Percolation Threshold of Red-Bed Soft Rock during Damage and Destruction

Abstract

The critical damage point of the red-bed soft rock percolation phenomenon can be described as the percolation threshold. At present, there are insufficient theoretical and experimental studies on the percolation phenomenon and threshold of red-bed soft rock. In combination with theoretical analysis, compression experiment and numerical simulation, the percolation threshold and destruction of red-bed soft rock are studied in this paper. The theoretical percolation threshold of red-bed soft rock was obtained by constructing a renormalization group model of soft rock. Based on damage mechanics theory, rock damage characterization and strain equivalent hypothesis, a constitutive model of red-bed soft rock percolation damage was obtained. The percolation threshold of red-bed soft rock was determined by compression test and a damage constitutive model, which verified the rationality of the theoretical percolation threshold, and we numerically simulated the percolation of red-bed soft rock under triaxial compression. The results showed that the percolation threshold increases as the confining pressure rises, but decreases significantly with the action of water. In this study, the critical failure conditions and percolation characteristics of red-bed soft rock under different conditions were obtained. The relationship between percolation and soft rock failure was revealed, providing a new direction for studying the unstable failure of red-bed soft rock.

1. Introduction

Percolation theory is one of the commonly used theoretical methods to deal with systems with strong disorder and random geometric structure. The concept of percolation was first introduced by S.R. Broadbent and J.M. Hammersley in 1957 to describe the flow of fluid in disordered porous media [1]. Percolation refers to the process in which a medium outside the system enters the system through a certain path in a unitary or multivariate system. It is a widespread physical phenomenon, which exists in both the micro and macro worlds. For example, liquids can pass through disordered media through diffusion and percolation processes. Percolation theory has been applied to other problems in different fields. People have discovered simultaneous common features, that is, when a certain density, occupation number, concentration and proportion gradually change (increase or decrease) to a certain value, there will be a certain change in the macro properties of organisms, that is, the so-called emergence or disappearance of long-range connectivity. This sudden change of long-range connectivity is called percolation transition, and the equivalent density, occupation, concentration and proportion of percolation transition are called critical probability (critical damage point), also known as the percolation threshold. The process of generation, expansion, clustering and penetration of micro cracks caused by rock fracture is actually a process of sudden generation of long-range connectivity [2].

Red beds, widely distributed from the early Proterozoic to the Neogene, are particularly notable for their red color. Red beds are also widely distributed in space and have been found on seven continents and in four oceans [3,4]. Red-bed soft rock is widely distributed and relatively complete, hard and good in its mechanical properties when under natural conditions (relative to the excavation environment) [5,6]. However, in an engineering excavation, the red-bed soft rock rapidly expands, disintegrates and softens in a short time after coming into contact with water, and its mechanical properties are greatly reduced, which leads to engineering disasters. When red-bed soft rock meets water, it forms a complex three-directional stress state under the action of water-stress coupling. In a short time, the rock rapidly expands, softens, disintegrates, and internal damage develops continuously until, finally, the rock suddenly breaks. This is typical of the percolation phenomenon, and the critical point of damage of red-bed soft rock can be described as the percolation threshold. Red-bed soft rock is a common type of soft rock with special properties in engineering construction [5,6,7,8]. After project excavation, the red-bed soft rock, under the action of water and water-stress coupling, will form the complex state of three-directional stress. Under the action of these forces, damage will start gradually from the inside of the rock before developing and expanding. When the damage reaches a certain critical value, a large number of cracks will occur and the rock will gradually reach a state of unstable failure [9,10,11]. When the amount of damage is less than the critical threshold, the soft rock is in a stable state. When the amount of damage reaches the critical threshold, its state changes and leads to destruction.

From the perspective of percolation, the sudden change in mechanical properties is a percolation phenomenon [12,13,14,15]. According to percolation theory, the critical point of damage and destruction of red-bed soft rock can be described as the percolation threshold. Under different stress states, red-bed soft rock has different percolation characteristics and percolation thresholds. Therefore, the influence of different stress states on percolation should be taken into account when studying the percolation of red-bed soft rock during its damage and destruction. However, theoretical and experimental research on the percolation phenomenon and percolation threshold has been insufficient, especially in relation to soft rocks.

The study of percolation theory with regard to rock damage has typically taken two approaches: one is to study percolation phenomenon by calculating the percolation threshold; the other is to study the percolation characteristics of physical quantities of rocks [1,16,17,18]. The percolation threshold for rock damage can be calculated by the theoretical method. Among these methods, the renormalization group method is an effective method that has clear physical meaning, and can be used not only to calculate the theoretical percolation threshold but also to analyze the nature of its destruction. Therefore, this method is widely used [19,20,21,22,23,24]. Percolation characteristics can be studied through the characteristics of energy damage. The process of rock damage and destruction is monitored in experiments that study changes in external radiant energy during the whole process of destruction. It is generally believed that energy which radiates from inside the rock can represent the damage to the rock. When percolation occurs during rock damage, the radiative energy apparently increases [25,26,27,28,29,30,31,32,33]. However, existing research and percolation models have mainly focused on brittle rocks, which often differ in their properties from red-bed soft rocks. Furthermore, the experimental determination of percolation thresholds has yet to mature, and the factors that influence the percolation threshold of rocks require further study.

At the same time, the effective application of percolation theory to rock damage should be based on two aspects, namely, the constitutive and the quantitative description of the damage process. Most damage constitutive studies combine different assumptions to construct different damage constitutive structures, so as to obtain the damage evolution model [34,35,36,37,38,39,40]. In quantitative descriptive research of damage in rocks, various techniques are used, but there is no unified method for describing damage [41,42,43,44,45,46,47]. An effective quantitative method for describing damage in the percolation of red-bed soft rock is urgently needed.

In conclusion, while there have been some achievements in the study of percolation and rock damage, some problems remain: (1) research into rock percolation has been insufficient and research into damage criticality based on the rock percolation needs to be built; (2) most studies of percolation phenomena and models have been based on brittle rocks, but few studies have been undertaken on the percolation of soft rocks; (3) effective quantitative description methods of the damage process need to be studied under laboratory compression test conditions.

To address the above problems, this paper employs the renormalization group theory to study the percolation of red-bed soft rock, including the method of calculating its theoretical percolation threshold. The constitutive model of percolation damage of red-bed soft rock is derived and, combined with the indoor uniaxial compression acoustic emission (AE) test, the percolation phenomenon in soft rock is studied and its threshold calculated [23]. The percolation phenomenon and threshold under different stress conditions—such as uniaxial compression, conventional triaxial compression and water-stress coupling triaxial compression—are studied by numerical simulation. The influences of confining pressure, water and other factors affecting the percolation threshold are analyzed. Effective quantitative description of damage was carried out in the experiment. Combined with theory and simulation, the critical failure conditions and the percolation characteristics of red-bed soft rock under different states are obtained. At the same time, the relationship between the percolation and soft rock failure is obtained, which can generate new directions for studying the instability and failure of red-bed soft rock.

2. Research Contents and Methods

2.1. Renormalization Model of Percolation Threshold of Red-Bed Soft Rock Damage and Destruction

Renormalization models are based on self-similarity. Similar skeleton particles and the filler between skeleton particles can be found in the rock from microscopic, mesoscopic and macroscopic levels. Brittle rocks with higher strength generally have stronger fillings between skeletons. In a renormalized group model, the filler can be considered part of the skeleton, so the model can be constructed in the form shown in Figure 1a. In Figure 1a, the small cube with scale 1 is a basic unit, and four units of scale 1, forms a basic unit of scale 2. In scale 3, the four basic units of scale 2 constitute a basic unit of scale 3. When a scale 1 unit is damaged in some way and satisfies the transformation criterion for destruction, it is also damaged under observation in scale 2, as shown in Figure 1b (white blocks are non-damaged and black blocks are damaged). The elements that cannot reach the transformation criterion for destruction are not destroyed under observation in scale 2, as shown in Figure 1c.

Compared with brittle rocks, the cementite strength between the skeleton particles of red-bed soft rocks is low, so the cementing member and the skeleton particles should be considered separately in the renormalization group model. Take silty mudstone, a type of red-bed soft rock, as an example: its microstructure is shown in Figure 2a. Silty particles (spherical in the figure) and clay minerals (rod-shaped in the figure) are seen in the microstructure. Silty particles act as the framework while the clay minerals act as the cementing substance in the structure. Due to the special structure of soft rock, renormalized model units can be established, as shown in Figure 2b. The skeleton particles are connected by a cementing substance, and the strength of the skeleton particles is higher than that of the cementing substance. In the process of change at each level of scale, the material acting as skeleton and the cementing substance can always be found in red-bed soft rock. The failure of the basic unit at the higher levels of the scale only affects the cementation component of the basic unit at the current scale.

As can be seen, the basic units at different scales can be combined to establish the soft rock renormalization model, as shown in Figure 2c. The renormalization model at each level is the same, that is, there is a model unit between each cementing substance and the connection mode between the skeleton particles and the cementing substance is cube shaped. The self-similarity of soft rock shows that the unit models at different scales are the same, and properties in all directions are the same; that is, the failure patterns are the same. When the soft rock is damaged, the microscopic cementation component damages and accumulates to cause mesoscopic cementite damage, and the mesoscopic cementation component damages and accumulates to form macroscopic damage.

To simplify the calculation, the following assumptions were made of the renormalization model of soft rock: (1) the renormalized model is isotropic, (2) at the same scale, the failure probability of the cementing component is equal.

The rules of renormalization transformation can be determined according to the failure of rocks under load compression, as shown in Figure 3.

The transformation rules are as follows: (1) the cementation component of each basic unit can be divided into three groups as shown in Figure 3a; (2) each group has four cementation components and when three of these components are damaged, the group is deemed to be damaged; (3) when one group of cementation components in a basic unit is damaged, that unit is deemed to be damaged, as shown in Figure 3b,c.

2.2. Constitutive Model of Percolation in Rock Damage

AE counts were used to characterize the damage of rock under external load. Assuming that the damaged amount of rock caused by external load is D_b when percolation occurs, the cumulative amount of AE reaches Ω₀ [25]. If the cumulative count of AE at a certain time is Ω, then the variable for rock damaged by the process (characterized by AE count) can be expressed as follows:

$$D^{\prime }=D_b\frac{\Omega }{{\Omega }_0}$$

(1)

In addition to the damage caused by external load, the rock has initial damage at the time of formation. The damage from percolation is severance or disconnection of connection, so porosity can be used to roughly characterize the disconnection of cementing material at the initial state of soft rock. The initial damage and the damage during the process are added together to obtain the overall damage to the rock when percolation occurs under external load, namely:

$$D=D_a+D^{\prime }=D_a+D_b\frac{\Omega }{{\Omega }_0}$$

(2)

where D_a is the variable for initial damage. The percolation threshold, then, in the process of rock damage is:

$$D_p=D_a+D_b$$

(3)

In the triaxial compression test, confining pressure was first added to the rock sample, and then axial pressure was added to carry out the compression test. The stress state is shown in Figure 4.

According to the principle of elastic mechanics, the stress–strain relationship after added confining pressure becomes:

$$\left\{\begin{array}{c}{\sigma }_1^0=\left(\lambda +2G\right){\epsilon }_1^0+\lambda {\epsilon }_2^0+\lambda {\epsilon }_3^0\\ {\sigma }_2^0=\lambda {\epsilon }_1^0+\left(\lambda +2G\right){\epsilon }_2^0+\lambda {\epsilon }_3^0\\ {\sigma }_3^0=\lambda {\epsilon }_1^0+\lambda {\epsilon }_2^0+\left(\lambda +2G\right){\epsilon }_3^0\end{array}\right.$$

(4)

where the stress in the initial state (a) is ${\sigma }_1^0={\sigma }_2^0={\sigma }_3^0$. As $\lambda =\frac{\nu E}{\left(1+\nu \right)\left(1-2\nu \right)}$ and $G=\frac{E}{2\left(1+\nu \right)}$, the strain in state (a) can be obtained as:

$${\epsilon }_1^0={\epsilon }_2^0={\epsilon }_3^0=\frac{{\sigma }_3^0\left(1-2\nu \right)}{E}$$

(5)

After adding axial pressure, the stress–strain relationship in state of (b) is as follows.

$$\left\{\begin{array}{c}{\sigma }_1=\left(\lambda +2G\right){\epsilon }_1+\lambda {\epsilon }_2+\lambda {\epsilon }_3\\ {\sigma }_2=\lambda {\epsilon }_1+\left(\lambda +2G\right){\epsilon }_2+\lambda {\epsilon }_3\\ {\sigma }_3=\lambda {\epsilon }_1+\lambda {\epsilon }_2+\left(\lambda +2G\right){\epsilon }_3\end{array}\right.$$

(6)

From the above formula, it can be deduced that:

$$\left\{\begin{array}{c}{\sigma }_1=\left(\lambda +2G\right){\epsilon }_1+2\lambda {\epsilon }_3\\ {\sigma }_3=\lambda {\epsilon }_1+2\left(\lambda +G\right){\epsilon }_3\end{array}\right.$$

(7)

$${\sigma }_1=\frac{\lambda }{\left(\lambda +G\right)}{\sigma }_3+\left(\lambda +2G-\frac{{\lambda }^2}{\lambda +G}\right){\epsilon }_1$$

(8)

Substituting λ and G into the above equation, we obtain:

$${\sigma }_1=2\nu {\sigma }_3+E{\epsilon }_1$$

(9)

Because ${\sigma }_1={\sigma }_3+\Delta {\sigma }_1$, ${\epsilon }_1={\epsilon }_3^0+\Delta {\epsilon }_1$ and ${\sigma }_2^0={\sigma }_3^0={\sigma }_2={\sigma }_3$, then $\Delta {\sigma }_1=E\Delta {\epsilon }_1$ or

$${\sigma }_1-{\sigma }_3=E\left({\epsilon }_1-{\epsilon }_3^0\right)$$

(10)

Following confined pressure loading, the axial strain is zeroed, so ${\epsilon }_3^0=\frac{{\sigma }_3^0\left(1-2\nu \right)}{E}$ is cleared and the triaxial compression is recorded as:

$${\sigma }_1-{\sigma }_3=E{\epsilon }_1$$

(11)

Then, according to the strain equivalence hypothesis, $\epsilon =\frac{\tilde{\sigma }}{E}=\frac{\sigma }{\left(1-D\right)E}$, then

$${\epsilon }_1=\frac{{\tilde{\sigma }}_1-{\tilde{\sigma }}_3}{E}=\frac{{\sigma }_1-{\sigma }_3}{E\left(1-D\right)}$$

(12)

That is,

$${\sigma }_1-{\sigma }_3=E{\epsilon }_1\left(1-D\right)$$

(13)

By substituting the amount of damage in the percolation process into the above equation, we obtain:

$${\sigma }_1-{\sigma }_3=E{\epsilon }_1\left(1-D_a+D_b\frac{\Omega }{{\Omega }_0}\right)$$

(14)

The elastic modulus in the above formula is the elastic modulus in the lossless state. However, the elastic modulus measured by the stress–strain curve obtained from the indoor compression test entailed initial damage. Therefore, let the elastic modulus containing the initial damage be $\tilde{E}$, so that

$$\tilde{E}=E\left(1-D_a\right),$$

(15)

where D_a is the initial damage.

Therefore, the constitutive model of percolation damage under triaxial compression is:

$${\sigma }_1-{\sigma }_3=E{\epsilon }_1\left(1-D\right)=E{\epsilon }_1\left[1-D_a-D_b\frac{\Omega }{{\Omega }_0}\right]=\tilde{E}{\epsilon }_1\left[1-\frac{D_b}{1-D_a}\frac{\Omega }{{\Omega }_0}\right]$$

(16)

By applying the same method, we can find the constitutive model of percolation damage under uniaxial compression:

$$\sigma =E\epsilon \left(1-D\right)=E\epsilon \left[1-D_a-D_b\frac{\Omega }{{\Omega }_0}\right]=\tilde{E}\epsilon \left[1-\frac{D_b}{1-D_a}\frac{\Omega }{{\Omega }_0}\right]$$

(17)

Using the constitutive model of percolation damage, the percolation threshold of red-bed soft rock can be obtained experimentally. The method is as follows:

(1) conduct compression test with rock samples of red-bed soft rock and record the stress–strain curve and AE data during the compression test; (2) determine the cumulative AE number Ω₀ at the time of red-bed soft rock percolation; (3) use porosity as the initial damage D_a of soft rock. Then, the amount damaged by the percolation process (D_b) is adjusted to make the compressive strength equal, and the constitutive model curve for percolation damage is fitted to the stress–strain curve to obtain the amount damaged by the percolation process (D_b). The percolation threshold of the red-bed soft rock sample is then D_P = D_a + D_b.

2.3. Percolation Test for Compression Damage of Red-Bed Soft Rock

2.3.1. Uniaxial Compression AE Test

The rock sample used is typical of red-bed soft rock: silty mudstone, purplish-red in color, argillaceous and calcareous in cementation, as shown in Figure 2a. In general, the clay minerals of such soft rocks are mainly illite and kaolinite and the XRD chart is shown in Figure 2d. The samples were prepared into cylinders with a diameter φ of 50 mm and a height H of 100 mm. There were three samples in total, as shown in Figure 5a. The test was performed according to ASTM standards [48].

The microcomputer-controlled pressure testing machine was used as the loading system in the test. The testing machine can control the loading rate by computer, automatically calculate the stress value in the compression process, read the displacement data, and store it to the computer. The AE acquisition and analysis system used was the DS2 series full information AE signal analyzer, which can measure 16 channels simultaneously and reach a maximum sampling frequency of 10 MHz. A USB high-speed data transmission was adopted, with a maximum transmission speed of 384 Mbps. According to the requirements of the test, the number of AE events was recorded to characterize the degree of damage in the rock. Data was collected through a single channel with a frequency of 1 MHz and an amplifier of 40 dB. In order to make the sensor and the sample as close to each other as possible, a coating of special silicone grease coupling agent was inserted between the sample and the sensor, and a rubber band was used to tighten the sample and the sensor. The location of the sensor is shown in Figure 5c. The sampling threshold of the AE instrument was 100 mV. The starting mode of signal acquisition was set externally, primarily to synchronize the start time of data acquisition to the start time of loading. The loading process adopted a displacement control mode, and loaded at a speed of 0.04 mm/s.

2.3.2. Determination of Percolation Threshold in Uniaxial Compression Test of Red-Bed Soft Rock

Using AE data and the derivation outlined in Section 2.2, the stress–strain curve of the rock sample was obtained according to the percolation damage constitutive model. The percolation threshold of the rock sample was then determined. The specific steps are as follows:

(1) record the cumulative number of AE events to determine when AE events occur most frequently. Namely, record the cumulative value of the ringing count Ω₀ at the time when the ringing count is at its maximum during a burst period; (2) establish the damage model according to Equation (17); (3) ascertain the amount of damage during the percolation process (D_b) and adjust D_b so that the two compressive strengths are equal and the damage model is as close as possible to the test curve; (4) add the initial amount of damage (D_a) and the amount damaged in the process (D_b) to obtain total damage during percolation; that is, the percolation threshold D_p. When the percolation threshold of red-bed soft rock under the failure conditions for loads is D_p, the inner structural unit of the red-bed soft rock is destroyed by D_p × 100%, the perforation failure occurs.

2.4. Numerical Simulation of Percolation in Compression Test of Red-Bed Soft Rock

In this section, the uniaxial compression test is numerically simulated from aspects of the stress–strain curve, the damage characteristics and the percolation threshold, and by using real fracture process analysis (RFPA) based on the finite element principle. The characteristics of the simulation process include: (1) the heterogeneity of rock and randomness of damage distribution were considered and calculated by a statistical method; (2) it was assumed that the elastic modulus and compressive strength of the rock conformed with a certain statistical law, such as the Weibull distribution, and the elastic modulus and compressive strength were randomly selected for each unit according to the selected distribution in order to simulate the fracture process of the rock; (3) it was assumed that the number of units damaged is in direct proportion to the amount of AE, and the AE energy is in direct proportion to the compressive strength of the failure unit. In the process of rock failure, AE data will be generated when the unit is constantly destroyed. The simulation method meets the requirements of the model for damage for calculating the percolation threshold in Section 2.2.

The numerical simulation of the uniaxial compression percolation test of red-bed soft rock was carried out according to the following requirements. The model adopted a 50 mm × 100 mm model consistent with an experimental situation, the elastic modulus and compressive strength of experimental rock samples, a displacement loading control method, with a speed of 0.04 mm/s. The numerical simulation was carried out using the Mohr–Coulomb strength criterion and 20,000 computing units.

Based on numerical simulation results of uniaxial compression, the percolation characteristics of red-bed soft rock under triaxial compression were further studied by numerical simulation compression test, and the factors affecting the percolation threshold were analyzed. The numerical simulation included two parts: conventional numerical simulation of triaxial compression and numerical simulation of triaxial compression under water-stress coupling.

First, based on the conventional triaxial compression test, the experimental stress–strain curve of the rock sample was taken as the numerical simulation object, and numerical simulation was judged to be consistent with the compression test by realizing the coincidence of the stress–strain curve in the numerical test and the stress–strain curve in the compression test. Based on this, the AE data from the numerical simulation was obtained in order to calculate the percolation threshold under numerical simulated conditions. In the conventional triaxial compression test, the rock sample is separated from water, but a characteristic of red-bed soft rock is that it will disintegrate and its strength will decrease rapidly when it meets with water. The conventional triaxial compression test alone will therefore not adequately capture the percolation of red-bed soft rock.

Instead, it is necessary to consider the triaxial compression test under the action of water-stress coupling in order to better describe the percolation characteristics of red-bed soft rock. In the triaxial compression test under the action of water-stress coupling, silty mudstone samples were filled with water for 48 h and then directly inserted into the pressure chamber with water as the confining pressure for the loading test. The 48 h saturation and confining pressure environment formed the water-stress coupling condition. Under this condition, the stress–strain curve obtained is the water-stress coupling stress–strain curve.

3. Results and Discussion

3.1. Theoretical Determination of Percolation Threshold for Damage of Red-Bed Soft Rock

The renormalization transformation rule corresponds to the nonlinear iteration. According to the renormalization model of red-bed soft rock established in Section 2.1, the probability of failure of a basic unit can be calculated. A basic unit is composed of 8 skeleton members and 12 cementing members. Therefore, there are 13 cementing member failure situations, namely, 0, 1, 2, …, 12 cementing member failure cases, and a total of 2765 cases that can lead to unit failure. If the probability of damage for the first-order cementing member is set at P, then the probability of non-damage of the cementing member is 1 − P. Then, the failure probability for each unit given the 13 types of failure conditions is shown in

Table 1. Failure probability of a unit under the 13 types of failure conditions for cementing members.

Damaged Number of Cementing Members	Probability of Unit Failure	Damaged Number of Cementing Members	Probability of Unit Failure
0	0	7	792P7 (1 − P)5
1	0	8	495P8 (1 − P)4
2	0	9	220P9 (1 − P)3
3	12P3 (1 − P)9	10	66P10 (1 − P)2
4	99P4 (1 − P)8	11	12P11 (1 − P)
5	360P5 (1 − P)7	12	P12
6	708P6 (1 − P)6

.

According to Table 1, the probability of the first-level unit failing is:

$$\begin{gathered} P_1=12P^3{\left(1-P\right)}^9+99P^4{\left(1-P\right)}^8+360P^5{\left(1-P\right)}^7+216P^6{\left(1-P\right)}^6+792P^7{\left(1-P\right)}^5+495P^8{\left(1-P\right)}^4 \\ +220P^9{\left(1-P\right)}^3+66P^{10}{\left(1-P\right)}^2+12P^{11}\left(1-P\right)+P^{12} \\ =-27P^{12}+108P^{11}-144P^{10}+64P^9-27P^8+72P^7-48P^6-9P^4+12P^3 \end{gathered}$$

(18)

Under renormalization transformation, according to renormalization theory, the probability of the second level unit failing can be obtained as follows:

$$P_2=-27P_1{}^{12}+108P_1{}^{11}-144P_1{}^{10}+64P_1{}^9-27P_1{}^8+72P_1{}^7-48P_1{}^6-9P_1{}^4+12P_1{}^3$$

(19)

When renormalization reaches all the way to level n:

$$P_n=-27P_{n-1}^{12}+108P_{n-1}^{11}-144P_{n-1}^{10}+64P_{n-1}^9-27P_{n-1}^8+72P_{n-1}^7-48P_{n-1}^6-9P_{n-1}^4+12P_{n-1}^3$$

(20)

According to renormalization theory, when n is large enough (with a sufficient number of renormalization iterations), a fixed point will be reached which satisfies the following:

$$P_{n+1}=P_n .$$

(21)

Substituting Equation (20) into Equation (21), we obtain:

$$P_n=-27P_n^{12}+108P_n^{11}-144P_n^{10}+64P_n^9-27P_n^8+72P_n^7-48P_n^6-9P_n^4+12P_n^3$$

(22)

To calculate Equation (22), there are three solutions in the 0~1 range: 0, 0.3636 and 1, that is, there are three fixed points as shown in Figure 6.

The derivative with respect to P_n on the right side of Equation (22) can be obtained as follows:

$$g^{\prime }\left(P_n\right)=-324P_n^{11}+1188P_n^{10}-1440P_n^9+576P_n^8-216P_n^7+504P_n^6-288P_n^5-36P_n^3+36P_n^2.$$

(23)

By substituting the three solutions into Equation (23), the first derivative value of the transformation function at the fixed point can be obtained as $g^{\prime }\left(0\right)=0$, $g^{\prime }\left(0.3636\right)=2.241$ and $g^{\prime }\left(1\right)=0$. Since the critical point is the unstable fixed point among the fixed points, and the absolute value of the derivative of the transformation function at the critical point is greater than 1, the percolation threshold of red-bed soft rock under the failure conditions for loads is 0.3636. In other words, when under load conditions the inner structural unit of the red-bed soft rock is destroyed by 36.36%, the perforation failure occurs. At this point, the strength of the red-bed soft rock decreases sharply and disordered damage changes into ordered damage. As the abrupt change occurs in an overall stable state, rock failure occurs. At this point, percolation occurs.

3.2. Percolation Test Results of Compression Damage of Red-Bed Soft Rock

3.2.1. AE Test Results under Uniaxial Compression

Damage to the rock sample after compression is shown in Figure 5b. By extracting the ringing count in the AE of rock samples, the relationships between strain, ringing count, accumulative ringing count and stress were obtained, as shown in Figure 7. According to the ringing count curves in Figure 7a1–a3, there are almost no AE events in silty mudstone at the initial stage of the test, and only a small amount of AE events were scattered. With increasing strain, the number of AE events gradually increases. After a peak number of AE events, their number begins to decrease. Therefore, the damage characteristics of silty mudstone can be summarized into the following five stages:

(1)

Initial stage of damage: this stage roughly includes the compaction stage and the early elastic stage of the rock sample. At this stage, the amount of AE is zero or small, indicating that no damage or minor damage has occurred to the sample. The initial microfracture in the rock sample is squeezed by external force and closes gradually. Most of the energy exerted on the rock sample by external forces is converted into elastic potential energy, which is stored in the rock sample. Only a small amount of energy cannot be converted effectively and causes a small amount of damage, so most damage to the rock sample at this point is initial damage. The measured porosity value of soft rock is 0.084 and the permeability value of soft rock is 1.02 × 10⁻⁹ m/s.

(2)

Stable development of damage stage: this stage includes the middle and later stages of the elastic stage. During this period, the amount of AE increases and is relatively stable, indicating that under external force, the internal damage of the rock sample begins to occur and new microfractures appear. With the stable increase in microfractures, damage also develops steadily.

(3)

Sharp damage stage: this stage includes the yield stage. During this stage, the amount of AE increases rapidly, and the rise of the cumulative number curve of ringing count accelerates. This acceleration indicates that under the action of external force and the addition of new microfractures in the rock, the microfractures are partially connecting to form macroscopic fractures. However, the macroscopic fractures have yet to connect.

(4)

Peak of damage stage: this stage includes the stress softening stage. During this period, the number of AE reaches a peak, and the cumulative number curve for the ringing count rises in an almost straight line. This result indicates that under the action of external forces, more microfractures are formed in the rock, these microfractures are connecting to form macrofractures, and the macrofractures are also connecting with each other. This then leads to a sharp decline in the strength of the rock, and eventually its damage. Percolation then occurs.

(5)

Residual damage stage: this stage includes the post-destruction stage. At this stage, the AE ringing count will gradually fall, showing that some residual energy continues to cause fracture to the rock sample after its destruction, although this fracture is small.

The three rock samples can be divided into two types, according to the relation between stress, ringing count and strain in Figure 7b1–b3. First, when the rock sample reaches peak stress, the stress–strain curve drops rapidly, such as that seen for the No. 3 rock sample. The second type of rock sample did not show rapid decline in the stress–strain curve after reaching peak value. Its stress–strain curve dropped in one stage and then dropped again. The second decline was more rapid than the first, and the moment of rapid decline occurred shortly after the stress peak, as with samples No. 1 and No. 2. It can also be seen from Figure 7b1–b3 that when the amount of damage appears is at a small peak, the stress–strain curve falls once, and when the damage amount appears at maximum peak, the stress–strain curve falls the most. In other words, when damage occurs, the stress–strain curve of the rock sample declines; when the damage reaches the percolation threshold, the degree of decline is at its greatest.

3.2.2. Percolation Threshold for Uniaxial Compression Test of Red-Bed Soft Rock

Through the method outlined in Section 2.3.2, the curve of the stress–strain relationship can be obtained from the damage constitutive model and compared with the stress–strain relationship curve obtained from the test, as shown in Figure 8. Several characteristics of the curve predicted by the damage constitutive model can be seen in this figure:

(1)

Based on the AE phenomenon, the stress–strain relationship calculated by the damage constitutive model accurately simulates the stress–strain relationship of rock samples under uniaxial compression.

Figure 8. Comparison of stress–strain curves of test results and those predicted by the damage model.

Figure 8. Comparison of stress–strain curves of test results and those predicted by the damage model.

(2)

The curves based on laboratory test of rocks generally show a compaction stage during which there is not much AE signal. As the constitutive model of percolation damage is based on AE data, it cannot accurately simulate compaction.

(3)

In the laboratory test curve, the stress may fall, rise, and then fall again in the yield stage and strain softening stage, however, the calculated curve from the constitutive model of percolation damage cannot simulate this fluctuation in stress completely and accurately.

Using the established damage constitutive model and data from the AE test, the process of damage and percolation threshold in red-bed soft rock or silty mudstone can be obtained as shown in

Table 2. Damage and percolation threshold of rock samples.

Sample Number	Initial Damage Da	Process Damage Db	Percolation Threshold DP
1	0.084	0.320	0.404
2	0.084	0.250	0.334
3	0.084	0.350	0.434
Mean value	0.084	0.3067	0.3907

.

According to percolation theory, porosity can be used to roughly describe the initial degree of damage to the cement inside soft rock. Therefore, initial damage D_a of the rock is characterized by porosity. For the same batch of samples, the porosity is similar. Therefore, the same initial damage value can be used, that is, the measured porosity value of 0.084 for soft rock. The amount of damage in the process (D_b) was 0.25 minimum, 0.35 maximum and 0.3067 on average; the percolation threshold was 0.334 minimum, 0.434 maximum and 0.3907 on average. In Section 3.1, the percolation threshold of red-bed soft rock or silty mudstone calculated according to the theory of renormalization group was 0.3636, with an error of 6.9% compared with an error of 0.3907 here. The small error verifies the rationality of the theoretical percolation threshold and indicates that the renormalization group model is in good agreement with the experimental results.

3.3. Numerical Simulation of Percolation in Compression Test of Red-Bed Soft Rock

3.3.1. Numerical Simulation of Uniaxial Compression Percolation Test

The numerical simulation of uniaxial compression percolation test of red-bed soft rock was carried out according to the simulation method outlined in Section 2.4. Figure 9 compares the stress–strain curve from the numerical simulation and the stress–strain curve from indoor compression test. It can be seen that, on the whole, the two stress–strain curves are in good agreement. However, because there is no compaction stage in the numerical simulation, this stage was not well simulated.

In terms of damage characteristics, the numerical compression test can be divided into initial stage of damage, stage of stable development of damage, stage of sharp damage, stage of peak damage and stage of residual damage. The simulated position of maximum damage was in good agreement with that obtained from the indoor compression test. The damage characteristics during compression were simulated well by numerical experiments.

The percolation threshold results are shown in

Table 3. Comparison of percolation thresholds obtained by numerical simulation and by laboratory compression test.

Sample Number	Percolation Threshold from Laboratory Compression Test	Percolation Threshold from Numerical Compression Test	Error (%)
1	0.404	0.460	13.9
2	0.334	0.300	–10.2
3	0.434	0.430	–0.9
Mean value	0.391	0.397	1.5

. The percolation threshold from numerical compression test was also consistent with the percolation threshold from the laboratory compression test. Because there are individual differences between samples, the error in the No. 1 and No. 2 rock sample was relatively large at more than 10%, but the average error was only 1.5%. That is, there may have been a deviation in the calculated percolation threshold of single rock samples, but statistically speaking, the mean percolation threshold can be more accurately simulated.

3.3.2. Numerical Simulation of Triaxial Compression Percolation of Red-Bed Soft Rock

(1)

Conventional triaxial compression numerical test

Previous conventional triaxial compression tests of red-bed soft rock [49] have been numerically simulated to obtain its damage characteristics under different confining pressures, as shown in Figure 10.

It can be seen from Figure 10 that when the confining pressure is 0, it becomes a common uniaxial compression test and there are five stages of damage characteristics as described in Section 3.2.1. When the confining pressure is greater than 0, the five stages are not obvious. The initial stage of damage is obviously shortened, the stage of stable development of damage occupies almost the entire process, the stages of sharp damage and peak damage are not obvious, and the stage of residual damage is prolonged. The damage caused by an external force is more evenly distributed in all stages, and the amount of damage in each stage is not extensive.

Under the action of external force, the soft rock continues to suffer minor damage which accumulates and eventually leads to its failure. However, due to the existence of confining pressure, the rock’s compressive strength is enhanced and its sudden damage prevented, such that there is no obvious moment of sudden damage in the whole damage process. However, the damage still showed a trend of first rising and then falling, and the position of maximum extent of damage occurred after the stress peaks.

(2)

Triaxial compression numerical test under the action of water-stress coupling

The stress–strain curve of water-stress coupled triaxial compression test [50] was numerically simulated to obtain soft rock damage under different confining pressures, as shown in Figure 11.

As can be seen in Figure 11, the damage characteristics of red-bed soft rock under coupled water-stress conditions differ from those achieved by the conventional triaxial compression test. On the whole, the damage characteristics obtained by the water-stress coupled triaxial compression test are similar to those found under uniaxial compression. There are five stages of damage but the extent of damage at each stage was found to be lower than that under uniaxial compression.

Compared with the effects of the conventional triaxial compression test, the softening effect of water on soft rock was apparent when water is taken as the confining pressure and under the condition of 48 h saturation. Although the confining pressure increased the compressive strength of soft rock, it failed to provide sufficient support for soft rock to counteract the effects of water. There was still a sudden moment of rock damage.

3.4. Analysis of the Influencing Factors on Percolation Threshold

3.4.1. Confining Pressure

According to the theory outlined in Section 2.2 and Section 2.3.2 and the numerical simulation data in Section 3.3, the percolation threshold of red-bed soft rock under the conventional triaxial compression test can be calculated, as shown in

Table 4. Percolation threshold and failure strain under two kinds of triaxial compression.

Confining Pressure (MPa)	Conventional Triaxial Compression		Water-Stress Coupled Triaxial Compression
	Failure Strain	Percolation Threshold	Percolation Threshold
0	0.0081	0.320	0.250
1	0.0146	0.464	0.304
3	0.0153	0.504	0.350

. This table shows that when the confining pressure is 0, the percolation threshold is 0.32; when confining pressure rises to 1 MPa, the percolation threshold rises to 0.464 (an increase of 0.144); when the confining pressure rises to 3 MPa, the percolation threshold continues to rise to 0.504 (an increase of 0.04). When the confining pressure increases from 0 to 1mpa, the threshold increases by 45%. When the confining pressure increases from 1 MPa to 3 MPa, the threshold increases by only 8.6%. It can be seen that the percolation threshold increasing speed decreases with the confining pressure increase, as shown in Figure 12. When there is no confining pressure, the soft rock as a whole will be destroyed when one-third of the cement inside is damaged. However, when there is confining pressure and when one-third of the cement inside the soft rock is damaged, the strength from the confining pressure means that the soft rock is not destroyed but continues to be damaged. On the other hand, with increase in confining pressure, the strain at which soft rock failure occurs also increases, as shown in Table 4. This pattern indicates that the ductility of the soft rock improves; that is, the degree of damage can be greater and the percolation threshold increased.

3.4.2. Water

To determine the influence of water on the percolation threshold of red-bed soft rock, the percolation threshold under water-stress coupled triaxial compression was compared with that under conventional triaxial compression. It can be seen from Table 4 that the percolation threshold increases with increases in confining pressure, but the percolation threshold is lower than that under the conventional triaxial compression test. When the confining pressure was 0 (without the direct effect of confining water), the percolation threshold under uniaxial compression after 48 h of water saturation was approximately 20% lower than that of the sample that was unsaturated. When the confining pressure was 1 MPa (the confining water is in direct contact), the percolation threshold of soft rock under water-stress coupling triaxial compression after 48 h of water saturation was about 34% lower than that under traditional anhydrous triaxial compression. Similarly, when the confining pressure was 3 MPa, the percolation threshold decreased by about 31%, as shown in Figure 12.

It can be seen that the percolation threshold of red-bed soft rock decreased by about 20% after 48 h of water saturation alone. Under the joint action of 48 h of water saturation and direct contact with confining water pressure, the percolation threshold of red-bed soft rock can be decreased by 34%, which shows that water has a great influence on the percolation threshold of red-bed soft rock. This decrease in percolation threshold is mainly due to the following reasons: when red-bed soft rock has been saturated with water for 48 h, water enters the soft rock and reduces the strength of its cement. When water acts as a confining pressure, it directly acts to further erode and soften the soft rock, reducing its compressive strength as well as the percolation threshold at which damage occurs.

4. Conclusions

At present, the theory and experimental research of percolation in soft rock is insufficient. The experimental determination of the percolation threshold and effective quantitative description of damage in soft rock have yet to be fully developed. Further, the influencing factors on the percolation threshold of red-bed soft rock are not clear or understood. In view of this, percolation in the process of damage and destruction of red-bed soft rock was studied in this paper by taking red-bed soft rock as the research object of theoretical analysis, compression testing and numerical simulation. The damage characteristics of soft rock were analyzed, the percolation threshold of soft rock under different conditions calculated, and the essential law of percolation of red-bed soft rock revealed. The main conclusions are as follows:

• The feasibility of carrying out damage and destruction research on red-bed soft rock, based on percolation theory has been established. Based on microstructural analysis of typical red-bed soft rock, the model of the soft rock renormalization group can be constructed. The constitutive model of percolation damage and a method of calculating the percolation threshold of red-bed soft rock under different compression conditions is presented.
• According to percolation theory, red-bed soft rock damage is quantitatively determined using AE measurement. Through AE tests of uniaxial compression of red-bed soft rock, the compression process of red-bed soft rock is revealed as having five stages of damage: initial damage, stable development of damage, sharp damage, peak damage, and residual damage. According to the constitutive model of percolation damage established in this paper, the average percolation threshold of red-bed soft rock is first obtained experimentally, and the rationality of the theoretical percolation threshold obtained by renormalization group theory is then verified.
• The numerical simulation of percolation in red-bed soft rock as described in this paper has been executed well. Based on this, the percolation of red-bed soft rock under conventional triaxial compression and water-stress coupled triaxial compression were simulated. The influence of confining pressure and water on the percolation threshold of red-bed soft rock is then obtained. The results show that the percolation threshold increases with an increase in confining pressure, but decreases significantly after rock–water interaction.