1 Introduction

As China’s economy grows, so does its demand for energy. Despite the nation’s ambitious “carbon peak, carbon neutral” energy policy, coal remains a cornerstone of China’s energy portfolio, particularly in the coal-rich western regions of Inner Mongolia. These areas are predominantly covered by the Pleistocene Sala Wusu Formation (Q3S), a thick layer of eolian sand. The mining of coal in these regions is associated with the risk of water-sand inrush disasters (Han et al. 2023), which are the primary geological hazards encountered during mining. The construction of a shield tunnels near a water-rich sandy stratum, where sand can easily inrush, can damage to ground-level buildings or pose a serious threat to workers in the tunnel (Zheng et al. 2017), such as in the case of the catastrophic failure of Shanghai Metro Line 4(Shen et al. 2014). Therefor, a comprehensive understanding of the inrush process and the principles governing water-saturated thick sand formations under stress and flow field influences is essential.

Water-sand inrush events involve the sudden and violent intrusion of high-sand-content mixtures into underground workspaces, leading to potentially catastrophic consequences for both property and human safety. The phenomenon of water and sand mixed inrush is caused by the combined effects of the fluidity and stability properties of groundwater and sand solid particles (Sui 2023). Characterized by their intensity, perilous nature, and unpredictability, these events pose substantial challenges in terms of prevention and control, thereby critically impacting the safety performance and economic sustainability of coal mines, and preliminary statistics of water-sand inrush disasters are shown in Table 1, and the actual footage of water-sand inrush disasters is shown below as Fig. 1. In response, many researchers have meticulously studied the triggering conditions and fundamental mechanisms of water-sand inrush (Chen et al. 2020; Fan et al. 2018; Li et al. 2020; Liu and Liu 2020), aiming to formulate effective preventive measures and management strategies.

Table 1 Preliminary statistics of water–sand inrush disasters of China
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Fig. 1
figure 1

Actual footage of the water–sand disaster in various scenarios a Water–sand Inrush through boreholes in a sloping tunnel excavation face, b Water–sand and subsidence at Yuan Datan Coal Mine

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Abundant sand deposits (sedimentary source), pressurised groundwater (dynamic force) and interconnected structural fractures or fissure (flow conduits) are the main controlling factors for water-sand inrush hazards (Yang 2016). Naturally, a certain amount of clay is present within real-world sand deposits. Some scientists (Castro et al. 2017; Vallejos et al. 2017) believe that intense water inflow is one of the main causes of water-sand inrush. Geological structures within the underlying layers of water-rich sand strata, such as fractures, folds, and bedding planes, serve as inherent factors conducive to the formation of inrush channels, impacting both groundwater flow dynamics and the stability of the geological layers. Construction disturbances during underground engineering, such as excavation-induced unloading leading to rock mass fracturing and possible collapse, disrupt the equilibrium status of the water-sand layers, thereby creating pathways for inrush events. Cochard and Ancey (2009) found that the inclination of fractures was the main factor leading to sand and mudflows, and Haza et al. (2013) found experimentally that this was an important factor influencing sand erosion. Kendorski (2006) divided the overburden into a caved zone, fractured zone, dilated zone, constrained zone and surface zone, and the caved zone and the fractured zone were the main influence areas leading to water inrush. Xu et al. (2018) believed that the double influence of high water pressure and collision under high water pressure conditions led to water-sand inrush. The catastrophic process of a water-sand inrush event is characterized by a coupled interaction between soil pressure within the sand layer and hydraulic pressure. Undoubtedly, higher water pressures and wider fractures generally increase the likelihood of water-sand inrush events and their severity. However, the specific patterns of water-sand inrush behaviour under different combinations of clay contents, ground water stress, soil stress and different fissure widths have yet to be systematically investigated and quantified.

In the realm of water-sand inrush risk assessment, Fan et al. (2018) studied a mine with thin bedrock as an example and found that water-sand inrush is prone to occur in the area of the washout channels/fissures. Zhang et al. (2020) used the Factor Analysis (FA) and Fisher discriminant methods to analyze and assess the risk of a water-sand inrush under such conditions. Yang et al. (2016) developed a fuzzy comprehensive estimation model for judging mine water-sand inrush, revealed the basic conditions for such disasters under thin bedrock and thick loose aquifer. And used AHP-weighted factors and qualitative-quantitative analysis, quantified risks into four levels. Miao et al. (2022) have investigated the water inrush risks in North China Plain coal mines, where complex aquifer structures overlying coal seams play a significant role. By analyzing data from Zhaogu Mining Area, it identifies key factors affecting safety pillars and highlights the vulnerability of fractured layers under high water stress. The study also utilized a bespoke funnel test apparatus to investigate water-sand inrush and devised a calculation method for roof support resistance under conditions of high hydraulic pressure. Fu et al. (2012) addressed the challenges of water-sand inrush in thick coal seams under water-bearing unconsolidated strata, utilized GIS technology to analyze hydro-geological conditions and predict the heights of caving and water-conducted zones, the study highlights the importance of considering the mechanical properties of the strata and the impact of mining methods on the risk of water-sand inrush. In summary, through practical experience and related experimental findings, the formation conditions for water-sand inrush disasters induced by coal mining in water-rich sandy layers primarily include high hydraulic head, millimetre-scale inrush fissures/channels, abundant sand content, and sufficient subsurface cavities. The influencing factors during the inrush process are mainly attributed to water stress, ground stress, and the width of the channel or fissure openings.

In the realm of experimental investigations into the evolution of overburden fractures and the mechanisms governing water-sand outburst phenomena, the physical model tests (Yang et al. 2011, 2019, 2020,) were conducted to analyze the overburden failure and the development of water-conducting fractured zones, examined the two-zone failure mechanism in overburden strata during shallow buried thick seam mining at Halagou coal mine’s 22407 working face. It reveals a four-stage evolution process—gestation, formation, fracture-to-caving transformation, and stabilization—using a combination of physical modeling, theoretical analysis, and field data. The transport of sand particles through various mining-induced caving zones was investigated experimentally under different initial infiltration water heads, that were performed in a laboratory-scale caving zone packed with glass beads, using four different sizes of sand grains (Liang et al. 2022). Jiang et al. (2023) explored the evolution mechanism of tunnel water-sand inrush considering water-rich sandy dolomite hazard-causing structures, used fluid-solid coupling model tests to identify different types of water-sand inrush mechanisms and the factors affecting the inrush disaster. Zhao et al. (2023) investigated the process of water-sand inrush through vertical karst conduits, employed a simulated testing system to study the migration and inrush of water-sand mixtures, revealing the dynamic behavior of water stress and flow rate in relation to conduit size.

Overall, the formation of water-sand inrush disasters is currently well-understood, yet there is a lack of research on the underlying mechanisms and predictive theories driving these events. Presently, assessments often rely on the concept of critical hydraulic gradient for pipe surges to estimate the critical head conditions for sand inrush, without a dedicated theory for predicting critical fissure openings. Moreover, no quantitative relationship has been established between the three triggering factors—the width of the fissure opening widths, water stress, and sand stress—which limits the reflection of the varying critical fissure width required for inrush under different groundwater heads and burial depths. Additionally, the “Peck Formula” predominantly governs predictions of surface settlement curves but overlooks the impact of stratigraphic parameters and the dynamic outflow during an inrush (Peck 1969). The water-sand inrush disaster experimental research lacks readily available commercial equipment, mostly based on self-research and self-produced test system, hydraulic and ground stress coupling is not enough, and the model test scale is too small (Peng et al. 2023; Jiang et al. 2023; Chen et al. 2021a, 2021b).

Hence, to effectively anticipate and manage such disasters, a comprehensive understanding of the inrush process and the principles governing water-saturated thick sand formations under stress and flow field influences is essential. This study investigates the mechanisms underlying water-sand mixed inrush events within water-rich viscous sand layers subject to hydraulic coupling effects. The objective is to establish a inrush theoretical and technical basis for the development of water-sand inrush prevention techniques and the advancement of technology for mitigating and controlling water-sand inrush disasters.

2 Experimentation

This section defines scaling and similarity firstly, followed by an explanation of water-sand mixed material within viscous. The experimentation setup and measurement devices are then presented. Finally, the experimental scheme and procedure are discussed.

3 Scaling and similarity

Water-sand mixed inrush is complex and catastrophic phenomena in water-rich sand layers, with enormous volumes of mixed mass that buried all useful underground space. Therefore, scaling issues arise to replicate natural water-sand mixed inrush on a laboratory scale, scale reduction and replication of the actual inrush flow mechanism and features. If one uses the real original stratigraphic sand samples, actual ground stress and groundwater stress, the effective earth pressure on the strata must be increased by using a pressure device to maintain the stress similarity.

The sand samples used in the experiments were taken from the Kubuzi Desert, located in the suburbs of Ordos, Inner Mongolia Autonomous Region. The grain size of the sand samples ranged from 0.075 mm to 0.5 mm after sieve testing. The sand samples exhibited an Uneven coefficient of 1.71 and a Curvature coefficient of 0.97, indicating poor grading and uniformity. The tested properties of the sand were as follows: a clay content of 0.909%, an internal friction angle of 26.29°, and a cohesion of 14.22 kPa in the saturated state. The dry density of the sand samples was 1.47 g/cm3, and the permeability coefficient was 6×10−3cm/s. As for the natural dry clay used, its bulk density was measured at 1.25 g/cm3, additional sand samples with varying clay content can be produced through the proportional mixing of the natural eolian sand and dry clay specimens. The specific morphology of the materials used in the test is shown in Fig. 2.

Fig. 2
figure 2

Morphology of the materials a Natural eolian sand b Dry clay

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4 Experimental configuration

In this study, a series of experiments were conducted to investigate the effects of varying clay content, water stress, sand ground stress, and fissure opening widths on viscous sand samples. The samples were prepared by mixing eolian sand and clay at four different initial clay contents: 0%, 1%, 5%, and 10% by weight. The grain size distribution shown by Figure 3 was the experiments sand with nature clay content (0.909%). The fissure openings were set to widths of 1 mm, 3 mm, 5 mm, 7 mm, 11 mm, 15 mm and 20 mm, sand stress was varied between 0.2 MPa, 0.3 MPa, 0.4 MPa, 0.5 MPa, while the water stress was set at 0 MPa, 0.05 MPa, 0.1 MPa and 0.15 MPa. The densities of the sand layer ranged from 1875 kg/m3 to 2050 kg/m3 in the experiments.

Fig. 3
figure 3

Particle size distribution of the sand sample with nature clay content used in the experimentation

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5 Experimental setup and procedure

The modeling experiment assumes that the water-sand inrush disaster is triggered by the coupling of geo-stress and water stress. This occurs when the sand layer in the fallout zone surges into the mine through the thin bedrock fissure of the fracture above the mine shaft during the mining of coal mine shafts under the viscous water-rich sand layer in the northern Shaanxi Province of China.

At the Shaanxi Concrete Structure Safety and Durability Laboratory, a customized cylindrical water-sand inrush tank (with an internal diameter of 400 mm and a depth of 400 mm) was meticulously designed, as shown in Fig. 4. This experimental apparatus consisted of four integral components: a data acquisition system, a sand tank with a bottom inrush fissure, a pneumatic loading system and a measurement system specifically designed for water-sand inrush phenomena. The pneumatic loading system included a cylinder to simulate soil stress and a pneumatically operated diaphragm pump to precisely control the water stress, allowing the simulation of hydraulically coupled water-sand inrush events. The central component of this set-up, the sand tank used in the experiments, featured a 90 mm long bottom fissure with an adjustable width from 1 mm to 20 mm. This innovative feature facilitated the accurate simulation of different fracture widths that could potentially occur in real engineering projects.

Fig. 4
figure 4

Schematic and physical drawings of the test equipment a device physical drawing b Schematic diagram of test device

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The model was rested on a rectangular steel truss attached with an adjustable pulley, as shown in Fig. 4. Additionally, instruments were employed in the experimental setup to record the various experimental observations. A two 50 L rectangular tank was placed on an electronic scale, under the water-sand inrush fissure to receive the water-sand inrush mass in each run. BW-type miniature earth pressure sensors and BWK-1-type pore water stress sensors were installed at 0, 50, 100, and 150 mm directly above the water-sand inrush fissures to monitor the changes of pore water stress and earth pressure. Then, the quality of water-sand inrush mass versus the inrush time history will be recorded when water-sand inrush occurs.

The experimental procedures were as follows:

  1. (1)

    Calculate and weigh the appropriate amounts of aeolian sand and clay to achieve the desired clay content (0%, 1%, 5%, and 10%), mix the weighed sand and clay thoroughly. The experiment involved sand samples with varying amounts of clay. The sand was sieved and weighed to ensure cleanliness, and the clay was added in its natural, dry state according to weight percentage.

  2. (2)

    To achieve the target fissure widths of 1 mm, 3 mm, 5 mm, 7 mm, 11 mm, 15 mm and 20 mm, fissures of the appropriate width are installed interchangeably at the bottom of the tank and secured with gaskets and bolts to ensure no water or sand leakage.

  3. (3)

    Fill the experiment tank with 200 mm of a specific type of sand samples, place the micro earth pressure sensors and micro pore pressure sensors layer by layer at 0, 50, 100, and 150 mm while loading sand samples.

  4. (4)

    Connect the sensors to the DH3818Y Data Acquisition Instrument, cover the lid of the sand tank and open the ventilation hole.

  5. (5)

    Fill the experiment tank with water and ensure that the total height of the water–sand mixture reaches 350 mm. Allow the mixture to rest for a duration of 2 h to ensure complete saturation.

  6. (6)

    When saturation is achieved, close the ventilation hole and activate the pressure control system to maintain the water stress consistently at the pre-determined target pressures (0 MPa, 0.05 MPa, 0.1 MPa, 0.15 MPa) throughout the test.

  7. (7)

    Engage the testing apparatus such that the loading head contacts the surface of the sand layer. Maintain a constant loading pressure to simulate ground stress conditions, ensuring the soil pressure remains within the predetermined set points (0.2 MPa, 0.3 MPa, 0.4 MPa, 0.5 MPa) in sequence.

  8. (8)

    Switch on the monitoring equipment to capture the quality and rate of water–sand outflow, along with real-time fluctuations in both water and soil pressures during the experiment.

  9. (9)

    Open the water–sand inrush fissure and start the inrush experiment until the water–sand in the tank is exhausted. The standard protocol including basic test conditions for this physical model test following Fang et al. (2022) is shown in Table 2 in Appendix.

Each experiment began with meticulous calibration of all measuring equipment to ensure measurement accuracy. Strict adherence to standardised operating procedures minimised errors. A pneumatically controlled diaphragm pump in the sand tank precisely controlled water pressure and soil tension. Every step, from sample preparation to instrument calibration and measurement of critical parameters, was carefully confirmed and monitored. This meticulous attention to detail enhanced the reliability and reproducibility of the experimental design, ensuring the accuracy and validity of the data collected.

6 Results

6.1 Influence of inrush fissure width on water–sand inrush

This section analyzes the experimental process for saturated sand with 1% clay content under groundwater stress of 0 MPa and 0.025 MPa. The analysis covers five different fissure widths: 1 mm, 3 mm, 7 mm, 11 mm, 15 mm, and 20 mm, each loaded with a ground stress of 0.2 MPa, 0.3 MPa, 0.4 MPa, and 0.5 MPa.

During the experimental process, we observed a phenomenon (Fig. 5) where a fissure at the bottom of the sand tank opened, releasing water-sand. As the sand layer above the water layer had gradually reduced, the pressurized system had compressed downward, resulting in a decrease in the velocity of sand collapse until stabilization, after this, the sand was no longer routed or flowed out, and sinkholes formed in the sand tank. This entire process lasted a short period of time, ranging from 20 s to 200 s.

Fig. 5
figure 5

The inrush phenomenon a water–sand inrush, b sinkhole in sand tank

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The resulting curves illustrate the inrush of the sand and water mass, and the inrush rate was calculated for each fissure width. Figure 6 illustrates the relationship between the cumulated inrush mass and inrush time (Fig. 6a, b) and the inrush rate with different fissure widths (Fig. 6c). Define the slope of the curves as the inrush rates, the unit is kg/s.

Fig. 6
figure 6

The curves of the inrush process for various fissure widths. a, b: The relationship between cumulative inrush mass and inrush time (a No water stress but saturated with sand stress 0.2 MPa, b water stress 0.025 MPa and sand stress 0.5 MPa), c: The relationship between inrush rate and fissure width

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Referring to Fig. 6, it is evident that in cases where the water-rich sand layer is fully saturated with no additional water stress applied, the relationship curves of the cumulative inrush mass and the fissure width tend to be predominantly logarithmic trend (Model 1). However, the relationship appears to be linear (Model 2) when the sand layer experiences groundwater stress.

The inrush rate of water-sand is utilized to assess the inrush intensity under varying conditions. The inrush rates are determined by the slope of the curves, as previously stated. It has been observed that curves of the cumulative inrush mass and the inrush time tend to slow down towards the end of the curve (II section in Fig. 6a, b) when there is no water stress or at 0.025MPa water stress. The corresponding phenomenon is that the saturated sand layer inside the sand tank forms a inrush channel to the surface, and the water and ground stress is no longer maintained. So only the I interval section is taken into account to calculate the slope of the curves (inrush rates). Then, the experimental findings reveal that a critical start-up fissure width threshold of 3 mm exists for the inrush rate of sand-water mixtures under varying soil and water stress conditions. This threshold is substantiated by the observation that a sudden increase in the inrush rate of water-sand occurs when the fissure width transitions from 1 to 3 mm, consistently seen across scenarios with no applied water stress and those at a water stress of 0.025 MPa. When the fissure width is less than approximately 3 millimeters, water seepage occurs only within the sand layer, and the solid-phase (sand particles) do not undergo fluidization.

7 Influence of sand stress on water–sand inrush

This section aims to investigate the relationship between sand stress and the rate of water-sand inrush, with the goal of elucidating the underlying mechanisms and quantifying the extent to which changes in sand stress influence the dynamics of water-sand inrush. The experimental process record for saturated sand (with 1% clay content) under water stress of 0 MPa and 0.1 MPa, with a sand ground stress of 0.2, 0.3, 0.4 and 0.5 MPa and a fissure width of 5mm.

Figure 7 illustrates the relationship between the cumulated inrush mass and the inrush time with different sand ground stress (Fig. 7a and b). Furthermore, it shows the relationship between the inrush rate and the sand ground stress (Fig. 7c). Which demonstrates the inrush process of the sand and water mixture.

Fig. 7
figure 7

The curves of the inrush process for various sand ground stress in fissure width of 5 mm a, b: The relationship between cumulative inrush mass and inrush time (a  No water stress but saturated, b water stress 0.1 MPa), c The relationship between inrush rate and sand ground stress with water stress of 0.1 MPa and fissure width of 5 mm

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In accordance with Fig. 7a, b, the curves representing cumulative inrush mass against inrush time display a three-segmented feature (designated as segments I, II, and III), which is particularly notable during the inrush process of a saturated layer with no water stress, high ground stress (more than 0.4 MPa), and the inrush process under water stress (e.g. 0.1 MPa) conditions. The first segment (segment I) is the inrush gestation stage, wherein the fissure is formed (fissure is opening) and the sand stratum stress is redistributed and expanded through hydraulic coupling. It is worth noting that with the sand ground stress becomes greater as the depth of burial increases, this segment is more significant when the water stress is zero of the sand layers, while the higher sand ground stress result in a longer development of the stage when the water stress is greater than zero. The second segment (segment II) is the inrush stage, during which the mixture of water-sand flows through the fissure. The rate of inrush is directly proportional to the sand stress when the sand layer is only saturated with no water stress. While under the influence of water stress, the increase in stratum stress has little effect on the stage, the slopes of the curves under a certain amount of water stress are almost equal for the various layers of pressure, indicating that the inrush rate is also almost equal 1.606 kg/s. This conclusion is further confirmed in Fig. 7c. The third segment (segment III) is the end of the routing stage. During this stage, the water-sand mixture is completely drained when the sand layer is no water stress, the duration of this stage is longer when the sand ground stress is greater (i.e. when the depth of burial is greater). In contrast, the sudden change of the inrush mass in this stage occurs due to the action of water stress, resulting in the rapid outflow of the last sand.

The experimental findings also reveal that there is no start-up critical threshold of the sand stress on water-sand inrush.

8 Influence of water stress on water–sand inrush

This section focuses on the effects exerted by variations in water stress on the occurrence and characteristics of water-sand inrush events, through a meticulous examination of experimental data under diverse water stress conditions, elucidating the mechanisms governing the response of the inrush rate and dynamics of the water-sand mixture. The experimental process record for saturated sand (with 1% clay content) under a sand ground stress of 0.5 MPa and a fissure width of 1 mm, 3 mm, 5 mm, 11 mm, each was loaded with a water stress of 0 MPa, 0.025 MPa, 0.05 MPa, 0.1 MPa and 0.15 MPa.

Figure 8 illustrates the relationship between cumulated inrush mass and inrush time with different water stress (Fig. 8a). Furthermore, it shows the relationship between the inrush rate and the water stress (Fig. 8b).

Fig. 8
figure 8

The curves of the inrush process for various water stress under sand ground stress of 0.5 MPa and fissure width of 5 mm a: The relationship between cumulative inrush mass and inrush time, b: The relationship between inrush rate and water stress

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Based on the data presented in Fig. 8, it is evident that under conditions of zero water stress, saturated sand exhibited a sluggish flow through a 5 mm fissure/channel, with an approximate flow rate of 0.02 kg/s. Upon increasing water stress, a marked intensification of the inrush process was observed, as evidenced by the steepening of the inrush curve (refer to Fig. 8a), notably, at a fissure width of 11 mm, when the water stress reached 0.1 MPa, the inrush rate escalated beyond 3 kg/s. Throughout the experimental range of fissure widths, the water-sand inrush rate displayed a linear increment with escalating water stress (see Fig. 8b, Eq. (1), the linear proportionality coefficient k, however, is in a natural logarithmic relationship with the fissure width, as Eq. (2).

$$v_{si} = k\sigma_{s} + b{\text{w}}_{f} + c$$
(1)
$$k = m_{{\text{w}}} ln\left( {{\text{w}}_{f} } \right){ + }n$$
(2)

where: \(v_{si}\) is water–sand inrush rate, kg/s. \(\sigma_{s}\) is sand ground stress, MPa. \({\text{w}}_{f}\) is fissure width, mm. b, c, mw, and n are fitting parameters. In the results of this experiment, the fitted values are: mw = 8.3587, n = 5.7148, b = 0.0179, and c = -0.0065.

The significance of water stress for inrush disasters and their intensity is evident. When the inrush fissure/channel beneath the sand layer (at the bottom of the sand layer) and the opening reaches a critical width, a inrush disaster occurs. The disaster becomes more intense as water stress increases. Therefore, it is difficult to determine a specific value for the critical water stress.

9 Influence of clay content on water–sand inrush

The presence of clay can significantly affect the hydraulic and mechanical properties of a sand layer, which can alter its susceptibility to water-sand inrush events. Therefore, it is crucial to investigate the impact of clay content on the occurrence and intensity of such events in a systematic manner. The experimental process record details the testing of saturated sand with varying clay content (0%, 1%, 5%, and 10%) under specific conditions including a sand ground stress of 0.5 MPa, a fissure width of 5 mm, and a water pressure of 0.1 MPa.

Figure 9 illustrates the relationship between cumulated inrush mass and inrush time for different clay contents (Fig. 9a) and shows the relationship between inrush rate and clay content (Fig. 9b).

Fig. 9
figure 9

The curves of the inrush process for various clay contents under sand ground stress of 0.5 MPa, fissure width of 5 mm, and water pressure of 0.1 MPa. a: The relationship between cumulative inrush mass and inrush time, b: The relationship between inrush rate and clay content

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Figure 9 shows that the inrush process becomes more violent as the clay content decreases. Similarly, the inrush curve becomes steeper. Conversely, the inrush rate decreases as the clay content increases. The test data fitting indicates a negative exponential function relationship between the inrush rate and the clay content in the sand, as shown in Eq. (3).

$$v_{si} = k_{c} e^{{ - m_{c} C_{s} }}$$
(3)

where: \(C_{s}\) is clay content of sand layer, %. kc and mc are fitting parameters. In the results of this experiment, the fitted values are: kc = 2.8434, mc = 0.211, and R2 = 0.9006.

The negative exponential relationship reflects the regulatory effect of clay content on water-sand inrush rate and provides an important quantitative basis for analysing the effect of clay content on water-sand inrush. This means that clay content influences the hydraulic and mechanical properties of sand layers, thereby affecting water-sand inrush characteristics. Lower clay content reduces the stability of the sand layer and increases permeability, promoting inrush. Conversely, higher clay content increases stability and inhibits inrush. This is evident from the decrease in inrush rate with increasing clay content. These results highlight the importance of clay content in the assessment of inrush risk and indicate the need for consideration in engineering practice. It is clear that clay content has significant practical implications for guiding strategies to control water-sand intrusion hazards. The need for strategies to stabilise the sand layer by adding clay or other means to increase viscosity.

10 Sensitivity analysis of factors influencing the evolution process of water–sand inrush

In the foregoing sections of this paper, experiments have solely analyzed the initiation conditions and inrush disaster evolution patterns of water-sand inrush phenomena under a single variable. However, further research is necessary to determine the most influential factor in the event of an inrush disaster, considering all relevant factors in combination by adopting appropriate theories. The Gray Correlation Theory (GCT) has become a useful tool for assessing geological hazards and evaluating the sensitivity of various factors that contribute to their occurrence (Gao et al. 2018). This theory, originally developed by Professor Deng Julong (Deng 2002), is particularly adept at analyzing systems with incomplete or uncertain data, offering a quantitative method to measure the degree of correlation between factors and outcomes (Wang and Pu 2018), GCT constitutes a methodological approach designed to analyze and appraise the degree of interdependence among various factors within a system. It is grounded in the concept of similarity in the evolving trends between internal system components, wherein it computes correlation coefficients between these factors to ascertain their respective degrees of influence on a specific outcome variable. This theory is particularly well-suited for application in “grey systems,” characterized by incomplete data sets or insufficient information, where traditional methods may prove inadequate.

The application of GCT in the context of geological water-sand inrush involves several key steps (Chai et al. 2015). Initially, collect the data on the factors that are believed to influence the water-sand inrush events, such as sand stress, water stress, fissure widths, and viscosities (clay contents), and determine the reference sequence and comparative sequence. Subsequently, Matrix dimensionless, the data is then normalized to ensure comparability across different units and scales, which is a critical step given the diverse nature of geological data (Qian et al. 2018). Thirdly, the gray correlation difference information of the matrix is determined by calculating the correlation coefficients for each factor, which quantify the degree of correlation between the factors and the inrush event. The coefficients range from 0 to 1, with higher values indicating a stronger correlation. Step Four, calculate correlation degree. To make it easier to compare the large number of correlation coefficients, it is necessary to gather them and find their average value as the correlation degree. This allows for a comparison of the relevance of the influencing factors. The correlation degree represents the external performance between factors of interaction effect and ranges from 0 to 1. The correlation value indicates the degree of influence of a factor on the stability of the water-sand inrush. A higher correlation value indicates a larger effect and greater sensitivity, while a lower value indicates less sensitivity. The sensitivity of the influencing factors can be ranked based on the correlation degree. The correlation value indicates the degree of influence of a factor on the stability of the inrush. For a detailed procedure of the sensitivity assessment theory based on the Grey system theory, see Gao et al. (2018).

In the present experimental investigation, a comprehensive set of 80 trials was tested to simulate the whole process from start to finish of a real inrush disaster under different conditions. Each group had a constant value for three out of the four variables: soil stress, water stress, fissure width, and clay content respectively. The rate of water-sand inrush of each group was calculated separately. Subsequently, the inrush rates were used as the primary sequences for analysis. The corresponding soil stress, water stress, fissure width, and clay content values were included as the associated sub-sequences. The obtained correlation degrees for water stress, sand ground stress, clay content and fissure width were determined to be 0.886, 0.792, 0.776, and 0.754. The data indicates that water stress has the greatest impact on water-sand inrush intensity, followed by soil stress, viscosity, and seam width, which has the least impact.

The test results and sensitivity analysis conclusions show that the fissure width and water stress are the most significant factors in determining the occurrence of a water-sand inrush disaster. Once an inrush occurs, the intensity of the process is most influenced by water pressure and ground stress.

11 Discussion

11.1 Influencing factors of water–sand inrush disasters

Based on the hydrological environment of water-rich sand layers and their engineering impacts, the occurrence and development of water-sand inrush disasters within such layers are determined by the combined effects of multiple factors, including geological conditions, hydrogeological conditions, and engineering interventions (Wang and Pu 2018; Zhao et al. 2023). Geological conditions, such as the thickness, grain size, and clay content of the sand layer, the presence of weak layers or faults, can significantly affect the stability of the surrounding rock mass and the potential for inrush events. Hydrogeological conditions, including groundwater stress and flow patterns, play a critical role in the initiation and progression of inrush disasters. Excavation and engineering interventions, such as tunnelling and mining under water-rich layers, can alter the stress distribution and stability of the sand, water, and rock mass, potentially triggering or exacerbating inrush events. Thus, the influencing factors of water-sand inrush disasters are specifically categorized into the following aspects: fissure aperture/fissure width, ground stress, water pressure, and clay content.

12 Critical conditions for the occurrence of sand and water inrush disasters

Water-sand inrush disasters can occur due to the destruction of underground structures within water-saturated sand layers, such as tunnel linings (Zhao et al. 2023), or the fracturing of rock strata, like overlying strata in coal mine roofs (Xu et al. 2018). However, what are the critical conditions and specific values that trigger such events? These parameters are essential prerequisites for initiating disaster forecasting and prevention measures. But there is no normative methods and standards have been developed at present.

In practical work, the Hole Erosion Test (HET) is commonly employed for internal erosion experiments, the experiments in this paper are comparable to scholars’, even with a stronger focus on the interaction between water pressure and ground stress. A critical hydraulic gradient experimental method have proposed for water-sand inrush due to mining-induced fractures(Sui et al. 2007, 2017; Li 2008), a detection method for seepage failure has be introduced (Xu 2008), the simulations of underground borehole water-sand inrush experiments have been conducted (Liang et al. 1996), and refined the calculation formula for the critical hydraulic gradient, the value ranges from 1.0 to 65.0 under applied water pressures between 25 and 175 kPa (Moffat and Fannin 2011a, 2011b). Yang and Yang (2021) and Yang et al. (2020) investigated the initiation and migration of water-sand mixed fluids. They showed that the critical velocity of sand particle motion and the interaction between water-sand grains are key factors. These dynamics can either promote or hinder the inrush, depending on the hydraulic conditions and the structural stability of the porous skeleton.

The critical hydraulic gradient characterises only the intrinsic properties of sandy soils and is widely used in engineering projects such as free-flowing embankments to assess seepage failure. However, for water-saturated sandy layers, a sudden water-sand intrusion occurs under the action of water pressure following the formation of a permeable fissure or fracture through the underlying rock or underground engineering structures. Consequently, the critical fissure (or fracture) width serves as a key criterion for identifying the catastrophic water-sand inrush problem. The experimental results of the present study have shown that the critical condition for the occurrence of water-sand inrush disasters is determined solely by the fissure width. This critical condition is reached when the fissure width is 3 millimeters, which is equivalent to six times the diameter of the largest particle, the water-sand inrush occurred. No explicit threshold values are identified for other influencing factors (sand stress, water stress, and clay contents) to trigger inrush events. Instead, those conditions modulate the rate and intensity of the inrush process, rather than determining whether an inrush disaster will occur.

13 The evolutionary patterns of water–sand inrush disasters under the influence of various factors

Each factor not only determines the occurrence of a disaster but also influences the inrush process and intensity (evolutionary patterns) of the surge disaster. The post-inrush development of water-sand inrush can be characterized by distinct stages, as identified by Li et al. (2023), these stages include the initial rapid flow, followed by a period of instability, migration, deposition, and eventual stabilization, as shown in Figs. 6 and 7 of this paper. The primary determinants of water-sand inrush risk are the sand layer and the presence of water-sand conducting channels. Dong et al. (2023) and Gui et al. (2017) have emphasized the significance of these factors in the Yuxi section of the Water Diversion Project in Central Yunnan and China’s coalmines, respectively. The characteristics of the water-rich sand layer and the existence of primary and man-made channels contribute to the initiation and migration of water-sand mixtures. This study reveals that under the influence of water stress, inrush rates increased linearly with rising fissure width (Fig. 6c Model 2). As meticulously investigated by Jiang et al. (2023) and Li et al. (2020), the excavation of coal seams and tunnel construction activities indeed modify the stress distribution and permeability characteristics within the surrounding rock, thereby engendering water-sand inrush incidents. The stability of sand layers is significantly affected by mining-induced fractures and fluctuations in groundwater pressure, which increases the risk of inrush events. Water stress and the thickness of sand aquifers, significantly influence the risk of water-sand inrush, Zhang et al. (2020) and Gui et al. (2019) highlight that the water pressure and the thickness of the bottom clay layer are major factors affecting the risk under loose aquifer conditions. High water pressure can lead to increased seepage forces, which can exacerbate the inrush process. This research refines the conclusions, showing that when water-sand inrush occurred, the significance of water stress for inrush disasters and their intensity is evident, the water-sand inrush rate displayed a linear increment with escalating water stress (see Fig. 8b). Furthermore, the clay content in loose layers and the compaction of the sand layer have been shown to affect the occurrence of water-sand inrush. Chen et al. (2020) and Liu et al. (2020) demonstrate that both factors can influence the seepage-erosion process and the stability of the sand layer. Higher clay content can stabilize the sand layer, inhibiting the inrush. Soil compaction, on the other hand, can increase the seepage-erosion ability, potentially accelerating the inrush process. This study further refines and advances the understanding by revealing that the inrush rate exhibits a negative exponential relationship with increasing clay content.

14 Sensitivity analysis of factors influencing the evolution process of water–sand inrush

Sensitivity analysis of factors influencing the evolution process of water-sand inrush disasters has been a central focus across multiple studies. Dong et al. (2023) conducted an assessment for sandy dolomite tunnels, using Fuzzy-AHP to identify 13 key factors. The sandy composition of dolomite and rich water are particularly sensitive indicators for potential water-sand inrush incidents. Liu’s (2020) research revealed that soil properties, such as compactness and hydraulic pressure, exhibit high sensitivity. Gui’s teams (2019, 2017) emphasized the sensitivity of water-sand inrush to human-induced alterations to natural water pathways in coal mining contexts. They highlighted that even minor disruptions could drastically affect stability and risk levels. Yang et al.’s (2020) experimental study investigated the initiation of water-sand mixture flows. The study found that particle size and external stress are highly sensitive parameters that determine the onset of motion. The delicate balance between water-particle interactions plays a decisive role when thresholds are exceeded. In summary, sensitivity analyses consistently indicate that geological features, hydrological conditions, and anthropogenic influences are pivotal factors governing the progression of water-sand inrush. In this study, the Grey Relational Analysis method has been employed to examine the influence of four factors on the inrush intensity (inrush rate) of inrush processes. The sensitivity hierarchy determined reveals that water stress, ground stress, clay content, and fissure width have descending degrees of impact. It is important to note that the width of the fissure is the most significant factor in determining the occurrence of an inrush disaster. However, once such a disaster occurs, its impact on the intensity of the inrush is not the most sensitive among the factors considered.

In summary, this study provides a wealth of essential insights into the fundamental mechanisms driving water-sand inrush hazards and suggests promising avenues for their mitigation. In particular, it identifies the critical fracture width that triggers inrush events and, by establishing a quantitative correlation between inrush velocity and water pressure, provides critical data for the design of effective mitigation measures. Furthermore, the in-depth investigation of the varying sensitivities of different factors to inrush intensity addresses knowledge gaps in the current research landscape. The results of this study have significant practical implications for geotechnical engineering, particularly in areas susceptible to water-sand inrush hazards. By sealing fissures, draining excess groundwater and increasing sand viscosity, the likelihood of such disasters can be significantly reduced. The study’s thorough understanding of the dynamic processes that govern inrush events also provides a solid scientific foundation on which to build targeted geotechnical prevention and mitigation strategies. Ultimately, this research makes a significant contribution to the anticipation and management of water-sand inrush hazards, providing actionable insights for engineering applications.

15 Conclusion

This study provides a comprehensive analysis of the factors influencing the initiation and evolution of water-sand inrush disasters. It meticulously examines the impact of fissure width, sand ground stress, water stress, and clay content on the inrush process, offering valuable insights into the dynamics of these geological hazards.

A fissure width of 3 mm is a critical threshold for water-sand inrush, with a sudden increase in inrush rate beyond this width. Water stress is the most influential factor on inrush intensity, with a direct correlation to inrush acceleration, especially when fissure width exceeds the threshold. Sand ground stress contributes to inrush dynamics but lacks a critical threshold effect. Reduced clay content in the sand layer enhances inrush, while increased clay content provides protection. The study highlights the critical role of fissure width and water stress in inrush initiation and intensity, and the fitting equations of inrush rate and the variables of fisure width, water stress, soil stress and clay content were established. Although the physical meaning of these parameters may not be directly interpretable, they are essential for practical application and predictive purposes. Effective strategies for preventing and controlling water-sand inrush include fissure closure, groundwater drainage and increasing sand viscosity. Grey correlation theory, which is used to determine the sensitivity of the effects of different factors on the inrush rate, shows that water stress, soil stress, clay content and fissure width have descending degrees of influence, and that fissure width is the most important factor in determining the occurrence of an inrush catastrophe, not the inrush rate. This can guide the prioritisation of risk mitigation efforts in real engineering projects.