Study on the mechanism and simulation of slurry diffusion in fractured rock mass based on detour effect

The grout diffusion law and grouting effect of fractured rock mass are affected by the roundabout effect. In this paper, based on the power law fluid theory, the formula of slurry diffusion mechanism in fractured rock mass considering the circuitous effect is derived; Further, based on COMSOL numerical simulation software, the influence of grouting pressure and grouting time on the slurry diffusion range is studied; Finally, through comparative analysis, the correctness and feasibility of the theoretical analysis are verified based on the numerical simulation results. The research results can provide theoretical reference and technical support for the design and construction of grouting engineering.


Introduction
Grouting is one of the common methods for reinforcement of engineering fractured rock mass, and is widely used in traffic, water conservancy, mining and other fields.However, the development of grouting theory is relatively backward, which leads to certain experience and blindness in grouting construction.Therefore, the research on the slurry diffusion law has become a hot issue of concern at present.
Scholars at home and abroad have carried out a lot of research work on the slurry diffusion law and achieved fruitful research results.Shang Hongbo et al. [1] built an equivalent fractured rock mass model, studied the diffusion law of grouting slurry in horizontal holes of fractured rock mass, and deduced the theoretical relationship between grouting pressure and grouting time and diffusion distance.Based on fractional Brownian motion, Liu Yang et al. [2] established a numerical model of rough fracture grouting in rock mass through COMSOL numerical simulation software, and analyzed the influence of slurry viscosity and fracture surface roughness on slurry diffusion.M Zhai [3] studied the mechanism of mud diffusion in horizontal fractures of fractured aquifers by establishing a theoretical model.Zhang Guohua et al. [4] described the diffusion law slurry in inclined plane fractures by the finite element method.Li Shucai et al. [5], based on the theoretical model of percolation effect, established the slurry motion equation considering the percolation effect, established the slurry percolation diffusion model using numerical simulation software, and revealed the influence mechanism of percolation effect on the slurry diffusion and effective reinforcement range.Based on Herschel-Bulkley rheological model with timevarying viscosity, Liu Haiming et al. [6] deduced the formula of splitting grouting diffusion radius of Herschel-Bulkley rheological model, and analyzed the influencing factors of slurry diffusion.Liu Zhaoxing et al. [7] established the slurry diffusion model of Newtonian fluid and viscous-time variable fluid in the horizontal grouting hole of inclined fracture, and calculated and analyzed the change law of the slurry diffusion trace and diffusion distance of inclined fracture under different factors.In order to solve the problem of water gushing accident when the high-temperature deep-buried tunnel passes through the water-rich stratum, Niu Jiandong et al. [8] adopted the method of indoor test and numerical simulation to study the diffusion law of slurry in the fissure at different temperatures.
To sum up, scholars at home and abroad have carried out a lot of research work in the field of grout diffusion mechanism in fractured rock mass, and obtained relatively rich research results, but these theories have not considered the influence of detour effect on the grout diffusion law.Based on the power law fluid theory, this paper establishes a columnar mechanical model of grout diffusion microflow unit in fractured rock mass, and carries out the study of grout diffusion mechanism in fractured rock mass considering detour effect, It is verified by numerical simulation, which provides theoretical reference and technical support for the design and construction of grouting engineering.

Analysis of slurry diffusion process
It is very important to study the slurry diffusion mechanism for achieving good grouting effect.The filling of slurry in rock mass is a dynamic filling process in space.Due to the natural micro-fractures, pores and joint surfaces in rock mass, the diffusion process of slurry in rock mass is actually complex and uneven.When grouting, the slurry flows along the grouting pipe, first fills the gap of the grouting pipe, and then compresses and diffuses along the longitudinal and transverse directions under sufficient pressure.Based on this, the slurry diffusion process can be analyzed in two parts, as shown in Figure 1.The first is to spread around along the circumference of the cross section of the grouting pipe; Second, each point on the cross section of the grouting pipe is squeezed and diffused in the longitudinal direction.In order to simplify the study, this paper only analyzes the transverse diffusion of slurry.The detour effect refers to that the liquid does not move along a straight line in the actual flow process, but moves forward in a curve shape in a detour way [9].As shown in Figure 2, the actual flow path of the slurry is shown in bdue to the influence of the naturally existing micropores and cracks in the rock mass, but the current calculation of the fluid is generally based on a.Therefore, in order to better study the actual flow of the slurry, this paper introduces the roundabout effect to study its impact on the slurry diffusion, and the research results are more authentic and accurate.

Equation of slurry diffusion motion
As the transverse diffusion process of slurry is uneven, the following assumptions are made to study its diffusion mechanism: (1) In the process of transverse diffusion, the slurry is uniformly distributed in the longitudinal direction; (2) The transverse and longitudinal diffusion of slurry are two independent processes, and they do not interfere with each other; (3) The longitudinal and transverse pressure at any point in the slurry diffusion range is equal; (4) The bedding surface of the rock mass is smooth, the friction coefficient is ignored, and the velocity of the slurry diffusion process is basically no loss.
In view of the above slurry diffusion process in fractured rock mass, based on the power law fluid theory [10], take any point in the circumferential direction of the slurry diffusion cross section as the research basis, and establish the columnar mechanical model of the slurry diffusion micro-flow unit in fractured rock mass, as shown in Figure 3.As shown in the figure above, the radius of the microfluidic unit is R 0 , and the pressure at both ends of the unit is p 0 and p respectively, so its change rate dp=p-p 0 .According to the power-law fluid theory, the fluid flow mechanism conforms to the following equation: Where, τ is shear stress, c is consistency coefficient, γ is the shear rate, n is the rheological index, v is the flow velocity, and r is the flow radius.
Under the condition of neglecting gravity, atmospheric pressure and other external forces, the stress relationship of the microfluidic unit can be expressed as: Where, dL e is the unit length of the actual flow of the microfluidic unit.
From this, the shear stress of the microfluidic unit body can be obtained τ by 2 e R dp dL When the slurry has not yet diffused, the radius of the microfluidic unit is R 0 , and the flow rate is 0. Based on this, the flow rate v of the power-law fluid can be obtained from the simultaneous equations ( 1), ( 2) and ( 4) Based on the above formula, the flow Q d of the microfluidic unit in unit time meets Therefore, the average flow velocity of the microfluidic unit within the unit distance is In consideration of the seepage velocity and circuitous characteristics of grout in rock mass, the following laws exist [11] V v Where, V is the seepage velocity of the slurry, ε is porosity, K is permeability coefficient, μ is the viscosity of water, ξ is the degree of detour, Le is the length of the actual flow path of the slurry, and L is the length of the actual flow path of the slurry.
According to the simultaneous equations ( 6)~( 9), the diffusion equation of slurry under the detour effect can be obtained as The slurry flow also meets the following requirements: FMIA-2023 Journal of Physics: Conference Series

(2023) 012040
Where, A is the theoretical surface area of the columnar diffusion of the microfluidic unit, and its value is equal to 2πLR 0 .
The formula (11)~(12) can be used to obtain the expression of grouting pressure change with time under the roundabout effect: It should be noted that the derivation of the above mechanical mechanism is only limited to the laminar flow mode of slurry diffusion movement, and whether the turbulent flow mode is suitable or not needs to be analyzed and discussed.

Model establishment and simulation scheme
In order to more intuitively display the slurry diffusion process and its laws, the large-scale finite element multi-physical field coupling analysis software COMSOL Multiphysics is used to study and analyze the slurry permeation and diffusion laws.Based on the lithological characteristics of fractured rock mass, and considering the roundabout effect, the slurry diffusion mechanism formula (formula 11) of fractured rock mass is established to establish a numerical model.Specific model dimensions and boundary conditions are shown in Figure 4, and the corresponding parameters are shown in Table 1, and the scheme of numerical simulation is shown in Table 2.As shown in the above figure, the diffusion radius shows a trend of increasing first and then steading, and when the grouting pressure is 4MPa，the diffusion radius basically no longer increases.In addition, the growth proportion of the slurry diffusion radius shows a trend of increasing first and then decreasing.When the pressure is greater than 4MPa, the proportion decreases obviously, the effect of grouting is not satisfied.

Grouting time.
Taking the grouting pressure is 4 MPa as an example, the slurry diffusion law of each grouting time period is studied.the borehole as the center point, the tunnel trend and inclination are respectively x and y axes, and the size of the internal slurry pressure in the surrounding rock is z axis, the contour map of the internal slurry pressure in the roof surrounding rock of each grouting time period is drawn, In order to express the change situation more intuitively and clearly, the slurry pressure contour map at the grouting time of 1 min, 2 min, 4 min, 6 min, 8 min and 10 min is selected for comparative analysis, as shown in Figure 7.As can be seen from Figure 4, as the slurry gradually moves away from the borehole wall, the internal pressure of the slurry gradually decreases, and the contour map of the internal slurry pressure in the surrounding rock shows a "suona shape" of fine at the top and thick at the bottom.Through comprehensive analysis of the contour map of the internal slurry pressure in the surrounding rock of each time period, it is concluded that the size of the "suona bowl" at the lower part of the "suona shape" contour map of the slurry pressure increases rapidly in the early stage of the grouting time growth.When the grouting time reaches a certain time, the size growth speed of the "suona bowl" at the lower part of the contour map gradually slows down.
Therefore, in order to reduce the impact of the uncertainty of grouting time on the grouting reinforcement effect, the following will conduct quantitative analysis of the slurry infiltration and diffusion effect in each grouting time period to determine a reasonable grouting time.By analyzing and sorting out the original data formed during the slurry diffusion simulation operation in each time period, the slurry diffusion radius under different grouting time is obtained, and the integration analysis is carried out, and finally the slurry diffusion evolution law under the time variable is obtained, as shown in Figure 8.As shown in Figure 8, the diffusion range of slurry is increasing according to increasing of the grouting pressure.The growth ratio shows a trend of increasing first and then decreasing, and when the grouting time exceeds 7 minutes, the growth ratio is less than 10%, that is, the slurry diffusion effect is not satisfied.

Theoretical model verification
In order to verify the grouting diffusion mechanism of fractured rock mass under the detour effect proposed in Chapter 2, based on the above COMSOL numerical simulation, monitor the change rule of the slurry pressure and time in the radial direction during the slurry diffusion process, and substitute the relevant data into Formula (13) to obtain the theoretical calculation results of the slurry pressure of fractured rock mass under the detour effect, and compare and analyze the results with the numerical simulation results, as shown in Figure 9.

Figure 9. Fitting results
The results show that the theoretical value of grout pressure based on formula (13) is basically consistent with the numerical simulation results, and the change rule is completely the same, and the correlation coefficient R 2 is 0.9263, which shows that the grouting diffusion mechanism of fractured rock mass under the roundabout effect proposed in this paper is correct and reasonable, and the mechanical model established is basically consistent with the actual diffusion flow mode of grout.

Conclusion
Using the research method of combining theoretical derivation, numerical simulation and comparative analysis, this paper studies the slurry diffusion mechanism of fractured rock mass based on detour effect, and obtains the following research conclusions: (1) Based on the theory of detour effect and power-law fluid, a columnar mechanical model of grout diffusion micro-flow unit in fractured rock mass is established, and the mechanism of grouting diffusion in fractured rock mass under detour effect is deduced theoretically; (2) Using COMSOL numerical simulation software, the variation of grouting pressure and grouting time in fractured rock mass is studied, and the flow characteristics of grout in fractured rock mass are more intuitively displayed; (3) The grouting diffusion mechanism of fractured rock mass based on roundabout effect is verified by comparing theoretical analysis and numerical simulation results.The verification results show that the slurry diffusion formula considering the detour effect can better reflect the true diffusion law of slurry in fractured rock mass.

Figure 3 .
Figure 3. Mechanical model of slurry diffusion column

Figure 4 .
Figure 4. Model mesh generation and boundary conditions

Figure 7 .
Figure 7. Internal slurry pressure contour map of surrounding rock in each period

Figure 8 .
Figure 8. Slurry diffusion effect at different grouting time

Table 1 .
The parameter of the model

Table 2 .
Scheme of numerical simulation