# Theoretical Research on Grouting in Deep Loose Layers Based on the Cylindrical Diffusion Model of Radial Tube Flow

Authors: Xuesong Wang; Hua Cheng; Zhishu Yao; Chuanxin Rong; Xianwen Huang; Longhui Guo
Journal: Geofluids
 (2022)

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## Abstract

Grouting in deep, loose layers are a complex process in which many modes such as infiltration, splitting, and compaction coexist. It is of great significance to establish a realistic, simplified physical model to study the law of slurry diffusion. Herein, a cylindrical diffusion model of radial tube flow is established, and the control differential equations of both the Bingham slurry diffusion velocity in a single tube and the diffusion velocity of the radial tube flow are deduced. Additionally, the calculation formulas for the diffusion radius and slurry pressure distribution function are obtained. The rationality of the theory is verified by combining our results with those of the field grouting test of the Guotun coal mine. The results show that the cylindrical diffusion model of radial tube flow can successfully characterize the slurry diffusion law of grouting in a deep, loose layer. The slurry pressure attenuation shows distinguishable stages: within the first 30% of the diffusion radius, the slurry pressure decreased sharply by approximately 70%, and the slurry pressure decreases slowly in the later stages. Furthermore, the diffusion radius has a nonlinear, negative correlation with the height of the grouting section and the comprehensive injection rate of formation; the change rate is relatively more gradual, and there are no distinguishable stages. The research results provide a theoretical basis for reasonably determining the grouting parameters of deep, loose layers in the future.


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## Body

## 1. Introduction

As a common treatment method for antiseepage, plugging, and reinforcement of underground engineering, grouting methods are widely used in mine water plugging and reinforcement, water and mud inrush treatment in tunnel fault fracture zones, foundation pit supports, dam reinforcement, and other fields [1–3]. Grouting in deep and loose layers has the typical “three high characteristics” of high ground stress, high permeability, and high grouting pressure [4, 5]. In addition, the uncertainty of the change in soil layer properties during the deposition process leads to many modes of slurry diffusion, such as splitting, permeability, and compaction, and its diffusion mechanism and design theory have significantly lag between engineering practice and research [6, 7].Regarding the diffusion of split grouting slurry, the main established mathematical models include the flat-plate diffusion model that does not consider the change in crack width and the plane radiation circle diffusion model that does consider the change in crack width [8–10]. The main research topics are slurry diffusion radius and velocity distribution, pressure distribution, and viscosity temporal and spatial distribution in the process of migration. Regarding infiltration grouting, numerous studies have been conducted with a combination of different flow patterns of slurry and different diffusion models. The slurry flow patterns of Newtonian, Bingham, and power-law fluids have been investigated. The established diffusion models mainly include the spherical, cylindrical, and cylindrical-hemispherical models [11–14]. Regarding compaction grouting, the fluid–solid coupling effect between the slurry and soil is mainly studied in combination with the theories of ball hole expansion or column hole expansion and soil nonlinear compaction. The research focus is mainly on considering different yield criteria, dilatancy, strain softening, different moduli of tension, and compression of soil [15–17].The above researches ignored the tortuous effect in the process of slurry diffusion and assumed that the diffusion channels of slurry were straight. There was a large deviation between the theoretical calculation results and the engineering practice. Through comparison, some scholars [18–20] have found that the slurry diffusion law considering the tortuous effect was more consistent with the grouting test results, and the calculation error was smaller. It is of great significance to consider the tortuous effect in the design of grouting parameters in porous media. In addition, the abovementioned research only discusses a particular diffusion mode of the slurry, but the process of grouting in deep, loose layers involves multiple diffusion modes [21, 22]. The grouting theory of the single diffusion mode cannot be applied when designing the grouting parameters in deep, loose layers. The grouting construction parameter design is not significantly influenced by the slurry diffusion mode but focuses on the design of parameters such as grouting pressure and diffusion radius [23, 24]. Therefore, it is necessary to establish a grouting diffusion theory suitable for deep, loose layers, which simultaneously weakens the slurry diffusion mode and can be better suited to guide the design of grouting parameters.Grotenhuis et al. [25, 26] characterized the slurry diffusion channel in sand with a straight circular tube and assumed the diffusion of the slurry in the sand to be represented by the flow of the slurry in the circular tube. The model can characterize both the split grouting and infiltration grouting mechanisms and can be used to calculate parameters such as the grouting pressure and diffusion radius. Based on this and considering the tortuous effect of the slurry diffusion path, this study considers the slurry diffusion channel in sand to be a tortuous circular tube and establishes the cylindrical diffusion model of the radial tube flow of the slurry in a deep, loose sand layer. First, the control differential equation of the Bingham slurry diffusion velocity in a single tube is derived, and the permeability coefficient considering tortuosity is used to replace the average radius of the seepage tubes in the injected layer within the differential equation. Subsequently, the diffusion velocity control equation of the radiation tube flow is derived, and the calculation formulas for the diffusion radius and slurry pressure distribution function are obtained. Finally, combined with the field grouting test results of the Guotun coal mine, the rationality of the theory is verified, and the attenuation law of the grouting pressure and the influential factors on the diffusion radius are discussed for the diffusion area. The research results provide a reference for future guidance of the design of grouting parameters in deep and loose layers.


## 2. Cylindrical Diffusion Model of Radial Tube Flow

### 2.1. Basic Assumptions and Model Establishment

To simplify the calculation and establish the diffusion model of the slurry in a deep, loose sand layer, the following assumptions are made within this study:(1)
The diffusion mode of the slurry in the layer is weakened, and the diffusion of slurry in splitting cracks and pores is regarded as the flow of slurry in tortuous tubes(2)
The slurry is an incompressible Bingham fluid, and the slurry flow pattern does not change during grouting(3)
The slurry velocity on the inner wall of the tube is 0, which satisfies the no-slip boundary condition(4)
The energy loss of the slurry flowing through the bend of the tube was ignoredBased on the abovementioned assumptions, a cylindrical diffusion model of the radial tube flow of the slurry in a deep, loose sandy soil layer is constructed, where the height of the grouting section ish, and the diffusion radius is R. N horizontal circuitous circular tubes with radius r0 radiate from the vicinity of the grouting tube, where r0 is the comprehensive average radius of the tubes, and the diffusion of the slurry in the sand layer is regarded as the flow of the slurry in circuitous circular tubes. The specific values of N and r0 of the tubes are difficult to determine, but it is possible to simplify the theoretical derivation process by eliminating these two parameters. Therefore, a cylindrical surface coaxial within the grouting tube is selected, where the height of the cylindrical surface is equal to the height h of the grouting section ,and the radius is set to R0, which satisfies the following requirement:
(1)Nπr02=2πR0h.It is worth noting thatR0 is neither the grouting tube radius nor the drilling radius. This cylindrical surface is an imaginary cylindrical surface selected for the convenience of the subsequent theoretical derivation, which is referred to as the virtual divergent surface in this study. According to Equation (1), the sum of the cross-sectional areas of all tubes is equal to the side surface area of the divergence surface, which is equivalent to the radial divergence of the slurry from the virtual divergence surface within the tortuous circular tube. A schematic of the diffusion model is shown in Figure 1.Figure 1
Cylindrical diffusion model of radial tube flow.


### 2.2. Diffusion Velocity Control Differential Equation of Bingham Slurry in a Single Tube

The Bingham fluid rheological equation is as follows [27]:
(2)τ=τ0+ηgγ̇,where τ is the shear stress between grout layers, τ0 is the shear yield strength of the slurry, ηg is the viscosity coefficient of the slurry at 25°C, and γ̇=−dv/dρ is the shear rate of the slurry.The actual length of the tortuous circular tube isχR−R0, the radius is r0, the slurry pressure upon entry is the initial grouting pressure, and the slurry pressure at the end is the hydrostatic pressure pw at this point. The x-axis of the natural coordinate system along the axis of the tortuous circular tube is established, and the intersection of the x-axis and the grouting tube axis is taken as the coordinate origin. The slurry flow direction is along the positive x-axis. 0,R0 is the straight-line section, and [R0, χR−R0+R0] is the tortuous section, as shown in Figure 2(a). The attenuation value of the slurry pressure along the x-axis was set as px. Note that the value of px represents the attenuation value of the slurry pressure, which is therefore distinguished from the slurry pressure. The boundary conditions are
(3)x=R0,p=0,x=χR−R0+R0,p=p0−pw.Figure 2
Coordinate system and fluid microelement stress diagram.
(a)(b)A section of the cylindrical fluid microelement with radiusρ and length dx and its axial force is shown in Figure 2(b), which is obtained from the equilibrium conditions of the axial force:
(4)p0−px+dx−p0−px⋅πρ2+τ⋅2πρ⋅dx=0,which can be simplified to
(5)px+dx−px=2τρdx.Letdp=px+dx−px, and then Equation (5) can be simplified to
(6)dp=2τρdx.Therefore, along the slurry flow direction, the pressure attenuation gradient can be expressed as follows:(7)λ=dpdx=2τρ.The pressure attenuation gradient,λ, is the attenuation speed of the pressure along the flow direction. It can be seen from Equation (7) that λ is positively correlated with the slurry shear stress, τ, and negatively correlated with radial distance, ρ. τ is positively correlated with the slurry viscosity. The greater the viscosity, the more difficult it is to overcome the shear stress, the greater τ is, and the faster the pressure attenuation is. When the radial distance ρ is equal to the tube radius r0, the average starting pressure gradient of the slurry flowing in the tube is
(8)λ0=2τ0r0,where λ0 is the average starting pressure gradient of the slurry, τ0 is the shear yield strength of the slurry, and r0 is the comprehensive average radius of the tubes, which is positively correlated with the effective porosity of sand, and is the parameter after homogenizing the pore channel size of sand. According to [28], the calculation formula for r0 is
(9)r0=∑i=1nNiri4∑i=1nNiri3≈∑i=1nNiri4∑i=1nNi14,where Ni is the number of tubes with a cross-section radius of rii=1,2,⋯,n.From Equation (7), it is then possible to obtain the following:
(10)τ=ρ2⋅dpdx.Substituting Equation (10) into Equation (2) results in
(11)−dvdρ=1ηgρ2⋅dpdx−τ0.Whenτ≤τ0, the slurry moves as a piston, dv/dρ=0, with a radius of rp and a speed of vp. Using Equation (11), we obtain
(12)rp=2τ0dp/dx.Whenτ>τ0, the slurry moves in a laminar flow. From Equation (11), we can separate the variables and integrate, resulting in
(13)v=−1ηgρ24⋅dpdx−τ0ρ+C1,where C1 is the integration constant.Substituting the no-slip boundary condition, i.e.,ρ=r0, the integration constant is determined to be
(14)C1=1ηgr024⋅dpdx−τ0r0.Therefore, the velocity distribution of the laminar flow in [rp, r0] is
(15)v=1ηgr02−ρ24⋅dpdx−τ0r0−ρ,and the fluid velocity of the piston-like flow in [0, rp] is
(16)vp=1ηgr02−rp24⋅dpdx−τ0r0−rp.The velocity distribution in the circular tube is a truncated paraboloid shape, and the flow rate per unit time,q1, in a single tube is
(17)q1=πrp2vp+∫rpr02πρvdρ.Substituting Equations (15) and (16) into Equation (17), we obtain
(18)q1=πr048ηg⋅dpdx⋅1−43rpr0+13rpr04.The average velocity,v¯, within the tube section is
(19)v¯=r028ηg⋅dpdx⋅1−43rpr0+13rpr04.Substituting Equation (12) into Equation (19) results in
(20)v¯=r028ηg⋅dpdx⋅1−432τ0/r0dp/dx+132τ0/r0dp/dx4.From Equation (20), it can be seen that when 2τ0/r0/dp/dx=1, v¯=0. Specifically, when the pressure gradient λ=2τ0/r0=λ0, the average flow in the circular tube is 0.To satisfydp/dx=λ0 and v¯=0, Equation (20) is rewritten as
(21)v¯=r028ηg⋅dpdx⋅1−λ0dp/dx−13⋅λ0dp/dx⋅1−λ0dp/dx3.In the grouting process,dp/dx≫λ0 [28], the third term in the curly bracket of Equation (21) is ignored in order to obtain an approximate slurry diffusion velocity control differential equation, as follows:
(22)v¯=r028ηg⋅dpdx−λ0.Owing to the uneven spatial distribution of the pore structure in the injected layer, the average radius,r0, of the seepage tube in the injected layer in Equation (22) is difficult to measure directly. However, the permeability coefficient can be easily obtained through laboratory tests. Therefore, the expression of the permeability coefficient was used to replace r0 within this study. The relationship between the permeability coefficient and r0 is derived as follows:The relationship between the permeability coefficientK and permeability k is [29]
(23)K=kγη,where γ is the gravity of the fluid, and η is the viscosity coefficient of the fluid.The permeability of porous media is independent of the fluid characteristics and only depends on the porosity, capillary diameter, and tortuosity of the porous media. The existing theory only uses porosity and capillary diameter to characterize the permeability of porous media, without considering the nonnegligible parameter of tortuosity [28]. It is worth noting that because permeability is independent of the fluid characteristics, to simplify the calculation, the fluid selected in the process of deriving permeability is the laminar Newtonian fluid, which does not conflict with the Bingham fluid studied within this research.According to the generalized Hagen Poiseuille equation [30], the flow through all tubes per unit time is
(24)q=nAπr048ηχ⋅dpdl,where q is the flow through the pore section, n is the number of circular tubes per unit area, A is the cross-sectional area, and dp/dl is the pressure gradient.In the laminar flow state, the flow of fluid in porous media satisfies Darcy’s law, and the flow through the pore section per unit time is(25)q=kAη⋅dpdl.Through theoretical derivation, Li et al. [31] highlighted how the tube flow and seepage flow are unified, and the formulas of the two mathematical models are equivalent; they are simply expressed by different parameters. In combination with Equations (24) and (25), the permeability considering tortuosity is
(26)k=nπr048χ.The porosity of the tube model is(27)φ=nπr02χ.When combining Equations (26) and (27), we find
(28)k=φr028χ2.Therefore, the permeability coefficient considering tortuosity is(29)K=φr02γ8ηχ2.Setting the viscosity ratio of slurry to water toβ, we obtain
(30)β=ηgηw,where ηw is the viscosity coefficient of water at 25°C.Combining Equations (29) and (30) results in
(31)r028η=χ2Kwβφγw,where Kw is the permeability coefficient of water in a porous medium.Therefore, Equation (22) can be transformed into
(32)v¯=χ2Kwβφγw⋅dpdx−λ0.Equation (32) is the approximate slurry diffusion velocity control differential equation of the Bingham slurry in a single tube.


### 2.3. Diffusion Velocity Control Equation of Radial Tube Flow

The flow and diffusion path of the slurry in the deep, loose layer was tortuous, as shown in Figure3. Tortuosity is used to characterize the tortuous effect of fluid particle motion diffusion [32, 33], which is commonly defined as
(33)χ=ltl,where χ is the tortuosity of the tube which commonly takes the values obtained by related scholars of 1.4 ~ 1.6 [28]. The average value used within this research is 1.5, lt is the actual length of the diffusion trace, and l is the effective length of the diffusion path.Figure 3
Schematic of slurry diffusion tortuous effect.The effective porosity of the injected layer is the ratio of the pore volume filled by the slurry to the volume of the injected soil layer, and the relationship with the porosity of the soil layer is(34)φ′=αφ,where φ′ is the effective porosity of the injected layer, φ is the actual porosity of the injected layer, and α is the pore injection coefficient of approximately 0.15~0.2 for silty clay and 0.3 ~ 0.5 for soft soil and fine sand [34].In the hollow cylindrical layer of [R0, x−R0/χ+R0], the effective porosity is
(35)φ′=Nπr02x−R0πhx−R0/χ+R02−R02.Combining Equations (1) and (35) results in
(36)φ′=2χ2R0x+2χ−1R0.It can be seen from Equation (36) that the effective porosity decreases with an increase in the diffusion distance, which is consistent with practice. An increase in the slurry diffusion distance means that the slurry pressure gradient gradually decreases. When it is less than the starting pressure gradient of the slurry flow in small tubes, the slurry will not enter these small tubes. When the distance from the grouting tube increases, the number of pores that the slurry can enter gradually decreases; that is, the effective porosity gradually decreases.The slurry column radius can be obtained by combining Equations (34) and (36), such that
(37)R0=αφ2χ2−2χ−1αφx.It can be seen from Equation (37) that during grouting, the slurry column radius changes with the change in effective porosity and diffusion distance, which is not a fixed value.Let(38)ξ=αφ2χ2−2χ−1αφ,where ξ is a parameter selected to simplify the formula, which is related to α, φ, and χ, and is calculated by Equation (38).Then,(39)R0=ξx.Because the slurry is incompressible, the slurry injection flow is equal to the outflow flow of the cylindrical surface outside the slurry column,(40)q=v¯χ⋅2πR0h.Combining Equations (39) and (40) results in
(41)v¯=χq2πξhx.


### 2.4. Slurry Diffusion Radius and Pressure Distribution

Through combining Equations (32) and (41), we obtain
(42)dpdx=qβφγw2πξhχKw1x+λ0.Subsequently, integrating Equation (42) results in
(43)px=qβφγw2πξhχKwlnx+λ0x+C.When the following boundary conditions are implemented, we have(44)x=R0,p=0.The integration constant is determined to be(45)C=−qβφγw2πξhχKwlnR0−λ0R0By substituting Equation (45) into Equation (43), we obtain
(46)px=qβφγw2πξhχKwlnxR0+λ0x−R0.Equation (46) describes the relationship between the pressure attenuation value and the path distance of the diffusion curve. Three transformations are required to transform this into the relationship between the pressure attenuation value and diffusion distance. First, the function image is shifted to the right by the a factor ofR0. Then, the ordinate of the function is unchanged, and the abscissa becomes 1/χ the original. Finally, the function image is shifted to the left by a factor of R0. The transformed function is pχx−R0+R0, and to clarify the difference, the abscissa of the new function is represented by the letter r, representing the slurry diffusion distance, and the ordinate is represented by p~, representing the slurry pressure attenuation value at the diffusion distance R. Then, the relationship between the pressure attenuation value and the diffusion distance is
(47)p~r=qβφγw2πξhχKwlnχr−R0+R0R0+λ0χr−R0.The pressure attenuation value at the diffusion radiusR is
(48)p~R=qβφγw2πξhχKwlnχR−R0+R0R0+λ0χR−R0.Therefore, the initial grouting pressure is(49)p0=pw+qβφγw2πξhχKwlnχR−R0+R0R0+λ0χR−R0.The slurry pressure distribution function,Pr, in the diffusion area is
(50)Pr=p0−p~r=pw+qβφγw2πξhχKwlnχR−R0+R0χr−R0+R0+λ0χR−r.However, the grouting diffusion radius,R, is usually unknown, and its calculation formula needs to be deduced. According to Equation (41), when the diffusion radius is R, the slurry column radius is
(51)R0=ξR.The volume of the slurry filled in the layer is equal to the total amount of slurry injected, such that(52)αφπR2−R02h+πR02h=Q.Namely,(53)ξ2+αφ1−ξ2πR2h=Q.Let(54)η=ξ2+αφ1−ξ2,where η is the comprehensive injection rate of the layer.The grouting diffusion radius can be calculated according to the grouting volume, grouting section height, and comprehensive injection rate of the layer as follows:(55)R=Qπhη.Therefore, the initial grouting pressure is(56)p0=pw+qβφγw2πξhχKwlnχQ/πhη−R0+R0R0+λ0χQπhη−R0,and the slurry pressure distribution function, Pr, in the diffusion area is
(57)Pr=pw+qβφγw2πξhχKwlnχQ/πhη−R0+R0χr−R0+R0+λ0χQπhη−r.


## 2.1. Basic Assumptions and Model Establishment

To simplify the calculation and establish the diffusion model of the slurry in a deep, loose sand layer, the following assumptions are made within this study:(1)
The diffusion mode of the slurry in the layer is weakened, and the diffusion of slurry in splitting cracks and pores is regarded as the flow of slurry in tortuous tubes(2)
The slurry is an incompressible Bingham fluid, and the slurry flow pattern does not change during grouting(3)
The slurry velocity on the inner wall of the tube is 0, which satisfies the no-slip boundary condition(4)
The energy loss of the slurry flowing through the bend of the tube was ignoredBased on the abovementioned assumptions, a cylindrical diffusion model of the radial tube flow of the slurry in a deep, loose sandy soil layer is constructed, where the height of the grouting section ish, and the diffusion radius is R. N horizontal circuitous circular tubes with radius r0 radiate from the vicinity of the grouting tube, where r0 is the comprehensive average radius of the tubes, and the diffusion of the slurry in the sand layer is regarded as the flow of the slurry in circuitous circular tubes. The specific values of N and r0 of the tubes are difficult to determine, but it is possible to simplify the theoretical derivation process by eliminating these two parameters. Therefore, a cylindrical surface coaxial within the grouting tube is selected, where the height of the cylindrical surface is equal to the height h of the grouting section ,and the radius is set to R0, which satisfies the following requirement:
(1)Nπr02=2πR0h.It is worth noting thatR0 is neither the grouting tube radius nor the drilling radius. This cylindrical surface is an imaginary cylindrical surface selected for the convenience of the subsequent theoretical derivation, which is referred to as the virtual divergent surface in this study. According to Equation (1), the sum of the cross-sectional areas of all tubes is equal to the side surface area of the divergence surface, which is equivalent to the radial divergence of the slurry from the virtual divergence surface within the tortuous circular tube. A schematic of the diffusion model is shown in Figure 1.Figure 1
Cylindrical diffusion model of radial tube flow.


## 2.2. Diffusion Velocity Control Differential Equation of Bingham Slurry in a Single Tube

The Bingham fluid rheological equation is as follows [27]:
(2)τ=τ0+ηgγ̇,where τ is the shear stress between grout layers, τ0 is the shear yield strength of the slurry, ηg is the viscosity coefficient of the slurry at 25°C, and γ̇=−dv/dρ is the shear rate of the slurry.The actual length of the tortuous circular tube isχR−R0, the radius is r0, the slurry pressure upon entry is the initial grouting pressure, and the slurry pressure at the end is the hydrostatic pressure pw at this point. The x-axis of the natural coordinate system along the axis of the tortuous circular tube is established, and the intersection of the x-axis and the grouting tube axis is taken as the coordinate origin. The slurry flow direction is along the positive x-axis. 0,R0 is the straight-line section, and [R0, χR−R0+R0] is the tortuous section, as shown in Figure 2(a). The attenuation value of the slurry pressure along the x-axis was set as px. Note that the value of px represents the attenuation value of the slurry pressure, which is therefore distinguished from the slurry pressure. The boundary conditions are
(3)x=R0,p=0,x=χR−R0+R0,p=p0−pw.Figure 2
Coordinate system and fluid microelement stress diagram.
(a)(b)A section of the cylindrical fluid microelement with radiusρ and length dx and its axial force is shown in Figure 2(b), which is obtained from the equilibrium conditions of the axial force:
(4)p0−px+dx−p0−px⋅πρ2+τ⋅2πρ⋅dx=0,which can be simplified to
(5)px+dx−px=2τρdx.Letdp=px+dx−px, and then Equation (5) can be simplified to
(6)dp=2τρdx.Therefore, along the slurry flow direction, the pressure attenuation gradient can be expressed as follows:(7)λ=dpdx=2τρ.The pressure attenuation gradient,λ, is the attenuation speed of the pressure along the flow direction. It can be seen from Equation (7) that λ is positively correlated with the slurry shear stress, τ, and negatively correlated with radial distance, ρ. τ is positively correlated with the slurry viscosity. The greater the viscosity, the more difficult it is to overcome the shear stress, the greater τ is, and the faster the pressure attenuation is. When the radial distance ρ is equal to the tube radius r0, the average starting pressure gradient of the slurry flowing in the tube is
(8)λ0=2τ0r0,where λ0 is the average starting pressure gradient of the slurry, τ0 is the shear yield strength of the slurry, and r0 is the comprehensive average radius of the tubes, which is positively correlated with the effective porosity of sand, and is the parameter after homogenizing the pore channel size of sand. According to [28], the calculation formula for r0 is
(9)r0=∑i=1nNiri4∑i=1nNiri3≈∑i=1nNiri4∑i=1nNi14,where Ni is the number of tubes with a cross-section radius of rii=1,2,⋯,n.From Equation (7), it is then possible to obtain the following:
(10)τ=ρ2⋅dpdx.Substituting Equation (10) into Equation (2) results in
(11)−dvdρ=1ηgρ2⋅dpdx−τ0.Whenτ≤τ0, the slurry moves as a piston, dv/dρ=0, with a radius of rp and a speed of vp. Using Equation (11), we obtain
(12)rp=2τ0dp/dx.Whenτ>τ0, the slurry moves in a laminar flow. From Equation (11), we can separate the variables and integrate, resulting in
(13)v=−1ηgρ24⋅dpdx−τ0ρ+C1,where C1 is the integration constant.Substituting the no-slip boundary condition, i.e.,ρ=r0, the integration constant is determined to be
(14)C1=1ηgr024⋅dpdx−τ0r0.Therefore, the velocity distribution of the laminar flow in [rp, r0] is
(15)v=1ηgr02−ρ24⋅dpdx−τ0r0−ρ,and the fluid velocity of the piston-like flow in [0, rp] is
(16)vp=1ηgr02−rp24⋅dpdx−τ0r0−rp.The velocity distribution in the circular tube is a truncated paraboloid shape, and the flow rate per unit time,q1, in a single tube is
(17)q1=πrp2vp+∫rpr02πρvdρ.Substituting Equations (15) and (16) into Equation (17), we obtain
(18)q1=πr048ηg⋅dpdx⋅1−43rpr0+13rpr04.The average velocity,v¯, within the tube section is
(19)v¯=r028ηg⋅dpdx⋅1−43rpr0+13rpr04.Substituting Equation (12) into Equation (19) results in
(20)v¯=r028ηg⋅dpdx⋅1−432τ0/r0dp/dx+132τ0/r0dp/dx4.From Equation (20), it can be seen that when 2τ0/r0/dp/dx=1, v¯=0. Specifically, when the pressure gradient λ=2τ0/r0=λ0, the average flow in the circular tube is 0.To satisfydp/dx=λ0 and v¯=0, Equation (20) is rewritten as
(21)v¯=r028ηg⋅dpdx⋅1−λ0dp/dx−13⋅λ0dp/dx⋅1−λ0dp/dx3.In the grouting process,dp/dx≫λ0 [28], the third term in the curly bracket of Equation (21) is ignored in order to obtain an approximate slurry diffusion velocity control differential equation, as follows:
(22)v¯=r028ηg⋅dpdx−λ0.Owing to the uneven spatial distribution of the pore structure in the injected layer, the average radius,r0, of the seepage tube in the injected layer in Equation (22) is difficult to measure directly. However, the permeability coefficient can be easily obtained through laboratory tests. Therefore, the expression of the permeability coefficient was used to replace r0 within this study. The relationship between the permeability coefficient and r0 is derived as follows:The relationship between the permeability coefficientK and permeability k is [29]
(23)K=kγη,where γ is the gravity of the fluid, and η is the viscosity coefficient of the fluid.The permeability of porous media is independent of the fluid characteristics and only depends on the porosity, capillary diameter, and tortuosity of the porous media. The existing theory only uses porosity and capillary diameter to characterize the permeability of porous media, without considering the nonnegligible parameter of tortuosity [28]. It is worth noting that because permeability is independent of the fluid characteristics, to simplify the calculation, the fluid selected in the process of deriving permeability is the laminar Newtonian fluid, which does not conflict with the Bingham fluid studied within this research.According to the generalized Hagen Poiseuille equation [30], the flow through all tubes per unit time is
(24)q=nAπr048ηχ⋅dpdl,where q is the flow through the pore section, n is the number of circular tubes per unit area, A is the cross-sectional area, and dp/dl is the pressure gradient.In the laminar flow state, the flow of fluid in porous media satisfies Darcy’s law, and the flow through the pore section per unit time is(25)q=kAη⋅dpdl.Through theoretical derivation, Li et al. [31] highlighted how the tube flow and seepage flow are unified, and the formulas of the two mathematical models are equivalent; they are simply expressed by different parameters. In combination with Equations (24) and (25), the permeability considering tortuosity is
(26)k=nπr048χ.The porosity of the tube model is(27)φ=nπr02χ.When combining Equations (26) and (27), we find
(28)k=φr028χ2.Therefore, the permeability coefficient considering tortuosity is(29)K=φr02γ8ηχ2.Setting the viscosity ratio of slurry to water toβ, we obtain
(30)β=ηgηw,where ηw is the viscosity coefficient of water at 25°C.Combining Equations (29) and (30) results in
(31)r028η=χ2Kwβφγw,where Kw is the permeability coefficient of water in a porous medium.Therefore, Equation (22) can be transformed into
(32)v¯=χ2Kwβφγw⋅dpdx−λ0.Equation (32) is the approximate slurry diffusion velocity control differential equation of the Bingham slurry in a single tube.


## 2.3. Diffusion Velocity Control Equation of Radial Tube Flow

The flow and diffusion path of the slurry in the deep, loose layer was tortuous, as shown in Figure3. Tortuosity is used to characterize the tortuous effect of fluid particle motion diffusion [32, 33], which is commonly defined as
(33)χ=ltl,where χ is the tortuosity of the tube which commonly takes the values obtained by related scholars of 1.4 ~ 1.6 [28]. The average value used within this research is 1.5, lt is the actual length of the diffusion trace, and l is the effective length of the diffusion path.Figure 3
Schematic of slurry diffusion tortuous effect.The effective porosity of the injected layer is the ratio of the pore volume filled by the slurry to the volume of the injected soil layer, and the relationship with the porosity of the soil layer is(34)φ′=αφ,where φ′ is the effective porosity of the injected layer, φ is the actual porosity of the injected layer, and α is the pore injection coefficient of approximately 0.15~0.2 for silty clay and 0.3 ~ 0.5 for soft soil and fine sand [34].In the hollow cylindrical layer of [R0, x−R0/χ+R0], the effective porosity is
(35)φ′=Nπr02x−R0πhx−R0/χ+R02−R02.Combining Equations (1) and (35) results in
(36)φ′=2χ2R0x+2χ−1R0.It can be seen from Equation (36) that the effective porosity decreases with an increase in the diffusion distance, which is consistent with practice. An increase in the slurry diffusion distance means that the slurry pressure gradient gradually decreases. When it is less than the starting pressure gradient of the slurry flow in small tubes, the slurry will not enter these small tubes. When the distance from the grouting tube increases, the number of pores that the slurry can enter gradually decreases; that is, the effective porosity gradually decreases.The slurry column radius can be obtained by combining Equations (34) and (36), such that
(37)R0=αφ2χ2−2χ−1αφx.It can be seen from Equation (37) that during grouting, the slurry column radius changes with the change in effective porosity and diffusion distance, which is not a fixed value.Let(38)ξ=αφ2χ2−2χ−1αφ,where ξ is a parameter selected to simplify the formula, which is related to α, φ, and χ, and is calculated by Equation (38).Then,(39)R0=ξx.Because the slurry is incompressible, the slurry injection flow is equal to the outflow flow of the cylindrical surface outside the slurry column,(40)q=v¯χ⋅2πR0h.Combining Equations (39) and (40) results in
(41)v¯=χq2πξhx.


## 2.4. Slurry Diffusion Radius and Pressure Distribution

Through combining Equations (32) and (41), we obtain
(42)dpdx=qβφγw2πξhχKw1x+λ0.Subsequently, integrating Equation (42) results in
(43)px=qβφγw2πξhχKwlnx+λ0x+C.When the following boundary conditions are implemented, we have(44)x=R0,p=0.The integration constant is determined to be(45)C=−qβφγw2πξhχKwlnR0−λ0R0By substituting Equation (45) into Equation (43), we obtain
(46)px=qβφγw2πξhχKwlnxR0+λ0x−R0.Equation (46) describes the relationship between the pressure attenuation value and the path distance of the diffusion curve. Three transformations are required to transform this into the relationship between the pressure attenuation value and diffusion distance. First, the function image is shifted to the right by the a factor ofR0. Then, the ordinate of the function is unchanged, and the abscissa becomes 1/χ the original. Finally, the function image is shifted to the left by a factor of R0. The transformed function is pχx−R0+R0, and to clarify the difference, the abscissa of the new function is represented by the letter r, representing the slurry diffusion distance, and the ordinate is represented by p~, representing the slurry pressure attenuation value at the diffusion distance R. Then, the relationship between the pressure attenuation value and the diffusion distance is
(47)p~r=qβφγw2πξhχKwlnχr−R0+R0R0+λ0χr−R0.The pressure attenuation value at the diffusion radiusR is
(48)p~R=qβφγw2πξhχKwlnχR−R0+R0R0+λ0χR−R0.Therefore, the initial grouting pressure is(49)p0=pw+qβφγw2πξhχKwlnχR−R0+R0R0+λ0χR−R0.The slurry pressure distribution function,Pr, in the diffusion area is
(50)Pr=p0−p~r=pw+qβφγw2πξhχKwlnχR−R0+R0χr−R0+R0+λ0χR−r.However, the grouting diffusion radius,R, is usually unknown, and its calculation formula needs to be deduced. According to Equation (41), when the diffusion radius is R, the slurry column radius is
(51)R0=ξR.The volume of the slurry filled in the layer is equal to the total amount of slurry injected, such that(52)αφπR2−R02h+πR02h=Q.Namely,(53)ξ2+αφ1−ξ2πR2h=Q.Let(54)η=ξ2+αφ1−ξ2,where η is the comprehensive injection rate of the layer.The grouting diffusion radius can be calculated according to the grouting volume, grouting section height, and comprehensive injection rate of the layer as follows:(55)R=Qπhη.Therefore, the initial grouting pressure is(56)p0=pw+qβφγw2πξhχKwlnχQ/πhη−R0+R0R0+λ0χQπhη−R0,and the slurry pressure distribution function, Pr, in the diffusion area is
(57)Pr=pw+qβφγw2πξhχKwlnχQ/πhη−R0+R0χr−R0+R0+λ0χQπhη−r.


## 3. Results and Discussion

### 3.1. Field Test Verification and Analysis

To verify the applicability of the cylindrical diffusion model of radial tube flow established in this study, we compared the theoretical calculations with the field test results of ground drilling grouting in the Guotun coal mine. The main, auxiliary, and air shafts of the Guotun coal mine in Shandong Province pass through the loose layer with a thickness of 561.20~587.40 m. After the mine was put into operation, owing to the drainage of the mine, the above three shafts deviated to the asymmetric mining working face of the mine. The maximum deflection displacements of the main shaft and auxiliary shaft to the west were 284 mm and 299 mm, respectively, and the maximum deflection displacements to the north were 30 mm and 103 mm, respectively, which seriously threaten the production safety of the mine. After many demonstrations, it was decided to adopt the scheme of ground drilling grouting around the shaft to control the shaft deflection displacements. The engineering site diagram and schematic diagram of the ground drilling grouting of the vertical shaft are shown in Figure4. To scientifically formulate the grouting implementation scheme, reasonably select the grouting parameters, and to ensure the safety of the existing shaft, 11 water injection tests and 6 grouting tests were completed in different layers in 7 test holes around the main, auxiliary, and air shafts of the Guotun coal mine. 5 grouting tests were conducted on the water-bearing sand layers over different layers of the loose layer within the 3rd wind inspection hole.Figure 4
Vertical shaft ground drilling grouting.Single cement slurry was used for grouting. The main grouting material is 42.5 ordinary portland cement, which is prepared by adding water to the cement. The water cement ratio of slurry used on site is 1.25 : 1 and 1 : 1, and the slurry with water cement ratio of 1 : 1 is mainly used. The physical parameters of the core soil sample taken from the 3rd wind inspection hole are listed in Table1, and the grouting test results of each grouting section are listed in Table 2.Table 1
Physical parameters of the core soil samples taken from the 3rd wind inspection hole.
Soil sample no.Soil sample typeh (m)φK01Fine sand180.60~180.750.31061.13802Silty sand281.20~281.350.33331.12603Silty sand352.15~352.300.31541.07904Fine sand444.15~444.300.36741.08005Fine sand552.00~552.350.25381.82406Silty sand560.20~560.350.36370.981Note:h is the coring depth (m), φ is the porosity, and K is the permeability coefficient, 10-5 m/s.Table 2
Grouting test results of the 3rd wind inspection hole.
Grouting section no.H (m)t (h)p0 (MPa)Q (m3)R (m)Diffusion radius calculation error (%)ZJ5175.05~196.1131.304.24~4.802906.83-14.63ZJ4247.34~282.0026.404.89~6.692206.34-20.75ZJ3332.27~364.2723.507.18~9.992066.57-17.88ZJ2430.33~462.4322.408.73~13.40140——ZJ1540.46~574.9020.5011.82~13.92166——Note:H is the depth of the grouting section, t is the grouting time, p0 is the injection point pressure, Q is the grouting amount, and R is the theoretically calculated value of the diffusion radius. Owing to the first grouting in the ZJ1 grouting section, the grouting was stopped when only 166 m3 of slurry was injected, and the grouting in the ZJ2 grouting section was stopped when only 140 m3 of slurry was injected, owing to the slurry leakage from the stop-grouting plug.Substituting the relevant parameters from Tables1 and 2 into Equation (56), the theoretically calculated values of the initial grouting pressure of the ZJ3, ZJ4, and ZJ5 grouting sections were 9.05 MPa, 7.31 MPa, and 5.46 MPa, respectively, and the associated errors of the initial grouting pressure values in Table 2 were -9.41%, 9.27%, and 13.75%, respectively. Substituting the relevant parameters from Tables 1 and 2 into Equation (55), the theoretically calculated values of the diffusion radius of the ZJ3 to ZJ5 grouting sections were 6.57 m, 6.35 m, and 6.84 m, respectively, and the errors of the design value were -17.88%, -20.63%, and -14.50%, respectively, which are all within the engineering acceptable range [35–37], thus verifying the rationality of the theoretical model proposed in this study.


### 3.2. Slurry Pressure Attenuation Law

Substituting the relevant parameters from Tables1 and 2 into Equation (57), the slurry pressure distribution curves in the diffusion zone of the ZJ3 to ZJ5 grouting sections were obtained, as shown in Figure 5. The attenuation of the slurry pressure has distinguishable stages, which decrease sharply near the grouting tube and gradually away from the grouting tube. We calculated the slurry pressure value, pressure attenuation value, and attenuation percentage at different diffusion distances, where the calculation results are listed in Table 3. At a diffusion distance of 1 m, the slurry pressure attenuation of the ZJ3 to ZJ5 grouting sections were 60.32%, 60.52%, and 51.67% of the total pressure attenuation within the diffusion range, respectively. At a diffusion distance of 2 m, the slurry pressure attenuation of the ZJ3 to ZJ5 grouting sections were 74.51%, 74.89%, and 68.61% of the total pressure attenuation within the diffusion range, respectively. Beyond a diffusion distance of 2 m, the attenuation rate of the slurry pressure slows down, which is approximately a low-speed linear attenuation. Within the first 30% of the diffusion radius, the slurry pressure decreases sharply by approximately 70%; within the last 70% of the diffusion radius, the slurry pressure decreases gradually to the remaining 30%. This relationship is consistent with the engineering practices and trends of the measured data within the relevant literature [38–40]. The reason for this is that there is a high velocity of the slurry in the area close to the grouting tube, and a lower velocity of the slurry in the area far away from the grouting tube. According to Equation (32), the slurry pressure gradient is linearly and positively correlated with the slurry velocity; so, the pressure gradient decreases gradually along the diffusion direction; specifically, and the pressure decreases more slowly.Figure 5
Pressure attenuation law of slurry.Table 3
Slurry pressure attenuation.
ParametersGrouting section01 m2 mRPr (MPa)ZJ39.055.694.903.48ZJ47.314.493.822.65ZJ55.463.602.991.86p~r (MPa)ZJ303.364.155.57ZJ402.823.494.66ZJ501.862.473.60p~r/p0−pw (%)ZJ3060.3274.51100ZJ4060.5274.89100ZJ5051.6768.61100


### 3.3. Analysis of Influencing Factors on the Diffusion Radius

When considering the field grouting parameters of the Guotun coal mine combined with Equation (55), the effects of grouting volume, grouting section height, and comprehensive layer injection rate on the grouting diffusion radius are discussed. The comprehensive layer injection rates were 0.05, 0.06, 0.07, 0.08, and 0.09, the grouting volume was 200 m3, 250 m3, 300 m3, 350 m3, and 400 m3, and the heights of the grouting sections were 20 m, 25 m, 30 m, 35 m, and 40 m.When the grouting section height is taken to be 30 m, the influence of the grouting volume and the comprehensive layer injection rate on the slurry diffusion radius was studied, as shown in Figure6(a). When the comprehensive layer injection rate is taken to be 0.07, the influence of the grouting volume and grouting section height on the slurry diffusion radius is studied, as shown in Figure 6(b). When the grouting volume was 300 m3, the influence of the grouting section height and comprehensive layer injection rate on the slurry diffusion radius was studied, as shown in Figure 6(c).Figure 6
Analysis of the influencing factors of the grouting diffusion radius. (a)R−Q−η 3D function image. (b) R−Q−h 3D function image. (c) R−h−η 3D function image.
(a)(b)(c)As can be seen from Figures6(a) and 6(b), when the same grouting section height and comprehensive layer injection rate are considered, the diffusion radius increases with an increasing grouting volume, but the growth range gradually decreases. The possibility of improving the slurry diffusion radius solely by increasing the grouting volume gradually decreases. According to Figures 6(a)–6(c), the diffusion radius has a nonlinear negative correlation with the grouting section height and the comprehensive layer injection rate, the rate of change is relatively slow, and there are no distinguishable stages. Under the condition of the same grouting volume and grouting section height, it was found that increasing the comprehensive layer injection rate reduces the diffusion radius, while the grouting volume in the unit volume soil layer increases, which can effectively improve the mechanical properties of the injected layer. In engineering, multiple grouting parameters are usually coordinated and controlled to achieve the designed diffusion radius and reinforcement strength. For example, before formal grouting on site, water injection tests are often carried out to increase the hydraulic permeability of the layer, which can improve the comprehensive layer injection rate. Accordingly, the design grouting volume should also be increased to achieve the desired grouting diffusion radius.


## 3.1. Field Test Verification and Analysis

To verify the applicability of the cylindrical diffusion model of radial tube flow established in this study, we compared the theoretical calculations with the field test results of ground drilling grouting in the Guotun coal mine. The main, auxiliary, and air shafts of the Guotun coal mine in Shandong Province pass through the loose layer with a thickness of 561.20~587.40 m. After the mine was put into operation, owing to the drainage of the mine, the above three shafts deviated to the asymmetric mining working face of the mine. The maximum deflection displacements of the main shaft and auxiliary shaft to the west were 284 mm and 299 mm, respectively, and the maximum deflection displacements to the north were 30 mm and 103 mm, respectively, which seriously threaten the production safety of the mine. After many demonstrations, it was decided to adopt the scheme of ground drilling grouting around the shaft to control the shaft deflection displacements. The engineering site diagram and schematic diagram of the ground drilling grouting of the vertical shaft are shown in Figure4. To scientifically formulate the grouting implementation scheme, reasonably select the grouting parameters, and to ensure the safety of the existing shaft, 11 water injection tests and 6 grouting tests were completed in different layers in 7 test holes around the main, auxiliary, and air shafts of the Guotun coal mine. 5 grouting tests were conducted on the water-bearing sand layers over different layers of the loose layer within the 3rd wind inspection hole.Figure 4
Vertical shaft ground drilling grouting.Single cement slurry was used for grouting. The main grouting material is 42.5 ordinary portland cement, which is prepared by adding water to the cement. The water cement ratio of slurry used on site is 1.25 : 1 and 1 : 1, and the slurry with water cement ratio of 1 : 1 is mainly used. The physical parameters of the core soil sample taken from the 3rd wind inspection hole are listed in Table1, and the grouting test results of each grouting section are listed in Table 2.Table 1
Physical parameters of the core soil samples taken from the 3rd wind inspection hole.
Soil sample no.Soil sample typeh (m)φK01Fine sand180.60~180.750.31061.13802Silty sand281.20~281.350.33331.12603Silty sand352.15~352.300.31541.07904Fine sand444.15~444.300.36741.08005Fine sand552.00~552.350.25381.82406Silty sand560.20~560.350.36370.981Note:h is the coring depth (m), φ is the porosity, and K is the permeability coefficient, 10-5 m/s.Table 2
Grouting test results of the 3rd wind inspection hole.
Grouting section no.H (m)t (h)p0 (MPa)Q (m3)R (m)Diffusion radius calculation error (%)ZJ5175.05~196.1131.304.24~4.802906.83-14.63ZJ4247.34~282.0026.404.89~6.692206.34-20.75ZJ3332.27~364.2723.507.18~9.992066.57-17.88ZJ2430.33~462.4322.408.73~13.40140——ZJ1540.46~574.9020.5011.82~13.92166——Note:H is the depth of the grouting section, t is the grouting time, p0 is the injection point pressure, Q is the grouting amount, and R is the theoretically calculated value of the diffusion radius. Owing to the first grouting in the ZJ1 grouting section, the grouting was stopped when only 166 m3 of slurry was injected, and the grouting in the ZJ2 grouting section was stopped when only 140 m3 of slurry was injected, owing to the slurry leakage from the stop-grouting plug.Substituting the relevant parameters from Tables1 and 2 into Equation (56), the theoretically calculated values of the initial grouting pressure of the ZJ3, ZJ4, and ZJ5 grouting sections were 9.05 MPa, 7.31 MPa, and 5.46 MPa, respectively, and the associated errors of the initial grouting pressure values in Table 2 were -9.41%, 9.27%, and 13.75%, respectively. Substituting the relevant parameters from Tables 1 and 2 into Equation (55), the theoretically calculated values of the diffusion radius of the ZJ3 to ZJ5 grouting sections were 6.57 m, 6.35 m, and 6.84 m, respectively, and the errors of the design value were -17.88%, -20.63%, and -14.50%, respectively, which are all within the engineering acceptable range [35–37], thus verifying the rationality of the theoretical model proposed in this study.


## 3.2. Slurry Pressure Attenuation Law

Substituting the relevant parameters from Tables1 and 2 into Equation (57), the slurry pressure distribution curves in the diffusion zone of the ZJ3 to ZJ5 grouting sections were obtained, as shown in Figure 5. The attenuation of the slurry pressure has distinguishable stages, which decrease sharply near the grouting tube and gradually away from the grouting tube. We calculated the slurry pressure value, pressure attenuation value, and attenuation percentage at different diffusion distances, where the calculation results are listed in Table 3. At a diffusion distance of 1 m, the slurry pressure attenuation of the ZJ3 to ZJ5 grouting sections were 60.32%, 60.52%, and 51.67% of the total pressure attenuation within the diffusion range, respectively. At a diffusion distance of 2 m, the slurry pressure attenuation of the ZJ3 to ZJ5 grouting sections were 74.51%, 74.89%, and 68.61% of the total pressure attenuation within the diffusion range, respectively. Beyond a diffusion distance of 2 m, the attenuation rate of the slurry pressure slows down, which is approximately a low-speed linear attenuation. Within the first 30% of the diffusion radius, the slurry pressure decreases sharply by approximately 70%; within the last 70% of the diffusion radius, the slurry pressure decreases gradually to the remaining 30%. This relationship is consistent with the engineering practices and trends of the measured data within the relevant literature [38–40]. The reason for this is that there is a high velocity of the slurry in the area close to the grouting tube, and a lower velocity of the slurry in the area far away from the grouting tube. According to Equation (32), the slurry pressure gradient is linearly and positively correlated with the slurry velocity; so, the pressure gradient decreases gradually along the diffusion direction; specifically, and the pressure decreases more slowly.Figure 5
Pressure attenuation law of slurry.Table 3
Slurry pressure attenuation.
ParametersGrouting section01 m2 mRPr (MPa)ZJ39.055.694.903.48ZJ47.314.493.822.65ZJ55.463.602.991.86p~r (MPa)ZJ303.364.155.57ZJ402.823.494.66ZJ501.862.473.60p~r/p0−pw (%)ZJ3060.3274.51100ZJ4060.5274.89100ZJ5051.6768.61100


## 3.3. Analysis of Influencing Factors on the Diffusion Radius

When considering the field grouting parameters of the Guotun coal mine combined with Equation (55), the effects of grouting volume, grouting section height, and comprehensive layer injection rate on the grouting diffusion radius are discussed. The comprehensive layer injection rates were 0.05, 0.06, 0.07, 0.08, and 0.09, the grouting volume was 200 m3, 250 m3, 300 m3, 350 m3, and 400 m3, and the heights of the grouting sections were 20 m, 25 m, 30 m, 35 m, and 40 m.When the grouting section height is taken to be 30 m, the influence of the grouting volume and the comprehensive layer injection rate on the slurry diffusion radius was studied, as shown in Figure6(a). When the comprehensive layer injection rate is taken to be 0.07, the influence of the grouting volume and grouting section height on the slurry diffusion radius is studied, as shown in Figure 6(b). When the grouting volume was 300 m3, the influence of the grouting section height and comprehensive layer injection rate on the slurry diffusion radius was studied, as shown in Figure 6(c).Figure 6
Analysis of the influencing factors of the grouting diffusion radius. (a)R−Q−η 3D function image. (b) R−Q−h 3D function image. (c) R−h−η 3D function image.
(a)(b)(c)As can be seen from Figures6(a) and 6(b), when the same grouting section height and comprehensive layer injection rate are considered, the diffusion radius increases with an increasing grouting volume, but the growth range gradually decreases. The possibility of improving the slurry diffusion radius solely by increasing the grouting volume gradually decreases. According to Figures 6(a)–6(c), the diffusion radius has a nonlinear negative correlation with the grouting section height and the comprehensive layer injection rate, the rate of change is relatively slow, and there are no distinguishable stages. Under the condition of the same grouting volume and grouting section height, it was found that increasing the comprehensive layer injection rate reduces the diffusion radius, while the grouting volume in the unit volume soil layer increases, which can effectively improve the mechanical properties of the injected layer. In engineering, multiple grouting parameters are usually coordinated and controlled to achieve the designed diffusion radius and reinforcement strength. For example, before formal grouting on site, water injection tests are often carried out to increase the hydraulic permeability of the layer, which can improve the comprehensive layer injection rate. Accordingly, the design grouting volume should also be increased to achieve the desired grouting diffusion radius.


## 4. Conclusions

(1)
The cylindrical diffusion of grouting in the deep, loose layer is regarded as the flow diffusion of slurry in a certain number of tortuous circular tubes radiated by the slurry column. The cylindrical diffusion model of the radial tube flow in a deep, loose layer is established, and the variation laws of key parameters such as slurry pressure and diffusion radius in the process of grouting are realized(2)
The permeability coefficient considering tortuosity is derived, and the average radius of the seepage tubes in the injected layer is equivalently replaced by the permeability coefficient. The diffusion velocity approximate control differential equation of the Bingham slurry in a single tube, the diffusion velocity control equation of the radiation tube flow, and the slurry pressure distribution function in the diffusion area are obtained(3)
By combining the model results with the field grouting test results in the Guotun coal mine, the rationality of the cylindrical diffusion model of radial tube flow was verified, and the attenuation law of the grouting pressure in the diffusion area is obtained. It was found that the attenuation of the slurry pressure exhibits distinguishable stages. Within the first 30% of the diffusion radius, the slurry pressure decreases sharply by approximately 70%; within the last 70% of the diffusion radius, the slurry pressure decreases gradually to the remaining 30%(4)
The calculation formula of the grouting diffusion radius is deduced, and the effects of the grouting volume, grouting section height, and comprehensive layer injection rate on the grouting diffusion radius are discussed. It is found that the diffusion radius has a nonlinear positive correlation with the grouting volume, but the corresponding growth range gradually decreases. The diffusion radius has a nonlinear negative correlation with the grouting section height and the comprehensive layer injection rate; the rate of change is relatively low, and no distinguishable stages are observed



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*Source: 1302260-2022-03-25.xml*