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

: 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.


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 2 of 21 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 quasithree-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.

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.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].

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.

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.

Test Device and Procedure
The polymer slurry fracture grouting test device shown in Figure 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.

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.

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 Figures 3 and 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.

Chemical Reaction Kinetic Equation
The kinetic equation of the gelling reaction is as follows: 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 Figures 3 and 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.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 Figures 3 and 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.

Chemical Reaction Kinetic Equation
The kinetic equation of the gelling reaction is as follows:

Chemical Reaction Kinetic Equation
The kinetic equation of the gelling reaction is as follows: The kinetic equation of the blowing reaction is as follows: 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].

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.
Sustainability 2023, 15, x FOR PEER REVIEW 6 of 21 The kinetic equation of the blowing reaction is as follows: 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.AOH and AW are the pre-exponential factors, EOH and EW are the activation energies; these four parameters are collectively referred to as chemical reaction kinetic parameters (CRKP).ci,0 is the initial concentration of each component, and ci is the current concentration of each component.Xi is their fractional conversion.R is the universal gas constant.T is the polymer temperature.ri is the mass fraction of each component, and ρi is their density [44,45].

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 vi = (vi1, vi2, vi3, vi4).Its current position is xi = (AOH, EOH, AW, EW).Its corresponding fitness value is fi.The optimal position of particle i is pi = (AOH, EOH, AW, EW).The optimal position of the swarm is pg = (AOH, EOH, AW, EW).The fitness values of pi and pg are expressed as fpi and fg, respectively.
In each iteration, particles move according to the following equation: For the polymer CRKP identification PSO method, assuming there is a single particle i in the swarm n, its searching velocity is 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: where t and t − 1 represent iteration times, r 1 and r 2 are the random factors, and c 1 and c 2 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: 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: 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.
Sustainability 2023, 15, x FOR PEER REVIEW 7 of 21 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: The polymer temperature and density variation with time are calculated when particle i reaches a new position xi = (AOH, EOH, AW, EW), and its fitness value fit is calculated as follows: where Tk is the test polymer temperature, Tk′ is the calculated polymer temperature at the same time, Dj is the test polymer density, Dj′ 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 pg = (AOH, EOH, AW, EW) are shown in Table 2.The temperature and density variation calculated from pg = (AOH, EOH, AW, EW) are compared with test results, which are shown in Figure 7.

CRKP Searching Space Inversion Value Minimum Value
Maximum Value  The inversion CRKP results corresponding to p g = (A OH , E OH , A W , E W ) are shown in Table 2.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.

CRKP Searching Space Inversion Value Minimum Value
Maximum Value 30,000 34,000 32,450.557 25.676 29.676 27.457 37,410 41,000 39,375.422It 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.

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: where subscript BG is the gaseous physical blowing agent and CO2 is carbon dioxide.ri,0 is the initial mass fraction.
At the beginning, the physical blowing agent exists in the slurry as a liquid (rBL,0), and gradually vaporizes when heated.The relationship between rBG and rBL is as follows: Figure 8 shows the change of rBL with temperature during the polymer chemical reaction process, which refers to the method in [46].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.

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: where subscript BG is the gaseous physical blowing agent and CO 2 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: 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: where M i is the molecular weight and r CO 2 ,D is the initial mass fraction of carbon dioxide.rW and rCO 2 are calculated as follows: where Mi is the molecular weight and rCO 2 ,D is the initial mass fraction of carbon dioxide.

Polymer Flowing Control Equation
The polymer slurry satisfies the following mass conservation equation during the process of reaction: 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: where ρ is density, p is the pressure of the fluid microelement, μ is the dynamic viscosity, and Su, Sv, and Sw 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: where T is the temperature.For the polymer slurry, the source term ST can be expressed as the following:

Polymer Flowing Control Equation
The polymer slurry satisfies the following mass conservation equation during the process of reaction: ∂ρ ∂t 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: 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: where T is the temperature.For the polymer slurry, the source term S T can be expressed as the following: where α is the volume fraction of polymer slurry in solution domain, ρ F,0 is the initial density of polymer slurry, and ∆H i is the reaction heat.C F represents the specific heat capacity of the slurry, and its calculation method is as follows: 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: The mass conservation equation and the momentum conservation equation can be expressed as the flowing control equation: 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: 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: The component mass conservation equation of the blowing reaction is as follows: where the source term describes the rate of change in the conversion rate of water.According to Formula (2), its expression is as follows:

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: 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.

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: 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: The following symbols are introduced to sort out the control integral equation for the momentum discrete equation: For the energy discrete equation, the symbols are as follows: where Fi is the flow rate through the interface, Di is the diffusion conductivity of the interface, Ai is the interface area, and δ is the width of the control volume.
Therefore, Equation ( 23) becomes the following: where a P , a W , a E , a S , a N , and b are coefficients.They are expressed as follows: The quasi-three-dimensional FVM model can greatly improve the model calculation efficiency compared with the traditional FVM model.

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:

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: where

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

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.The material parameters used in this model are shown in Table 3.

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.

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.

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.

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.
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.

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.

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.
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.

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.

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%.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 • 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.
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 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.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.

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.

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 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  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.

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.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 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  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 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.
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.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  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.

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-threedimensional coordinate grid FVM was used to discretize the slurry flow control equation,

Figure 1 .
Figure 1.Polymer material and its reaction principle.

Figure 1 .
Figure 1.Polymer material and its reaction principle.

Figure 3 .
Figure 3. Diffusion range of polymer slurry with different preheating temperatures.

Figure 4 .
Figure 4. Variation of polymer diffusion radius with time.

Figure 3 .
Figure 3. Diffusion range of polymer slurry with different preheating temperatures.

Figure 4 .
Figure 4. Variation of polymer diffusion radius with time.

Figure 4 .
Figure 4. Variation of polymer diffusion radius with time.

Figure 7 .
Figure 7.Comparison results between the inversion result and the test data.

Figure 7 .
Figure 7.Comparison results between the inversion result and the test data.

Figure 8 .
Figure 8.The change of rBL with temperature.

Figure 8 .
Figure 8.The change of r BL with temperature.

Figure 11 .
Figure 11.Comparison of polymer diffusion radius variation with time.
(a) 0.1 m from the center point (b) 0.2 m from the center point

Figure 12 .
Figure 12.Comparison of polymer temperature variation with time at different locations.

Figure 11 .
Figure 11.Comparison of polymer diffusion radius variation with time.

Figure 11 .
Figure 11.Comparison of polymer diffusion radius variation with time.
(a) 0.1 m from the center point (b) 0.2 m from the center point

Figure 12 .
Figure 12.Comparison of polymer temperature variation with time at different locations.Figure 12.Comparison of polymer temperature variation with time at different locations.

Figure 12 .
Figure 12.Comparison of polymer temperature variation with time at different locations.Figure 12.Comparison of polymer temperature variation with time at different locations.

FigureFigure 13 .
Figure13ais the polymer density variation with time at the center point, and Figure13bis the density distribution from the center point along the radius direction.It can be seen from Figure13athat the density of the slurry gradually decreases over time.For Figure13b, 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.

Figure 14 .
Figure 14.Slice diagram of the polymer-consolidated body.

Figure 14 .
Figure 14.Slice diagram of the polymer-consolidated body.

Figure 14 .
Figure 14.Slice diagram of the polymer-consolidated body.

Figure 15 .
Figure 15.Comparison of polymer final density at different locations.

Figure 15 .
Figure 15.Comparison of polymer final density at different locations.
• 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 15 .
Figure 15.Comparison of polymer final density at different locations.

Figure 17 .
Figure 17.Variation of the slurry diffusion radius with time under different preheating temperatures.

Figure 16 .
Figure 16.Variation of the polymer slurry diffusion range under different preheating temperatures.

Figure 16 .
Figure 16.Variation of the polymer slurry diffusion range under different preheating temperatures.

Figure 17 .
Figure 17.Variation of the slurry diffusion radius with time under different preheating temperatures.

Figure 17 .
Figure 17.Variation of the slurry diffusion radius with time under different preheating temperatures.
(a) Variation of slurry temperature (b) Variation of peak temperature

Figure 18 .
Figure 18.Variation of the slurry temperature change under different preheating temperatures.

Figure 19 .
Figure 19.Variation of the slurry diffusion radius under different preheating temperature conditions.

Figure 18 .
Figure 18.Variation of the slurry temperature change under different preheating temperatures.
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.

Figure 18 .
Figure 18.Variation of the slurry temperature change under different preheating temperatures.

Figure 19 .
Figure 19.Variation of the slurry diffusion radius under different preheating temperature conditions.

Figure 19 .
Figure 19.Variation of the slurry diffusion radius under different preheating temperature conditions.

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

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

Table 2 .
Chemical reaction kinetic parameters of the polymer slurry.
• C was much lower than the corresponding value in the preheating temperature range of 35-39 • C.