Grouting is an effective method to solve the problem of water inrush in tunnel and underground engineering. However, rock fractures are often simplified as horizontal and smooth fractures in most grouting studies, while studies on vertical inclined fractures are still rare. To investigate the diffusion law in vertical inclined fractures, a vertical inclined fracture grouting simulation device was developed. A new type of cement slurry with low weight and high flowing water resistance was developed by combining carbon nanotube (CNT) slurry with foamed cement. Physical simulation experiments were conducted to investigate various factors (initial flowing water, inclination angle, sand content, and grouting rate) on the sealing efficiency of grouting. Results show that the high foam content has a negative effect on the compressive strength of the slurry, and has a positive effect on the fluidity and water resistance. The optimum ratio of slurry is 30% foam content, 1.0% CNT content, 1.3 water/cement ratio, and 3% additive content. The inclination angle and inclination direction of the fracture have a great influence on the sealing efficiency of grouting. Foam-CNT composite grouts can meet the requirement of flowing water grouting in vertical inclined fractures.

Water inrush is a common problem in tunneling and underground engineering. Water inrush will delay the project and result in high personal injury and property damage [1, 2]. Grouting is an effective method to solve the problem of water inrush [3]. Grouting can improve the strength and reduce the permeability of the rock by injecting slurry into the rock fractures [4-7]. Considerable progress has been made in the area of flowing water grouting in recent years [8-10]. However, the theory of grouting still cannot meet the requirements of practical engineering.

Many scholars studied the sealing and diffusion law of flowing water grouting [11-14]. Sui et al. [15] investigated the effects of fracture width, initial flowing rate, grouting time, and grouting amount on the sealing efficiency of flowing water grouting through laboratory simulation experiments. Liang et al. [16] demonstrated that the inclination of fracture has a significant effect on the sealing efficiency and the diffusion law of flowing water grouting. Depending on the relation between the direction of flowing water and fracture, rock fractures can be divided into horizontal fractures, horizontally inclined fractures, and vertically inclined fractures (Figure 1). However, in most studies on flowing water grouting, rock fractures are simplified as horizontal fractures [17, 18], and studies on the vertical inclined fracture grouting are still rare.

Many scholars have proved that the diffusion law of liquid in inclined fractures and horizontal fractures is completely different [19]. Graf et al. [20] proposed a numerical method to discretize inclined nonplanar two-dimensional (2D) fractures within a three-dimensional (3D) finite element grid for subsurface flow and transport simulations. Mustapha et al. [21] introduced a new method to discretize inclined nonplanar 2D fractures in 3D fractured media for subsurface flow and transport simulations. Some scholars investigate the sealing and diffusion law of grouts in vertical inclined fractures through numerical methods [22]. Mu et al. [23] employed a space stepwise method (SSM) using multi-direction sectors (MDS) for calculating the diffusion of slurry in a single inclined rough fracture. Li et al. [24] used numerical methods to investigate the diffusion of C–S grout with time-varying viscosity in inclined rock fracture under seawater conditions. Results show that the grouting difficulty of vertical fractures is much higher than that of horizontal fractures. It is difficult for traditional grout to achieve effective treatment of water inrush in vertical fractures, and reliable grouting materials are urgently needed.

Suitable grouting materials are a key factor for the success of grouting in practical engineering [25]. Scholars have developed grouting materials with different functions to meet the grouting requirements of various projects [26, 27], such as cement slurry with high durability and strength [28], chemical slurry with high flowability and pumpability [29-31], composite slurry with balanced grouting performance [32], and environment-friendly slurry with less environmental pollution [33]. Cement-based slurry is the most widely used grouting material in practical engineering. Some scholars have improved the properties of grouts through various methods, such as reducing the particle size [34], replacing part of cement using fly ash [35], mixing with additive [36], and so on. However, most studies on grouting materials also focus on horizontal fracture grouting, and studies on grouting materials suitable for vertical inclined fractures are still rare. The grouting environment of vertical inclined fractures and horizontal fractures are pretty different. The slurry is easily affected by gravity and accumulates at the bottom of the fracture in the vertical inclined fracture, resulting in the decrease of sealing efficiency. Therefore, there is an urgent need for a kind of grout which could improve the grouting efficiency in vertical inclined fractures.

The author has compared the grouting performance of slurry with different nanomaterials (nan-SiO2, carbon nanotube (CNT), nano-Al2O3, and graphene oxide) [37, 38]. Results prove that CNT cement-based grouts possess the best flowing water resistance and can avoid being washed out from fractures. The sealing efficiency of CNT cement-based grouts is pretty well in horizontal inclined fracture [39]. However, the density of CNT cement-based grouts is great, thereby having a negative effect on the vertical inclined fracture grouting.

Foam concrete becomes a hot research topic due to its light weight [40], long durability [41], cost-effectiveness [42], and high water resistance [31]. Previous studies indicated that foam concrete can reduce the weight and viscosity of grouts due to its special porosity structure [43, 44]. Yao et al. [45] added an expansion agent into the grouting material to provide compressive stress and make rock mass more stable. Wang et al. [46] combine foam and reinforcement polyurethane/water glass to examine the interaction performance of grout seepage. Theoretically, the application of foam cement in flowing water grouting can offset the negative effects of gravity to a certain extent and improve the flowability of grouts. Therefore, based on the basic properties of CNT cement-based grouts and foam cement, a new kind of grout is acquired through mixing CNT cement-based grouts and foam cement. Theoretically, the foam-CNT composite cement grouts may possess light weight and high flowing water resistance in vertical inclined fracture grouting.

The basic properties of the foam-CNT composite grouts were measured by laboratory tests, and the optimum constituents of the grouts were determined by the gray relational analysis (GRA) method. In addition, a simulation setup for vertical inclined rough fracture grouting was established according to the authors' previous studies [37, 39]. The effects of various factors (initial flow rate, inclination angle, sand content, and grouting rate) on the diffusion pattern and sealing efficiency of foam-CNT composite grouts were investigated by grouting simulation experiments, which provide technical guidance for practical work.

2.1 Materials

Ordinary Portland cement (P.O 42.5) was supplied by the Hailuo company. The additive consists of sodium aluminate (60%), sodium silicate (20%), calcium chloride (14%), and cellulose (6%). According to previous literature [47], the additive with this component can accelerate the solidification rate of grouts and thereby improving the sealing efficiency of grouting.

The foaming agent used in foamed cement is anionic surfactant (sodium dodecyl benzene sulfonate and sodium alkenyl sulfonate). The foam stabilizer is modified silicone polyether microemulsion and cellulose. Physical foaming method is used to generate foam. The air was forced into the tank with the foaming liquid to realize the gas–liquid mixture for foam generation. The foaming efficiency of this method is high and the bubble diameter is uniform. The physical properties of CNTs are summarized in Table 1.

In construction practice, water inrush is usually accompanied by sand, which may have an impact on the sealing quality of the grouting [35]. Therefore, experimental studies on dynamic water grouting with sand were conducted and the grain composition of sand was summarized in Table 2.

2.2 Basic Property Tests of Grouts

Compressive strength (consolidation strength), flowability, and flowing water resistance are basic properties of grouts in flowing water grouting. To investigate the effects of grout composition [water/cement ratio (W/C), additive content, foam content (ratio of foam volume to total volume), and CNT content] on the physical properties of the grouts, orthogonal experiments (Table 3) were carried out.

2.2.1 Compressive Strength Tests

The flow of the compressive strength tests is shown in Figure 2. Cement samples (70.7 × 70.7 × 70.7 mm) were using the material component in Table 3 (Figure 2(a)). Then, samples were stored in a constant temperature and humidity environment for 24 hours (Figure 2(b)). Finally, the specimens were demolded (Figure 2(c)) and their compressive strength (unconfined) was measured.

2.2.2 Flowability Tests

A simple simulation setup was created to investigate the flowability and flowing water resistance of grout. The setup consists of two acrylic plates with a thickness of 10 mm, as shown in Figure 3. The length of fracture is 80 cm and the width is 30 cm. The side of fracture is fixed and sealed.

Water was injected into the fracture at a constant flow rate (1 cm/s). According to the experimental design (Table 3), additives, foam, CNT, and water were mixed with cement and stirred for 50 seconds. Then the grouts were allowed to stand for 100 seconds and injected into the fracture through the grouting hole. The diffusion area of grouts after a certain time (2 seconds) was recorded with a high-speed camera, indicating the flowability of grouts.

2.2.3 Flowing Water Resistance Tests

The retention rate is an important indicator for evaluating the flowing water resistance of grouts. If less grout is washed out by flowing water, the sealing quality of grouting can be improved. A higher retention rate means a higher resistance to flowing water.

Flowing water resistance tests were carried out to study the effects of grouts components on the retention rate. Water was injected into the simple grouting simulation setup (Figure 3), with a flowing velocity of 3 cm/s. Grouts were prepared according to Table 3 and injected into the experimental setup. The injection took 60 seconds and the water injection took 150 seconds. After grouting, the grouts that remained in the fracture were collected, dried, and measured. The ratio of grouts’ weight before and after grouting is the retention of grouts [16, 39].

2.3 Theory on the Diffusion of Grouts

In the 1950s, scientists established the cubic law of motion for water flow through a large number of laboratory tests. The flow in unit fracture width is proportional to the cube of the fracture opening. The motion of flow in a single fracture satisfies the one-dimensional Navier–Stokes equation (equation. 1).

ρDuDt=ρfx-px+μ(2ux2+2uy2+2uz2)
(1)

The cubic law is derived from the assumption that the fracture is smooth and straight without fillings. In fact, most fractures in nature are rough and filled with fillings, which are uneven in fracture opening and connectivity. Under this condition, the use of the cubic law to describe the seepage in the fracture leads to a certain error. Therefore, the cubic law is corrected as follows:

q=ge212μJ11+σ(Δe)1.5 (R<Rcr)
(2)
q=egJe[2.6+5.11g(2Δe)1]  (RRcr)
(3)

Re is the Reynolds coefficient of water flow. Re = (ve)/2μ. Rcr is the critical Reynolds coefficient of laminar and turbulent flow. and e are the absolute roughness and average hydraulic width of fractures.

2.3 Flowing Water Grouting Simulation Experiments

2.3.1 Inclined Rough Fracture Grouting Simulation Set-up

Flowing water grouting simulation setup (Figure 4(a) and (b)) has been used in previous publications [37] to simulate the grouting process in horizontal and smooth fractures. However, most fractures in practical engineering are inclined and rough, which increase the difficulties of flowing water grouting. To investigate the diffusion law of grouts in vertical inclined fractures, a grouting simulation setup (Figure 4(c)) was created based on the simulation set-up in authors’ previous studies (Figure 4(a) and (b)). The vertical inclined rough fracture grouting simulation set-up comprises a rough fracture simulation device, angle adjustment holders, a grouting system, a water supply system, and a data acquisition system (Figure 4(c) and (d)).

An angle adjustment holder was used to adjust the angle of inclination of the fracture, as shown in Figure 4. An acrylic plate was used to simulate the upper surface of the fracture for real-time monitoring of the grouting process. The surface of the rock fracture was simplified as smooth in previous literatures, which is different from practical engineering [15]. Therefore, a fracture structure with a rough surface was used in this study. The roughness of the fracture was designed according to the Barton curve and the fabrication process was described in detail in the author’s previous publications [37, 39, 47].

2.3.2 Experimental Scheme and Experimental Process

The effects of different factors (initial flowing rate, inclination angle, sand content, and grouting rate) on the sealing efficiency of grouting were investigated by orthogonal experiments. The experimental scheme is shown in Table 4.

According to the experimental design (Table 4), the sand was poured into the water tank and stirred to ensure that the sand was well proportioned. The water was injected into the fracture at appropriate initial flowing rate and constant pressure (25 kPa). Additives, CNT, and cement were poured into the water and stirred for 50 seconds. The foam is made by mixing, diluting, and foaming the foam liquid. Then foam is mixed with grouts for 60 seconds. Then the grouts were allowed to stand for 40 seconds to improve the strength and flowing water resistance. The grouts were injected into the fracture at a rate of 60 mL/s and the grouting lasted for 5 minutes. The flowing water was supplied by a pump and the grout was supplied by a grouting machine. Finally, the velocity of the flowing water before and after grouting was measured with a velocity meter.

2.4 Horizontal Fracture Grouting Versus Inclined Fracture Grouting

In order to compare the influence of fracture inclination angle and direction, comparative experiments were carried out. The experimental scheme is shown in Table 5. In the experiment, the flow velocity is 2 cm/s and the sand content is 4%. The grouting material used in the experiments is ordinary cement slurry. To adapt to the flowing water conditions and improve the solidification rate, additives were used. Other materials (such as foaming agent, CNT, etc.) are not used in the comparative tests. The grouting process is consistent with Section 2.3.2.

3.1 Basic Properties of Grouting Materials

The effects of various factors (W/C, additive content, foam content, and CNT content) on the basic properties of grouting materials were investigated and the trend charts are shown in Figure 5.

As shown in Figure 5(a), the compressive strength of the grouts decreased with an increase in W/C and additive content. This indicates that although the additive can accelerate the setting rate of the grout, it has a negative effect on the compressive strength. Compared with W/C and the additive content, the influences of the foam content were much smaller. Previous studies [48] indicate that CNT can effectively improve the strength of cement, which is consistent with the experimental results.

CNT has a great influence on the viscosity of grouts and therefore the diffusion distance of grouts decreased with an increase in CNT content. The diffusion distance of the grouts increased with an increase in foam content. This indicates that foam can play the role of lubricant in grouts. Higher W/C can improve the diffusion distance of grouts which is consistent with previous studies [47]. The effects of additive content on the flowability of slurries are smaller than other factors.

As shown in Figure 5, W/C has the greatest influence on the retention rate (flowing water resistance) of grouts. The retention rate of grouts first increased and then decreased as W/C and foam content increased. The retention rate was highest at a W/C of 1.3 and a foam content of 35%. Previous studies have demonstrated that foam content within a certain value can increase the flowing water resistance of slurries due to its special pore structures [49]. Excessive foam content and high W/C decreased the viscosity and solidification rate of grouts. Then grouts are easier to be washed out from the fracture. Therefore, the retention rate of the grout decreases when W/C value and the foam content exceed a certain value.

3.2 Component Choice of Grouts Based on GRA

Theoretically, the sealing efficiency of grouts increases with the increase in compressive strength, diffusion distance, and flowing water resistance. However, the results of the basic properties tests suggest that one factor (W/C, foam content, additive content, or CNT content) can have different, even opposite, effects on properties. For example, when the W/C value increases, the compressive strength decreases rapidly, and the retention rate (flowing water resistance) increases. Therefore, there is an urgent need for a reliable method to select the optimum components of grouts.

GRA is a widely used method for solving the problem of multifactor analysis. It can address the shortcomings of the statistics-based systemic analysis method. One of the advantages of GRA is that the data size and the calculated quantity are small [50]. Therefore, GRA has been used to analyze the effects of various factors on the basic properties of grouts and to optimize the components of grouts. The steps of GRA are summarized as follows.

First, GRA was used to calculate the gray relationships between the various properties (compressive strength, diffusion distance, and flowing water resistance) of grout, as shown in equations (4) and (5).

Δik=1-rik
(4)
ζik=miniminkΔik+ρmaximaxkΔikΔik+ρmaximaxkΔik
(5)

where ri(k) is the normalized results of measured data; ζi(k) is the gray relational coefficient; ρ is the difference coefficient, which ranges from 0 to 1. The difference between gray relational coefficients increases with a decrease in ρ value.

Then, the weighting of the gray relationship coefficient was calculated by the objective weighting method according to the maximum deviation, as shown in equations (6) and (7)

ω`(k)=i=1mj=1mdij(k)i=1mj=1mk=1ndij(k)
(6)
gi=k=1nωkζi(k)
(7)

where ω' (k) is the normalized weight vector, which could distinguish the difference between various component schemes; gi is the weight gray relation coefficient.

Finally, the average and extremum of the influences of various factors on the weighted gray coefficient can be calculated. The trend diagram of the weighted gray correlation coefficient is shown in Figure 6.

As shown in Figure 6, the correlation coefficient initially increased and then decreased with increasing W/C, additive content, and foam content. The correlation coefficient peaked when the constitution of grouts is 1.3 W/C, 3% additive content, 30% foam content, and 1.0% CNT content. Therefore, the above component is the optimal component of the grouts. The correlation coefficient for this component is 0.955.

3.3 Result of Flowing Water Grouting Simulation Experiments

3.3.1 Diffusion Law of Grouts in Vertical Inclined Fractures

The final distribution pattern of residual grouts is taken photo after the grouting simulation tests. The pictures were processed with the software Photoshop and the outlines of slurry were delineated, as shown in Figure 7.

As shown in Figure 7 (Tests 1, 5, 9, and 13), the grouts diffuse in “U” shape when the inclination angle is 3°. This phenomenon is consistent with diffusion law of grouts in horizontal fractures [41]. When the inclination angle is 6° and 9°(Tests 3, 7, 10, 11, 14, and 15), the distribution pattern presents asymmetry due to the influence of inclination. The diffusion distance of grouts in the bottom part is greater than that in the top part. This is because the slurry accumulates at the bottom of fractures due to the effect of gravity and then more grouts at the bottom can avoid being washed out by the flowing water. When the inclination angle is 12° (Tests 4, 8, 12, and 16), the influence of gravity increases and the diffusion pattern of grouts evolved into a comet shape.

In horizontal fractures, the slurry tends to be spread in a U-shape and the maximal diffusion distance can be easily distinguished. However, the test results show that there is no obvious regularity for the distribution of slurry in vertical fractures, and it is difficult to determine the diffusion distance of the slurry. Therefore, the fitting formula cannot be used to simulate the diffusion of grouting.

3.3.2 Sealing Efficiency of Foam-CNT Composite Grouts in Vertical Inclined Fractures

According to previous studies [15], sealing efficiency is one of the most effective indicators to evaluate the grouting quality of flowing water grouting. Sealing efficiency is the ratio of flowing rate before and after grouting, as shown in equation (8).

SE=Q0-QgQ0×100%
(8)

Where SE is sealing efficiency, Q0 is the initial flowing rate and Qg is the flowing rate after grouting.

The grouting quality can be classified into 6° according to SE, as shown in Table 6 [15].

SE of grouting simulation experiment is shown in Figure 8. The results indicate that the sealing efficiency of the grouts ranges from 50% (fair) to 100% (excellent). The sealing efficiency of Test 2 is the highest and reaches about 100% (excellent). The sealing efficiency of Test 16 is the lowest and reaches 50.59% (fair). Figure 8 indicates that the foam-CNT composite grouts can meet the requirement for flowing water grouting in vertical inclined fractures.

The trend chart of influences of different factors (initial flowing rate, inclination angle, sand content, and grouting rate) on the sealing efficiency is shown in Figure 9.

As shown in Figure 9, the initial flowing rate has the greatest influence on the flowing water grouting in inclined fracture. The sealing efficiency decreased rapidly with an increase in initial flowing rate. A higher flow velocity results in a scattering of the grout material and a lower concentration of the grout material, leading to a decrease in sealing efficiency.

As shown in Figure 9, sealing efficiency first increased and then decreased as the fracture angle increased. Sealing efficiency reached the maximum value when the angle was 6°. In a horizontal fracture, grouts may aggregate in the center of the fracture and flow channels may form at the edge of the fracture, resulting in a decrease in sealing efficiency. A slight inclination favors the increase in diffusion area, and the expansion capacity of the foam cement can offset the negative effects of the inclination to a certain extent, increasing the sealing efficiency of the grout. However, if the inclination exceeds a certain value, the grouts clump together at the bottom, resulting in a decrease in sealing efficiency.

Figure 9 indicates that sand content has limited effect on sealing efficiency. Sealing efficiency increased slightly with increasing sand content. Sand can fill up the fracture and prevent the scattering effects of flowing water. Therefore, increasing sand content is beneficial to the increase in sealing efficiency.

Sealing efficiency increased rapidly at first and then increased slowly as the grouting rate increased. The reason is that more grouts are injected into the fracture, which improves the flowing water resistance of grouts. However, the space of fracture is limited. When the grouting rate exceeds a certain value, the injected grout exerts a pressure on the solidified grout in the fracture and may offset the positive effects of the high grouting volume. It is worth noting that the size of fracture in practical engineering is much larger than in simulation. Therefore, the limitation of grouting rate (grouting volume) will be less, and the sealing efficiency will increase continuously as the grouting rate increases in practical engineering. Then the foam cement can play a more important role due to its high pumpability and the foam-CNT composite grouting may have better performance in practical engineering.

According to the test results, the software 1stOpt is used to fit the sealing efficiency. The fitting results are shown in equation (9) and the correlation coefficient is 0.88, which can provide a quantitative reference for practical engineering.

S=0.9C+0.14G8.75V1.53I+93
(9)

where C is sand content; G is grouting velocity; V is flowing velocity; and I is inclination angle.

3.4 Effect of Fracture Inclination on Sealing Efficiency

The results of the comparison tests are shown in Figure 10. It can be seen that the sealing efficiency of cement slurry in horizontal fractures is good. When the inclination direction of the fracture is consistent with the flowing direction (horizontal inclination), the slurry is quickly washed out from the fracture, and the sealing efficiency is reduced. This could be due to the overlapping effect of gravity and flowing water. When the inclination direction of the fracture is perpendicular to the direction of the water flow, the sealing efficiency will decrease with an increase in the inclination angle. This differs from the results in Figure 9, where sealing efficiency first increases and then decreases. The main reason is that the CNT slurry has a high viscosity, and a suitable inclination can increase its diffusion speed. However, due to the lack of adhesion, ordinary cement accumulates at the bottom in the vertical fracture. This leads to a rapid decrease in the sealing efficiency.

An experimental setup was developed to simulate the process of flowing water grouting in vertical inclined fractures. Foam cement was used to modify the CNT composite grouts and a new type of grouts which is suitable to the vertical inclined fractures was acquired. Laboratory tests were conducted to investigate the basic properties (compressive strength, flowability, and flowing water resistance) of the foam-CNT composite grouts. Orthogonal experiments were conducted to investigate the influences of various factors on the sealing efficiency of grouting. The main conclusions from this study are as follows:

  1. The flowing water resistance of grouts first increases and then decreases as the foam content increases. The increase in CNT content can improve the flowing water resistance and decrease the fluidity of grouts. The foam cement can decrease the self-weight of grouts and offset the negative gravity on the vertical inclined fracture grouting to a certain extent.

  2. W/C has the most significant influence on the compressive strength of grouts and the CNT content has a positive effect on the increase in compressive strength of grouts. The increase in foam content and additive content decreases the compressive strength of grouts.

  3. Results from GRA indicate that the optimum ratio of grout is 1.3 W/C, 3% additive content, 30% foam content, and 1.0% CNT content.

  4. Results of flowing water grouting simulation experiments indicate that sealing efficiency initially increases and then decreases with increasing fracture angle. Sealing efficiency increases rapidly at first and then increases slowly as the grouting rate increases. The influence of gravity on the diffusion pattern of grouts increased and the diffusion pattern of grouts evolves into comet-shape with the increase in inclination angle. A fitting formula for sealing efficiency is acquired based on the experimental results.

  5. Results of comparative tests show that the inclination angle and inclination direction of the fracture have a great influence on the sealing efficiency of grouting. The sealing efficiency in vertical inclined fractures is lower than that in horizontal inclined fractures when the inclined angle is consistent.

In conclusion, foam-CNT composite grouts can improve the sealing efficiency of flowing water grouting in vertical inclined fractures and thus provide important guidance for practical engineering.

The data underlying this article will be shared on reasonable request to the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This investigation was supported by the National Natural Science Foundation of China (projects No. 42307198), National science Foundation of Jiangsu Province (BK20221148), China Postdoctoral Science Foundation (2023M733747), Fundamental Research Funds for the Central Universities (XJ2021008101), Open Fund of State Key Laboratory of Coal Mining and Clean Utilization (China Coal Research Institute) (Grant No. 2021-CMCU-KF019), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20231501), and the Key Research and Development Program of Xuzhou (Grant No. KC23294).

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