A method for controlling the diffusion radius of infiltration grouting

Abstract

This invention discloses a method for controlling the diffusion radius of permeable grouting. Addressing the shortcomings in current research on the influencing factors of the diffusion radius of permeable grouting, this method no longer focuses on the influence of the grout's inherent properties on the diffusion radius. Instead, it studies the changes in permeability of porous media caused by grout flow resistance and lateral forces. Using Bingham fluid as an example, a control equation for the diffusion radius of permeable grouting considering the influence of grout flow resistance and lateral forces on the permeability of porous media is derived. This method can reveal the mechanism by which the grout affects the permeability of porous media and effectively solves the problem that the theoretical diffusion radius is larger than the actual diffusion radius in traditional methods.

Description

Technical Field

[0001] This invention relates to the field of diffusion radius control technology for permeation grouting, and specifically to a method for controlling the diffusion radius of permeation grouting. Background Technology

[0002] Grouting is the most convenient and effective way to lift surface subsidence, reinforce strata, improve the bearing capacity of structures, and restore their original shape. Its forms include permeation grouting, fracturing grouting, and compaction grouting. Among these, permeation grouting is the most common method because the grout does not damage the surrounding soil during the grouting process; it only flows and diffuses along the existing pores between soil particles. The grouting process is relatively easy, and therefore it was the earliest discovered and established grouting method.

[0003] For infiltration grouting, studies on the influencing factors of grout diffusion radius have focused on external factors such as grouting time and pressure, as well as the properties of the grout itself, such as the time-varying viscosity. These studies have derived the controlling equation for the diffusion radius of infiltration grouting based on fluid rheological and equilibrium equations. However, the differences between various forms of fluid rheological equations lead to limitations and complexities in the theoretical derivation, and the results often have significant errors compared to actual values. Furthermore, existing technologies suggest that when grout flows within a porous medium, it exerts two forces on the solid boundary: a resistance along the velocity direction caused by the viscosity of the boundary surface, and a transverse force perpendicular to the velocity direction caused by changes in surface pressure. Both forces affect the solid boundary of the porous medium, thus changing its permeability. However, current research on infiltration grouting does not consider the influence of the grout on the permeability of the porous medium during the grouting process.

[0004] Therefore, it is necessary to study the effects of grout flow resistance and lateral force on the permeability of porous media during the grouting process, while studying the grout velocity equation. Summary of the Invention

[0005] Therefore, it is necessary to provide a method for controlling the diffusion radius of permeation grouting to address the existing problems.

[0006] This invention provides a method for controlling the diffusion radius of penetrating grout, the method comprising:

[0007] S1: Using the continuity equation and motion equation of viscous fluid, the partial differential equation of velocity of Bingham fluid in the X direction is obtained. The partial differential equation of velocity includes slurry viscosity and slurry pressure gradient.

[0008] S2: The width of the core region of Bingham fluid during flow is obtained by using the slurry pressure gradient and the equilibrium equation of slurry micro-elements;

[0009] S3: Based on the slurry viscosity, slurry pressure gradient, width of the core region, and boundary conditions of Bingham fluid in the grouting pipe, the velocity equation of the slurry in the pipe is obtained;

[0010] S4: Based on the velocity equation of the grout in the pipe, the formula for calculating the grouting volume per unit time is obtained;

[0011] S5: Based on the formula for calculating the grouting volume per unit time, the average velocity of the grout micro-clusters is obtained;

[0012] S6: The percolation velocity of Bingham fluid in porous media is obtained by the average velocity of the slurry micro-particles;

[0013] S7: Based on the seepage velocity calculation formula, the first permeability calculation formula is obtained through the resistance model, and the second permeability calculation formula is obtained through the generalized Hooke's law;

[0014] S8: Integrating the grout pressure gradient based on the seepage velocity and the second boundary condition;

[0015] Substituting the first permeability calculation formula into the slurry pressure gradient integral formula, we obtain the permeation grouting diffusion radius control equation that takes into account the influence of slurry flow resistance.

[0016] Substituting the second permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of the lateral force of the grout.

[0017] Preferably, in S1, the partial differential equation of the Bingham fluid velocity in the X direction is expressed as:

[0018]

[0019] in, (Since p = f(x), it can also be expressed as) The slurry pressure gradient is represented by ), P represents the slurry pressure at a diffusion radius of x, where x represents the diffusion radius, and μ represents the slurry viscosity. This represents the second partial derivative of the slurry flow velocity in the y-direction.

[0020] Preferably, in S2, the equilibrium equation for the slurry particles is expressed as:

[0021] P·πr0 2 -(P+ΔP)·πr0 2 =τ·2πr0dx

[0022] Where P1 represents the pressure at one end of the slurry particle, r0 represents the radius of the slurry particle; P1+ΔP represents the pressure at the other end of the slurry particle, τ represents the shear stress of the slurry particle, and dx represents the length of the particle.

[0023] When τ≤τ0, the slurry forms a core during the flow process. The formula for calculating the width of the core region is:

[0024]

[0025] Among them, h p The width of the core region is represented by τ, and τ0 represents the yield stress of the slurry micro-particles. This represents the slurry pressure gradient.

[0026] Preferably, in S3,

[0027] The boundary conditions for Bingham fluid within the grouting pipe include:

[0028]

[0029] The velocity equation of the slurry inside the pipe is expressed as:

[0030]

[0031] Where u1 represents the slurry flow velocity outside the core retention area, u2 represents the slurry flow velocity within the core retention area, y represents the region perpendicular to the slurry flow velocity, u represents the slurry flow velocity, and h p The width of the core retention area is represented by h, the diameter of the grouting pipe is represented by μ, and the viscosity of the grout is represented by μ. This represents the slurry pressure gradient.

[0032] Preferably, in S4, the formula for calculating the grouting volume per unit time is expressed as:

[0033]

[0034] Where q represents the grouting volume per unit time, and h p The width of the core region is represented by , h represents the diameter of the grouting pipe, μ represents the viscosity of the grout, and τ0 represents the yield stress of the grout particles. The gradient represents the slurry pressure gradient, u1 represents the slurry velocity outside the core region, u2 represents the slurry velocity in the core region, and m represents the region perpendicular to the slurry velocity.

[0035] Preferably, in S5, the infinitesimal quantity in the formula for calculating the grouting volume per unit time is ignored, and the average velocity of the grout particles is obtained. The formula for calculating the average velocity of the grout particles is expressed as follows:

[0036]

[0037] in, The average velocity of the slurry particles is represented by q, the grouting volume per unit time is represented by r0, the radius of the slurry particles is represented by μ, and the viscosity of the slurry is represented by μ. τ represents the slurry pressure gradient, and τ0 represents the yield stress of the slurry micro-elements.

[0038] Preferably, in S6, the seepage velocity of Bingham fluid in the porous medium is obtained based on the relationship between the average velocity of the slurry microparticles and the seepage velocity.

[0039] The relation is expressed as:

[0040]

[0041] The formula for calculating the seepage velocity of Bingham fluid in porous media is as follows:

[0042]

[0043] Where v represents the seepage velocity of Bingham fluid in porous media, and K represents the formation permeability. μ represents the viscosity of the slurry. τ represents the slurry pressure gradient, τ0 represents the yield stress of the slurry microparticles, r0 represents the radius of the slurry microparticles, and φ represents the porosity of the porous medium.

[0044] Preferably, in S7, the Iberall resistance model indicates that the first permeability is a function of the Reynolds number, and the derivation formula for the first permeability is:

[0045]

[0046] in,

[0047]

[0048] The slurry follows the law of spherical diffusion, therefore:

[0049]

[0050] A = 4πx 2 (4)

[0051] Combining equations (1) to (4), we obtain the formula for calculating the first permeability, which is:

[0052]

[0053] K1 represents the first permeability, φ represents the porosity of the porous medium, δ represents the initial pore diameter of the porous medium, Re represents the Reynolds number, ρ represents the slurry density, v represents the seepage velocity of Bingham fluid in the porous medium, q represents the grouting volume per unit time, A represents the diffusion area, Q represents the grouting volume, t represents the grouting time, and x represents the diffusion radius.

[0054] Preferably, in S7, according to the generalized Hooke's law:

[0055]

[0056]

[0057] Where σ represents the slurry pressure at the pore boundary of the porous medium, ε represents the strain of the porous medium under stress, dP represents the pressure change of the slurry along the boundary surface, dδ represents the deformation of the porous medium, l represents the initial average width of the porous medium, E represents the elastic modulus of the porous medium, φ represents the porosity of the porous medium, and δ represents the initial pore diameter of the porous medium.

[0058] Based on formation permeability achievable but

[0059]

[0060] Combining equations (5) to (7), we get:

[0061]

[0062] The formula for calculating the grouting volume within the grouting time t is as follows:

[0063] Q = v·4πx 2 ·t; (9)

[0064] By combining the formulas for calculating the seepage velocity of Bingham fluid in porous media and formula (9), we obtain:

[0065]

[0066] because Therefore, the part in equation (10) can be ignored. Substituting formula (3) and formula (10) ignoring the last term into formula (8), we get:

[0067]

[0068] Substituting the third boundary conditions: x = 0, K = K0 into formula (11), and then integrating formula (11), we get:

[0069]

[0070] Taking the square root of both sides of formula (12), we obtain the second permeability calculation formula, which is:

[0071]

[0072] K2 represents the second permeability, r0 represents the radius of the slurry particle, Q represents the amount of slurry injected within the injection time t, v represents the seepage velocity of Bingham fluid in the porous medium, x represents the diffusion radius, τ0 represents the yield stress of the slurry particle, and K0 represents the initial permeability of the porous medium.

[0073] Preferably, in S8, the grout pressure gradient is integrated based on the relationship between the grouting volume Q and the seepage velocity v, and in combination with the second boundary conditions: P = P0, l = r0;

[0074] Substituting the first permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of grout flow resistance, which is expressed as:

[0075]

[0076] Among them, P fr denoted by x, considering the effect of slurry flow resistance on the permeability of porous media, and with a slurry diffusion radius of x; φ represents the porosity of the porous media, μ represents the slurry viscosity, t represents the grouting time, δ represents the initial pore diameter of the porous media, ρ represents the slurry density, l represents the initial average width of the porous media, r0 represents the radius of the slurry particle, τ0 represents the yield stress of the slurry particle, and P0 represents the boundary value of the slurry pressure with a slurry diffusion radius of x.

[0077] Substituting the second permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of the lateral force of the grout, which is expressed as:

[0078]

[0079] Among them, P tr The value represents the slurry pressure when considering the influence of the lateral force of the slurry on the permeability of the porous medium, and the slurry diffusion radius is x; E represents the elastic modulus of the porous medium, and K0 represents the initial permeability of the porous medium.

[0080] Beneficial effects: The method provided by this invention addresses the shortcomings in current research on the influencing factors of the diffusion radius of permeable grouting slurries. Instead of focusing on the influence of the slurry's inherent properties on the diffusion radius, it studies the changes in slurry flow resistance and lateral quantity on the permeability of porous media. Using Bingham fluid as an example, it derives a control equation for the diffusion radius of permeable grouting that considers the influence of slurry flow resistance and lateral quantity on the permeability of porous media. This method can reveal the influence mechanism of slurry on the permeability of porous media and effectively solve the problem that the traditional theoretical diffusion radius is larger than the actual diffusion radius. Attached Figure Description

[0081] Exemplary embodiments of the present invention can be more fully understood by referring to the accompanying drawings. The drawings are provided to further illustrate the embodiments of this application and form part of the specification. They are used together with the embodiments of this application to explain the present invention and do not constitute a limitation thereof. In the drawings, the same reference numerals generally represent the same components or steps.

[0082] Figure 1 A flowchart illustrating a method provided according to an exemplary embodiment of this application.

[0083] Figure 2 This is a schematic diagram illustrating the effect of lateral force of slurry on the pores of porous media according to an exemplary embodiment of this application.

[0084] Figure 3 This is a schematic diagram illustrating the effect of slurry flow resistance on the pore size of porous media according to an exemplary embodiment of this application.

[0085] Figure 4a This is a schematic diagram showing the effect of grout flow resistance on the permeability of porous media with grouting time.

[0086] Figure 4b This is a schematic diagram showing the effect of lateral force of grout on the permeability of porous media with grouting time.

[0087] Figure 5a This is a schematic diagram showing the effect of grout flow resistance on the permeability of porous media caused by grouting pressure.

[0088] Figure 5b This is a schematic diagram showing the effect of lateral force of grout on the permeability of porous media as a function of grouting pressure. Detailed Implementation

[0089] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0090] It should be noted that, unless otherwise stated, the technical or scientific terms used in this application shall have the ordinary meaning as understood by one of ordinary skill in the art to which this application pertains.

[0091] Furthermore, the terms "first" and "second," etc., are used to distinguish different objects, not to describe a specific order. Additionally, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to those processes, methods, products, or apparatuses.

[0092] Example 1

[0093] This embodiment provides a method for controlling the diffusion radius of permeation grouting, which will be described below with reference to the accompanying drawings.

[0094] Please refer to Figure 1 The figure shows a flowchart of a method for controlling the diffusion radius of permeation grouting according to some embodiments of this application. As shown, the method may include the following steps:

[0095] S1: Using the continuity equation and motion equation of viscous fluid, the partial differential equation of velocity of Bingham fluid in the X direction is obtained. The partial differential equation of velocity includes slurry viscosity and slurry pressure gradient.

[0096] Specifically, the partial differential equation for the velocity of the Bingham fluid in the X direction is expressed as:

[0097]

[0098] in, (Since p = f(x), it can also be expressed as) The slurry pressure gradient is represented by ), P represents the slurry pressure at a diffusion radius of x, where x represents the diffusion radius, and μ represents the slurry viscosity. This represents the second partial derivative of the slurry flow velocity in the y-direction.

[0099] S2: The width of the core region of Bingham fluid during flow is obtained by using the slurry pressure gradient and the equilibrium equation of slurry micro-elements;

[0100] Specifically, the equilibrium equation for the slurry particles is expressed as:

[0101] P·πr0 2 -(P+ΔP)·πr0 2 =τ·2πr0dx

[0102] Where P represents the pressure at one end of the slurry particle, r0 represents the radius of the slurry particle; P+ΔP represents the pressure at the other end of the slurry particle, τ represents the shear stress of the slurry particle, and dx represents the length of the particle.

[0103] When τ≤τ0, the slurry forms a core during the flow process. The formula for calculating the width of the core region is:

[0104]

[0105] Among them, h p The width of the core region is represented by τ, and τ0 represents the yield stress of the slurry micro-particles. This represents the slurry pressure gradient.

[0106] S3: Based on the slurry viscosity, slurry pressure gradient, width of the core region, and boundary conditions of Bingham fluid in the grouting pipe, the velocity equation of the slurry in the pipe is obtained;

[0107] Specifically, the boundary conditions of Bingham fluid within the grouting pipe include:

[0108]

[0109] The velocity equation of the slurry inside the pipe is expressed as:

[0110]

[0111] Where u1 represents the slurry flow velocity outside the core retention area, u2 represents the slurry flow velocity within the core retention area, y represents the region perpendicular to the slurry flow velocity, u represents the slurry flow velocity, and h p The width of the core retention area is represented by h, the diameter of the grouting pipe is represented by μ, and the viscosity of the grout is represented by μ. This represents the slurry pressure gradient.

[0112] S4: Based on the velocity equation of the grout in the pipe, the formula for calculating the grouting volume per unit time is obtained;

[0113] Specifically, the formula for calculating the grouting volume per unit time is as follows:

[0114]

[0115] Where q represents the grouting volume per unit time, and h p The width of the core region is represented by , h represents the diameter of the grouting pipe, μ represents the viscosity of the grout, and τ0 represents the yield stress of the grout particles. The gradient represents the slurry pressure gradient, u1 represents the slurry velocity outside the core region, u2 represents the slurry velocity in the core region, and m represents the region perpendicular to the slurry velocity.

[0116] S5: Based on the formula for calculating the grouting volume per unit time, the average velocity of the grout micro-clusters is obtained;

[0117] Specifically, ignoring the infinitesimal quantity in the formula for calculating the grouting volume per unit time, the average velocity of the grout particles is obtained. The formula for calculating the average velocity of the grout particles is as follows:

[0118]

[0119] in, The average velocity of the slurry particles is represented by q, the grouting volume per unit time is represented by r0, the radius of the slurry particles is represented by μ, and the viscosity of the slurry is represented by μ. τ represents the slurry pressure gradient, and τ0 represents the yield stress of the slurry micro-elements.

[0120] S6: The percolation velocity of Bingham fluid in porous media is obtained by the average velocity of the slurry micro-particles;

[0121] Specifically, based on the relationship between the average velocity of the slurry microparticles and the seepage velocity, the seepage velocity of Bingham fluid in porous media is obtained;

[0122] The relation is expressed as:

[0123]

[0124] The formula for calculating the seepage velocity of Bingham fluid in porous media is as follows:

[0125]

[0126] Where v represents the seepage velocity of Bingham fluid in porous media, and K represents the formation permeability. μ represents the viscosity of the slurry. τ represents the slurry pressure gradient, τ0 represents the yield stress of the slurry microparticles, r0 represents the radius of the slurry microparticles, and φ represents the porosity of the porous medium.

[0127] S7: Based on the seepage velocity calculation formula, the first permeability calculation formula is obtained through the resistance model, and the second permeability calculation formula is obtained through the generalized Hooke's law;

[0128] Specifically, the process of obtaining the formula for calculating the first penetration rate is as follows:

[0129] The Iberall resistance model states that the first permeability is a function of the Reynolds number; therefore, the derivation formula for the first permeability is:

[0130]

[0131] in,

[0132]

[0133] The slurry follows the law of spherical diffusion, therefore:

[0134]

[0135] A = 4πx 2 (4)

[0136] Combining equations (1) to (4), we obtain the formula for calculating the first permeability, which is:

[0137]

[0138] K1 represents the first permeability, φ represents the porosity of the porous medium, δ represents the initial pore diameter of the porous medium, Re represents the Reynolds number, ρ represents the slurry density, v represents the seepage velocity of Bingham fluid in the porous medium, q represents the grouting volume per unit time, A represents the diffusion area, Q represents the grouting volume, t represents the grouting time, and x represents the diffusion radius.

[0139] The process of obtaining the formula for calculating the second permeability is as follows:

[0140] According to the generalized Hooke's Law:

[0141]

[0142]

[0143] Where σ represents the slurry pressure at the pore boundary of the porous medium, ε represents the strain of the porous medium under stress, dP represents the pressure change of the slurry along the boundary surface, dδ represents the deformation of the porous medium, l represents the initial average width of the porous medium, E represents the elastic modulus of the porous medium, φ represents the porosity of the porous medium, and δ represents the initial pore diameter of the porous medium.

[0144] Based on formation permeability achievable but

[0145]

[0146] Combining equations (5) to (7), we get:

[0147]

[0148] The formula for calculating the grouting volume within the grouting time t is as follows:

[0149] Q = v·4πx 2 ·t; (9)

[0150] By combining the formulas for calculating the seepage velocity of Bingham fluid in porous media and formula (9), we obtain:

[0151]

[0152] because Therefore, the part in equation (10) can be ignored. Substituting formula (3) and formula (10) ignoring the last term into formula (8), we get:

[0153]

[0154] Substituting the third boundary conditions: x = 0, K = K0 into formula (11), and then integrating formula (11), we get:

[0155]

[0156] Taking the square root of both sides of formula (12), we obtain the second permeability calculation formula, which is:

[0157]

[0158] K2 represents the second permeability, r0 represents the radius of the slurry particle, Q represents the amount of slurry injected within the injection time t, v represents the seepage velocity of Bingham fluid in the porous medium, x represents the diffusion radius, τ0 represents the yield stress of the slurry particle, and K0 represents the initial permeability of the porous medium.

[0159] S8: Based on the relationship between grouting volume Q and seepage velocity v, and combined with the second boundary conditions: P = P0, l = r0, integrate the grout pressure gradient;

[0160] Substituting the first permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of grout flow resistance, which is expressed as:

[0161]

[0162] Among them, P fr denoted by x, considering the effect of slurry flow resistance on the permeability of porous media, and with a slurry diffusion radius of x; φ represents the porosity of the porous media, μ represents the slurry viscosity, t represents the grouting time, δ represents the initial pore diameter of the porous media, ρ represents the slurry density, l represents the initial average width of the porous media, r0 represents the radius of the slurry particle, τ0 represents the yield stress of the slurry particle, and P0 represents the boundary value of the slurry pressure with a slurry diffusion radius of x.

[0163] Substituting the second permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of the lateral force of the grout, which is expressed as:

[0164]

[0165] Among them, P trThe value represents the slurry pressure when considering the influence of the lateral force of the slurry on the permeability of the porous medium, and the slurry diffusion radius is x; E represents the elastic modulus of the porous medium, and K0 represents the initial permeability of the porous medium.

[0166] BEAR proposed that when slurry flows within the pores of a porous medium, it generates a lateral force perpendicular to the slurry velocity direction and a flow resistance along the slurry velocity direction at the solid boundary.

[0167] like Figure 2 As shown, during the grouting process, the pores in the porous medium are unevenly distributed. The size and curvature of the pores are different, which will lead to different diffusion radii of the grout in different pores. Therefore, for pores with a large grout diffusion range, the lateral force of the grout will inevitably exert a squeezing effect on the boundary of the surrounding pores with a small grout diffusion range, resulting in changes in the diameter of some pores, thereby causing changes in the permeability of the porous medium.

[0168] like Figure 3 As shown, the flow resistance generated by the grout at the boundary of the porous medium is caused by the viscosity and velocity gradient of the grout boundary surface. In actual engineering, the pores in the porous medium are not uniformly distributed. For non-linear pipes, the medium particles at the bends will inevitably be transported due to the flow resistance of the grout, causing the pores to narrow, or even turning open pores into closed pores, thus affecting the permeability of the porous medium. The Iberall model uses a flow resistance model to derive a permeability variation model for porous media. This embodiment, based on the action of two forces on the solid boundary of the porous medium by the grout during the grouting process, obtains an analytical expression for the permeability of the porous medium as a function of the medium pore characteristics, grouting construction parameters, and grout diffusion radius. Based on this, a grout diffusion control equation considering the change in the permeability of the porous medium is established, and the trend of the permeability of the porous medium changing with grouting time and gradual pressure is obtained, such as... Figure 4a , Figure 4b , Figure 5a , Figure 5b As shown.

[0169] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of this application, and they should all be covered within the scope of the claims and specification of this application.

Claims

1. A method for controlling the diffusion radius of permeation grouting, characterized in that, include: S1: Using the continuity equation and motion equation of viscous fluid, the partial differential equation of velocity of Bingham fluid in the X direction is obtained. The partial differential equation of velocity includes slurry viscosity and slurry pressure gradient. S2: The width of the core region of Bingham fluid during flow is obtained by using the slurry pressure gradient and the equilibrium equation of slurry micro-elements; S3: Based on the slurry viscosity, slurry pressure gradient, width of the core region, and boundary conditions of Bingham fluid in the grouting pipe, the velocity equation of the slurry in the pipe is obtained; S4: Based on the velocity equation of the grout in the pipe, the formula for calculating the grouting volume per unit time is obtained; S5: Based on the formula for calculating the grouting volume per unit time, the average velocity of the grout micro-clusters is obtained; S6: The percolation velocity of Bingham fluid in porous media is obtained by the average velocity of the slurry micro-particles; S7: Based on the seepage velocity calculation formula, the first permeability calculation formula is obtained through the resistance model, and the second permeability calculation formula is obtained through the generalized Hooke's law; S8: Integrating the grout pressure gradient based on the seepage velocity and the second boundary condition; Substituting the first permeability calculation formula into the slurry pressure gradient integral formula, we obtain the permeation grouting diffusion radius control equation that takes into account the influence of slurry flow resistance. Substituting the second permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of the lateral force of the grout.

2. The method for controlling the diffusion radius of permeation grouting according to claim 1, characterized in that, In S1, the partial differential equation for the velocity of the Bingham fluid in the X direction is expressed as: in, Let P represent the slurry pressure gradient, where P represents the slurry pressure at a slurry diffusion radius of x, x represents the diffusion radius, and μ represents the slurry viscosity. This represents the second partial derivative of the slurry flow velocity in the y-direction.

3. The method for controlling the diffusion radius of permeation grouting according to claim 2, characterized in that, In S2, the equilibrium equation for the slurry particles is expressed as: P·πr0 2 -(P+ΔP)·πr0 2 = τ·2πr0dx Where P represents the pressure at one end of the slurry particle, r0 represents the radius of the slurry particle; P+ΔP represents the pressure at the other end of the slurry particle, τ represents the shear stress of the slurry particle, and dx represents the length of the particle. When τ≤τ0, the slurry forms a core during the flow process. The formula for calculating the width of the core region is: Among them, h p The width of the core region is represented by τ, and τ0 represents the yield stress of the slurry micro-particles. This represents the slurry pressure gradient.

4. The method for controlling the diffusion radius of permeation grouting according to claim 3, characterized in that, In S3, The boundary conditions for Bingham fluid within the grouting pipe include: The velocity equation of the slurry inside the pipe is expressed as: Where u1 represents the slurry flow velocity outside the core retention area, u2 represents the slurry flow velocity within the core retention area, y represents the region perpendicular to the slurry flow velocity, u represents the slurry flow velocity, and h p The width of the core retention area is represented by h, the diameter of the grouting pipe is represented by μ, and the viscosity of the grout is represented by μ. This represents the slurry pressure gradient.

5. The method for controlling the diffusion radius of permeation grouting according to claim 4, characterized in that, In S4, the formula for calculating the grouting volume per unit time is expressed as: Where q represents the grouting volume per unit time, and h p The width of the core region is represented by , h represents the diameter of the grouting pipe, μ represents the viscosity of the grout, and τ0 represents the yield stress of the grout particles. The gradient represents the slurry pressure gradient, u1 represents the slurry velocity outside the core region, u2 represents the slurry velocity in the core region, and m represents the region perpendicular to the slurry velocity.

6. The method for controlling the diffusion radius of permeation grouting according to claim 5, characterized in that, In S5, ignoring the infinitesimal quantity in the formula for calculating the grouting volume per unit time, the average velocity of the grout particles is obtained. The formula for calculating the average velocity of the grout particles is as follows: in, The average velocity of the slurry particles is represented by q, the grouting volume per unit time is represented by r0, the radius of the slurry particles is represented by μ, and the viscosity of the slurry is represented by μ. τ represents the slurry pressure gradient, and τ0 represents the yield stress of the slurry micro-elements.

7. The method for controlling the diffusion radius of permeation grouting according to claim 6, characterized in that, In S6, the seepage velocity of Bingham fluid in porous media is obtained based on the relationship between the average velocity of slurry microparticles and the seepage velocity. The relation is expressed as: The formula for calculating the seepage velocity of Bingham fluid in porous media is as follows: Where v represents the seepage velocity of Bingham fluid in porous media, and K represents the formation permeability. μ represents the viscosity of the slurry. τ represents the slurry pressure gradient, τ0 represents the yield stress of the slurry microparticles, r0 represents the radius of the slurry microparticles, and φ represents the porosity of the porous medium.

8. The method for controlling the diffusion radius of permeation grouting according to claim 7, characterized in that, In S7, the Iberall resistance model states that the first permeability is a function of the Reynolds number, and the derivation formula for the first permeability is: in, The slurry follows the law of spherical diffusion, therefore: A = 4πx 2 ; (4) Combining equations (1) to (4), we obtain the formula for calculating the first permeability, which is: K1 represents the first permeability, φ represents the porosity of the porous medium, δ represents the initial pore diameter of the porous medium, Re represents the Reynolds number, ρ represents the slurry density, v represents the seepage velocity of Bingham fluid in the porous medium, q represents the grouting volume per unit time, A represents the diffusion area, Q represents the grouting volume, t represents the grouting time, and x represents the diffusion radius.

9. The method for controlling the diffusion radius of permeation grouting according to claim 8, characterized in that, In S7, according to the generalized Hooke's law: Where σ represents the slurry pressure at the pore boundary of the porous medium, ε represents the strain of the porous medium under stress, dP represents the pressure change of the slurry along the boundary surface, dδ represents the deformation of the porous medium, l represents the initial average width of the porous medium, E represents the elastic modulus of the porous medium, φ represents the porosity of the porous medium, and δ represents the initial pore diameter of the porous medium. Based on formation permeability achievable but Combining equations (5) to (7), we get: The formula for calculating the grouting volume within the grouting time t is as follows: Q = v - 4πx 2 • t; (9) By combining the formulas for calculating the seepage velocity of Bingham fluid in porous media and formula (9), we obtain: because Therefore, the part in equation (10) can be ignored. Substituting formula (3) and formula (10) ignoring the last term into formula (8), we get: Substituting the third boundary conditions: x = 0, K = K0 into formula (11), and then integrating formula (11), we get: Taking the square root of both sides of formula (12), we obtain the second permeability calculation formula, which is: K2 represents the second permeability, r0 represents the radius of the slurry particle, Q represents the amount of slurry injected within the injection time t, v represents the seepage velocity of Bingham fluid in the porous medium, x represents the diffusion radius, τ0 represents the yield stress of the slurry particle, and K0 represents the initial permeability of the porous medium.

10. The method for controlling the diffusion radius of permeation grouting according to claim 9, characterized in that, In S8, based on the relationship between the grouting volume Q and the seepage velocity v, and combined with the second boundary conditions: P = P0, l = r0, the grout pressure gradient is integrated. Substituting the first permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of grout flow resistance, which is expressed as: where P fr represents the slurry pressure when the slurry diffusion radius is x, φ represents the porosity of the porous medium, μ represents the slurry viscosity, t represents the grouting time, δ represents the initial pore diameter of the porous medium, ρ represents the slurry density, l represents the initial average width of the porous medium, r0 represents the radius of the slurry micelle, τ0 represents the yield stress of the slurry micelle, and P0 represents the boundary value of the slurry pressure when the slurry diffusion radius is x. Substituting the second permeability calculation formula into the grout pressure gradient integral formula, we obtain the control equation for the diffusion radius of the permeation grouting considering the influence of the lateral force of the grout, which is expressed as: Among them, P tr The value represents the slurry pressure when considering the influence of the lateral force of the slurry on the permeability of the porous medium, and the slurry diffusion radius is x; E represents the elastic modulus of the porous medium, and K0 represents the initial permeability of the porous medium.

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