A grouting simulation method for quick-setting slurry in karst conduit:The sequential flow and solidification method

Zhenhao Xu, Dongdong Pan, Shucai Li, Yichi Zhang, Zehua Bu, Jie Liu

Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, 250061, China

Keywords:Karst conduit Sequential flow and solidification (SFS)Quick-setting slurry Grouting simulation method Grouting in flowing water

ABSTRACT It is difficult to temporally and spatially track and characterize the slurry viscosity in flowing water during grouting simulation.In this study, a sequential flow and solidification (SFS) method considering the spatial-temporal evolution of slurry viscosity in flowing water in karst conduit is proposed.First, a time-dependent model for the threshold function of slurry viscosity is established.During the grouting process, the spatial-temporal evolution of slurry viscosity is revealed by tracking the diffusion behavior of the slurry injected at different times.This method is capable of describing the gradual solidification process of the slurry during grouting.Furthermore, a physical model of grouting in a karst conduit is developed.Second, the effectiveness of the SFS method in grouting simulation is verified by the experiment of grouting conduit in flowing water.The SFS method enables real-time monitoring of fluid velocity and pressure during grouting in flowing water and provides a feasible calculation method for revealing the grouting plugging mechanism in complex karst conduits at different engineering scales.In addition, it can be used to guide the design of grouting tests in flowing water, improve cost efficiency,and provide theoretical basis for optimizing grouting design and slurry selection.

1.Introduction

Grouting is the most commonly used method in underground engineering projects to prevent water inrush and ensure the stability of the surrounding rock(Chepurnova,2014;Wang et al.,2016;Bahrani and Hadjigeorgiou, 2017; Thenevin et al., 2017;Vlachopoulos et al., 2018; Jafarpour et al., 2020; Mu et al., 2021).The flow characteristics of slurry are critical to engineering projects(Saeidi et al., 2013; Chen et al., 2014; Li et al., 2017; Funehag and Thörn, 2018; Jin et al., 2021).Although various grouting theories have been developed for the treatment of groundwater disasters,the research on the theory of grouting during water inflow is still in the exploratory stage (Sui et al., 2015; Zhou et al., 2017; Zou et al.,2020).Due to the nature of the slurry and the complexity of karst conduit structure, the diffusion behavior of the slurry in flowing water is difficult to be accurately described by geophysical methods.Moreover,the theory of flowing conduit grouting is rarely reported (Borghi et al., 2016; Fischer et al., 2018).The engineering and geological environments in karst areas are complex and often change rapidly (Bauer et al., 2003; Liedl et al., 2003; Birk et al.,2004; Pan et al., 2019, 2020).The diffusion and plugging mechanism of the slurry in flowing water remains poorly understood(Gustafson et al., 2013; Zhang et al., 2017; Zhu et al., 2019).Although numerical method allows for the visualization of the slurry flow (Eriksson et al., 2004), its application is limited due to lack of advancement in slurry-water interaction theory.Therefore,the study on the simulation method of grouting in flowing water is an important aspect of the grouting theory.

Grouting in flowing water involves many aspects, such as the calculation of the complex groundwater turbulence, slurry-water interaction, interactions within the slurry in different stages, interactions between formation fluids and the injected fluids,as well as the time-dependent characteristics of the slurry(Li et al.,2020;Xu et al., 2021).The solidification process of a slurry is often described by its viscosity (Chen et al., 2014; Deng et al., 2018).In addition,due to the spatial variation of the viscosity,it is difficult to simulate the dynamic grouting process for quick-setting slurry.For karst conduits with complex flow behavior,many studies have been carried out through physical model testing and numerical simulation(Capecelatro and Desjardins,2013;Chen et al.,2014;Marsooli and Wu,2014;Sun and Sakai,2015;Deng et al.,2018).To date,thedescription of conduit flow characteristics has mainly focused on the use of multiphase flow models such as the Eulerian-Lagrangian model (Capecelatro and Desjardins, 2013), the Eulerian-Eulerian model (Lee et al., 2016), and the mixture model (MagniniMatar,2019).These numerical methods show that the multiphase fluid interaction during grouting in flowing water can be simulated.However,another difficulty in the numerical description is how to reasonably characterize the time-dependent characteristics of slurry viscosity (Li et al.,2016; Zhang et al., 2017).

For studying the time-dependent characteristics of slurry viscosity,Zhang et al.(2017)proposed an analytical model to describe the diffusion behavior of slurry, while considering the spatialtemporal variation of slurry (Li et al., 2016; Zhang et al., 2017).Combining with the computational fluid dynamics (CFD) method,Deng et al.(2018) established a three-dimensional (3D) fracture network model to explore the flow behavior of slurry in fractures.Based on the Bingham constitutive model, Gustafson et al.(2013)developed a model to describe the motion behavior of slurry in a single fracture under constant pressure.Luo et al.(2009) demonstrated that grouting velocity is independent of fracture inclination based on the Bingham model.However, existing grouting simulation methods have yet considered the spatial-temporal distribution of slurry simultaneously (Eriksson et al., 2004).It is well known that the type of slurry affects the viscosity over time.Current numerical methods can be divided into 2 types.In type 1, the timedependent viscosity of the slurry is not considered, and the slurry is treated as a liquid with a viscosity higher than that of water.Twophase flow theory and numerical algorithms are used to study the slurry-water interaction.However, this method does not take into account the effect of viscosity change and solidification.In type 2,the slurry viscosity is regarded as unified time-dependent function and is space-independent.However, this is equivalent to the process of “first injecting a thin slurry, then injecting a viscous slurry,and finally injecting a slurry with very low fluidity”,which is quite different from the actual grouting process, and the relevant literature is only found in fracture grouting simulation (Gothäll and Stille, 2010; Zhang et al., 2011; Sui et al., 2015; Xiong et al., 2018).No literature exists on conduit grouting with complex interactions between slurry and water.

To this end, the threshold function of slurry viscosity is established,which can track and update the spatial position of the slurry after injection in different sequences.To solve the above problems,we propose the sequential flow and solidification (SFS) method in this study.The SFS method extends the idea of sequential calculation from the fracture medium to the conduit medium,focusing on the flow behavior of the slurry in karst conduits and replacing“diffusion”with“flow”(Li et al.,2020).The reason is that grouting process of quick-setting slurry in fast flowing water is dominated by the coupling process of flow rather than diffusion.Thus, our method is finally termed the SFS method, which is capable of realizing the numerical characterization of the uneven distribution of slurry viscosity, and describing the gradual transformation of slurry from flow to solidification in flowing water.As a result, the visualization of grouting in flowing water can be realized.In order to verify the effectiveness of the SFS method, we use a selfdeveloped grouting testing system to perform grouting experiments under static and flowing water conditions, which are simulated by the SFS method.We compare and analyze the velocity,pressure,and flow rate at the outlet of the pipeline,as well as the initial deposition position, down-flow diffusion distance,counter-flow diffusion distance, and overall diffusion deposition behavior of the slurry under different initial conditions.The calculation results show that the SFS method is consistent with the physical model testing results,which verifies the effectiveness and feasibility of the SFS method in the grouting simulation of karst conduits.Furthermore, we study the applicability of the SFS method to the diffusion mechanism of grouting in rough conduits of engineering scale, and analyze the characteristics of slurry flow and solidification, as well as the distribution of velocity and pressure in the conduit,which shows that the SFS method is effective in simulating the grouting process in flowing water.It is expected that the SFS method can further guide the design of grouting in flowing water, thereby saving testing cost, optimizing the grouting design,and providing a theoretical basis for reasonable slurry selection and grouting practices.

Fig.1.Time-dependent slurry viscosity in the SFS model(Ttot represents the total time of grouting).

2.Numerical method of conduit grouting

2.1.Time-dependent viscosity

Commonly used quick-setting grouting materials are mainly divided into chemical slurry and cement-based quick-setting slurry.In grouting engineering projects, the most suitable grouting material is often selected according to the site conditions.The chemical slurry can achieve good results in specific situations,but due to their high price and certain toxicity, its application is limited.The cement-based quick-setting slurry mainly includes cement-sodium silicate(C-S)slurry and polymer-modified cement slurry.The C-S slurry is widely used in grouting projects because of its cost efficiency and environmental protection.

A large number of experimental studies have demonstrated that the reaction process of the slurry can be divided into three phases:low-viscosity, viscosity rising, and solidification phases (Li et al.,2016; Zhang et al., 2017).A time-dependent model with a threshold function is then proposed to consider the characteristics of the slurry during the viscosity rising and solidification phases.The model of the slurry viscosity(t) is expressed as

where t is the transient time of grouting; Tconrepresents the sequence interval of the grouting time, Tini.represents the setting time at the velocity threshold μmax; and A, B, and C are the fitting parameters.The slurry to be injected is divided into i different sequences, and the corresponding grouting times of different sequences are T1, T2, …, TN.

To describe the spatial-temporal variation of viscosity, the sequential representation of viscosity can simultaneously characterize the process of slurry from flow to solidification.The slurry injected into the karst conduit follows different time-varying viscosity functions according to different sequences.The viscosity variation of the slurry in different sequences is shown in Fig.1.

2.2.Numerical implementation of the SFS method

The Navier-Stokes equation of incompressible fluids is used to describe the flow behavior of the slurry phase.When using the fluid volume method to calculate the multiphase fluid(Hirt and Nichols,1981), the governing equations are as follows.

The mass conservation equation can be expressed as

The momentum equation is

where ρ is the density of the fluid,is the velocity vector of the fluid,μ represents the kinematic viscosity coefficient, P represents pressure,andis the gravitational acceleration.

For the slurry system, when the volume fraction (VF) function F = 1, it indicates that the computational domain is filled by the slurry phase.And when F = 0, it indicates that the computational domain is filled by the water phase.When 0

Fig.2.The flowchart of SFS method.

where ρqis the density of q, and μqis the viscosity of q.

The increase of slurry sequence means the increase in the number of equations.To reduce the amount of computation, the number of equations is controlled by the sequential fusion method.Since the grouting time lasts until tj= Tini+jTcon(j = 1, 2,…,n),the fusion and modification of the jth slurry sequence are carried out during the solidification phase.

At the moment tj, the jth sequential slurry is modified.The regions occupied by water and slurry in different sequences can be represented asrespectively, whereis the region of water.The fusion and modification method of the jth sequential slurry can be expressed as

The phase transformation equation can be expressed as

where Fqis the VF of q,Uqis the corresponding flow velocity,and q is the phase sequence.

The VF of any phase satisfies the following condition:

In addition, the viscosity and density of the slurry mixture can be calculated by

Similarly, the state of the slurry-water mixture after the jth sequential fusion can be expressed as

Furthermore, the sequential fusion method is used to reconstruct the slurry parameters in the slurry solidification phase.The fused time-dependent function of the slurry viscosity is represented as(t).Among them,Snrepresents the different sequential slurry, and Ωjis the overall distribution of the slurry after the jth sequential fusion in the solidification phase, where Ωj= Ωμmax+Ωμj(t).The specific slurry viscosity correction can be expressed as

where An= ⎿Tini/Tcon」.

The density and viscosity of the fused fluid can be further calculated using the following formulae:

After the sequential fusion, the changes in the slurry-water interface are characterized by solving the following governing equations:

Furthermore, the final grouting pressure is set to Pterand the outlet flow threshold is set to Qplu.During the grouting, the flow rate at the outlet of the conduit and the grouting pressure in the hole are monitored in real-time to determine whether Pterand Qpluare reached.After a continuous grouting time Tini+ jTcon, the jth sequential fusion of the slurry is completed.If the requirements to stop grouting are not met,the next grouting sequence is continued.The sequential fusion is repeated until the requirements for stopping grouting are met, and the sealing requirements of flowing water are satisfied.The flowchart of the SFS method is shown in Fig.2.

3.Validation of the SFS method

In this study, we independently developed a visual conduit grouting testing system.The system was used to carry out the conduit grouting under static and flowing water conditions.The system can intuitively compare the flow characteristics of the slurry and grout plugging processes under different conditions.Moreover,the feasibility and effectiveness of the SFS method in conduit grouting have also been verified and analyzed.

3.1.The design of verification test

The grouting testing system used to verify the effectiveness of the SFS method mainly consists of four parts, as shown in Fig.3,including a double liquid grouting system, a stable pressure water supply system, a pipeline system, and a data acquisition and processing system.

The physical testing system is shown in Fig.4.The stable water head supply can be applied using the testing system.The initial flowing water fields are obtained by controlling the water supply.The system can provide operating conditions consistent with the simulation analysis of the SFS.The 3D visible pipeline system enables real-time observation of slurry flow behavior and water plugging process during grouting.The data acquisition and processing system allows for real-time monitoring of fluid velocity,fluid pressure, grouting pressure, and grouting rate during slurry flow.

To verify the effectiveness of the SFS method proposed in this study, the flow behaviors of the slurry under static and flowing water conditions were analyzed using the testing system.The universal C-S double slurry was selected for the test.The testing parameters and working conditions are shown in Table 1.The selection of the pressure monitoring points is consistent with the monitoring scheme of the model test.The locations of the fluid pressure monitoring points(Nos.1 and 3) are at the bottom of thepipe and 0.5 m on both sides of the grouting pipe,respectively.The bottom of the pipe directly under the grouting pipe was selected as the monitoring point No.2.The specific locations are marked in Fig.5a.

Table 1Property parameters of the simulation model.

Fig.3.Schematic diagram of the grouting testing system.1-Pipeline support;2.-Pipeline;3-Flange;4-Flow control valve;5-Flowmeter;6-Pressure sensor;7-Paperless recorder; 8 - Single slurry grouting pump; 9 - Water glass grouting pump; 10 - #1 water supply device; 11 - #2 water supply device; 12 - High speed camera.

Fig.4.Visual testing system of grouting in flowing water:(a)The grouting test system and (b) The grouting pump.

3.2.Verification of the effectiveness of the SFS method under static water condition

As shown in Fig.5a, with the grouting pipe as the central axis,the slurry flow behavior is basically symmetrical diffusion under the static condition.Compared with the model testing results, the numerical results show that the flow diffusion process is relatively stable,and basically shows a regular symmetrical distribution.Due to different flatness of the model site and the double liquid grouting pump, there are some errors in the left-right diffusion shape and flow distance.

After being injected into the pipe,the slurry is first deposited at the bottom of the pipe.The results show that the left-right diffusion distances exhibit a short-term nonlinear trend,and the results are basically consistent.As shown in Fig.6,the grouting lasted for 14 s and the pipe was completely filled.The slurry diffused to left and right sides,and the diffusion distance of stable deposition increased linearly.The slurry flowing distances at 26 s, 57 s, and 62 s are selected for further comparison.The maximum errors between the SFS method and the experimental results are 10.7%, 11.9%, and 11.5%, respectively, all of which are less than 15%, and the right diffusion distance is basically the same.All these agreements suggest that the SFS method can well simulate the grouting process of the slurry in the static water.

Fig.5.Comparison between the testing results and the SFS calculation results in static water condition: (a) Slurry diffusion behaviors; and (b) SFS calculation results.

Due to the reciprocating movement of the piston, the grouting pressure and the pressure at each monitoring point show clear pulse characteristics.The fluid pressure changes at different measuring points are basically consistent.The pressure increases slowly at the initial stage of grouting,but increases rapidly with the blockage of the pipeline.As shown in Fig.6, both the grouting pressure (No.2) and the slurry diffusion pressure (Nos.1 and 3)have obvious sharp-rising points, which are related to the monitoring position.In the initial stage of grouting,there is a short-term stable phase.In the later stage of plugging, the pressure changes differently and increases linearly.For grouting pressure, the pressure increases steadily after 14 s.The steady growth rate of the pressure is about 0.061 kPa/s.The overall change in pressure is similar, indicating that the SFS simulation method can achieve pressure analysis of grouting under static conditions.

3.3.Verification of the effectiveness of the SFS method under flowing water condition

Due to the impact of flowing water,it is difficult for the slurry to deposit at the bottom of the grouting pipe during the initial stage of injection.At the interface between the slurry and water,the model testing results show that the slurry and water are miscible.As shown in Fig.7,the suspension behavior of slurry and water likely causes the illusion of effective deposition visually.Thisphenomenon is more pronounced in static water.The entire grouting process can be divided into two parts: the down-flow diffusion region and the counter-flow diffusion region.In the initial stage of grouting, the thickness of the slurry deposit decreases with the increase of the diffusion distance, showing a streamline diffusion pattern.The thickness of the slurry deposit near the grouting pipe gradually increases, and the subsequent slurry movement along the flow direction is hindered, and it gradually diffuses in the reverse direction until the pipeline is plugged.

Fig.6.Comparison of (a) grouting pressure and (b) slurry diffusion pressure in static water condition.

The calculation results of the SFS method are shown in Figs.7 and 8, and the slurry-water interaction behavior is basically consistent with the experimental results.With continuous grouting, the slurry-water interaction shows an obvious down-flow diffusion region and a counter-flow diffusion region.The downflow diffusion region shows a trend of first nonlinear diffusion and then linear diffusion.After grouting for about 14 s, apparent counter-flow diffusion began to occur, which was later than the model testing results.The counter-flow diffusion process is basically linear.After grouting for 35 s, the down-flow diffusion increases linearly.Comparing the two results, it is found that the slurry loss caused some measurement errors.The experimental results show that the down-flow diffusion distance is greater than the SFS calculation result,while the counter-flow diffusion distance is smaller than the SFS calculation result.

Fig.7.Comparison between the testing results and the SFS calculation results under flowing water condition: (a) Slurry diffusion behavior and (b) SFS calculation results.

The grouting pressure fluctuates in varying degrees under the influence of grouting pulses, as shown in Fig.9.However, the results of the SFS method are stable.In the initial stage of grouting,the grouting pressure change behaviour was stable.As the slurry is deposited and solidified,the flowing water in the pipe tends to get blocked, and the grouting pressure begins to rise.The monitoring information is divided into the change of grouting pressure and the change of slurry diffusion pressure.

For grouting pressure, the SFS numerical results show that the grouting pressure remains stable for about 26 s,and then begins to increase steadily.The constant increase rate of pressure is about 0.088 kPa/s.For slurry diffusion pressure, the results of the SFS method show that the grouting lasts for about 21 s, and the pressure begins to increase steadily.The steady increase rate of pressure is approximately 0.087 kPa/s.The time difference of steady increase of pressure on the left and right sides of the grouting pipe is about 3 s, both showing a trend of pressure firstly increasing and then decreasing.Fig.9 shows the comparison of the model testing results with the SFS numerical results.Under the influence of pulses,the experimental results fluctuate considerably, and a pressure rising stage is still visible.The numerical results are in good agreement with the experimental results in the initial groutingstage.However, the numerical results are slightly larger than the experimental results in the pressure rising stage.The consistency of the slurry diffusion pressure is better than that of the grouting pressure, and the difference of numerical results between the left and right monitoring points is small.In the model test, the downflow diffusion pressure is slightly higher than that in the counterflow diffusion, and the down-flow diffusion pressure first shows a rising trend.In summary, the overall trend of pressure change is consistent, indicating that the SFS simulation method can be used to conduct pressure analysis of grouting in flowing water.

Fig.8.Comparison of slurry diffusion behavior: (a) The test results and (b) The SFS calculation results.

4.Further application of the SFS method

4.1.Development of the calculation model for rough conduits

The karst conduits usually do not have smooth walls and often appear a very irregular state under the long-term erosion of groundwater.Due to cost and operation difficulty, physical model testing is usually simplified,thus it is difficult to carry out grouting tests considering the roughness of the walls, and the numerical simulation method for rough conduits is also rarely reported.Therefore,based on the SFS method,the applicability of slurry flow behavior considering the conduit roughness was studied in this work.First, a rough conduit model was constructed, as shown in Fig.10.The parameters of commonly used quick-setting slurry were selected to conduct the verification.The outlet velocity of the karst conduit was 0.6-0.7 m/s.Fixed pressure boundary conditions were used to create the initial flow field.The parameters are shown in Table 2.

Table 2Input parameters.

4.2.Slurry flow characteristics and water reduction effect

The flow rates at the outlet of the karst conduit and the slurry flow characteristics in the conduit during grouting are shown in Fig.11.The slurry flow characteristics in the rough conduit also show obvious down-flow diffusion region and counter-flow diffusion region.Overall, it exhibits a three-phase attenuation characteristics, namely the slow decline phase, the rapid decline phase,and steady-state change phase,corresponding to phases A,B,and C,respectively.Taking 30 s as the time interval, the water reduction rates of the conduit outlet are 3.6%,17.39%,43.04%,58.95%,69.14%,and 79.93%, respectively.

Fig.9.Comparison of (a) grouting pressure and (b) slurry diffusion pressure under flowing water condition.

Fig.11.The outlet flow and slurry diffusion characteristics.

The change in slurry deposition during the grouting process is shown in Fig.12.Under flowing water conditions,with the gradual deposition of slurry,the water reduction effect at the outlet of the conduit becomes more profound.Specifically, when grouting lasted 27.2 s, the slurry deposited with a length of 1.3 m near the grouting conduit, the maximum down-flow deposition distance was 5.4 m, and the maximum deposition thickness was about 0.12 m,which was located near the position of y=12.5 m.At this time, the slurry deposition was dominated by the flowing water,and the corresponding water reduction rate was 15.3%.When grouting lasted 65.2 s,the slurry deposited with a length of 0.8 m near the grouting pipe, the maximum down-flow deposition distance was 6.9 m, and the maximum deposition thickness was about 0.17 m, which was located near the position of y = 13.8 m.This moment was the threshold time for counter-flow diffusion,and the corresponding water reduction rate was 46.7%.When grouting lasted 138.8 s,the counter-flow diffusion distance of the slurry was 1.7 m, the maximum down-flow deposition distance was 9.4 m, and the maximum deposition thickness was about 0.35 m, which was located near the position of y = 12 m.At this time, the corresponding water reduction rate was 74.1%.

Fig.10.Illustration of the complex 3D karst conduit and the grouting pipe.

Fig.12.The evolution of slurry deposition during grouting.

As shown in Fig.13, when grouting lasted 129.5 s, a “concave type” punching groove was formed at 1.2 times the diameter on the right side of the grouting pipe.Thereafter, due to the resistance of the previous sequential slurry deposited on both left andright sides, the slurry would spread to both sides simultaneously as if it touches the bottom of the karst conduit,and form a vortex motion at the location where the slurry first contacts the conduit bottom.

Fig.13.“Concave type” punching groove and local streamline diagram: (a) The“concave type” behavior of slurry diffusion; (b) The vertical section; and (c) The axial section.

Fig.14.The velocity distribution and slurry deposition state at the monitoring section:(a) 30.2 s and (b) 78.9 s.

Fig.15.Velocity distribution of vertical survey lines at (a) 105.7 s and (b) 134.8 s.

4.3.Variation of fluid velocity during grouting

What has been shown so far is mainly aimed at the macro phenomenon of slurry-water interaction, and the revelation of grouting mechanism requires a quantitative characterization of the flow rate.For velocity analysis, the typical time points of different stages were selected.The vertical and axial velocity distributions in the conduit are described as follows.

(1) Velocity distribution in vertical profile

To describe the velocity change during grouting, the section at y = 12 m was taken as the monitoring section, which was located on the right side of the“concave type”punching groove and slurryinitial deposition position.The velocity is relatively stable.According to the actual flow behavior of the slurry,the flow velocities when grouting lasted 30.2 s in the down-flow diffusion stage and 78.9 s in the counter-flow diffusion stage were selected to analyze the flow behavior.

Fig.16.Evolutions of axial velocity at different time:(a)Axial upper line at 105.7 s;(b)Axial lower line at 105.7 s;(c)Axial upper line at 134.8 s and(d)Axial lower line at 134.8 s.

As shown in Fig.14,the flow velocity in the pipe is larger in the middle, and smaller near the pipe wall and slurry mixing area,which is consistent with the general law of fluid flow.With the gradual deposition of slurry, the cross-section of the conduit decreases at the slurry deposition position, and groundwater flow velocity increases in a short time, and the maximum flow velocity appears in the area with the largest slurry deposition thickness.With the deposited slurry increasing at the bottom of the conduit,the viscous resistance of the slurry increases, and the velocity decreases at the cross-section of the karst conduit.In addition,through the 3D velocity contour, it can be seen that slurry in the lower part of the conduit is not completely solidified,the transport velocity from top to bottom is gradually reduced, and the slurry velocity near the wall is almost zero.The“push and wash”effect of sequential slurry and the “wrapped” flow behavior can also be reflected by the distribution characteristics of the velocity contour.

In order to describe the relationship between the slurry flow rate and the groundwater flow rate in detail, we drew different vertical survey lines on the left and right sides of the grouting conduit within the slurry deposition scope for comparative analysis of the vertical velocity.The survey lines are located at y=7 m,8 m,9 m,10 m, and 11 m.Groundwater velocity at the cross-section of the karst conduit is clearly larger near the axis,and smaller near the wall and phase interface.

Fig.15a shows the velocity distribution at 105.7 s.On the survey line of y = 9 m, the velocity varies greatly.With the further deposition of the slurry, the velocity above the conduit axis increases noticeably which is essentially the same with the survey line on the right side of the grouting conduit.At 134.8 s,the counter-flow range of the slurry reaches the survey line of y=8 m,the bottom velocity is clearly lower than that at the survey line of y = 7 m, and the velocity above the deposition area remains 0.2 m/s.The maximum counter-flow velocity of the slurry can reach 0.08 m/s.For the counter-flow slurry, its velocity is large near the axis and close to y= 0 m near the wall.

(2) Velocity distribution along the conduit axis

Previous analysis has shown that the slurry deposited at the bottom of the pipe is not a completely rigid solid.The slurry can still move at a certain speed in the flowing water, but its speed is very low compared to the water speed.Slurry stops moving until it is completely solidified.

Fig.17.Volume flow rate of slurry loss at the outlet of the conduit.

As shown in Fig.16, a comparative analysis of the velocity distributions at 105.7 s and 134.8 s in the counter-flow diffusion stage was performed.At these two moments,the thickness of the slurry deposition is almost two-thirds of the pipe diameter.Moreover,the velocity difference between the inlet and the outlet is small, and the four velocity curves are basically consistent.At 105.7 s, the outlet velocity is approximately 0.38 m/s, slightly higher than the inlet velocity of 0.2 m/s.After 29 s of grouting, the flow velocity is reduced by 20%.Compared with the velocity change in the axonometric line, the velocity curve is smoother, indicating that with the slurry deposition, the position of the central axis is in contact with the relatively homogeneous slurry deposition at 134.8 s.However,the difference of time-dependent characteristics of slurry viscosity can still be seen from local fluctuations.

Compared with the upper line,the velocity of the lower line at the inlet and outlet of the pipe varies greatly.The closer the inlet is to the grouting pipe,the smaller the velocity is.At the position of the pipe axis y=9.4 m,the velocity near the pipe outlet fluctuates sharply.In addition,a“concave type”punching groove appears at the bottom of the pipe, and the flow cross-section is further reduced, so that the flow velocity changes suddenly near y = 9.4 m.

4.4.Calculation and analysis of slurry loss rate

For the conduit grouting in the flowing water,the slurry loss rate has always been the key affecting the plugging effect.The roughness of the conduit acts as a natural barrier to slurry loss,reducing the slurry loss rate and accelerating the plugging effect.The slurry loss rate was calculated and analyzed using the SFS method.As shown in Fig.17,the initial stage with a large amount of slurry loss in grouting was analyzed.The slurry flow rate at the outlet of the karst pipe from 0 s to 10 s was selected as the calculation parameter for the slurry loss rate.From the calculation, the slurry loss rate is 98.2%.

5.Limitations and discussion

The SFS method can be used for engineering scale grouting simulation, however, there are some limitations for its application to addressing complex grouting engineering problems in conduits.

The grouting pump is an extremely complex system.Specifically,there is a complex chemical reaction process in the mixing of two liquids,and it is difficult to simulate chemical changes by using existing numerical methods.In order to reasonably simplify the boundary conditions, a fixed grouting rate is assumed in the simulation.Thus,the boundary conditions in existing literature are constant grouting rates similar to the actual grouting conditions.And the error caused by the fixed grouting rate is within the reasonable pressure fluctuation range.

The geometric condition (roughness) that affects grouting performance is only one aspect of our study.The geometric model shown in Fig.10 does not mean that it can completely represent the karst conduit, but rather explains the calculation ability of the complex geometric model shown in Fig.18.The applicability of the SFS method in irregular geometric models shows that the method has the computational capability for complex geological models,as shown in Fig.13.It is challenging to accurately detect the real karst conduits,which is why using multi-source exploration data to build a more realistic 3D geological model has become a hot topic.Therefore, our study focuses on the introduction of grouting simulation methods.

Fig.18.Sketch map of grouting in large flow karst conduits: (a)The grouting process in karst conduit, (b) The behavior of slurry diffusion, and (c)The behavior of slurry filling in karst conduit.

In summary,the SFS method can be used to simulate grouting of conduits in flowing water.Furthermore,it can be used to guide the design of grouting test in flowing water, which can improve cost efficiency and provide theoretical basis for grouting design.

6.Conclusions

Aiming at solving the difficulty of spatial-temporal tracking and numerical characterization of slurry viscosity, the SFS method for grouting simulation in flowing water was proposed, and the following conclusions can be drawn:

(1) A time-dependent threshold function of slurry viscosity was established to track and update the spatial position of the slurry after different sequences of injection.This method allows for the numerical simulation of the spatial-temporal distribution of slurry viscosity, and is capable of describing the gradual transition of slurry from flow to solidification in flowing water, while enabling the visualization of the process.

(2) In order to verify the effectiveness of the SFS simulation method, grouting experiments were performed under static and flowing water conditions.The SFS simulation method was used to solve the flow process of slurry.We compared and analyzed the flow velocity, the pressure, and the outlet flow rate in the pipe,as well as the initial deposition position,down-flow diffusion distance, counter-flow diffusion distance and overall diffusion deposition behavior of the slurry under different initial conditions.The calculation results obtained by the SFS simulation method are in good agreement with the physical model testing results, which proves the effectiveness and feasibility of the SFS method in the grouting simulation in flowing water.

(3) The applicability of the SFS method to the grouting diffusion mechanism in a rough conduit was studied at the engineering scale.And the characteristics of slurry flow and solidification, as well as the distribution of velocity and pressure in the conduit were analyzed.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We would like to acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos.52022053 and 51879153) and the China National Postdoctoral Program for Innovative Talents (Grant No.BX2021172).