radius between the calculation and test are −2.23%. The average relative error of slurry temperature changes is 1.63% at a distance of 0.1 m from the center point and 3.05% at a distance of 0.2 m from the center point. The average relative error of polymer final density is 1.44%. The results show that the CF model results are closer to the test results, which can be used to simulate and analyze the slurry diffusion process in fractures;

(3) The influence of preheating temperature and ambient temperature on the slurry diffusion behavior in fractures was analyzed. The results show the following:

• Under the same ambient temperature conditions, as the slurry preheating temperature increases, the reaction rate, diffusion rate, and heating rate of the slurry increase, and the slurry peak temperature increases, reaching the peak point earlier.
• Under the same preheating temperature conditions, the higher the ambient temperature, the faster the slurry diffusion rate and the larger the diffusion radius. When the preheating temperature of the slurry increases to a certain value, the final diffusion range of the slurry no longer changes significantly.