Deformation behavior and damage-induced permeability evolution of sandy mudstone under triaxial stress

The mechanic and permeability behavior in sandy mudstone is crucial to hazard prevention and safety mining. In this study, to investigate the evolution and characteristic of permeability and mechanical properties of sandy mudstone subjected to loading of in-site stress, a series of triaxial compression–seepage experiments are performed. The increase in permeability and decrease in mechanical strength gradually transformed to the decrease in permeability and increase in mechanical strength responding to the increase in confining stress from 5 to 15 MPa, which corresponds to the transformation from brittleness to ductility in sandy mudstone, and the transformation threshold of 10 MPa confining stress was determined. The penetration shear fracture generated at brittle regime, while plastic flow behavior presented at semibrittle and ductile state. The critical value of the yielding stage in axial strain increases with the increase in confining stress. The relatively higher permeability corresponds to the higher pore pressure during the increase in confining stress. The increase in confining stress promoted the increase in volumetric strain, while increased pore pressure reduced the volumetric strain, and the lower permeability occurred at higher volumetric strain. In addition, an improved permeability model was developed to describe the loading-based permeability behavior considering the Klinkenberg effect.


Introduction
The mechanical behaviors and seepage properties of sandy mudstone have been widely studied in the field of radioactive waste disposal, shale gas exploration, cap-rock behavior of hydrocarbon reservoirs, carbon geo-sequestration, underground coal gasification, and coal mining (Gale et al. 2007;Fu et al. 2015;Rezaeyan et al. 2015;Wilson et al. 2021). With increasing depth, the transitional deformation behavior from brittleness to ductility and complex transport characteristics occurred in the rock (Bishop 1967;Renner et al. 2000;Nygård et al. 2006;Zhang et al. 2019;Wu et al. 2021). At the macroscopic scale, brittle rock is characterized by a macroscopic fracture with much localized deformation, whereas ductile rock is distinguished by uniformly distributed deformation (Evans et al. 1990). Furthermore, the brittle-ductile transition deformation behavior can be obtained with a mixed microscopic transfiguration mechanism (Evans et al. 1990), while the permeability change varies with the mining activity. An abundant work focused on this field, and outstanding findings were reported.
Existing studies in this area mainly focused on the destruction properties, which includes the bearing capacity and fracturing patterns (Petley 1999;Renner et al. 2000;Boulin et al. 2013;Rezaeyan et al. 2015;Wu et al. 2021). Meanwhile, the stress depended brittle or ductile rock was also a hot issue, and it was evaluated using triaxial compression strengths and residual strengths (Rutter 1986;Wang et al. 2019), as well as Young's modulus and Poisson's ratios during the linear elastic deformation stage (Rickman et al. 2008;Bai and Wierzbicki 2010;Memon et al. 2020), of rocks under various stress situations. The two key parameters of internal friction angle and cohesion derived using the Mohr-Coulomb failure criterion may be used to define it for brittle rocks (Hucka and Das 1974;Singh et al. 2011;Nooraiepour et al. 2017). The smaller disparity between the maximum strength and residual strength, or the lower Young's modulus and internal friction angle of rocks, on the other hand, indicates a higher degree of ductility (Evans et al. 1990;Brantut et al. 2011;Wu et al. 2021).
The seepage property influences the mechanical behavior in mudstone. The permeability evolution of mudstone described by different deformation modes, such as elastic deformation, brittle failure, and ductile deformation, is extensively investigated (Djeran-Maigre et al. 2000;Zhang and Rothfuchs 2008;Zhang 2016;Wu et al. 2021). The permeability is observed to drop fast during the early loading stage and later rise, and the permeability turning point associates well with the alteration in volumetric strain known as dilatancy . Therefore, the permeability of mudstone in brittle regime increases considerably as dilatancy and failure develop, while the permeability of ductile mudstone keeps constant at the strain hardening stage Alkan 2009). Additionally, the dynamic response of permeability is reported by the pattern of localized or distributed deformation determined by the effective stress (De Paola et al. 2009). Wu et al. (2021) conducted a series of experiments, composed of the conventional triaxial compression tests, the hydrostatic loading-unloading tests, and the triaxial loading-unloading tests, on mudstone to study the permeability evolution at various stress conditions. Zhang et al. (2021) characterized the permeability variation of the fractured sandy mudstone by considering the impact of hydro-mechanical coupling, fracture morphology, and mineral composition. In addition, the permeability models are usually developed on different empirical functions, such as exponential functions (Seidle et al. 1992;Li et al. 2013;Tan et al. 2019), cubic functions (Gangi 1978;Kwon et al. 2001), and logarithmic functions (Kranzz et al. 1979;Walsh 1981). To predict the coal permeability at the yielding and post-failure stage, Xue et al. (2015) developed a permeability model considering the damage process. To define the variation in coal permeability when effective stress changes from the elastic deformation stage to the post-peak stage, Chen et al. (2016) introduced an effective stressdependent exponential permeability model. To investigate the permeability behavior with stress varies throughout the unloading process, Zhang et al. (2017) put forward an analytical permeability model that includes the influence of damage evolution.
As mentioned above, most of the existing experiments are concentrated on rock mechanical properties especially failure modes, and the established permeability models are usually used to predict the mining-induced permeability alteration in the resource reservoir. However, little has been done for sandy mudstone mechanical behavior and permeability evolution under the effect of elastic and plastic deformation. Moreover, the change in sandy mudstone permeability and mechanical properties caused by mining activity is complex; it is of great significance to investigate the mechanical behavior and transport properties of sandy mudstone subjected to the change in in-site stress. Therefore, in this study, the triaxial compression-seepage experiments were performed, and the relationship of damage behavior and seepage characteristics and the influence of deformation on permeability response at brittle and ductile states were analyzed. An improved permeability model was developed, and the permeability evolution of mudstone was discussed.

Sample preparation
The dark grayish sandy mudstone block used in this study was collected from a coal mine in the Ordos Basin, which has bedding and clastic texture, showing foliated fabric described by light and dark bands. Some cylindrical samples, with a diameter of 50 mm, were drilled parallel to the bedding planes of the sandy mudstone block by a core drilling machine. Then nine cylinders were shaped to Φ 50 × 100-mm cylindrical samples by a grinding device with a high-precision standard. The X-ray diffraction (XRD) results show that the sandy mudstone mainly consists of clay minerals (24.3%), quartz (20.7%), plagioclase (15%), ankerite (15%), calcite (11.3%), and other minerals (13.7%). The averaged density and porosity of the sandy mudstone are 1.99 g/cm 3 and 8.1%, respectively.

Experimental apparatus
The triaxial seepage tests were conducted using a hydromechanical coupling triaxial permeation apparatus (Fig. 1). A servo-hydraulic double pump booster was utilized to control the confining pressure up to 70 MPa with an accuracy of 0.01 MPa, and a servo-hydraulic booster was applied to ensure the axial force up to the maximum force of 1000 kN. A gas pressure reducing valve and two different gas mass flow controllers were employed to regulate the gas pressure and flow. The axial strain and the radial strain were measured by a linear variable differential transducer (LVDT) and an extensometer chain, respectively. The apparatus can provide an axial and confining stress acting on the samples with a stable temperature during measuring permeability in this work.

Experimental procedure
In this work, N 2 with a purity of 99.999% was chosen as the testing gas, and different confining stresses (5 MPa, 10 MPa, and 15 MPa) and gas pressures (2 MPa, 3 MPa, and 4 MPa) were considered in the experimental tests. Axial stress was applied at a rate of 0.1 mm/min using a displacement control, and all the experimental tests were carried out at 25 °C. Moreover, the sample information and experimental conditions used in each test are listed in Table 1. The specific test procedure is described below: (1) The sandy mudstone samples were firstly dried at a constant temperature of 50 ℃ for 48 h. And then the dried sandy mudstone sample was sealed with the base pedestal and top cap by the heat shrinkable sleeve. Subsequently, the sample was placed into the triaxial pressure cell. (2) After completing the installation of the triaxial cell, the air compressor was opened to discharge air and fill the triaxial cell with silicone oil. And then, the axial stress and confining stress were set to predefined values. (3) Afterward, the outlet valve was closed and the vacuum pump was turned on to vacuum degassing the sample for 2 h. And then, N 2 with specified pressure was injected to the lower end of the sample and finally flowed to the outlet via a stainless steel pipe which is connected to the gas mass flow controller. Finally, the axial stress controlled by displacement control mode was loaded to the higher end of the sample until destruction.
Considering Darcy's law, the permeability of sandy mudstone was obtained (Somerton et al. 1975): where k is the permeability (m 2 ); q is the gas permeation rate (m 3 /s); P 0 is the standard atmospheric pressure (Pa); μ is the gas kinematic viscosity (Pa s); L is the length of the rock sample (m); A is the cross-sectional area of the rock sample (m 2 ); P 1 is the inlet or upper stream gas pressure (Pa); P 2 is the outlet or downstream gas pressure (Pa).

Mechanical behavior and permeability evolution
The bearing capacity and transport property of rock are prominently determined by its mineral grains composition and pore fracture system. With the change of stress condition, Triaxial mechanicalseepage coupling experimental apparatus the response of deformation and permeability behavior keeps a dynamic process. The pore fracture structure and texture of rock are the main reasons for the difference in initial permeability. The deformation behavior, both of brittle and ductile rock, can be divided into five typical stages (Fig. 2), and the most difference between brittle and ductile rock is at the fifth stage, termed strain softening stage and strain hardening stage, respectively. Correspondingly, this distinction is related to different deformation mechanisms, further leading to the variable permeability behaviors.

The impact of confining stress
The relationships of stress-strain-permeability of sandy mudstone are shown in Fig. 3 and the volumetric strain-axial strain behaviors are shown in Fig. 4. In this work, the deviatoric stress as the differentials in axial and confining stress is employed to describe the axial compression effect on the experimental sample. And the volume strain method is used to characterize mechanic evolution in sandy mudstone. In detail, the starting and ending points of the linear section in the axial stress-volumetric strain curve were determined as the critical threshold of Stages I and II in the experimental samples based on the volume strain method (Brace et al. 1966). The sandy mudstone performs brittle, brittle-ductile transition, and ductile under the studied confining stresses. The deformation behavior of sandy mudstone characterized brittle deformation under a low confining pressure of 5 MPa, and its deformation behavior consists of five classical stages: the compaction stage (I), the elastic deformation stage (II), the yielding stage consisting of the stable cracks growth and the unstable cracking (III and IV), and the post-destruction stage (V) (Fig. 3a-c), which leads to different corresponding permeability evolution (Fig. 2a). At the beginning of Stage I, the total volume and permeability decrease fastly with the deviatoric stress increasing due to the pre-existing microcracks closed continuously. Once reach Stage II, only restricted microcracks generate so the permeability decreases slowly without increase. And the elastic modulus can be altered as primary cracks close continuously in this stage. In Stages III and IV, the fractures start to grow and coalesce to each other, while the permeability did not increase, but continued to decrease gradually which presumably because the macrofracture is not formed and the cracks are still compressed. In these stages, it leads to plastic strain which is followed by a volume increase (Fig. 4a) or dilatancy (Fig. 4b, c) due to microfracturing, and reduced bearing capacity and elastic modulus of sandy mudstone. In Stage V, the macroscopic fracture which cut through the end of specimen formed by the coalescence of grain boundary and intragranular cracks, also called cataclastic flow behavior (Evans et al. 1990), results in the permeability behavior of sudden increase and then a gradual increase (Paterson and Wong 2005). After the sample failure, the permeability normally remains constant or slightly undulates, which is presumably because of the constriction and blocking by dislodging particles of the formerly established gas flow paths (Hangx et al. 2010). Furthermore, as shown in Fig. 3a-c, the permeability evolution after the sample failure does not increase a lot, and the incremental permeability is even lower than the original permeability for sandy mudstone. The existence of high content of clay minerals blocks the flow in seepage channels composed of cracks and macro-fractures. Under higher confining pressure of 10-15 MPa, the deformation behavior of sandy mudstone shows brittle-ductile transition and ductile deformation characteristics, and its deformation characteristic is also composed of five stages (Fig. 2b). In Stages I and II, the permeability decreases because the pre-existing microcracks closed continuously, which is similar to brittle deformation characteristics. Despite the presence of microcracks in Stages III and IV, the fractures are prevented and the dislocation mobility is enhanced. Hence, Fig. 3 Relationship for stress-strain-permeability of sandy mudstone in triaxial compression and permeability tests. Δσ e and Δσ y are the critical deviatoric stress threshold of sandy mudstone at Stages I and II, respectively. Δσ p is the peak deviatoric stress the permeability gradually decreases. Note that the volumetric strain curve of brittle-ductile transition sandy mudstone approximatively keeps symmetric parabola distribution characteristics before the dilatancy stage ( Fig. 4d-f). In Stage V, with the deviatoric stress increasing, the shear bands generate, nucleate, and develop to form a shear macro-fracture at a high angle with the compacted stress for brittle-ductile transition sandy mudstone and it does not penetrate the two end surfaces. For ductile sandy mudstone, strain localization can occur, involving mainly cracking, in which case the rock accumulates distributed microcracks without any macroscopic rupture. Therefore, the brittle-ductile and ductile deformation of sandy mudstone at normal temperature is mainly attributed to the plastic flow deformation mechanism (Evans et al. 1990). Different from brittle sandy mudstone, the permeability of brittle-ductile transition sandy mudstone keeps stabilized or decreases a little with the formation of high-angle shear fractures. In contrast with the ultimate permeability evolution of damaged brittle-ductile transition sandy mudstone, the permeability of ductile sandy mudstone has a slightly reduced trend at the strain hardening process because of the plastic flow deformation. In general, the permeability decreases as an exponential pattern under the whole compaction process for both brittle-ductile transition and ductile sandy mudstone.
The comparison of stress-strain and permeability-strain behavior of sandy mudstone under different confining pressures is presented in Fig. 5. As shown in Fig. 6, when the confining pressure is small, the rock is brittle and prone to shear failure. With the confining pressure increasing, the test results gradually show the characteristics of weakening brittleness, increasing ductility, and transformation from brittleness to ductility. In this study, as the gas pressure is constant, the sandy mudstones transform from strain softening at the post-peak stage to strain hardening state with confining pressure increasing (Fig. 5). In light of Cao et al. (2005) and Yu et al. (2013), the confining pressure Fig. 5 Differential stress, permeability, and volumetric strain data are plotted versus axial strain for sandy mudstone in different gas pressures triaxial compression and permeability tests (a−c is gas pressure of 2 MPa, 3 MPa, and 4 MPa, respectively) threshold of brittle-ductile transition critical state is finally determined as 10 MPa for studied sandy mudstone.

The impact of pore pressure
The comparison of stress-strain and permeability-strain behavior of sandy mudstone under different gas pressures is shown in Fig. 7. When the confining pressure is small and constant (σ 3 = 5 MPa), the deformation and permeability evolution were similar and the increased permeability of sample S1, S2, and S3 at the destruction stage are 2.89, 3.12, and 4.63% with the increase in pore pressure, respectively, indicating that enlarging the gas pressure can increase the seepage channel at the post-peak stage. Different from the brittle sandy mudstone, the permeability of both brittle-ductile transition and ductile sandy mudstone at Stage V maintains constant or decreases a little, mainly because they did not develop shear fracture penetrating the top and bottom surfaces, even the pore pressure is increased to reduce the effective stress. Overall, the alteration of gas pressure has little effect on sandy mudstone with different deformation regimes.

Failure modes of sandy mudstone
The failure characteristics of sandy mudstone are presented in Fig. 8, and the corresponding SEM results are shown in Fig. 9. The samples S1, S2, and S3 (brittle sandy mudstone) formed intuitionistic macroscopic shear fractures without cohesion after destruction. The semibrittle sandy mudstone (at brittle-ductile transition) developed a high-angle shear macro-fracture with a few cohesive and it does not pierce the top and bottom end surfaces, while the ductile sandy mudstone showed fully plastic flow deformation, which undergoes homogeneous large strains without marcofracturing. At the microscopic scale, the brittle regimes of sandy mudstone are mainly composed of cataclastic flow, which involves microfracturing (either intragranular or intergranular) and frictional sliding, and is distinctly different from the initial state. There is a mixed behavior (cataclastic and plastic flow) that can be observed for brittle-ductile transition sandy mudstone in Fig. 9c. Comparing the microscopic observation of brittle sandy mudstone, the micro-processes of ductile sandy

Model development
As the internal microstructure of sandy mudstone consists of cracks and matrices, the sandy mudstone can be presumed to be the matchstick geometry. As a result, the classical exponential permeability model for sandy mudstone at elastic state can be presented as (Seidle et al. 1992): where k i is the permeability of sandy mudstone, m 2 ; k 0 is the initial permeability of sandy mudstone, m 2 ; C f is the compression coefficient of fractures, MPa −1 ; σ e is the effective stress, MPa; and the σ e0 is the initial effective stress, MPa. Klinkenberg (1941) pointed out that the gas slippage phenomenon will be significant as the flow path size in fracture closes to the mean free path of the gas molecules at a lower gas pressure, which will further affect the permeability. So, by considering the Klinkenberg effect, the exponential permeability equation can be represented as: In this study, the Boit's coefficient is considered to remain constant at 1, and the effective stress in deviatoric stress loading can be expressed as (Chen et al. 2016): where P eff is the effective confining stress which equals to the confining stress σ 3 minus the pore pressure P, MPa; Δσ is the deviatoric stress, MPa.
According to the experimental results, the permeability decreases continuously in the yielding stage, and it remains unchanged or slightly decreases at strain hardening process for ductile and brittle-ductile transition sandy mudstone with increasing of the effective stress, so the exponential permeability model (Eq. 3) can depict the variation process of permeability well in all stages. However, for brittle and some semibrittle sandy mudstone, the permeability shows a large increase with the decrease in effective stress in the postpeak stage. Hence, the exponential permeability model could not describe the change process of permeability in the post-peak stages for brittle and some semibrittle sandy mudstone (see Fig. 10).
To better describe permeability behavior under the overall triaxial compression process for brittle and some semibrittle sandy mudstone, a correction factor β is imported to enhance the prediction accuracy of permeability. The correction factor β can be introduced as the modified coefficient for the difference between the predicted permeability and the (4) e = 3P eff + Δ 3 test permeability, corresponding to the effective stress. Furthermore, the peak point of the effective stress is regarded as a distinguishing point for brittle and some semibrittle sandy mudstone due to the permeability before and after failure varies dramatically, which is defined as σ s . Based on the thinking of symmetry (see the symmetrical permeability curve represented by the red dotted line in Fig. 10), the improved piecewise permeability model before and after σ s , combining the correction factor β, can be updated as: The relationship between the correction factor β and the deviatoric effective stress can be acquired using Eq. 3 and experimental permeability data. Then, the correction function of β can be obtained by fitting the results. Figure 11 illustrates the relationship between the modification factor β and the effective stress after symmetry for studied brittle and some semibrittle sandy mudstone. And the correction function of β can be gained by fitting, which is expressed as Eq. 6.
where a and b are the fitting parameters, and the analogous results of fitting parameters are used at the residual stress stage.

Model validation
To verify the revised permeability model, the analytical permeabilities acquired using Eqs. 3, 5, and 6 are compared with the test results for the sandy mudstone in triaxial compression tests (see Fig. 12). And the corresponding parameters applied in the improved models are summarized in Tables 2 and 3. In this work, the testing gas is N 2 , which belongs to non-adsorbing gas, so Klinkenberg constant B and fracture compressibility C f can be obtained by fitting the permeability data with respect to both confining pressure and gas pressure (Pan et al. 2015;Tan et al. 2019).
As listed in Table 3, the fitted parameter a for correction factor β is in the range of 0.042-0.079 with a mean value of 0.059, and the fitted parameter b is in the range of Fig. 10 Illustration of permeability-effective stress curve for brittle and some semibrittle sandy mudstone, and the evolution of permeability and effective stress based on symmetry 0.106-0.569 with a mean value of 0.293, indicating that the average relative errors between the experimental data and model prediction for sandy mudstone decrease linearly as the effective stress decreases at residual stress stage, and can be estimated by 0.059 (2 σ eσ s ) + 0.293. As shown in Fig. 12, the permeability increases slightly with the effective stress further decreasing at the residual stress stage for brittle sandy mudstone (Fig. 12a-c). Figure 12d, e presents that the permeability keeps constant or decreases a little after peak effective stress for brittle-ductile transition sandy mudstone. For ductile sandy mudstone, the permeability decreases continuously with the effective stress increases (Fig. 12g-i). For brittle and semibrittle sandy mudstone, the results can be modeled well, except Sample S2 (Fig. 12b), using Eqs. 5 and 6. The results of Sample S4 after peak effective stress are not modeled because Sample S4 does not reach the residual stress stage. Equation 3 is Fig. 11 Relationship between modified factor and effective stress after symmetry at σ e = σ s for a Sample S1, b Sample S2, c Sample S3, d Sample S5, and e Sample S6 used for ductile sandy mudstone in triaxial compression tests, and it matches well between the experimental data and model ( Fig. 12g-i). Notably, the mentioned permeability model focuses on characterizing the permeability evolution of different deformation regimes of sandy mudstone with the effective stress varying from elastic deformation to failure, and some empirical parameters are used in the model. Therefore, this model is an exploration for clay rock and necessitates further validation and improvement for better presentation and broader applications in future studies. ductile state in sandy mudstone was determined. The permeability of sandy mudstone gradually decreases in the pre-peak compression stage. The slight permeability increase presents in brittle sandy mudstone; even a decrease in final permeability is observed at brittle state, resulting from the limitation of clay minerals on flow in seepage channels created by the cracks generated in the yield stage and the post-peak macro-fracture. The permeability usually keeps constant or slight decreases for brittle-ductile transition and ductile sandy mudstone during the strain softening process. 2. The different failure modes present responding to the change of in-site stress. The macroscopic shear fracture penetrates the whole sample and the cataclastic flow regime was generated in brittle sandy mudstone, while the large-inclination shear region, characterized by microscopic cataclastic and plastic flow, was developed in semibrittle sandy mudstone. And homogeneous large strains characterized by plastic flow at the strain hardening stage were presented in ductile sandy mudstone. 3. The improved permeability model incorporating the correction factor β is developed based on the classical exponential permeability model and considering the slippage effects. The correction factor β related linearly to the effective stress can well reflect the difference between the measured permeability and the permeability obtained from the improved model. And the damage-based permeability of brittle and ductile sandy mudstones is well described.