# Estimation of Aperture and Stiffness of Fractures under High Water Pressure Using Hydrological Data and Slurry Consolidating Body

Authors: Jianbo Li; Yanchun Xu
Journal: Geofluids
 (2022)

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## Abstract

The fractured limestone aquifers under coal seams bring great threat to coal production, and grouting engineering before mining is usually used to seal the fractures and prevent water inrushes. However, the grouting mechanism is still a difficult issue since the hidden fissures cannot be observed. In this paper, two characteristics of fractures filled with high-pressure water, deformation, and stiffness are studied to seek some good explanations for grouting. First, an ideal flow model composed of a borehole and a fracture is established with the fracture aperture as the main variable. The estimation methods for the aperture of fractures with high-pressure water before grouting and the stiffness of the fracture filled with slurry consolidating body during high-pressure grouting are proposed, and the millimeter-scale solution of deformation is obtained by using hydrological data and slurry consolidating body. Then, the methods are used for the field grouting practice, and its applicable conditions are given. The estimated apertures of the fractures with high-pressure water range from 2.08 to 2.56 mm. Under the grouting pressure of 15 MPa and the designed slurry diffusion radius of 30 m, the normal stiffness of the fracture filled with slurry consolidating body ranges from 0.5 to 14.4 GPa/m. The results meet the positive relationship between normal stiffness and fracture aperture got by the literature data and can be used for the study of grouting effect of the hidden fractures, including the evaluation of strength and water resistance of the slurry consolidating body. Our method of estimating the aperture and stiffness of fractures is applicable when grouting is used in the fractured limestone aquifers with high water pressure under coal seams before mining.


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## Body

## 1. Introduction

Grouting reinforcement of the deep buried confined aquifers under coal seams is an important technology for the safety of coal production in northern China. However, in practice, although the aquifers in some mining areas are considered to be fully grouted, there are still water accidents occurring in the working face during mining. One main reason is that the fracture characteristics of hidden aquifers can not be seen, and grouting only depends on experience in practice, which has seriously troubled researchers. In order to study the fracture characteristics of grouted aquifers for the better implementation of grouting, this research on aperture and stiffness of fractures in deep buried limestone aquifers with high-pressure water is carried out. The related literatures on fracture apertures, fracture stiffnesses, fracture models, and estimating methods of apertures are first retrieved and analyzed, consistent with the main research content of this manuscript.In discussion of fracture models, the equivalent model is usually used to simplify the study of the complexed fractured aquifers, such as discrete equivalent geometric model of nonlinear flow, fine pipe flow model [1], network model for fractures [2], and plane parallel plate model for a single fracture. According to the dip angles of fractures, the models for grouting [3, 4] and water movement [5] can be divided into horizontal fracture models and vertical fracture models in theoretical and similar experiment researches. The model in this paper is an approximate vertical fracture model with an inclined borehole.In terms of fracture apertures, according to the International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests, an aperture between 0.5 and 10 mm is classified to have “gapped” features, which means that the aperture has a width that is open to moderately wide. Fractures with such an aperture are usually grouted to control groundwater inrush. In the latest study [6], aperture structures from 0.01 to 2 mm are selected for the study of fracture deformation and fluid flow. Specifically, there are many researches on fracture aperture relating to grouting, and the range of fracture aperture is large from 10 μm to 5 mm, including micron scale values [7–13] and millimeter scale values [14–18]. Obviously, the range of apertures is large from microns to millimeters, and the precise law has not been obtained. Moreover, the natural water pressure of the fractures and the grouting pressure are both smaller than that of this paper. Similar to the deformation study, the range of stiffness is also very wide. The estimating values of the normal stiffnesses of carbonate rock range from about 1.4 to 200 GPa/m [13]. For apertures ranging from 12 to 30 μm, the normal stiffnesses can vary from 10 to 1100 GPa/m [19, 20], which are smaller than the experimental values. For apertures ranging from 100 to 250 μm, the normal stiffnesses range from 2 to 10 GPa/m [21, 22]. The relationship between stiffness and deformation shows a positive correlation with fracture apertures rather than a nonlinear relationship, which provides space for research. In addition, the relationship between grouting and apertures is also used for the fracture dilation during grouting [23, 24], groutability prediction [25], and grouting effect [26–30]. However, it is difficult to determine the aperture of hidden fractures with high-pressure water in field application.In terms of estimating methods, the equivalent hydraulic aperture is often used to evaluate the width of fractures and usually estimated by using the cubic law, according to the data including water inflow and hydraulic head [31, 32]. However, the natural water pressure of the fractures and the grouting pressure are both smaller than that of this paper (more than 6 MPa), while high pressure has an effect on the fracture apertures. When the fracture was just formed in the geological history, it had an original aperture. Then, under the action of high water pressure in the aquifer (more than 6 MPa), the fracture expands on the basis of the original aperture. The apertures studied in this paper are the expanded fracture apertures by high water pressure. In addition, there are many studies of fracture aperture and stiffness close to the tunnel during grouting, while the research on the aperture of the fractures with high-pressure water in the deep buried limestone aquifers under coal seams has not been studied.This paper presents a borehole-fracture model of grouting for deep buried limestone aquifers under high water pressure. Based on the monitored hydrological parameters, a simple estimation method for aperture and normal stiffness of fractures with high-pressure water was put forward and then applied in the field grouting project. At last, the applicable conditions of the calculation results are discussed in detail.


## 2. Characteristics of Limestone Aquifers and Grouting Engineering

The mined coal seam located in Henan Province of China is threatened by hidden limestone aquifers with high-pressure water under coal seams,L9, L8, and L2. The spatial layout of aquifers and mined coal seams is shown in Figure 1. The water pressure of L2 is up to 6.4 MPa. During coal exploitation, the confined water may penetrate vertically upwards from L2 into working face through fractures, which increases the risk of water inrushes. For the safety of mine production, grouting is selected to seal the fractures and reconstruct the fractured limestone aquifers. In this paper, L2 was selected as the main studied aquifer.Figure 1
Characteristics of aquifers under coal seams and spatial layout of grouting boreholes.Generally, when boreholes encounter wide fractures with high-pressure water during drilling, the water inflow from the borehole is large (e.g., Borehole 1#). The water inflow from the borehole is small when meeting narrow fractures (e.g., Borehole 4#), and there is no water flowing out when boreholes do not encounter fractures (e.g., Borehole 3#), which is in line with general cognition that the limestone aquifers contain a large number of inclined or vertical fractures and the distribution of fractures in these aquifers is uneven and irregular. The terminal of drilling depends on the water inflow, which put forward strict requirements for grouting technicians.


## 3. Analytic Model for Grouting in Fractures

### 3.1. Ideal Fracture-Borehole Flow Model

Usually, experimental research is used as an important means to study the fracture grouting. Two sketch maps of boreholes and inclined fractures are listed in Figure2. The fractures are regarded as plane-parallel plates, and the fracture apertures are given in advance.Figure 2
Sketch maps of boreholes and inclined fractures: (a) a steep fracture penetrated by boreholes [4]; (b) four fractures with steep dip angle penetrated by a borehole in tunnel excavation [1].
(a)(b)However, it is difficult to determine the hidden fracture aperture with high-pressure water. When the fractures in the deep buried aquifers are exposed by drilling holes, the confined groundwater can flow out along the borehole through fractures under the action of high water pressure. Based on grouting practice, the fracture-borehole systems with different apertures are drawn as shown in Figure3. The ideal flow models under high water pressure contain a fracture and a borehole with diameter of 75 mm and length of L. In both cases with different fracture widths (δw and δn), water inflow q and water pressure p can be measured at the borehole outlet.Figure 3
Borehole-fracture flow models with (a) large apertureδw for free flow and (b) small aperture δn. The narrow fracture is penetrated by drilling borehole. H is aquifer height.
(a)(b)In the model, the fracture aperture is regarded as the main variable to determine the flow patterns while other parameters remain unchanged. Different flow patterns may occur, such as free flow and fissure flow. Supposeδb is the aperture of the critical state of free flow and fracture flow. When the aperture δw is larger than δb, free flow occurs (Figure 3(a)); when the aperture δn is less than δb, fracture flow occurs (Figure 3(b)). vb is the critical boundary velocity corresponding to δb.For a fracture-borehole flow system, if pure water is injected from the borehole outlet instead of slurry, the pure water can continuously flow into the fracture as long as the inject pressure is selected appropriately. The process of pure water injection is completely opposite to the fracture water inflow process. However, the grouting of clay-cement slurry is different from the injection of water due to its fluid characteristics, such as viscosity, deposition, and solidification, which is the focus of this paper.


### 3.2. Theoretical Solution of Fracture-Borehole Flow Model

#### 3.2.1. General Equations for Free Flow

The free flow pattern with critical aperture ofδb is selected to be studied. It is easy to get the hydrodynamic equations for free flow in a wide fracture-borehole system, as shown in Figure 4. In the system, confined water can flow out freely at a certain velocity when the borehole outlet is open. When the borehole outlet is closed, the water pressure of aquifers can be monitored.Figure 4
(a) Schematic of free flow for fracture-borehole flow system and (b) cross section.v is fluid velocity, and p is water pressure. H is the height of aquifer, and δ is the fracture aperture.
(a)(b)Applying Bernoulli equation in sectionsM and O of Figure 4, the basic equations in the fracture-borehole flow system with critical aperture at cross section M and O can be easily written down as follows:
(1)ZM+pMγ+α1vM22g=ZO+pOγ+α2vO22g+hf,O+hj,O.PointM is in the aquifer, and point O is at the borehole outlet. zM, pM, and vM are, respectively, water head height, water pressure, and flow velocity at cross section M; zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at cross section O; hf,O is the friction resistance of drilling hole; hj,O is the local resistance at the intersection of fracture and borehole. α1 and α2 are the kinetic energy correction factors, usually taken as 1.Another two different forms of Bernoulli equation are easily got through transformation as follows:(2)vM2=1α12gZO−ZM+2pO−pMρ+α2vO2+2ghf,O+hj,O,vO2=2pM−pO/ρ−2gZO−ZM+α1vM2α2+λL/d+ζ.The friction resistancehf,O and local resistance hj,O can be, respectively, calculated by equations hf,O=λL/dv2/2g and hj,O=ζv2/2g, where λ is the friction factor, L is the borehole length, d is the diameter of the borehole, and v is the water flow velocity in the borehole. ζ is the local resistance coefficient.


#### 3.2.2. Flow Model of a Wide Fracture with High-Pressure Water

For wide fracture, free flow occurs, and the flow at the junction from a wide fracture not penetrated by a borehole is shown in Figure5. The model of this paper shown in Figure 5 is similar to orifice outflow in the hydrodynamic textbooks. The water in the fracture is sufficient and far greater than the water output from the borehole. Cross section C-C is set as a reference plane. Cross section S-S is another virtual cross section not far away from C-C, and the continuity equation is applicable to these two cross sections.Figure 5
Sketch map of the flow at the junction from a wide fracture not penetrated by a borehole.Thus, the energy conservation equation at cross sections C-C and S-S is written as follows:(3)zs+psρg+vs22g=zc+pcρg+vc22g+ζvc22g,where zS, pS, and vS are, respectively, water head height, water pressure, and flow velocity at section S-S and zC, pC, and vC are, respectively, head height, water pressure, and flow velocity at cross section C-C. hj,O=ζvc2/2g is the local resistance at the intersection of the fracture and the borehole. The kinetic energy correction coefficient is temporarily taken as 1.The borehole diameter is very small compared with the aquifer thickness; the friction resistance can be ignored but the local resistance needs to be considered.Similarly, the energy equation is also applied at both ends of the borehole, written as follows:(4)ZC+pcρg+αCvc22g=ZO+pOγ+α2vO22g+hf,O,where zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at the cross section of borehole outlet; hf,O=λL/dvO2/2g is the friction resistance of drilling hole; αC is the kinetic energy correction factors, taken as 1.Therefore, according to Equations (3) and (4), the following Equation (5) with vO as the dependent variable can be obtained. vo is the flow velocity at the cross section of borehole outlet and can be monitored by instruments.
(5)vO2=2pS−pO/ρ−2gZO−ZS+αSvS2α2+λL/d+ζ.The term,ps−po, in Equation (5) has a great influence on vo, especially when the water pressure is very high. The term of vs in Equation (5) has a positive correlation effect on vo. The greater the aquifer velocity is, the greater the borehole outlet velocity will be. However, in practice, the water pressure is much larger than the square of the velocity of the aquifer. So the influence of aquifer velocity on the flow velocity at the borehole outlet is small, and the theoretical verification is completed as follows.


#### 3.2.3. Estimation of the Free Flow Velocity at the Borehole Outlet

Take borehole D23-8 as an example, the case when the water flow in the fracture is static is first considered with the water pressure of 5.8 MPa.According to the basic theory of fluid mechanics [33], when the inlet section of boreholes is mounted into the end-face wall at an angle, the inlet local resistance increases. The local resistance coefficient ζ can be calculated as follows:
(6)ζ=0.5+0.3sinθ+0.2sin2θ.The inclination angle of borehole D23-8 is 35°, so the local resistance coefficientζ is 0.74. Use the Moody diagram to determine the friction factor. Referring to the roughness of the cement pipe, the relative roughness of the borehole wall is obtained by Δ/d=0.33/75=0.005333. From the Moody diagram, the friction factor is preliminarily determined as λ=0.032.Therefore, when the fracture is filled with static high pressure water, the velocity of borehole outletvO is calculated as 12.69 m/s by substituting parameters into Equation (5).The flow velocity of aquifers sometimes may be greater than or equal to zero (vM≥0). As the aquifer velocity vM gradually increases from 0 to 11.3 m/s, the velocity of borehole outlet vO changes very slowly from 12.69 m/s to 12.77 m/s. The detailed relationship between vS and vO for free flow is obtained by theoretical calculation, as shown in Figure 6. vO is the velocity at the borehole outlet, and vS is the aquifer velocity. The aquifer velocity has little effect on the outlet velocity under high water pressure.Figure 6
Changing curve of flow velocity at the borehole outlet with aquifer velocity of D23-8.It is worth noting that the velocity difference,ΔvO, is smaller than the aquifer velocity difference ΔvS, and the approximate derivative function of the vO−vS curve is 12E-06x. Compared with high water pressure (the order of magnitude of the item containing water pressure is 103), the flow velocity of the aquifer (the order of magnitude of the item containing flow velocity is 102) has little effect on the velocity at the borehole outlet.For free flow, when the flow velocity of the aquifer changes fromvS to vS+ΔvS, the flow velocity at borehole outlet vO changes very little, and the difference of flow velocity at borehole outlet, ΔvO, can be ignored especially from engineering scale. That is, when the flow velocity of the aquifer changes from vS to vS+ΔvS, the minimum free flow velocity vmin for free flow at the borehole outlet is got by
(7)vmin=lim△vO⟶0vO+△vO=vO.This formula for the minimum free flow velocityvmin is important and can be used in engineering scale. In many cases, the flow velocity does not reach this value, which is mainly attributed to the small crack width in this paper.


#### 3.2.4. Velocity Coefficient of Narrow Fractures

In practice, the flow velocity at the borehole outlet does not reachvmin, far less than vmin, although the water pressure is very high. There is a big gap between the actual velocity and the theoretical velocity. This paper mainly studies the decrease of fracture water inflow due to the fracture width. High water pressure and low water inflow conform to the jet flow characteristics. A jet model of a fracture penetrated by a borehole is shown in Figure 7. The junction between the fracture and the borehole is tentatively determined as regular circle.Figure 7
Flow at the junction from a narrow fracture penetrated by a borehole.
(a)(b)For a narrow fracture, cross section A-A and cross section B-B are selected. Section B-B is close to the hole wall. Section A-A is an ideal section not far away from section B-B, and the mass conservation satisfies on sections A-A and B-B.Similarly, according to the conservation of energy, write the equations of cross section A-A and cross section B-B(8)pAρg+vA22g=pBρg+vj22g+ζnvj22g,where vj is the flow velocity of section B-B very close to the hole wall.Ignore the frictional resistance between section A-A and section B-B.ζn is the local resistance coefficient to be solved when the fracture water jets into the borehole in case of narrow fractures.According to the continuity equation, the formula,vAAA=vjAB=vOAO, can be got, where A is the section area. So, vA=d/D2vj. D is the diameter of circle marked on section A-A.The velocityvj can be obtained by transforming Equation (8) as follows:
(9)vj=11+ζn−d/D42Δp/ρ.Ford≪D, d/D4≈0, then
(10)vj=11+ζn2Δpρ=Cv2Δpρ,where Cv=1/1+ζn is the velocity coefficient.The velocity coefficientCv is the ratio of actual velocity to theoretical velocity of free flow,
(11)Cv=vOvmin.The actual velocity is got by drilling boreholes, and the theoretical velocity has been calculated above. The velocity coefficient is obtained, and then,vj can be obtained. Finally, the fracture aperture can be calculated.


#### 3.2.5. Estimation of Fracture Aperture

The areaA of the annular junction of borehole and aquifer is calculated by A=πd∗δn, where d is the diameter of the borehole and δn is the narrow fracture aperture. According to the continuity equation, the fracture aperture δn can be deduced as follows:
(12)δn=q/3600/vj/πd=q/3600/Cv2Δp/ρ/πd,where q is the water inflow monitored at the borehole outlet and d is the borehole diameter.So far, the fracture aperture has been determined, and its application is carried out in the following sections.


#### 3.2.6. Estimation of Stiffness of the Fractures Filled with Slurry Consolidating Body

Under the action of high grouting pressure, fracture expansion occurs. In order to explain the problem easily, the average deformation of fractures is used in the next calculation. After grouting is completed, the fractures affected by high injection pressure include consolidation zone and water separating zone, and the fracture filled with consolidating body of slurry is the effective area to prevent water inrush, which is regarded in this paper as an important parameter for calculating the stiffness.δ0 is the fracture aperture under high water pressure before grouting, and δj is the expanded average aperture of the fracture filled with consolidating body after grouting. Δun is the difference of the deformation before and after grouting and can be obtained as follows:
(13)Δun=δj−δ0,whereδj is determined by the quantity of the consolidating body (effective grouting slurry stone) in the fracture Qs. The final grouting slurry stone is the key part that can block water. For the convenience of calculation, it is assumed to be ideally and evenly distributed in the fracture. Here, δ0=δn and
(14)δj=QsLdH,where Ld is the distance of fracture effectively filled with slurry consolidating body.The difference between grouting pressure and water pressure,Δσ, can be got by
(15)Δσ=pj−pw,where pj is the grouting pressure and pw is the water pressure in the fracture.Then, the fracture normal stiffness under the action of water pressure,kn, is expressed [13] as follows:
(16)kn=Δσ△un=pj−pwQs/LdH−δ0.


## 3.1. Ideal Fracture-Borehole Flow Model

Usually, experimental research is used as an important means to study the fracture grouting. Two sketch maps of boreholes and inclined fractures are listed in Figure2. The fractures are regarded as plane-parallel plates, and the fracture apertures are given in advance.Figure 2
Sketch maps of boreholes and inclined fractures: (a) a steep fracture penetrated by boreholes [4]; (b) four fractures with steep dip angle penetrated by a borehole in tunnel excavation [1].
(a)(b)However, it is difficult to determine the hidden fracture aperture with high-pressure water. When the fractures in the deep buried aquifers are exposed by drilling holes, the confined groundwater can flow out along the borehole through fractures under the action of high water pressure. Based on grouting practice, the fracture-borehole systems with different apertures are drawn as shown in Figure3. The ideal flow models under high water pressure contain a fracture and a borehole with diameter of 75 mm and length of L. In both cases with different fracture widths (δw and δn), water inflow q and water pressure p can be measured at the borehole outlet.Figure 3
Borehole-fracture flow models with (a) large apertureδw for free flow and (b) small aperture δn. The narrow fracture is penetrated by drilling borehole. H is aquifer height.
(a)(b)In the model, the fracture aperture is regarded as the main variable to determine the flow patterns while other parameters remain unchanged. Different flow patterns may occur, such as free flow and fissure flow. Supposeδb is the aperture of the critical state of free flow and fracture flow. When the aperture δw is larger than δb, free flow occurs (Figure 3(a)); when the aperture δn is less than δb, fracture flow occurs (Figure 3(b)). vb is the critical boundary velocity corresponding to δb.For a fracture-borehole flow system, if pure water is injected from the borehole outlet instead of slurry, the pure water can continuously flow into the fracture as long as the inject pressure is selected appropriately. The process of pure water injection is completely opposite to the fracture water inflow process. However, the grouting of clay-cement slurry is different from the injection of water due to its fluid characteristics, such as viscosity, deposition, and solidification, which is the focus of this paper.


## 3.2. Theoretical Solution of Fracture-Borehole Flow Model

### 3.2.1. General Equations for Free Flow

The free flow pattern with critical aperture ofδb is selected to be studied. It is easy to get the hydrodynamic equations for free flow in a wide fracture-borehole system, as shown in Figure 4. In the system, confined water can flow out freely at a certain velocity when the borehole outlet is open. When the borehole outlet is closed, the water pressure of aquifers can be monitored.Figure 4
(a) Schematic of free flow for fracture-borehole flow system and (b) cross section.v is fluid velocity, and p is water pressure. H is the height of aquifer, and δ is the fracture aperture.
(a)(b)Applying Bernoulli equation in sectionsM and O of Figure 4, the basic equations in the fracture-borehole flow system with critical aperture at cross section M and O can be easily written down as follows:
(1)ZM+pMγ+α1vM22g=ZO+pOγ+α2vO22g+hf,O+hj,O.PointM is in the aquifer, and point O is at the borehole outlet. zM, pM, and vM are, respectively, water head height, water pressure, and flow velocity at cross section M; zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at cross section O; hf,O is the friction resistance of drilling hole; hj,O is the local resistance at the intersection of fracture and borehole. α1 and α2 are the kinetic energy correction factors, usually taken as 1.Another two different forms of Bernoulli equation are easily got through transformation as follows:(2)vM2=1α12gZO−ZM+2pO−pMρ+α2vO2+2ghf,O+hj,O,vO2=2pM−pO/ρ−2gZO−ZM+α1vM2α2+λL/d+ζ.The friction resistancehf,O and local resistance hj,O can be, respectively, calculated by equations hf,O=λL/dv2/2g and hj,O=ζv2/2g, where λ is the friction factor, L is the borehole length, d is the diameter of the borehole, and v is the water flow velocity in the borehole. ζ is the local resistance coefficient.


### 3.2.2. Flow Model of a Wide Fracture with High-Pressure Water

For wide fracture, free flow occurs, and the flow at the junction from a wide fracture not penetrated by a borehole is shown in Figure5. The model of this paper shown in Figure 5 is similar to orifice outflow in the hydrodynamic textbooks. The water in the fracture is sufficient and far greater than the water output from the borehole. Cross section C-C is set as a reference plane. Cross section S-S is another virtual cross section not far away from C-C, and the continuity equation is applicable to these two cross sections.Figure 5
Sketch map of the flow at the junction from a wide fracture not penetrated by a borehole.Thus, the energy conservation equation at cross sections C-C and S-S is written as follows:(3)zs+psρg+vs22g=zc+pcρg+vc22g+ζvc22g,where zS, pS, and vS are, respectively, water head height, water pressure, and flow velocity at section S-S and zC, pC, and vC are, respectively, head height, water pressure, and flow velocity at cross section C-C. hj,O=ζvc2/2g is the local resistance at the intersection of the fracture and the borehole. The kinetic energy correction coefficient is temporarily taken as 1.The borehole diameter is very small compared with the aquifer thickness; the friction resistance can be ignored but the local resistance needs to be considered.Similarly, the energy equation is also applied at both ends of the borehole, written as follows:(4)ZC+pcρg+αCvc22g=ZO+pOγ+α2vO22g+hf,O,where zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at the cross section of borehole outlet; hf,O=λL/dvO2/2g is the friction resistance of drilling hole; αC is the kinetic energy correction factors, taken as 1.Therefore, according to Equations (3) and (4), the following Equation (5) with vO as the dependent variable can be obtained. vo is the flow velocity at the cross section of borehole outlet and can be monitored by instruments.
(5)vO2=2pS−pO/ρ−2gZO−ZS+αSvS2α2+λL/d+ζ.The term,ps−po, in Equation (5) has a great influence on vo, especially when the water pressure is very high. The term of vs in Equation (5) has a positive correlation effect on vo. The greater the aquifer velocity is, the greater the borehole outlet velocity will be. However, in practice, the water pressure is much larger than the square of the velocity of the aquifer. So the influence of aquifer velocity on the flow velocity at the borehole outlet is small, and the theoretical verification is completed as follows.


### 3.2.3. Estimation of the Free Flow Velocity at the Borehole Outlet

Take borehole D23-8 as an example, the case when the water flow in the fracture is static is first considered with the water pressure of 5.8 MPa.According to the basic theory of fluid mechanics [33], when the inlet section of boreholes is mounted into the end-face wall at an angle, the inlet local resistance increases. The local resistance coefficient ζ can be calculated as follows:
(6)ζ=0.5+0.3sinθ+0.2sin2θ.The inclination angle of borehole D23-8 is 35°, so the local resistance coefficientζ is 0.74. Use the Moody diagram to determine the friction factor. Referring to the roughness of the cement pipe, the relative roughness of the borehole wall is obtained by Δ/d=0.33/75=0.005333. From the Moody diagram, the friction factor is preliminarily determined as λ=0.032.Therefore, when the fracture is filled with static high pressure water, the velocity of borehole outletvO is calculated as 12.69 m/s by substituting parameters into Equation (5).The flow velocity of aquifers sometimes may be greater than or equal to zero (vM≥0). As the aquifer velocity vM gradually increases from 0 to 11.3 m/s, the velocity of borehole outlet vO changes very slowly from 12.69 m/s to 12.77 m/s. The detailed relationship between vS and vO for free flow is obtained by theoretical calculation, as shown in Figure 6. vO is the velocity at the borehole outlet, and vS is the aquifer velocity. The aquifer velocity has little effect on the outlet velocity under high water pressure.Figure 6
Changing curve of flow velocity at the borehole outlet with aquifer velocity of D23-8.It is worth noting that the velocity difference,ΔvO, is smaller than the aquifer velocity difference ΔvS, and the approximate derivative function of the vO−vS curve is 12E-06x. Compared with high water pressure (the order of magnitude of the item containing water pressure is 103), the flow velocity of the aquifer (the order of magnitude of the item containing flow velocity is 102) has little effect on the velocity at the borehole outlet.For free flow, when the flow velocity of the aquifer changes fromvS to vS+ΔvS, the flow velocity at borehole outlet vO changes very little, and the difference of flow velocity at borehole outlet, ΔvO, can be ignored especially from engineering scale. That is, when the flow velocity of the aquifer changes from vS to vS+ΔvS, the minimum free flow velocity vmin for free flow at the borehole outlet is got by
(7)vmin=lim△vO⟶0vO+△vO=vO.This formula for the minimum free flow velocityvmin is important and can be used in engineering scale. In many cases, the flow velocity does not reach this value, which is mainly attributed to the small crack width in this paper.


### 3.2.4. Velocity Coefficient of Narrow Fractures

In practice, the flow velocity at the borehole outlet does not reachvmin, far less than vmin, although the water pressure is very high. There is a big gap between the actual velocity and the theoretical velocity. This paper mainly studies the decrease of fracture water inflow due to the fracture width. High water pressure and low water inflow conform to the jet flow characteristics. A jet model of a fracture penetrated by a borehole is shown in Figure 7. The junction between the fracture and the borehole is tentatively determined as regular circle.Figure 7
Flow at the junction from a narrow fracture penetrated by a borehole.
(a)(b)For a narrow fracture, cross section A-A and cross section B-B are selected. Section B-B is close to the hole wall. Section A-A is an ideal section not far away from section B-B, and the mass conservation satisfies on sections A-A and B-B.Similarly, according to the conservation of energy, write the equations of cross section A-A and cross section B-B(8)pAρg+vA22g=pBρg+vj22g+ζnvj22g,where vj is the flow velocity of section B-B very close to the hole wall.Ignore the frictional resistance between section A-A and section B-B.ζn is the local resistance coefficient to be solved when the fracture water jets into the borehole in case of narrow fractures.According to the continuity equation, the formula,vAAA=vjAB=vOAO, can be got, where A is the section area. So, vA=d/D2vj. D is the diameter of circle marked on section A-A.The velocityvj can be obtained by transforming Equation (8) as follows:
(9)vj=11+ζn−d/D42Δp/ρ.Ford≪D, d/D4≈0, then
(10)vj=11+ζn2Δpρ=Cv2Δpρ,where Cv=1/1+ζn is the velocity coefficient.The velocity coefficientCv is the ratio of actual velocity to theoretical velocity of free flow,
(11)Cv=vOvmin.The actual velocity is got by drilling boreholes, and the theoretical velocity has been calculated above. The velocity coefficient is obtained, and then,vj can be obtained. Finally, the fracture aperture can be calculated.


### 3.2.5. Estimation of Fracture Aperture

The areaA of the annular junction of borehole and aquifer is calculated by A=πd∗δn, where d is the diameter of the borehole and δn is the narrow fracture aperture. According to the continuity equation, the fracture aperture δn can be deduced as follows:
(12)δn=q/3600/vj/πd=q/3600/Cv2Δp/ρ/πd,where q is the water inflow monitored at the borehole outlet and d is the borehole diameter.So far, the fracture aperture has been determined, and its application is carried out in the following sections.


### 3.2.6. Estimation of Stiffness of the Fractures Filled with Slurry Consolidating Body

Under the action of high grouting pressure, fracture expansion occurs. In order to explain the problem easily, the average deformation of fractures is used in the next calculation. After grouting is completed, the fractures affected by high injection pressure include consolidation zone and water separating zone, and the fracture filled with consolidating body of slurry is the effective area to prevent water inrush, which is regarded in this paper as an important parameter for calculating the stiffness.δ0 is the fracture aperture under high water pressure before grouting, and δj is the expanded average aperture of the fracture filled with consolidating body after grouting. Δun is the difference of the deformation before and after grouting and can be obtained as follows:
(13)Δun=δj−δ0,whereδj is determined by the quantity of the consolidating body (effective grouting slurry stone) in the fracture Qs. The final grouting slurry stone is the key part that can block water. For the convenience of calculation, it is assumed to be ideally and evenly distributed in the fracture. Here, δ0=δn and
(14)δj=QsLdH,where Ld is the distance of fracture effectively filled with slurry consolidating body.The difference between grouting pressure and water pressure,Δσ, can be got by
(15)Δσ=pj−pw,where pj is the grouting pressure and pw is the water pressure in the fracture.Then, the fracture normal stiffness under the action of water pressure,kn, is expressed [13] as follows:
(16)kn=Δσ△un=pj−pwQs/LdH−δ0.


## 3.2.1. General Equations for Free Flow

The free flow pattern with critical aperture ofδb is selected to be studied. It is easy to get the hydrodynamic equations for free flow in a wide fracture-borehole system, as shown in Figure 4. In the system, confined water can flow out freely at a certain velocity when the borehole outlet is open. When the borehole outlet is closed, the water pressure of aquifers can be monitored.Figure 4
(a) Schematic of free flow for fracture-borehole flow system and (b) cross section.v is fluid velocity, and p is water pressure. H is the height of aquifer, and δ is the fracture aperture.
(a)(b)Applying Bernoulli equation in sectionsM and O of Figure 4, the basic equations in the fracture-borehole flow system with critical aperture at cross section M and O can be easily written down as follows:
(1)ZM+pMγ+α1vM22g=ZO+pOγ+α2vO22g+hf,O+hj,O.PointM is in the aquifer, and point O is at the borehole outlet. zM, pM, and vM are, respectively, water head height, water pressure, and flow velocity at cross section M; zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at cross section O; hf,O is the friction resistance of drilling hole; hj,O is the local resistance at the intersection of fracture and borehole. α1 and α2 are the kinetic energy correction factors, usually taken as 1.Another two different forms of Bernoulli equation are easily got through transformation as follows:(2)vM2=1α12gZO−ZM+2pO−pMρ+α2vO2+2ghf,O+hj,O,vO2=2pM−pO/ρ−2gZO−ZM+α1vM2α2+λL/d+ζ.The friction resistancehf,O and local resistance hj,O can be, respectively, calculated by equations hf,O=λL/dv2/2g and hj,O=ζv2/2g, where λ is the friction factor, L is the borehole length, d is the diameter of the borehole, and v is the water flow velocity in the borehole. ζ is the local resistance coefficient.


## 3.2.2. Flow Model of a Wide Fracture with High-Pressure Water

For wide fracture, free flow occurs, and the flow at the junction from a wide fracture not penetrated by a borehole is shown in Figure5. The model of this paper shown in Figure 5 is similar to orifice outflow in the hydrodynamic textbooks. The water in the fracture is sufficient and far greater than the water output from the borehole. Cross section C-C is set as a reference plane. Cross section S-S is another virtual cross section not far away from C-C, and the continuity equation is applicable to these two cross sections.Figure 5
Sketch map of the flow at the junction from a wide fracture not penetrated by a borehole.Thus, the energy conservation equation at cross sections C-C and S-S is written as follows:(3)zs+psρg+vs22g=zc+pcρg+vc22g+ζvc22g,where zS, pS, and vS are, respectively, water head height, water pressure, and flow velocity at section S-S and zC, pC, and vC are, respectively, head height, water pressure, and flow velocity at cross section C-C. hj,O=ζvc2/2g is the local resistance at the intersection of the fracture and the borehole. The kinetic energy correction coefficient is temporarily taken as 1.The borehole diameter is very small compared with the aquifer thickness; the friction resistance can be ignored but the local resistance needs to be considered.Similarly, the energy equation is also applied at both ends of the borehole, written as follows:(4)ZC+pcρg+αCvc22g=ZO+pOγ+α2vO22g+hf,O,where zO, pO, and vO are, respectively, water head height, water pressure, and flow velocity at the cross section of borehole outlet; hf,O=λL/dvO2/2g is the friction resistance of drilling hole; αC is the kinetic energy correction factors, taken as 1.Therefore, according to Equations (3) and (4), the following Equation (5) with vO as the dependent variable can be obtained. vo is the flow velocity at the cross section of borehole outlet and can be monitored by instruments.
(5)vO2=2pS−pO/ρ−2gZO−ZS+αSvS2α2+λL/d+ζ.The term,ps−po, in Equation (5) has a great influence on vo, especially when the water pressure is very high. The term of vs in Equation (5) has a positive correlation effect on vo. The greater the aquifer velocity is, the greater the borehole outlet velocity will be. However, in practice, the water pressure is much larger than the square of the velocity of the aquifer. So the influence of aquifer velocity on the flow velocity at the borehole outlet is small, and the theoretical verification is completed as follows.


## 3.2.3. Estimation of the Free Flow Velocity at the Borehole Outlet

Take borehole D23-8 as an example, the case when the water flow in the fracture is static is first considered with the water pressure of 5.8 MPa.According to the basic theory of fluid mechanics [33], when the inlet section of boreholes is mounted into the end-face wall at an angle, the inlet local resistance increases. The local resistance coefficient ζ can be calculated as follows:
(6)ζ=0.5+0.3sinθ+0.2sin2θ.The inclination angle of borehole D23-8 is 35°, so the local resistance coefficientζ is 0.74. Use the Moody diagram to determine the friction factor. Referring to the roughness of the cement pipe, the relative roughness of the borehole wall is obtained by Δ/d=0.33/75=0.005333. From the Moody diagram, the friction factor is preliminarily determined as λ=0.032.Therefore, when the fracture is filled with static high pressure water, the velocity of borehole outletvO is calculated as 12.69 m/s by substituting parameters into Equation (5).The flow velocity of aquifers sometimes may be greater than or equal to zero (vM≥0). As the aquifer velocity vM gradually increases from 0 to 11.3 m/s, the velocity of borehole outlet vO changes very slowly from 12.69 m/s to 12.77 m/s. The detailed relationship between vS and vO for free flow is obtained by theoretical calculation, as shown in Figure 6. vO is the velocity at the borehole outlet, and vS is the aquifer velocity. The aquifer velocity has little effect on the outlet velocity under high water pressure.Figure 6
Changing curve of flow velocity at the borehole outlet with aquifer velocity of D23-8.It is worth noting that the velocity difference,ΔvO, is smaller than the aquifer velocity difference ΔvS, and the approximate derivative function of the vO−vS curve is 12E-06x. Compared with high water pressure (the order of magnitude of the item containing water pressure is 103), the flow velocity of the aquifer (the order of magnitude of the item containing flow velocity is 102) has little effect on the velocity at the borehole outlet.For free flow, when the flow velocity of the aquifer changes fromvS to vS+ΔvS, the flow velocity at borehole outlet vO changes very little, and the difference of flow velocity at borehole outlet, ΔvO, can be ignored especially from engineering scale. That is, when the flow velocity of the aquifer changes from vS to vS+ΔvS, the minimum free flow velocity vmin for free flow at the borehole outlet is got by
(7)vmin=lim△vO⟶0vO+△vO=vO.This formula for the minimum free flow velocityvmin is important and can be used in engineering scale. In many cases, the flow velocity does not reach this value, which is mainly attributed to the small crack width in this paper.


## 3.2.4. Velocity Coefficient of Narrow Fractures

In practice, the flow velocity at the borehole outlet does not reachvmin, far less than vmin, although the water pressure is very high. There is a big gap between the actual velocity and the theoretical velocity. This paper mainly studies the decrease of fracture water inflow due to the fracture width. High water pressure and low water inflow conform to the jet flow characteristics. A jet model of a fracture penetrated by a borehole is shown in Figure 7. The junction between the fracture and the borehole is tentatively determined as regular circle.Figure 7
Flow at the junction from a narrow fracture penetrated by a borehole.
(a)(b)For a narrow fracture, cross section A-A and cross section B-B are selected. Section B-B is close to the hole wall. Section A-A is an ideal section not far away from section B-B, and the mass conservation satisfies on sections A-A and B-B.Similarly, according to the conservation of energy, write the equations of cross section A-A and cross section B-B(8)pAρg+vA22g=pBρg+vj22g+ζnvj22g,where vj is the flow velocity of section B-B very close to the hole wall.Ignore the frictional resistance between section A-A and section B-B.ζn is the local resistance coefficient to be solved when the fracture water jets into the borehole in case of narrow fractures.According to the continuity equation, the formula,vAAA=vjAB=vOAO, can be got, where A is the section area. So, vA=d/D2vj. D is the diameter of circle marked on section A-A.The velocityvj can be obtained by transforming Equation (8) as follows:
(9)vj=11+ζn−d/D42Δp/ρ.Ford≪D, d/D4≈0, then
(10)vj=11+ζn2Δpρ=Cv2Δpρ,where Cv=1/1+ζn is the velocity coefficient.The velocity coefficientCv is the ratio of actual velocity to theoretical velocity of free flow,
(11)Cv=vOvmin.The actual velocity is got by drilling boreholes, and the theoretical velocity has been calculated above. The velocity coefficient is obtained, and then,vj can be obtained. Finally, the fracture aperture can be calculated.


## 3.2.5. Estimation of Fracture Aperture

The areaA of the annular junction of borehole and aquifer is calculated by A=πd∗δn, where d is the diameter of the borehole and δn is the narrow fracture aperture. According to the continuity equation, the fracture aperture δn can be deduced as follows:
(12)δn=q/3600/vj/πd=q/3600/Cv2Δp/ρ/πd,where q is the water inflow monitored at the borehole outlet and d is the borehole diameter.So far, the fracture aperture has been determined, and its application is carried out in the following sections.


## 3.2.6. Estimation of Stiffness of the Fractures Filled with Slurry Consolidating Body

Under the action of high grouting pressure, fracture expansion occurs. In order to explain the problem easily, the average deformation of fractures is used in the next calculation. After grouting is completed, the fractures affected by high injection pressure include consolidation zone and water separating zone, and the fracture filled with consolidating body of slurry is the effective area to prevent water inrush, which is regarded in this paper as an important parameter for calculating the stiffness.δ0 is the fracture aperture under high water pressure before grouting, and δj is the expanded average aperture of the fracture filled with consolidating body after grouting. Δun is the difference of the deformation before and after grouting and can be obtained as follows:
(13)Δun=δj−δ0,whereδj is determined by the quantity of the consolidating body (effective grouting slurry stone) in the fracture Qs. The final grouting slurry stone is the key part that can block water. For the convenience of calculation, it is assumed to be ideally and evenly distributed in the fracture. Here, δ0=δn and
(14)δj=QsLdH,where Ld is the distance of fracture effectively filled with slurry consolidating body.The difference between grouting pressure and water pressure,Δσ, can be got by
(15)Δσ=pj−pw,where pj is the grouting pressure and pw is the water pressure in the fracture.Then, the fracture normal stiffness under the action of water pressure,kn, is expressed [13] as follows:
(16)kn=Δσ△un=pj−pwQs/LdH−δ0.


## 4. Application in Grouting Engineering for Fractured Aquifers

### 4.1. Grouting Engineering

The drilling boreholes for grouting in a drilling site of a working face are shown in Figure8. The total length of boreholes is up to 20,000 m, and the injection consumptions of cement and clay are more than 3000 t. The designed slurry diffusion radius is less than 30 m, according to the grouting engineering practice. At the termination stage of grouting, minimum flow criterion is used. Maintain a minimum flow rate of 58 L/min for 30 minutes with the grouting pressure of 15 MPa and then stop grouting.Figure 8
Sketch map of boreholes during grouting engineering.


### 4.2. Aperture Estimation of the Fracture under High Water Pressure by Using Hydrological Data of Typical Boreholes

First, take borehole D23-8 as an example. The key fluid parameters of borehole D23-8 are listed in Table1.Table 1
Key fluid parameters of borehole D23-8.
ParametersValueBorehole diameter0.075 mKinematic viscosity (20°C)1.0067×10−6m2/sWater density1000 kg/m3Borehole length141.5 mFlow velocity at the outlet of borehole0.0629079 m/sDrilling angle35°The water inflow of borehole D23-8 is 1 m3/h, and the water flow velocity in the borehole is 6.29×10−2 m/s. The water pressure po monitored at the borehole is 5.0 MPa. By substituting the monitored data of borehole D23-8 into Equations (5), (6), (7), (10), (11), and (12), the fracture aperture penetrated by borehole D23-8 is calculated as 2.38×10−3 m (2.38 mm).Literature retrieval is completed for this theoretical research on the grouting of fractures, and the research results are listed together in Table2, which is conducive to quickly understand the research of fractures and make a rapid judgment or comparison. The research methods include laboratory research and field test. Grouting pressure varies from 0.2 MPa to 15 MPa. The aperture scale ranges from microns to millimeters. The apertures of fractures with high-pressure water of this paper belong to millimeter scale.Table 2
Fracture apertures and its application conditions.
ApertureReferenceGrouting pressureMethods1-3 mmThis paper15.0 MPaField1-5 mmLee (2020)<1 MPaLab1-4 mmSui (2015, 2019)1.5 MPaLab0.1-0.025 mmFunehag (2018)0.2 MPaLab10-230μmGhafar (2017)1.5 MPaLab0.3 mmTani (2017)0.6 MPaLab5 mmLi (2016)350 KPaLab and field100μmPlace (2016)1.5 MPaLab100-500μmMohammed (2015)1.5 MPaLab0.1-2 mmHakami (1995)Lab0.2-2 mmJung (1989)1.9 MPaLabThen, substitute the monitored parameters of 9 typical boreholes into Equations (11) and (12), the corresponding theoretical flow velocity vmin of free flow at the borehole outlet, the velocity coefficient of the fracture-borehole system Cv, and the fracture aperture δn in the limestone aquifers are got and listed in Table 3. In practice, compared the theoretical water inflow with the actual water inflow, the water inflow monitored by boreholes does not reach the theoretical calculation value, and the fracture aperture is less than 3 mm and far less than the borehole diameter 75 mm.Table 3
Monitored data, velocity coefficient, and apertures of typical boreholes.
BoreholeLength/mqm3/hpw/MPavmin/m/sCvvj/m/sApertureδn/mmD17-131871511.085.68E-030.572.08D18-31472.55.513.061.20E-021.262.34D18-816214.811.645.40E-030.532.23D19-9n12214.813.354.71E-030.462.56D20-10n1421.24.512.026.28E-030.602.38D21-31471.23.810.866.95E-030.612.34D23-101221.2513.635.54E-030.552.56D23-314745.212.701.98E-022.022.34D23-8141.51512.694.96E-030.502.38


### 4.3. Stiffness of the Fracture Filled with Slurry Consolidating Body

High grouting pressure can affect the normal deformation of fractures, and the normal deformation is related to the damage degree of rock masses, which can be used to evaluate the quality of rock mass to a certain extent. When the grouting amountQf is given, the effective filling volume, Qs (slurry stone), can be obtained, and the grout-induced fracture aperture, δj, can be calculated. Next, the normal deformation difference due to grouting, Δδ, is calculated by subtracting δn from δj. At last, the normal stiffness of the fracture can be determined. Qs is the volume of the slurry stone, pg is the grout pressure, and Qf is the volume of slurry only in the fracture.The volume of the slurry consolidating body depends on the water separating proportion. The water separating proportion of clay-cement slurry is approximately linear with the cement content. For the slurry with the same specific gravity, the water separating proportion decreases with the increase of cement content. In this paper, the mass fractionC/S of cement in the total solid material is 19.2%, and the specific gravity of slurry is 1.16 g/cm3. According to the experimental results of the relationship between cement content and water separating proportion for the clay cement slurry with different specific gravity [34], the water separating proportion Wsepa is taken as 70%.Substitute the monitored data of 9 boreholes into Equation (16), the corresponding normal stiffness of fractures in the limestone aquifers is got and listed in Table 4.Table 4
Normal stiffness of fractures using the data of typical grouting boreholes.
Bhpg (MPa)△p (MPa)C/S (%)W/SWsepa (%)Qs (m3)Δδ (m)δj (m)Qf (m3)kn (GPa/m)D17-131510.023.83.39653.610.0023510.00442910.304.25D18-3159.519.23.73705.210.0040610.00639617.362.34D18-81510.219.23.73706.460.0057050.00793321.531.79D19-9n1510.230.63.54503.560.001820.0043757.125.61D20-10n1510.519.23.73708.710.0083230.01069829.031.26D21-31511.219.23.737011.880.0122580.01459339.610.91D23-101510.019.23.73703.650.001930.00448512.175.18D23-3159.819.23.737012.520.0130380.01537441.720.75D23-81510.01003.62702.500.0006940.0030738.3414.4The apertures of fractures filled with high-pressure water before grouting range from 2.08 to 2.56 mm. Under the grouting pressure of 15 MPa with the designed slurry diffusion radius of 30 m, the normal stiffnesses of the fractures filled with slurry consolidating body range from 0.75 to 14.4 GPa/m. The minimum normal stiffness of the fracture exposed by borehole D23-3 is 0.75 GPa/m, and the maximum normal stiffness calculated by using the data of borehole D23-8 is 14.4 GPa/m. In practice, the grouting of borehole D23-8 is more difficult than other boreholes, and pure cement slurry has to be used for better plugging effect during grouting.


### 4.4. Discussion

#### 4.4.1. Fracture Stiffness and Weathering Degree

The normal stiffness of fracture in the aquifers can be affected by weathering. According to the relative weathered degree of limestones, the fractures can be divided into several levels [35]: no weathered to slightly weathered, moderately weathered, and completely weathered. The corresponding lower limit values of stiffnesses are, respectively, 7.9, 4.9, and 3.8 GPa/m. The fractured limestone aquifers in this paper are weathered to a certain extent in some areas.


#### 4.4.2. Fracture Stiffness and Other Influencing Factors

Fracture stiffness is also affected by many other factors, including weathering during deposition in geological history, adjacent fractures, hydraulic erosion, and cyclic loading times. (1) In geological history, after the fracture was formed, it would first suffer from natural weathering during the deposition process, which will reduce the stiffness of the fracture. (2) Then, under the action of in situ stress field and high-pressure water, the local rock adjacent to the fracture may be eroded, and local large deformation will occur during high-pressure grouting, which will also reduce the normal stiffness of the crack. (3) The distance between fractures will also control the normal stiffness. When two fractures are far away, they have no influence on each other. When the spacing is not too large, the adjacent fracture may make the grouting relatively easy.


#### 4.4.3. Engineering Applicable Conditions

High-pressure water and drainage conditions are two important conditions for the estimating of fracture apertures in this paper. The apertures under high water pressure range from 2.08 to 2.56 mm, and the normal stiffnesses of the fractures filled with slurry consolidating body range from 0.75 to 14.4 GPa/m. The results meet the positive relationship between normal stiffness and fracture aperture got by the literature data. Our method of estimating the aperture and stiffness of fractures is applicable when grouting is used in the fractured limestone aquifers with high water pressure under coal seams before mining. It also lays a foundation for the theoretical evaluation of grouting effect of the hidden fractures, including the evaluation of strength and water resistance of the slurry stone.


## 4.1. Grouting Engineering

The drilling boreholes for grouting in a drilling site of a working face are shown in Figure8. The total length of boreholes is up to 20,000 m, and the injection consumptions of cement and clay are more than 3000 t. The designed slurry diffusion radius is less than 30 m, according to the grouting engineering practice. At the termination stage of grouting, minimum flow criterion is used. Maintain a minimum flow rate of 58 L/min for 30 minutes with the grouting pressure of 15 MPa and then stop grouting.Figure 8
Sketch map of boreholes during grouting engineering.


## 4.2. Aperture Estimation of the Fracture under High Water Pressure by Using Hydrological Data of Typical Boreholes

First, take borehole D23-8 as an example. The key fluid parameters of borehole D23-8 are listed in Table1.Table 1
Key fluid parameters of borehole D23-8.
ParametersValueBorehole diameter0.075 mKinematic viscosity (20°C)1.0067×10−6m2/sWater density1000 kg/m3Borehole length141.5 mFlow velocity at the outlet of borehole0.0629079 m/sDrilling angle35°The water inflow of borehole D23-8 is 1 m3/h, and the water flow velocity in the borehole is 6.29×10−2 m/s. The water pressure po monitored at the borehole is 5.0 MPa. By substituting the monitored data of borehole D23-8 into Equations (5), (6), (7), (10), (11), and (12), the fracture aperture penetrated by borehole D23-8 is calculated as 2.38×10−3 m (2.38 mm).Literature retrieval is completed for this theoretical research on the grouting of fractures, and the research results are listed together in Table2, which is conducive to quickly understand the research of fractures and make a rapid judgment or comparison. The research methods include laboratory research and field test. Grouting pressure varies from 0.2 MPa to 15 MPa. The aperture scale ranges from microns to millimeters. The apertures of fractures with high-pressure water of this paper belong to millimeter scale.Table 2
Fracture apertures and its application conditions.
ApertureReferenceGrouting pressureMethods1-3 mmThis paper15.0 MPaField1-5 mmLee (2020)<1 MPaLab1-4 mmSui (2015, 2019)1.5 MPaLab0.1-0.025 mmFunehag (2018)0.2 MPaLab10-230μmGhafar (2017)1.5 MPaLab0.3 mmTani (2017)0.6 MPaLab5 mmLi (2016)350 KPaLab and field100μmPlace (2016)1.5 MPaLab100-500μmMohammed (2015)1.5 MPaLab0.1-2 mmHakami (1995)Lab0.2-2 mmJung (1989)1.9 MPaLabThen, substitute the monitored parameters of 9 typical boreholes into Equations (11) and (12), the corresponding theoretical flow velocity vmin of free flow at the borehole outlet, the velocity coefficient of the fracture-borehole system Cv, and the fracture aperture δn in the limestone aquifers are got and listed in Table 3. In practice, compared the theoretical water inflow with the actual water inflow, the water inflow monitored by boreholes does not reach the theoretical calculation value, and the fracture aperture is less than 3 mm and far less than the borehole diameter 75 mm.Table 3
Monitored data, velocity coefficient, and apertures of typical boreholes.
BoreholeLength/mqm3/hpw/MPavmin/m/sCvvj/m/sApertureδn/mmD17-131871511.085.68E-030.572.08D18-31472.55.513.061.20E-021.262.34D18-816214.811.645.40E-030.532.23D19-9n12214.813.354.71E-030.462.56D20-10n1421.24.512.026.28E-030.602.38D21-31471.23.810.866.95E-030.612.34D23-101221.2513.635.54E-030.552.56D23-314745.212.701.98E-022.022.34D23-8141.51512.694.96E-030.502.38


## 4.3. Stiffness of the Fracture Filled with Slurry Consolidating Body

High grouting pressure can affect the normal deformation of fractures, and the normal deformation is related to the damage degree of rock masses, which can be used to evaluate the quality of rock mass to a certain extent. When the grouting amountQf is given, the effective filling volume, Qs (slurry stone), can be obtained, and the grout-induced fracture aperture, δj, can be calculated. Next, the normal deformation difference due to grouting, Δδ, is calculated by subtracting δn from δj. At last, the normal stiffness of the fracture can be determined. Qs is the volume of the slurry stone, pg is the grout pressure, and Qf is the volume of slurry only in the fracture.The volume of the slurry consolidating body depends on the water separating proportion. The water separating proportion of clay-cement slurry is approximately linear with the cement content. For the slurry with the same specific gravity, the water separating proportion decreases with the increase of cement content. In this paper, the mass fractionC/S of cement in the total solid material is 19.2%, and the specific gravity of slurry is 1.16 g/cm3. According to the experimental results of the relationship between cement content and water separating proportion for the clay cement slurry with different specific gravity [34], the water separating proportion Wsepa is taken as 70%.Substitute the monitored data of 9 boreholes into Equation (16), the corresponding normal stiffness of fractures in the limestone aquifers is got and listed in Table 4.Table 4
Normal stiffness of fractures using the data of typical grouting boreholes.
Bhpg (MPa)△p (MPa)C/S (%)W/SWsepa (%)Qs (m3)Δδ (m)δj (m)Qf (m3)kn (GPa/m)D17-131510.023.83.39653.610.0023510.00442910.304.25D18-3159.519.23.73705.210.0040610.00639617.362.34D18-81510.219.23.73706.460.0057050.00793321.531.79D19-9n1510.230.63.54503.560.001820.0043757.125.61D20-10n1510.519.23.73708.710.0083230.01069829.031.26D21-31511.219.23.737011.880.0122580.01459339.610.91D23-101510.019.23.73703.650.001930.00448512.175.18D23-3159.819.23.737012.520.0130380.01537441.720.75D23-81510.01003.62702.500.0006940.0030738.3414.4The apertures of fractures filled with high-pressure water before grouting range from 2.08 to 2.56 mm. Under the grouting pressure of 15 MPa with the designed slurry diffusion radius of 30 m, the normal stiffnesses of the fractures filled with slurry consolidating body range from 0.75 to 14.4 GPa/m. The minimum normal stiffness of the fracture exposed by borehole D23-3 is 0.75 GPa/m, and the maximum normal stiffness calculated by using the data of borehole D23-8 is 14.4 GPa/m. In practice, the grouting of borehole D23-8 is more difficult than other boreholes, and pure cement slurry has to be used for better plugging effect during grouting.


## 4.4. Discussion

### 4.4.1. Fracture Stiffness and Weathering Degree

The normal stiffness of fracture in the aquifers can be affected by weathering. According to the relative weathered degree of limestones, the fractures can be divided into several levels [35]: no weathered to slightly weathered, moderately weathered, and completely weathered. The corresponding lower limit values of stiffnesses are, respectively, 7.9, 4.9, and 3.8 GPa/m. The fractured limestone aquifers in this paper are weathered to a certain extent in some areas.


### 4.4.2. Fracture Stiffness and Other Influencing Factors

Fracture stiffness is also affected by many other factors, including weathering during deposition in geological history, adjacent fractures, hydraulic erosion, and cyclic loading times. (1) In geological history, after the fracture was formed, it would first suffer from natural weathering during the deposition process, which will reduce the stiffness of the fracture. (2) Then, under the action of in situ stress field and high-pressure water, the local rock adjacent to the fracture may be eroded, and local large deformation will occur during high-pressure grouting, which will also reduce the normal stiffness of the crack. (3) The distance between fractures will also control the normal stiffness. When two fractures are far away, they have no influence on each other. When the spacing is not too large, the adjacent fracture may make the grouting relatively easy.


### 4.4.3. Engineering Applicable Conditions

High-pressure water and drainage conditions are two important conditions for the estimating of fracture apertures in this paper. The apertures under high water pressure range from 2.08 to 2.56 mm, and the normal stiffnesses of the fractures filled with slurry consolidating body range from 0.75 to 14.4 GPa/m. The results meet the positive relationship between normal stiffness and fracture aperture got by the literature data. Our method of estimating the aperture and stiffness of fractures is applicable when grouting is used in the fractured limestone aquifers with high water pressure under coal seams before mining. It also lays a foundation for the theoretical evaluation of grouting effect of the hidden fractures, including the evaluation of strength and water resistance of the slurry stone.


## 4.4.1. Fracture Stiffness and Weathering Degree

The normal stiffness of fracture in the aquifers can be affected by weathering. According to the relative weathered degree of limestones, the fractures can be divided into several levels [35]: no weathered to slightly weathered, moderately weathered, and completely weathered. The corresponding lower limit values of stiffnesses are, respectively, 7.9, 4.9, and 3.8 GPa/m. The fractured limestone aquifers in this paper are weathered to a certain extent in some areas.


## 4.4.2. Fracture Stiffness and Other Influencing Factors

Fracture stiffness is also affected by many other factors, including weathering during deposition in geological history, adjacent fractures, hydraulic erosion, and cyclic loading times. (1) In geological history, after the fracture was formed, it would first suffer from natural weathering during the deposition process, which will reduce the stiffness of the fracture. (2) Then, under the action of in situ stress field and high-pressure water, the local rock adjacent to the fracture may be eroded, and local large deformation will occur during high-pressure grouting, which will also reduce the normal stiffness of the crack. (3) The distance between fractures will also control the normal stiffness. When two fractures are far away, they have no influence on each other. When the spacing is not too large, the adjacent fracture may make the grouting relatively easy.


## 4.4.3. Engineering Applicable Conditions

High-pressure water and drainage conditions are two important conditions for the estimating of fracture apertures in this paper. The apertures under high water pressure range from 2.08 to 2.56 mm, and the normal stiffnesses of the fractures filled with slurry consolidating body range from 0.75 to 14.4 GPa/m. The results meet the positive relationship between normal stiffness and fracture aperture got by the literature data. Our method of estimating the aperture and stiffness of fractures is applicable when grouting is used in the fractured limestone aquifers with high water pressure under coal seams before mining. It also lays a foundation for the theoretical evaluation of grouting effect of the hidden fractures, including the evaluation of strength and water resistance of the slurry stone.


## 5. Conclusions

(1)
An ideal flow model composed of a borehole and a fracture is established with the fracture aperture as the main variable. For critical free flow, when the flow velocity of the aquifer changes fromvS to vS+ΔvS, the minimum free flow velocity vmin for free flow at the borehole outlet is got by vmin=lim△vO⟶0vO+△vO=vO. The velocity coefficient Cv is the ratio of actual velocity to theoretical velocity Cv=vO/vmin. According to the continuity equation, the fracture aperture δn can be deduced as δn=q/3600/vj/πd=q/3600/Cv2Δp/ρ/πd. Then, the fracture normal stiffness, kn, is got as kn=pj−pw/Qs/LdH−δ0(2)
The estimated fracture apertures with high-pressure of typical boreholes range from 2.08 to 2.56 mm, and the fracture normal stiffness ranges from 0.75 to 14.4 GPa/m. The results meet the positive relationship between normal stiffness and fracture aperture got by the literature data. Compared with the method of determining apertures by cubic law, the method in this paper pays more attention to engineering application, considering the hydrogeological characteristics of fractured limestone aquifers. The influencing factors of fracture stiffness, including sedimentary weathering in geological history, adjacent fractures, and hydraulic erosion, are comprehensively discussed. The research results provide a foundation for the next theoretical evaluation of grouting effect in fractured limestone aquifers, including the evaluation of strength and water resistance of the slurry stone



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*Source: 8792040-2022-05-09.xml*