1 Introduction

The construction of tunnels are often encountered a variety of unfavorable hydrogeological condition, such as karst caves, faults, and weathered rock masses (Jeon et al. 2004; Li et al. 2016; Xue et al. 2021; Khan et al. 2021). When the engineering measures are not implemented in time in these water-rich zones, the construction of a tunnel will worsen the poor quality of the rocks, damage the integrity of the surrounding rocks, and affect the seepage path of groundwater, thereby increasing the local porosity and permeability. More importantly, high water pressures may cause water and mud/rock inrushes in the form of large volumes of water accompanied by mud, sand, or rock fragments bursting into the tunnel (Liang et al. 2015; Li et al. 2018, 2021; Gao et al. 2019). Compared with other unfavourable geological bodies, water and mud/rock inrushes are relatively more severe in the broken zones of water-rich fault, delaying the construction of tunnels and submerging these, destroying engineering equipment, and even potentially causing casualties (Xue et al. 2021; Wang et al. 2021a, b).

Existing studies have shown that the causes of water and mud inrushes in water-rich fault zones are the continuous adjustment of pore structures in broken zone rock masses under high water pressures, engineering disturbances, and particle migration (Wang et al. 2020; Kong et al. 2021). Under high water pressures and in the case of engineering disturbances, a large number of poorly cemented particles in a water-rich fault will be transported by water, and the porosity and permeability of the broken zone rock mass will increase (Ma et al. 2022). The pore structure of the broken rock mass will deform, which in turn will affect the stability of the rock surrounding a tunnel and further improve the flow of groundwater through the rock of the broken zone (Ma et al. 2017, 2022; Wang and Kong 2018; Liu et al. 2019; Kong and Wang 2019). Ma et al. (2022) developed a new rock testing system to investigate the process of particle migration and observed that water inflow events could be divided into four stages, i.e., rapid growth, decelerated increase, slow climbing, and a stable period. It has recently been suggested that the flow of water through granular rock materials is a complex process accompanied by a mechanical-hydrological-chemical (MHC) coupling effect and groundwater inrush hazards were further assessed using a proposed criterion related to the rate of mass-loss (Kong et al. 2020; Wang et al. 2019, 2020; Kong and Wang 2019).

Potential non-linear flow properties may appear with a consistent particle loss during this process and the inertia loss during fluid movement will be much greater than the cohesive loss (Wang et al. 2020). The classical Darcy equation no longer applies to the description of groundwater flow (Sedghi-Asl et al. 2014; Chen et al. 2015; Shi et al. 2020; Liu and Liu 2020; Li and He 2022). The non-Darcy seepage characteristics of groundwater in water-rich faults require further investigation (Kong et al. 2021). During the repeated crushing and grinding of the fault, rocks are crushed into broken rock, rock debris, rock powder, and fault gouges (consisting of montmorillonite, kaolinite, illite, chlorite, etc.) (Wang 2017). The non-Darcy flow behaviour of the core of the fault zone is seriously affected by the clay mineral contents in the fault gouges (Chen et al. 2013; Gui 2017; Tang 2017; Wang et al. 2021a, b; Xue et al. 2020). The clay minerals not only change the particle size distribution in the fault mixture (Gholamreza 1971) but also produce swelling when interacting with groundwater, thus changing the permeability of the fault (Boutin et al. 2011). Wu et al. (2015) explained the effects of clay particle size on the diameter of the permeability channel: the smaller the particle size, the smaller the permeability channel diameter, and the permeability coefficient is positively proportional to the diameter of the permeability channel. However, very few studies have investigated the non-Darcy flow behaviour of broken rock masses containing clay minerals, and it is unclear how clay minerals affect the characteristic non-Darcy parameters of the broken rock mass. Therefore, to better understand water and mud inrushes in water-rich faults, more focus should be directed at describing and explaining non-Darcy flow behaviour in granular rock masses with different porosities and clay contents.

In this paper, a seepage test using broken rock and soil containing clay under high water pressure was designed and constructed. The changes in the particle loss and flow field of broken rock and soil masses with different porosities (0.3, 0.35 and 0.4) and clay contents (5%, 10% and 15%) were studied. The Forchheimer equation was introduced to calculate the non-Darcy flow behaviour. The permeability and non-Darcy coefficient prediction models by the void ratio and liquid limit were established. Moreover, the critical non-Darcy flow velocity was discussed in depth. The results of this study have a positive theoretical significance and practical value for acquiring a better understanding of the disaster-causing mechanism of water and mud inrushes, water flux prediction, and disaster prevention in water-rich faults.

2 Experimental details

2.1 Experimental equipment

As shown in Fig. 1, the experimental system for testing the loss of quality of the broken rock mass during the process of seepage consists of five different parts: (a) a filling medium flow testing cell, air cylinder, and supporting frame, (b) sample porosity control device, (c) water pressure control equipment, (d) exhaust port and water pressure monitoring, and (e) the quality loss of broken rock mass collection module. Further details of each part of the testing system are provided below.

  1. (a)

    The filling medium flow testing cell (Fig. 1a) is the key component of the experimental system. The testing cell is a transparent cylinder with a diameter of 100 mm and a height of 400 mm; it consists of polymethyl methacrylate (PMMA) with a thickness of 10 mm. There are one scale mark and six outlets on the side wall of the container which are used to exhaust and connect the pore pressure sensor to test the pore pressure at different positions within the sample. A specific high barrel space is reserved at the water inlet of the testing cell to stabilise the water flow. At the bottom of the sample, the particular design of the water-flushing cap, together with the water-flushing funnel, ensures that the small rock particles can be washed out of the sample medium (Fig. 2).

  2. (b)

    The porosity control device (Fig. 1b) includes an air compressor, a pressure regulating valve, and a cylinder lifting switch. In order to control the porosity of the sample, a compressive pressure is applied by the air compressor and the air cylinder, and the sample height is then measured by the scale mark on the side wall of the container; this allows the calculation of the corresponding height and porosity of the sample.

  3. (c)

    The water pressure control equipment shown in Fig. 1c comprises a variable frequency booster constant water pump, a water supply tank, and a relief valve, which can provide the sample with a constant and stable water pressure. The maximum achievable water pressure of the loading system was 0.34 MPa.

  4. (d)

    The exhaust port and water pressure monitoring mainly measure the pressure gradient of the whole sample; the water outlet is connected to atmospheric pressure, so only one pressure sensor is installed at the upper end of the sample (Fig. 1d). The measurement accuracy of the pressure sensor was 0–0.4 Mpa ± 0.05% FS to ensure the accuracy of the test data. An exhaust port is connected through the outlet to allow venting during sample saturation and draining after the test.

  5. (e)

    The quality loss of the broken rock mass collection module (Fig. 1e) is made up of a fine strainer, bucket, electronic scale, and camera, and is used to collect and filter the small particles and clay mixed with the water flow. The quality of the effluent is collected by the electronic scale and then converted into the flow rate. The filter screen adopted a 325 mesh (0.045 mm).

Fig. 1
figure 1

Schematics of testing system

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Fig. 2
figure 2

Different components of the broken rock and soil mass flow apparatus

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2.2 Experimental materials and sample preparation

To study the seepage characteristics of broken rock and soil mass experiencing mass-loss under different clay contents, limestone samples obtained from the Beibei District Mine, Chongqing, China, were chosen as the experimental rock materials. The sieving method suggested by the ASTM (2009) then selects the particles with diameters ranging from 0.075 to 10.0 mm. The limestone samples were mechanically minced, screened, and grouped into four different size particles, as shown in Fig. 3a–d. The density of the limestone particles was 2720 kg/m3.

Fig. 3
figure 3

Composition of the filling medium materials different size: a 5–10 mm, limestone; b 2–5 mm, limestone; c 0.5–2 mm, limestone; d 0.075–0.5, limestone; and e 0.045 mm, Kaolin

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Compared with montmorillonite, illite, and chlorite, kaolinite is characterised by relatively strong hydrogen bonding forces between its crystal cells and a weak hydrophilicity (Mašín and Khalili 2016; Pötzl et al. 2018). Therefore, water-washed kaolin was selected as a component of the clay in the sample, with a plastic limit of 23%, a liquid limit of 39%, a particle size of less than 0.075 mm, and a density of 2.61 g/ml3. The mineral composition of the clay mainly consisted of kaolinite, montmorillonite, illite, and quartz, of which kaolinite accounted for the most abundant, as shown in Fig. 3e.

To investigate the effect of different clay contents on the seepage characteristics of the filling medium in a fault, tests were performed with three different kaolin contents: 5%, 10% and 15%. Particles with sizes of 0.075–0.5 mm, 0.5–2.0 mm, 2.0–5.0 mm and 5.0–10.0 mm were mixed in a 1:1:1:1 weight ratio. The particles with sizes ranging from 2.0 to 10.0 mm were considered as the main (large) rock fragments. The original particle size distribution (PSD) of the test material is shown in Fig. 4.

Fig. 4
figure 4

Particle size distributions of different samples

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Each sample was tested three times to achieve a greater reliability of the measurements. The details of each sample’s composition are summarised in Table 1. The sample density was calculated according to Formula 1:

$$\begin{aligned}{{\varvec{\rho}}}_{{\varvec{s}}{\varvec{c}}}&=\frac{{{\varvec{G}}}_{{\varvec{s}}{\varvec{c}}}}{{{\varvec{G}}}_{{\varvec{s}}{\varvec{c}}}\boldsymbol{*}(1-{{\varvec{w}}}_{{\varvec{c}}})/{{\varvec{\rho}}}_{{\varvec{s}}}+{{\varvec{G}}}_{{\varvec{s}}{\varvec{c}}}\boldsymbol{*}{{\varvec{w}}}_{{\varvec{c}}}/{{\varvec{\rho}}}_{{\varvec{c}}}}\\ &=\frac{{{\varvec{\rho}}}_{{\varvec{s}}}\boldsymbol{*}{{\varvec{\rho}}}_{{\varvec{c}}}}{{{\varvec{\rho}}}_{{\varvec{c}}}\boldsymbol{*}(1-{{\varvec{w}}}_{{\varvec{c}}})+{{\varvec{\rho}}}_{{\varvec{s}}}\boldsymbol{*}{{\varvec{w}}}_{{\varvec{c}}}}\end{aligned}$$
(1)

where ρsc is the mass density of the sample; Gsc is the quality of the sample; ρs is the mass density of the limestone; ρc is the mass density of the clay (Kaolin); and wc is the content of clay in the sample. The initial porosity of the sample was calculated using Formula 2:

$${\boldsymbol{\varphi }}_{0}=1-\frac{{{\varvec{M}}}_{{\varvec{s}}{\varvec{c}}}}{{\varvec{\pi}}{{\varvec{a}}}^{2}{{\varvec{\rho}}}_{{\varvec{s}}{\varvec{c}}}{{\varvec{h}}}_{{\varvec{s}}}}$$
(2)

where φ0 is the initial porosity of the sample; a is the radius of the cylinder barrel; hs is the height of the sample; and Msc is the mass of the sample.

Table 1 Details of each filling medium sample
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Fig. 5
figure 5

LD-100D combined liquid-plastic limit tester

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2.3 Experimental procedure

The experiments were carried out at room temperature and the testing fluid consisted of water (density ρw = 1000 kg/m3). The experimental procedures are summarised below:

  1. (1)

    Sample preparation and saturation. Each sample was prepared according to Table 1, and was first mixed with water to achieve an initial moisture content of 8% (Bendahmane et al. 2008). The single-layer proctor standard heavy compaction method was used to achieve a homogeneous filling medium sample. Each filling height was 50 mm. The sample was assembled into the penetration system and was saturated by injecting it water (Fig. 2). The saturation phase took 24 h to complete to make the kaolinite ultimately expand. According to the setting of the initial porosity of each sample, the axial displacement of the sample was adjusted to reach the set height of 200 mm.

  2. (2)

    Application of water pressure and data collectiona. Water pressure, 0.3 Mpa, was loaded to carry out the test. The lost sample mass was collected at the outlet every 1 min for each test. The flow velocity was recorded in real-time by a camera. The pressurisation process was terminated when the water inflow became stable and no more particles were flushed out.

  3. (3)

    Step-by-step pressure-reduced seepage test. After step (2), the small solid particles that lay in the broken sample were lost, the pore structure of the sample finally tended to be stable, and the permeability remained unchanged. In this test step, the non-Darcy water flow characteristics of the sample were calculated after particle migration. The seepage velocity s different water pressure could be obtained by reducing the water pressure step by step (0.25 Mpa, 0.2 Mpa, 0.15 Mpa, and 0.1 Mpa). Each pressure was maintained for 1 min. The change in the sample height was determined by the scale line on the side wall of the sample cylinder.

  4. (4)

    Sample weighing. After the process of unloading the water pressure was completed, the sample was recovered and dried in the drying oven; the particles from different sections were then screened and weighed, and the weight change of each particle size section was recorded. Finally, the clay and particles (0.075–05) were mixed and the liquid limit of each sample was measured.

3 Results

3.1 Changes in particle loss

During the test, the mass lost from the samples due to water flow flushing was collected every 1 min (Δt). The duration of each test was roughly 20 min. The total collected mass mt=j and the average mass migration rate m′t=j during any time interval can, respectively, be calculated as:

$${{\varvec{m}}}_{{\varvec{t}}={\varvec{j}}}=\Delta {{\varvec{m}}}_{{\varvec{t}}=1}+\Delta {{\varvec{m}}}_{{\varvec{t}}=2}+\cdots +\Delta {{\varvec{m}}}_{{\varvec{t}}={\varvec{j}}}({\varvec{j}}=\mathrm{1,2},\cdots ,20)$$
(3)
$${{\varvec{m}}}_{{\varvec{t}}={\varvec{j}}}^{\boldsymbol{^{\prime}}}=\frac{\Delta {{\varvec{m}}}_{{\varvec{t}}=1}}{\Delta {\varvec{t}}}({\varvec{j}}=\mathrm{1,2},\cdots ,{\varvec{n}})$$
(4)

Figure 6 shows the mass-loss and mass-loss rate of the sample. The mass-loss rate of the sample is defined as the maximum loss rate during the first minute, after which the mass-loss rate gradually decreases until there is no particle loss during the fifth minute. It should be noted that in samples I-1, II-1, II-3, III-2, and III-3, after a period of water flushing, the mass began to be lost again, which was particularly obvious in sample I-1. However, there was no loss of secondary mass in the other samples, which may be due to the different local skeletons formed by fine and coarse particles.

Fig. 6
figure 6

Variation in total mass and average mass loss rate of flushed out mass: a sample I, b sample II, c sample III

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In different samples having the same porosity, higher clay contents would result in less mass-loss. In different samples having the same contents, the more significant the porosity, the greater the mass-loss. For example, the loss-mass of sample I-1 was 206.4 g, that of sample II-1 was 266.9 g, and that of sample III-1 was 301.8 g. The original porosity and clay contents significantly affected the mass-loss of the sample.

Although the mass lost from the sample was collected by the screen, a small amount of kaolin and rock powder mixed with the water in the test. During the test, the water gradually changed from turbid to clear. Especially during the first one-two minute of the test, a small amount of kaolin was dissolved in water and thereby not collected by the screen.

3.2 Change of water inflow velocity

In the test, the time series of the mass of water was collected at equal intervals (the 30 s), and the flow velocity v wass calculated according to the change of the mass of water through the difference method. The expression is

$${{\varvec{v}}}_{{\varvec{t}}-{\varvec{i}}}=\frac{{{\varvec{M}}}_{{\varvec{w}}{\varvec{t}}={\varvec{i}}}-{{\varvec{M}}}_{{\varvec{w}}{\varvec{t}}={\varvec{i}}-1}}{{30{\varvec{\rho}}}_{{\varvec{w}}}{\varvec{\pi}}{{\varvec{a}}}^{2}}$$
(5)

where v is the flow velocity; Mw is the water quality; ρw is the mass density of water; and a is the radius of the cylinder barrel.

Figure 7 shows the variation in water velocity over time for each sample. The variation law of water velocity of the samples was consistent with their mass-loss because the lost mass was flushed from the samples of the filling medium, resulting in an increased permeability and porosity. The water velocities of the samples increased rapidly. Then, the water velocity remained stable. In samples I-1, II-1, II-3, III-2, and III-3, the water velocity increased again after a period of water flushing due to the sample mass-loss. The final steady flow velocity after the mass-loss test was positively correlated with the original porosity and negatively correlated with the original clay content.

Fig. 7
figure 7

Variation in water velocity over time: a sample I; b sample II; c sample III

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According to the experimental observations, the mass-loss and water velocity evolution during the tests could be divided into four stages: acceleration, stable with small fluctuations, reacceleration, and stable. In the first stage, the loss mass rate of the sample was at its maximum in the first minute before gradually decreasing. Major channels for water flow and particle migration were formed, resulting in erosion because of the migration of large amounts of clay and particles. The water velocity increased rapidly when the mass-loss rate changed. Figure 7 shows that during the first stage, the lower the original clay contents and the greater the original porosity of the sample, the longer the duration and stronger the variation in water velocity. The durations of the first stages for samples I-1, II-1, and III-1 were 2 min, 2.5 min, and 3 min, respectively. Indeed, in the case of a higher original porosity and lower clay contents, the weaker the bonding ability between the particles, the more easily the particles can migrate, and thus, it would take longer for particles of different sizes in the sample to stabilise again.

In the second stage, there was no particle loss, and the water velocity was stable with minor fluctuations. In the third stage, locally blocked particles were out of balance, and the mass began to be lost again. The corresponding flow rate increased again. This period lasted only 1–3 min. It should be noted that not all samples experienced secondary particle loss (the third stage). In the final stable water flow stage, there was no noticeable amount of small particles migrating out of the samples, and the water velocity also became constant.

3.3 Change in physical parameters after the tests

After the tests, the fine particles and the clay were flushed, and the PSDs of each sample were changed. Each sample was dried and weighed. The PSDs with 5.0–10.0 mm particle sizes changed litter in each sample. However, the PSDs changed significantly with other particle sizes. Figure 8 shows the PSDs of different samples after the tests. The changes in the sample with 5% clay contentss were the most significant out of the three original PSDs, and the larger the original porosity was, the larger the change of the PSDs with particle sizes of 0.075–5.0 mm. Table 2 details the physical parameters of each filling medium sample after the tests. As shown in Table 1, the changes in porosity, clay contents, liquid limit, and effective particle sizes (d10 and d60) were different for samples with different original clay contents and porosities. In general, the smaller the clay contents, the more obvious the changes. The original porosity, in turn, had the opposite effect on these changes. The porosity and clay contents together determined each filling medium sample's structural and permeability properties.

Fig. 8
figure 8

Particle size distributions of different samples after test

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Table 2 Details of each filling medium sample after the tests
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3.4 Non-Darcy flow behaviour of the samples after mass-loss

3.4.1 Variation of the non-Darcy hydraulic characteristics

The pore structure and seepage channels of each sample migrated were reorganized after the mass-loss tests. The water flow velocity rose rapidly under a constant water pressure. Numerous previous studies have found that the water flow behaviour changed from a Darcy flow to a non-Darcy flow (Hu et al. 2019; Shi et al. 2020; Kong et al. 2021; Li et al. 2022). The non-Darcy seepage characteristics of each sample was tested by the step-by-step pressure-reduced seepage test. The water pressure gradient \({{\varvec{G}}}_{{\varvec{p}}}\) could be obtained according to the water pressure difference and height at both ends of the sample

$${{\varvec{G}}}_{{\varvec{p}}}=\frac{{{\varvec{P}}}_{1}-{{\varvec{P}}}_{2}}{{{\varvec{h}}}_{{\varvec{s}}}}$$
(6)

where P1 is the pressure at the outlet end of the sample since the outlet end is connected with the atmosphere; P1 = 0; P2 is the pressure when the fluid enters the sample; and hs is the height of the sample.

Many empirical and theoretical equations have proposed to describe a non-linear flow behaviour (Sidiropoulou et al. 2007; Moutsopoulos et al. 2009). The most well-known relationship is the Forchheimer equation, which can be described as

$$-\frac{\partial P}{\partial x}=Av+{\varvec{B}}{v}^{2}$$
(7a)
$$A=\frac{1}{{\varvec{K}}}, B=\beta {{\varvec{\rho}}}_{{\varvec{w}}}$$
(7b)

where ∂P/∂x is the water pressure gradient on the upper and lower surfaces of the sample; v is the water velocity of fluid; ρw is the density of the fluid; and K and β are the viscous permeability coefficient and the inertial coefficient (non-Darcy coefficient), respectively.

By fitting the water pressure gradient with the seepage velocity under this gradient according to Eq. (7), the K and β of the viscous permeability coefficient and the non-Darcy coefficient of the sample could be obtained (Table 3). As shown in Fig. 9, the fitting coefficient R2 for all samples tested was close to 1. When the flow velocity increased with the increase of the pore pressure gradient, the fitting curve protruded to the velocity axis, that is, the increase of the seepage velocity was lower than the linear increase and presented significant non-linear characteristics. This phenomenon was obvious in samples III-1 and II-1, as shown in Fig. 9. Both the viscous permeability coefficient and the non-Darcy coefficient of each sample were significantly related to the porosity and the clay contents. When the porosity increased, the permeability coefficient also increased, and the non-Darcy coefficient decreased. However, the clay contents produced the opposite effect.

Table 3 Viscous permeability and non-Darcy coefficient of the samples
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Fig. 9
figure 9

Pressure gradient versus water velocity for flow through samples: a sample I; b sample II; c sample III

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3.4.2 Prediction models of the permeability and Non-Darcy coefficients

  1. (1)

    Prediction model of the permeability coefficient

Numerous models have been developed to predict the permeability coefficient of an aggregated medium (Ergun 1952; Ward 1964; Rogak and Flagan 1990; Lee et al. 1996; Sedghi-Asl et al. 2014; Mbonimpa et al. 2002). Among these, the empirical equation of porosity and permeability obtained by Carman-Kozeny (Carman 1956; Kozeny 1953) is the most widely applicable. However, since the equation assumes no electrochemical reactions between the solid particles and the permeant, it is not appropriate for broken rocks and soils including clay (David et al. 1962). Ren et al. (2016) introduced the concept of an effective void ratio to derive a hydraulic conductivity-void ratio relationship based on the Poiseuille's law. The equation is expressed as follows

$$K=\frac{1}{{C}_{F}}\cdot \frac{1}{{S}_{s}^{2}{\rho }_{m}^{2}}\cdot \frac{{\gamma }_{w}}{\mu }\cdot \frac{{e}^{3m+3}}{{(1+e)}^{\frac{5}{3}m+1}{[{\left(1+e\right)}^{m+1}-{e}^{m+1}]}^\frac{4}{3}}$$
(8)

where K is the permeability coefficient; CF is the coefficient related to the media tortuosity and the actual flow direction of the water; Ss is the specific surface area of the particles; γw is the unit weight of the fluid; μ is the fluid viscosity; ρm is the particle density of soils; e is the void ratio of soils; and m is related to the specific surface area which is related to the size and the shape of the soil grains. When m = 0, the equation goes back to the Kozeny-Carman equation. Furthermore, when m is 0 ~ 0.1, 1, and 1.5, it applies to the sandy soil, silt, and clay calculations.

In this research, the evolution equation of the Kozeny-Carman equation (Ren et al. 2016) was employed to predict the permeability coefficient change. Considering that the clay contents of each sample were not large relative to the total mass of the sample, the coefficient m was taken as 1. Considering an equivalent medium with particles having an equivalent diameter dh, Ss can be calculated by (Kovács 1981)

$${S}_{s}=\frac{\alpha }{{\rho }_{m}{d}_{h}}$$
(9a)
$${d}_{h}={{C}_{U}^{1/6}d}_{10}$$
(9b)

where α is a shape factor. To represent the particle distribution curve and the effective particle size, the uniformity coefficient CU (d60/d10) and d10 as the effective grain size were introduced for dh (Goldin and Rasskazov 1992).

Using the approach described above, the data required for the permeability coefficient evolution models (see Tables 2 and 3) were obtained. To compare the performances of the predictive models for the permeability evolution, Eq. (8), and to determine the empirical coefficient in Eqs. (8) and (9), a simple statistical efficiency criterion, based on the coefficient of determination factor R2, was used

$${R}^{2}=\mathrm{f}({C}_{p})=\frac{\sum_{n}{{(K}_{n}^{t})}^{2}-\sum_{n}{{(K}_{n}^{t}-{C}_{p}\cdot {k}_{n}^{m})}^{2}}{\sum_{n}{{(K}_{n}^{t})}^{2}}$$
(10)

where Cp is the empirical coefficient, which is related to the media tortuosity, the actual flow direction of water (CF), and the shape factor (α); n is the total number of test data, and the maximum number of n is 9; Ktn is the test measured value, and κmn is the model prediction associated with a test. Obviously, when the f(Cp) derivative was 0, we could obtain the optimal empirical coefficient Cp. After the calculation, when Cp was 0.002375, R2 reached its maximum of 0.935. For the R2 = 0.935, the predictive models for the permeability evolution, Eq. (10), could yield sufficiently accurate test results.

Substituting Eq. (9), m = 1 and Cp = 0.002375 into Eq. (8) yielded

$$K=0.002375\cdot {d}_{h}^{2}\cdot \frac{{\gamma }_{w}}{\mu }\cdot \frac{{e}^{6}}{{\left(1+e\right)}^\frac{8}{3}{\left(1+2e\right)}^\frac{4}{3}}$$
(11)

Equation (11) shows that the change of the permeability coefficient was affected by equivalent particle sizes, CU and d10, and the void ratio, e. To more intuitively confirm the influence of the clay contents on the permeability coefficient, a liquid limit was introduced to represent the change of clay contents and the equivalent particle sizes. Figure 10 shows the relationship between the liquid limit and effective particle sizes as a power function, \({d}_{h}=5.249\cdot {w}_{L}^{-1.0351}.\) The power function expression of the particle size and liquid limit can also be found in other research (Wetzel 1990; Jacques et al. 1984).

Fig. 10
figure 10

Relationship between liquid limit and effective particle size

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The effective particle size dh was converted to the liquid limit, while the weight of water γw, 9800 N/m3 and the fluid viscosity, 1e(-3) N s/m2 were substituted into Eq. (11), yielding

$$K=0.65436\cdot {w}_{L}^{-2.0702}\cdot \frac{{e}^{6}}{{\left(1+e\right)}^\frac{8}{3}{\left(1+2e\right)}^\frac{4}{3}}$$
(12)

Figure 11 shows the prediction model of the permeability coefficient in Eq. (12) and the experimental data in Tables 2 and 3. The experimental value point was very close to the predicted surface. The predicted surface showed that as the void ratio increased and the clay contents decreased, the permeability of the rock mass increased.

Fig. 11
figure 11

Predicted surface of permeability coefficient obtained through the liquid limit and void ratio

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  1. (2)

    Prediction model of the non-Darcy coefficient

Numerous scholars have tested thepermeability, porosity, and other seepage characteristics of rocks through experiments and summarised them and the non-Darcy coefficient relationships between these factors. Table 4 summarises the research results of some authors and provides the relational function form between the non-Darcy coefficient and permeability and porosity. Different scholars have presented different values for the coefficient Cβ because different materials were used, and the coefficient Cβ included the influence of particle shape, tortuosity, and other factors. Like in Eq. (11), the statistical efficiency criterion based on the coefficient of determination factor R2 was used. Combined with the test data provided in Tables 2 and 3, the optimal coefficient Cβ considering the maximum R2 was determined under different formula forms. Table 4 shows that the maximum R2 was 0.969 and the formal error of this function was the smallest.

Table 4 Summary of the relationship between the non-Darcy coefficient β and permeability and porosity
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The \(\beta ={C}_{\beta }/({k}^{0.5}{\varphi }^{1.5})\) and Cβ = 3.11*103 forms were substituted into Eq. (12), yielding

$$\beta =278278.1902\cdot {w}_{L}^{1.0351}\cdot \frac{{e}^{-\frac{9}{2}}}{{\left(1+e\right)}^{-\frac{17}{6}}{\left(1+2e\right)}^{-\frac{2}{3}}}$$
(13)

Figure 12 shows the prediction model of the non-Darcy coefficient in Eq. (13) and the experimental data in Tables 2 and 3. The non-Darcy coefficient was negatively correlated with the void ratio. The liquid limit was positively correlated with the change of the non-Darcy coefficient. This is because the higher the liquid limit or the greater the clay contents, the smaller the effective size of the broken rocks and the smaller the passage for water to pass through. Moreover, clay minerals absorb water and swell, and the swollen clay minerals may easily cause the blocking of the pore channels. When the void ratio and the pore channels were smaller, the water flow was more likely to produce an eddy current and cause energy damage under the action of high water pressures. The non-Darcy phenomenon of water flow would then be more and more noticeable.

Fig. 12
figure 12

Predicted surface of non-Darcy coefficient obtained through the liquid limit and void ratio

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3.4.3 Non-Darcy seepage behaviour criterion

Existing studies have mostly used the dimensionless critical Reynolds number Re to distinguish between Darcy and non-Darcy flows (Qian et al. 2005; Bagci et al. 2014) but the critical Reynolds number Re changes with changes to medium characteristics such as the characteristic length of the permeable media material. Zeng and Grigg (2006) recommended the Forchheimer number (Fo), the ratio of the second term (βρv2) of the Forchheimer equation to the first term (v/K), as the criterion for non-Darcy flows in porous media, which is expressed as

$${F}_{0}=\frac{\beta Kv}{g}$$
(14)

According to these authors, Zeng and Grigg (2006), the non-Darcy effect, E, can be defined as the ratio of the hydraulic gradient consumed in overcoming liquid–solid interactions to the total hydraulic gradient and be expressed by Fo

$$E=\frac{{F}_{0}}{1+{F}_{0}}$$
(15)

The non-Darcy effect denotes the error caused by ignoring non-Darcy flow behaviours. If a 10% non-Darcy effect is defined as the allowable error, then the critical Forchheimer number is 0.11, which marks the onset of non-Darcy flow behaviour as suggested by Zeng and Grigg (2006). The behaviour criterion of the non-Darcy flow could be evaluated by the critical flow velocity according to Eq. (14); Eqs. (12) and (13) were substituted into Eq. (14), yielding

$${v}_{c}=\frac{0.11\cdot g}{\beta K}=6.04081*{10}^{-6}\cdot {w}_{L}^{1.0351}\cdot \frac{{\left(1+e\right)}^{\frac{-1}{6}}{\left(1+2e\right)}^\frac{2}{3}}{{e}^{3/2}}$$
(16)

Figure 13 shows the prediction model, Eq. (16), of critical flow velocity obtained through the liquid limit and void ratio. In general, the lower the void ratio and the higher the liquid limit, the greater the critical flow velocity. Moreover, the porosity had a more obvious effect on the critical velocity than the clay contents. Non-Darcy flow usually occurred in broken rocks and soil masses with a high void ratio and a low liquid limit, which only needed a low critical flow velocity.

Fig. 13
figure 13

Predicted surface of critical flow velocity obtained through the liquid limit and void ratio

Full size image

4 Discussion

Water-rich fault zones, which contain granular rocks and fault gouges, can act as major water outburst channels during water and mud inrushes (Liu et al. 2019). Broken rocks, soil masses, and fault gouge contents comprising the existing faults could be crucial factors in determining tunnel construction safety and the risk of a water and mud inrush. In this study, the change of water inflow velocity could be divided into four stages, i.e., acceleration, stable with small fluctuations, reacceleration, and stable. During the first acceleration stage, major channels for water flow were formed and the water flow velocity rose rapidly (Fig. 7); this resulted in a large amount of lost sample mass (Fig. 6) and the clay contents and liquid limit decreased while porosity increased. The fluid velocity and mass-loss acceleration stages were generally coincident with the delayed phenomenon of water inflow (Hu et al. 2019). This means that there were probably multiple stage processes during the particle loss process. Therefore, in the actual construction, grouting reinforcements, which have become a general method to prevent water inflow hazards, should be implemented as soon as possible in the initial stages of water inflow to prevent further and multiple mass-loss from the fault zone.

Tunnel waterproofing and drainage design are also important measures to be considered for the disaster control of water and mud inrushes. The tunnel water influx prediction has a guiding significance for the tunnel waterproofing and drainage design (Xu et al. 2021). According to this study, when mass was lost from the broken zone and particles migrated from it, the void ratio increased and the liquid limit decreased, leading to a decrease of the critical flow velocity from a Darcy flow to a non-Darcy flow (Fig. 13). The mass-loss and particles migration behaviour during the seepage process of the broken rock would affect the non-Darcy seepage process. Indeed, the permeability coefficient and non-Darcy coefficient would change sharply (Figs. 11 and 12) where the permeability coefficient changed more significantly. The non-Darcy theory with dynamic changes in the permeability and non-Darcy coefficient must be considered in the tunnel water influx calculation formula.

Several studies have indicated that non-Darcy flow behaviours in porous media are complex and affected by many factors, such as particle size, porosity, granule shape, tortuosity, surface roughness, and wall effects (Sedghi-Asl et al. 2014; Yang et al. 2017; Chen et al. 2011). Among these factors, particle size and porosity are key factors which are used to describe the water flow and modify the flow theories (Sidiropoulou et al. 2007; Moutsopoulos et al. 2009). In this study, the contents of the fault gouges were characterized by a liquid limit index. The relationship between the liquid limit and effective particle size as a power function was built and is shown in Fig. 10. The test results and the evolution equation of the Kozeny-Carman equation (Ren et al. 2016) showed that the prediction performance of Eq. (12) for the permeability coefficient was acceptable. It should be noted that the fitting coefficients in Eqs. (12) and (13) must be adjusted accordingly for different geological situations. However, the connectivity of pores and the distribution of pore channels also affect the permeability properties of porous media such as broken rocks, as well as their anisotropy. The predicted models will require further modifications in future work.

5 Conclusions

A seepage testing system was designed and a series of tests were conducted to evaluate the mass loss and flow properties of broken rocks containing clay. The non-Darcy seepage characteristics associated with different porosities and clay contents after mass loss test were studied. Prediction models of the permeability and non-Darcy coefficients were put forward based on the evolution equation of the Kozeny-Carman equation. The main conclusions are summarised below.

  1. (1)

    The mass-loss and water velocity evolution during the water inrush could be divided into four stages: acceleration, stable with small fluctuations, reacceleration, and stable. The first acceleration stage corresponded to the main stage of mass-loss. The third acceleration stage was only observed in some samples and was related to a delayed water inflow migration phenomenon. The lost mass and change in water velocity were positively correlated with the clay contents and were negatively correlated with the original porosity.

  2. (2)

    The Forchheimer equation was applied to the non-Darcy flow behaviour in the broken rock and soil masses containing clay after the mass-loss test using the step-by-step pressure-reduced seepage test results. The evolution equation of the Kozeny-Carman equation, where the liquid limit index is introduced to characterise the effective particle size, fitted well with the experimental results. Six non-Darcy coefficient prediction models were verified against the testing results. It was shown that the \(\beta ={C}_{\beta }/({k}^{0.5}{\varphi }^{1.5})\) equation yielded the best prediction results. Prediction models were created for the critical flow velocity from a Darcy flow to a non-Darcy flow through the Forchheimer number. When mass was lost from the broken zones and particles migrated from it, the void ratio increased, the liquid limit decreased, and the critical flow velocity became smaller.

  3. (3)

    In the tunnel construction through a water-rich fault zone, grouting reinforcements should be implemented as soon as possible during the initial stages of water inflow to prevent multi-stage mass-loss processes. In addition, non-Darcy theory with dynamic changes in permeability and non-Darcy coefficient should be considered in calculation formula of tunnel water influx.