Study on the diffusion law of dynamic water grouting in fracture with slurry-rock stress coupling effect

Water inrush disasters in geotechnical engineering are mainly caused by seepage in fractures, and curtain grouting is the most common method to block water flow. To ensure the efficacy of the water blocking curtain, it is necessary to study the slurry’s diffusion pattern. In this study, by means of laboratory experiments and theoretical deductions, the grout diffusion morphology in fractures with the slurry-rock stress coupling effect is revealed, and the corresponding theoretical model is established. First, based on the failure of water-blocking curtain in a hydropower station in Yunnan Province, a four-level four-factor orthogonal table was set up, grouting experiments were conducted using the self-developed fracture grouting device, and the relationship between the main influencing factors and fracture deformation and grout diffusion distance was revealed. Then, based on the Bingham fluid constitutive model, fracture deformation equation and grouting model with flowing water, a new fracture grouting model considering the coupling effect of slurry-rock stress was established. Finally, based on the calculations and experiments, the morphology of slurry diffusion is described, thereby proving the validity of the new model.


Introduction
In recent years, the recurrent incidents of water inrush disasters during the construction of engineering have emerged as one of the main obstacles to the development of engineering science and technology.According to statistics, water inrush accounts for about 40% of all kinds of disasters.The hazards caused by water inrush are extremely severe, posing a threat not only to human life safety but also resulting in significant property losses.Taking the San Du Reservoir Dam as an example (Chen 2007), piping occurred at the elevation of 299.05 m on the left bank slope and at the elevation of 314.66 m on the right bank slope, with a maximum seepage rate of 0.27L/s, which led to casualties and substantial economic damages.In 1996, at Section 15 of the Panjiakou Dam (Han 1996), the seepage flow at the foundation continually increased due to the infiltration damage to the anti-seepage curtain.In addition, instances such as the St. Francis Dam in the United States, the Kamaraşa Dam in Spain, and the Kaibang Reservoir in Turkey (Ma and Sheng 2001;Zhang 2015;Huang 2014) have also been affected by similar issues.
Curtain grouting is currently the most widely employed water blocking method in geotechnical engineering.Its principle involves injecting slurry around structure to create a vertical or slanting grouting barrier.This aims to improve the groundwater flow conditions, prevent or reduce water or solute migration, thereby maintaining structural stability and environmental safety.This method is commonly applied in underground engineering, tunnels, excavations, waterproofing, and environmental restoration.Therefore, selecting the 512 Page 2 of 18 curtain grouting method as the primary focus of research is based on its significant application in waterproofing projects and the need for further optimization and in-depth study of this method (Lavrov 2023;Niu et al. 2022).
Due to the concealed and complex nature of grouting, they are considered as typical "black-box" problem.In practical construction, the effectiveness of grouting primarily relies on the experience of the construction personnel.To ensure that grouting materials can fully fill the target area, thus preventing the occurrence of water inrush disaster, it is necessary to make prudent choices of grouting parameters and estimate the diffusion range of the slurry in the design stage.Therefore, it is particularly important to study grouting theoretical model to guide construction.Through the study of grouting model, the relationship between various grouting factors and grouting diffusion radius can be comprehensively considered, and the design of grouting parameters can be optimized, so as to improve construction efficiency, reduce costs, and ensure engineering stability.To date, mature grouting models include infiltration grouting, compaction grouting, dynamic water grouting, split grouting, and fracture grouting (Zhou et al. 2019;Zhang et al. 2009;Yang et al. 2001Yang et al. , 2011;;Hao et al. 2001;Wang et al. 2023b;Zou et al. 2006;Zhang and Zou 2008).
In recent years, scholars have carried out further research on the basis of mature grouting models, and are committed to deepening the understanding and application of grouting (Wang et al. 2023a;Zhai et al. 2022;Ying et al. 2022;Huang et al. 2018;Li et al. 2019;Qie et al. 2021;Xie et al. 2022).For example, Shang and Wang (2022) conducted grouting research on the diffusion of slurry during horizontal hole grouting, considering the influence of gravity on the slurry diffusion, and established a model for horizontal hole grouting.Wei et al. (2020) derived the mass conservation equation and flow equation for slurries by analysing the internal mass change of the slurry-injected coal mass and established a variable mass percolation model of slurry for the plugging of fractured coal masses.To calculate the diffusion of C-S slurry in planar fractures, Zhang et al. (2018) applied the mass conservation equation to slurry viscosity that varied with time, and the results revealed the intrinsic link between the rheology of C-S slurries and the influence of other parameters on fracture grouting.An explicit algorithm model was proposed by Mohajerani et al. (2015) for slurry diffusion in rock joints, which can be used to simulate pore fluid pressure and describe its effect on permeability.Qi et al. (2022) used smooth and flat fractures with arbitrarily inclined finite boundaries and Bingham fluids as the object of study, established a high-level grouting model for fractures, and derived the analytical solution under working conditions with a constant grouting rate.
However, in practical grouting projects, the actual grouting often takes place within complex geological and engineering contexts, where the coupling effects of different factors can significantly affect grouting efficiency and engineering stability.Thus, a more precise consideration of multi-field coupling effects is needed.Researchers have gradually shifted their focus from single-field grouting to multi-field coupling grouting (Liu et al. 2019(Liu et al. , 2020(Liu et al. , 2022(Liu et al. , 2023;;Yan et al. 2021;Xu et al. 2023;Li et al. 2020;Yang et al. 2020;Cheng et al. 2017;Wang et al. 2018Wang et al. , 2021)).By introducing multi-field effects, establishing coupling models, and integrating experiments with numerical simulations, they have achieved a more comprehensive understanding and accurate prediction of the multi-field coupled grouting process.Among them, using theoretical analysis, Guo et al. (2020) and Zheng et al. (2022) established theoretical models for single fracture grouting under dynamic water condition.The results from the model were compared with test results, which verified the rationality of the theoretical model.The findings of the study can be of certain reference value for the design of fracture dynamic water grouting.Taking Bingham fluids as the research object, on the basis of Darcy's law, Wang et al. (2019) developed a formula for calculating the diffusion radius of paste considering its selfweight, and a model laboratory test was used to verify the calculation formula.The diffusion model of crack grouting under dynamic water conditions was developed by Liu et al. (2017) based on the time-varying viscosity characteristics of two commonly used fast-setting grouts, and the diffusion behaviour of crack grouting under the condition of dynamic water was studied.On the other hand, to study the diffusion behaviour of slurries during high-temperature injection in fractured rock, Yu et al. (2022) conducted several sets of fractured grouting tests.The sensitivity of slurry diffusion behaviour to various factors, including the formation temperature, flow rate per unit time, and water-cement ratio, was evaluated.
By continuously advancing the research in grouting theory and combining it with engineering practice, it is possible to drive innovation and optimization in grouting processes, leading to more efficient and reliable construction outcomes.For example, according to the U-shaped diffusion theory of horizontal fracture dynamic water grouting, Yu et al. (2014) developed a small-scale fracture dynamic water grouting model test system with variable inclination angle and performed model tests on the influence mechanism of the inclination angle on the diffusion pattern of fracture dynamic water grouting.The findings of this study are of certain value for the guidance of the management of sudden surges that cause water hazards.Through model tests and numerical simulations, Jian et al. (2012) investigated the diffusion pattern of cement slurry in planar fractures under hydrostatic and dynamic water conditions.In addition, suggestions were proposed to improve the conventional grouting process.A series of experiments were conducted by Zhang et al. (2011) on the diffusion pattern of slurries for chemical grouting in single fractures under different dynamic water flow velocity conditions.A theoretical model was used to calculate the effect of flow velocity on the diffusion of slurry along and against the direction of water flow.
Although significant progress has been made in the study of fracture grouting, the current researches mostly focus on the flow of slurry within the fractures, assuming that the fracture aperture remains constant during the grouting process.This assumption does not align with actual circumstances.The injection of high-pressure grout unavoidably causes deformation at the fracture, influencing slurry diffusion.Moreover, this impact becomes more pronounced as the scale of the engineering project increases.Currently, such deformation has been observed by researchers and has undergone some study (Liu et al. 2018;Xiao and Zhao 2020;Wang et al. 2021), but there is still no reliable and accurate grouting model proposed for grout diffusion patterns that accounts for fracture deformation.The influence of dilation during high-pressure grouting was discussed by Gothall and Stille (2010).In addition, both linear stiffness and nonlinear fracture stiffness are used in the modelling.Zheng et al. (2015) investigated the interaction between slurry and surrounding rocks during slurry injection based on the rock unloading-loading theory and the fracture slurry transport equation, findings of the study have certain guidance and reference value for fractured rock grouting projects.Rafi and Stille (2015) developed a grouting model based on the elastic jacking mechanism and Bingham fluids to investigate the effect of grouting pressure and fracture aperture variation on slurry diffusion and plugging efficiency in rock fractures.An explicit algorithm model was proposed.
This study aims to investigate the diffusion law of dynamic water grouting in fracture with slurry-rock stress coupling effect.An orthogonal grouting experiment was conducted with a self-developed fracture grouting device, and the relationship between the main influencing factors, fracture deformation, and grout diffusion was analyzed.A dynamic water grouting model with slurry-rock stress coupling effect was established through theoretical derivation, and the rationality and accuracy of the model were verified by comparison with the test results.The results of this study provides a new theoretical basis and technical support for grouting reinforcement of fractured rock masses.

Engineering background and water inrush disaster
Located in Yunnan Province, the hydroelectric power station project undertakes the responsibilities of power generation and farmland irrigation.Rock strata is N5°-20° W, NE or SW75°-90°, and the fracture dip angle of the rock stratum in the section with toppling deformation is 30°-70°.Rock mass faults, interlayer dislocation zones, and fractures are widely distributed in this area, and the dominant direction is NNW, followed by NE.The fractures in the right bank are mainly small-scale faults in the compression fracture zone with a width less than 1 m and a high fracture dip angle.
At 00:30 on 28 October 2016, the reservoir water level reached 1362.8 m elevation, and the flow rate of the measuring weir reached 115.53L/s.Water inrush occurred in many places in the right bank No. 3 grouting tunnel (1328 m elevation), of which the largest amount of water leaked at the top of the palm face with a flow rate of about 60L/s.The situation of water inrush is shown in Fig. 1 (Sun et al. 2019).
To comprehensively investigate the efficacy of the waterblocking curtain formed by grouting, holes for hydrogeological investigation were prepared within the impermeable curtain section of the grouted tunnel.Packer tests were performed to study the seepage characteristics and monitoring data were obtained.The test results indicated the presence of a highly permeable layer within the natural medium at the dam's base, and despite the implementation of curtain grouting, this permeable layer had not been completely and effectively sealed.The primary cause of the water inrush was attributed to the discontinuities within the water-blocking curtain.

Orthogonal experimental protocol
According to the engineering geological report, the rock mass of the project was intact, and the slurry and groundwater flowed mainly along the fractures between the large rock masses.By combining the engineering geological report with related references, four factors were selected for this test: grouting pressure, flow velocity, water-cement ratio and fracture deformation coefficient.Among the four factors, the fracture deformation coefficient could not be directly measured, and thus high-performance springs were installed within fractures.Combined with the loading device, the spring stiffness coefficient was used to replace the fracture deformation coefficient.
In addition, four levels were considered for each factor, and the factors and corresponding levels are shown in Table 1.The hierarchical selection of experimental design parameters was determined through comprehensive consideration, combining engineering practicality, geological reports, and relevant references, utilizing a preliminary testing approach.First, the geological report provided information on strata properties and hydrological conditions, forming the basis for the selection of parameter hierarchies.Second, consulting relevant references allowed for an understanding of the experience and recommendations regarding parameter hierarchy selection from previous similar studies.However, due to the unique geological conditions and requirements of each engineering, directly applying parameters from the literature may not be suitable.Therefore, in conjunction with the practical grouting design of the project, the preliminary testing approach was employed.By observing the results of initial tests and analyzing the diffusion of slurry, the selection of parameter hierarchies could be derived.
For the experiment with 4 factors and 4 levels, conducting a single-factor experiment would require 256 experimental groups, which was not optimal.Therefore, orthogonal experiment was more appropriate for investigating this issue.Orthogonal experimental design is a multi-factor, multi-level experimental method that assists researchers in systematically studying the influence of multiple factors on experimental results with a reduced number of experiments.The current experiment involves four factors that affect the grouting effect.Through orthogonal experimentation, these factors can be considered simultaneously, ensuring their independence and allowing analysis of their mutual interactions and effects.This approach identifies the main influencing factors, enhances experimental efficiency, and optimizes experimental results.
In this experiment, the orthogonal test scheme is shown in Table 2. Table 2 includes a total of 16 groups of orthogonal experiments.

Self-developed experimental equipment
To study the effects of different grouting parameters on fracture deformation and diffusion pattern.
A set of experimental equipment was designed by our research team for grouting with flowing water while  considering the coupling effect of slurry-rock stress.The experimental equipment is shown in Fig. 2, which mainly includes four major parts: grouting device, water supply device, loading device and data monitoring system.

Loading device
To simulate fracture deformation during grouting, the loading device was designed, as shown in Fig. 3.This apparatus comprised two layers: an upper layer and a lower layer.The upper layer was made of a 79.5 cm × 49.5 cm transparent glass plate with a thickness of 2 cm.An 0.5 cm-thick waterproof rubber seal was attached to all sides of the plexiglas to prevent leakage of slurry during the experiment.The principal structure of the lower layer was an 80 cm × 54 cm × 2 cm steel plate made of Q235 steel.Lateral restraining edges 2 cm in width and 5 cm in height were provided on both sides.
According to the experimental requirements, a certain amount of weight was placed on the loading hook and evenly distributed on the plexiglass plate through the pressure-bearing frame.High-pressure springs were placed between the plexiglass plate and the base to keep the fracture stable, so that the fracture as a whole was under pressure.The mass of the weight and the stiffness coefficient of the spring were adjusted to simulate the stable ground stress and different deformation coefficients of the fracture.

Grouting device
In the experiment performed in this study, the grouting device was a manual grouting pump with a range of 0 ~ 1 MPa, which was connected to the loading device through a 1.5 m grouting pipe, as shown in Fig. 4.

Water supply device
The water supply device was composed of an electric pump, water supply pipeline, water inlet and outlet, which could be used to provide stable water flow at different velocities.The water supply device is shown in Fig. 5.A water inlet box was equipped on the left side of the plexiglass plate.The front side of the box had a large hole as the water inlet, while a row of small holes was drilled on the opposite side to ensure laminar flow of the water, as shown in Figs. 6 and  7.This design ensured a stable water flow in the fracture during the test.

Data monitoring system
A monitoring point was set at the grouting hole to monitor the fracture deformation during grouting.In addition, the vibration string displacement metre and TST3826E static strain gauging box were used for data recording.Each box had 60 measuring points, thereby meeting the requirements of the experiment.The wiring methods included a full bridge, half bridge, 1/4 bridge, and other components.In this experiment, the vibrating string displacement metre was used to monitor the displacement by connecting the acquisition instrument based on the bridge method (Figs. 8 and 9).

Analysis of experimental results
Range analysis was employed to determine the influence of various factors on the fracture deformation and diffusion distance during grouting.Factors A, B, C and D corresponded to the grouting pressure, flow velocity, water-cement ratio and spring stiffness coefficient, respectively.

Range analysis of fracture deformation
Table 3 shows the range analysis process of fracture deformation under different working conditions.According to the data of factor A, K 4 > K 3 > K 2 > K 1 , it shows that the fracture deformation increases with the increase of grouting pressure.Using the same calculation method, the effects of the four factors on the fracture deformation during grouting were obtained, and the results are shown in Fig. 10. Figure 10 shows that the fracture deformation during grouting increased with increasing grouting pressure and flow velocity, decreased with increasing spring stiffness coefficient and fluctuates with increasing water-cement ratio.
The magnitude of the effect of each factor on fracture deformation could be judged by the range R, which was determined by the following equation: The ranges of the factors are shown in Fig. 11. Figure 11 shows that R D >R A >R B >R C , which indicated that factor D (spring stiffness coefficient) had the most significant influence on the fracture deformation, followed by factor A (grouting pressure), which also had a considerable effect.The influence of factor B (flow velocity) was the third, while the influence of factor C (water-cement ratio) was the smallest.Thus, the order of influence of the test factors on the fracture deformation was determined as D → A → B → C.

Range analysis of diffusion distance
Table 4 shows the diffusion distance under different working conditions.The results of range analysis are shown in Figs. 12,13,14,15,16 and 17.As shown in Fig. 12, under the condition perpendicular to the direction of water flow, the grout diffusion distance increased with increasing grouting pressure, water-cement ratio and spring stiffness coefficient, while it was basically unchanged with increasing flow velocity.Figure 13 shows that R A >R C >R D >R B , indicating that factor A (grouting pres- sure) had the most significant influence on the diffusion distance, followed by factor C (water-cement ratio), which also had a considerable effect.The influence of factor D (spring  As shown in Figs. 14 and 16, under the conditions of upstream and downstream, the grout diffusion distance increased with increasing grouting pressure, water-cement ratio and spring stiffness coefficient.In the downstream case, the grout diffusion distance increased with increasing flow velocity.In the upstream case, the grout diffusion distance decreased with increasing flow velocity. Figures 15 and 17 show that R A >R B >R C >R D , indicating that factor A (grouting pressure) had the most significant influence on the diffusion distance, followed by factor B (flow velocity), which also had a considerable effect.The influence of factor D (water-cement ratio) was the third, while the influence of factor C (spring stiffness coefficient) was the smallest.Thus, the order of influence of the test factors on the fracture deformation was determined as A → B → C → D.
According to Figs. 13, 15 and 17, the grouting pressure was the most important factor affecting the diffusion distance in three directions.The increase in grouting pressure improved the kinetic energy of the slurry itself and thus that the diffusion distance also increased.Second, the velocity of flow increased or decreased the grouting pressure to a certain extent in the upstream and downstream direction, which indirectly affected the diffusion distance of slurry.
Third, according to the available reference, the flow pattern of pure cement slurry could be divided into three different flow patterns depending on the water-cement ratio.The cement slurry with a water-cement ratio between 0.5 and 0.7 was power-law fluid, the cement slurry with a water-cement ratio between 0.8 and 1.0 was Bingham fluid, and the cement slurry with a water-cement ratio greater than 1.0 was Newtonian fluid.In the diffusion process of Bingham fluid, it was necessary for the grouting pressure to overcome the viscosity and shear yield strength of the slurry, while the diffusion of Newtonian fluid only needed to overcome the viscosity of the slurry.Therefore, as the water-cement ratio increased from 0.8 to 1.4, the slurry gradually changed from Bingham fluid to Newtonian fluid, and the shear yield strength gradually decreased with increasing water-cement ratio, so that the diffusion distance gradually increased with increasing water-cement ratio under the same conditions.In addition, according to the above section, the fracture deformation gradually decreased with increasing spring stiffness coefficient under the same working condition, and thus that the fracture space to be filled by the slurry decreased and the diffusion distance increased.

Reinforcement suggestion
The occurrence of water inrush is attributed to the failure of establishing a complete water-blocking curtain through grouting.The objective of reinforcement grouting is to maximize the diffusion range of the slurry while ensuring the filling of fractures in the bedrock.Therefore, based on the above experimental results, increasing grouting pressure and water-cement ratio is advantageous for slurry diffusion.However, an elevated water-cement ratio simultaneously decreases the concrete content in the slurry, leading to a reduction in the impermeability of the water-blocking curtain.Nevertheless, increasing grouting pressure can induce further deformation of fractures, enhancing grout filling efficiency.To some extent, this can offset the decrease in curtain impermeability caused by the increased water-cement ratio.
Considering the on-site construction conditions, it is recommended to introduce an additional row of reinforcement grouting holes between the existing holes on the right bank, with a spacing of 1.5 m.Meanwhile, the grouting pressure should be moderately increased to 0.4 MPa, and the water-cement ratio of the slurry can be adjusted to 4:1.According to the field monitoring results, the permeability of the dam foundation significantly decreased after the reinforcement grouting, which indicates that the reinforcement suggestion proposed in this paper is suitable and effective.

Model for fracture grouting
A qualitative analysis of the above experimental results was conducted to investigate the influence of various factors on grout diffusion to a certain extent, providing effective guidance for grouting design in practical engineering.In this section, on the basis of the above test results, a quantitative study on grout diffusion was performed, and a new fracture grouting model that considered the coupling effect of slurry-rock stress was established.

Basic assumptions of the model
After reviewing the relevant references (Gothall and Stille 2010;Zheng et al. 2015;Rafi and Stille 2015) and considering the actual experimental situation, the following assumptions were made to simplify the calculation process for grouting with flowing water while considering the coupling effect of slurry-rock stress: 1.The slurry was assumed to be an incompressible, homogeneous, isotropic Bingham fluid.
2. No slip occurred between the sidewalls of the fracture, and the slurry flow rate at the upper and lower sidewalls was assumed to be zero.3. The slurry was assumed to be in laminar flow during diffusion.4. It was assumed that the slurry only diffuses in the fracture.5. Grout was assumed to diffuse uniformly through horizontal fractures without considering the effect of gravity on the crack aperture.6.Only elastic deformation of the rock on either side of the fracture was assumed to occur.7. The static shear force generated by the flow of water on the slurry was ignored.

Model for fracture deformation
In engineering scenarios, the fracture's both sides experience compression due to the ground stress.Due to the roughness of the fracture, the rock masses on either side of the fracture can only connect through a narrow contact surface, resulting in a certain aperture between the fracture rock surfaces that provides space for the slurry.During grouting, the slurry flows within the fracture, exerting pressure on the adjacent rock in a direction counter to the rock's compression.Consequently, the slurry bears a portion of the compression force.
When the slurry pressure exceeds the critical pressure, the rock masses on either side of the fracture start to separate, leading to an enlargement of the fracture aperture.
The fracture aperture can be divided into two stages with the critical pressure as the boundary.When the slurry pressure is below the critical pressure, the fracture aperture is equal to the initial fracture aperture b 1 .When the slurry pres- sure exceeds the critical pressure, it is appropriate to use the Goodman model (Goodman et al. 1968), which describes the deformation process of the solid contact surface, and the fracture aperture is assumed to be linearly related to the slurry pressure, as illustrated in Fig. 18.
The equation for fracture deformation can be expressed as follows: ( where b refers to the fracture aperture; b 1 represents the ini- tial fracture aperture; p denotes the slurry pressure; p 1 stands for the critical pressure of fracture deformation; and k n is the elastic coefficient of fracture deformation. According to the Goodman model, the fracture aperture is assumed to be linearly and positively related to the slurry pressure.Consequently, it can be inferred that the fracture deformation and slurry diffusion distance are also linearly related.Figure 19 presents the fracture deformation and slurry diffusion pattern during the grouting process.The slurry pressure decreases along the direction of slurry diffusion within the slurry diffusion range.Therefore, within this range, the fracture can be divided into two regions, the deformation region and the non-deformation region, by taking the critical pressure p 1 as the threshold.The initial fracture aperture remains unchanged when the slurry pressure is below the critical pressure p 1 , the fracture starts to deform, and the fracture aperture gradually widens.Subsequently, the slurry infiltrates not only the initial fracture area but also the space formed by the fracture deformation.Within the deformation region, the coupling effect of the slurry-rock stress has a significant influence on the slurry diffusion pattern.Zhan et al. (2011) theoretically derived a single fracture dynamic water grouting diffusion model.In this study, the coupling effect of slurry-rock stress is considered, and the grout diffusion range is divided into a deformation region and non-deformation region for separate analysis on this basis.The proposed model can accurately describe the grouting diffusion pattern under the influence of the coupling effect of slurry and rock stress.

Grouting model for the deformation region
In the fracture, the slurry diffusion form is axisymmetric, and a plane perpendicular to the fracture is used in the study.The axis of symmetry of the fracture and the direction perpendicular to it are used to establish the coordinate system, as shown in Fig. 20.
The slurry continuously flows in the fracture, exerting vertical pressure on the upper and lower sides of the fracture, thereby inducing upward and downward movement.The fracture aperture gradually decreases from the grouting orifice towards the far terminus along the diffusion trajectory, resulting in an angle between the shear resistance of the slurry and the horizontal direction.Considering the sufficiently large radiation diffusion range of the slurry, the impact of this angle on the shear resistance is negligible.
The fracture aperture calculation model can be simplified as a physical model, where an elastic body in a semi-infinite space is subjected to uniformly distributed pressure.
The axis of symmetry is defined at the centre of the fracture to establish the slurry micro-elements for force analysis.In Fig. 20, v refers to the flow velocity; b 1 represents the initial aperture of the fracture; b 0 represents the aperture of the fracture after expansion; r 0 stands for the radius of the grouting hole, p c denotes the pressure at the grout hole; and where B refers to the shear yield strength and refers to the slurry viscosity.
Combining Eqs. ( 4) and ( 7) gives: When h 0 < |h| < b , the distribution of the slurry rate in the fracture aperture direction is where b refers to the aperture of the fracture.
When |h| ≤ h 0 , the distribution of the slurry rate in the frac- ture aperture direction is The average slurry rate u is Assuming that the slurry spreads in a semicircle in the direction of the flow, the unit flow rate of the slurry q is According to the law of mass conservation, the pressure gradient of the slurry in the diffusion direction can be obtained as follows: Combined with the Goodman model, the differential equation of fracture aperture b with grouting distance r can be obtained: Then MATLAB was used to calculate the solution of Eq. ( 16).The fracture aperture continuously decreased along the diffusion direction with the slurry pressure until it reached b 1 .According to the boundary condition, when r = r 1 and b = b 1 , the diffusion distance obtained.
According to the boundary conditions, when the direction of slurry diffusion is the same as that of the water flow: When they are reversed, the initial value of the calculation is When the direction of slurry diffusion is perpendicular to that of water flow: The amount of slurry injected in a unit time period should be equal to that required to increase the diffusion radius in that time period, and, therefore:

Grouting model for the non-deformation region
Differing from the deformation region, the fracture aperture is unchanged, since the slurry pressure is less than the critical pressure.In this case, the grout micro-element was taken as the symmetry axis in the fracture centre for force analysis, as shown in Fig. 21 As shown in Fig. 21, the slurry pressure at the starting point of the non-deformation region was always the critical pressure p 1 .Therefore, during the calculation of the diffusion distance in the non-deformation region, the diffusion was considered to be that of a Bingham fluid under the condition that the grouting hole radius was r 1 and the grouting pressure was p 1 .As shown in the following equation:

Validation of the model
To verify the correctness of the established model, the experimental results under working condition 7 in Sect. 3 above were chosen as the verification for the grouting model established in this study.The radius of the grouting hole was 2 mm, the width of the fracture was 2 mm, the grouting pressure was 0.2 MPa, the flow velocity was 0.3 m/s, the water-cement ratio was 0.8, and the grouting time was 30 s.The dynamic pressure conversion formula was used to transform the flow velocity and pressure.

Slurry diffusion pattern
The resulting grout diffusion distributions with and without considering the coupling effect of slurry-rock stress under the condition of downstream are shown in in Fig. 22: 1. Disregarding the slurry-rock coupling effect, the fracture aperture remained at 2 mm.Along the slurry diffusion trajectory, the slurry pressure underwent rapid attenuation from the grouting hole, culminating in a final diffusion distance of 164.9 mm under this grouting conditions.2. When the slurry-rock stress coupling effect was considered, the fracture aperture decayed from the grouting hole along the slurry diffusion path.The peak fracture aperture attained a measurement of 3.35 mm, representing 1.675 times the initial fracture aperture.3. When neglecting the slurry-rock stress coupling effect, at t = 30 s, the slurry diffusion radius is 164.9 mm.However, when the coupling effect was considered, it contracted to 127.18 mm.Within this, the length of deformation region is 100.48 mm, while the length of non-deformation region is 26.7 mm.Upon calculation, the slurry filling area encompassing the slurry-rock stress coupling effect was 1.3375 times larger than the area when the effect was disregarded for an equivalent diffusion distance, translating to a growth of 33.75%.The additional space generated by fracture deformation absorbed a portion of the slurry, leading to a reduction in the slurry diffusion radius During grouting, the direction of downstream flow, the direction of reverse flow and the diffusion distance perpendicular to the direction of water flow were recorded every 3 s with the results shown in Fig. 23.
As shown in Fig. 23, the slurry diffusion radius followed the order of downstream, vertical, and upstream, indicating that the water flow facilitated slurry diffusion.At the initial stage of grouting, the slurry diffusion radius in the three directions was almost the same, while the difference in diffusion radius gradually increased over time.In the later stage of grouting, the difference in the slurry diffusion radius in the three directions became apparent, suggesting that the shape of the slurry diffusion area was round during the early stage but became stream line form during the later stage.

Experimental validation and comparative analysis
The new model was used to calculate the diffusion distance, and the results were compared with the experimental data, as shown in Figs. 24,25,and 26.The figures display the slurry diffusion curves with and without considering the slurryrock stress coupling effect, and the experimental data were included for comparison.
As shown in Fig. 27, the diffusion radius calculated without considering the slurry-rock stress coupling effect was significantly larger than the actual diffusion radius.The proposed grouting model, which considered the slurryrock stress coupling effect and included the expansion of fractures, considered that the slurry must fill not only the initial fracture but also the space that developed after the fracture expansion.The calculated results were close to reality.Thus, the comparison demonstrated that the grouting model established in this study was reasonable and effective.
Traditional grouting models often overlook the potential deformations that fractures may undergo during the grouting process.However, fractures can experience deformations such as expansion and contraction during grouting, which can influence the effectiveness of grouting.The grouting model proposed in this paper takes into full consideration the deformation characteristics of fractures.It can more accurately describe the interrelationship between fracture deformation and slurry diffusion, leading to computation results that are closer to real-world engineering scenarios.This enhanced accuracy in depicting the grouting effects

Conclusion
On the basis of previous studies, this study incorporated fracture deformation into the process of fracture grouting.The fracture deformation and grout diffusion law in the grouting process were studied by orthogonal testing with a self-developed fracture grouting device.Through laboratory experiments and theoretical deductions, the corresponding theoretical model was established, and its validity was verified.The main conclusions were drawn as follows: 1.A four-level four factor orthogonal table was set up, and grouting experiments were conducted using the selfdeveloped fracture grouting device.The relationships between the selected factors (grouting pressure, flow velocity, water-cement ratio, spring stiffness coefficient) and fracture deformation and grout diffusion distance were investigated by range analysis.According to the research results and the engineering situation, effective reinforcement suggestions were put forward. 2. Based on the constitutive model for a Bingham fluid, the fracture deformation equation, and the grouting model with flowing water, a two-stage calculation was used to derive a model for grouting in the presence of flowing water and considering the coupling effect of slurryrock stress.By considering fracture deformation during grouting, the slurry diffusion pattern was effectively described.3. Based on a comparison of results from calculations and experiments, the behaviour of slurry diffusion during grouting was described, thereby proving the rationality and validity of the established model for grouting in the presence of flowing water and considering the coupling effect of slurry-rock stress.

Fig. 6
Fig. 6 Front side of plexiglass plate

Fig. 7
Fig. 7 Reverse side of plexiglass plate

Fig. 10
Fig. 10 Range analysis of fracture deformation under various factors

Fig. 12
Fig. 12 Range analysis of diffusion distance under various factors (perpendicular to the flow)

Fig. 14
Fig. 14 Range analysis of diffusion distance under various factors (downstream)

Fig. 16
Fig. 16 Range analysis of diffusion distance under various factors (upstream)

Fig. 19
Fig. 19 Fracture deformation and slurry diffusion pattern . The slurry diffusion equation simplified back to the original Bingham fluid flow formula, with the boundary condition: b = b 0 = b 1 , p = p 1 .

Fig. 21
Fig. 21 Force analysis of slurry motion in non-deformation region

Fig. 22
Fig. 22 Grouting diffusion pattern under different conditions (mm) Fig. 23 Curve of diffusion distance over time

Fig. 24
Fig. 24 Comparison of calculated and experimental results (perpendicular to the flow)

Table 1
Factors and corresponding levels in orthogonal tests

Table 4
Diffusion distance under different working conditions