Experimental Research into the Evolution of Permeability of Sandstone under Triaxial Compression

: Failure tests on sandstone specimens were conducted under di ﬀ erent conﬁning pressures and seepage pressures by using an MTS triaxial rock testing machine to elucidate the corresponding correlations of permeability and characteristic stress with conﬁning pressure and pore pressure during deformation. The results indicate that permeability ﬁrst decreases and presents two trends, i.e., a V-shaped increase and an S-shaped trend during the non-linear deformation stage. The greater the seepage pressure, the greater the initial permeability and the more obvious the V-shaped trend in the permeability. As the conﬁning pressure was increased, the trend in the permeability gradually changed from V- to S-shaped. Compared with the case at a high conﬁning pressure, the decrease of permeability occurred more quickly, the rate of change becomes greater, and the sudden increase observed in the permeability happened earlier under lower conﬁning pressures. Within the range tested, conﬁning pressure exerted a greater e ﬀ ect on the permeability than the seepage pressure. In comparison with the axial strain, volumetric strain better reﬂected changes in permeability during compaction and dilation of sandstone. The ratio of crack initiation stress to peak strength ranged from 0.37 to 0.50, while the ratio of dilation stress to peak strength changed from 0.58 to 0.72. Permeabilities calculated based on Darcy and non-Darcy ﬂow changed within the same interval, while the change in permeability was di ﬀ erent. the ﬂow Some new the and volumetric of local and inﬂuences of volumetric strain on the permeability.


Introduction
Microcrack initiation, propagation, and coalescence in a rock mass under the coupled effects of seepage and stress caused by engineering excavation are key factors threatening the stability of such rock masses [1]. According to statistical evidence [2], 90% of the failures of rock slopes are related to groundwater seepage, and 60% of mine accidents are correlated with groundwater action. Previous research shows that these factors (including: rock type, microcrack development, pore connectivity, load regime (stress path), stress state, and porosity) can affect the permeability [3][4][5][6][7][8][9][10], while the specific influence and mechanism underpinning such behaviors remain unclear.
At present, in most seepage-stress coupling tests of a rock mass, the axial stress is increased under certain confining pressures and water pressures and changes of permeability during deformation are measured on the premise of Darcy flow. The research objects include coal and rock masses, sandstone, salt rock, granite, etc. Yu et al. [11] and Jiang et al. [12] considered that change of permeability are commonly determined by microfracture initiation and propagation in rock and crushing of skeleton

Test Materials and Equipment
An MTS electro-hydraulic servo triaxial rock testing machine was used in the test and the machining precision of the samples conformed to the test requirements of rock mechanics. Considering the complexity and inhomogeneity of rock properties, to ensure uniformity and comparability of specimens, all specimens were sampled from adjacent positions within the same rock block. An image of sandstone samples and the triaxial cell are shown in Figure 1. The test process was as follows: A. The rock specimen was subjected to vacuum and loaded in a pressure chamber after saturation for 48 h in water. B. An axial force of 1 kN was applied to force the pressure heads at each end of the testing machine into tight contact with the specimen and a confining pressure was applied to pre-set values (10,20, and 30 MPa). C. pore pressure P1 was applied to both ends of the specimen and then pore pressure on the lower end was reduced to P2, thus forming stable differences in pore pressure P = P1 − P2 (1, 4, and 7 MPa). D. Axial load was applied (under axial displacement control) at a velocity of 0.02 mm/min and the positions of the test points were pre-set. Loading was applied to the first test point and then stopped when forming a stable difference in seepage pressures across the specimen. In this case, the changes in pore pressure difference over time were recorded. The other test points were measured (in increasing order of stress) until the onset of damage.

Failure Morphology
Typical failure morphologies of specimens under different confining pressures and a pore pressure of 4 MPa are shown in Figure 2 [33]. Mainly shear failure, shear and tensile failure, and a combination of local shear failure and compactive separately occur in such specimens under confining pressures of 10, 20, and 30 MPa. A combination of local shear failure and compactive the sample exhibits the combined characteristics of shear and compactive failure. By combining these with the data linking permeability, different local structures formed by triaxial compression and fracturing into fragments of size and sphericity may be seen under deformation as causing the different changes in permeability. The test process was as follows: A. The rock specimen was subjected to vacuum and loaded in a pressure chamber after saturation for 48 h in water. B. An axial force of 1 kN was applied to force the pressure heads at each end of the testing machine into tight contact with the specimen and a confining pressure was applied to pre-set values (10, 20, and 30 MPa). C. pore pressure P 1 was applied to both ends of the specimen and then pore pressure on the lower end was reduced to P 2 , thus forming stable differences in pore pressure P = P 1 − P 2 (1, 4, and 7 MPa). D. Axial load was applied (under axial displacement control) at a velocity of 0.02 mm/min and the positions of the test points were pre-set. Loading was applied to the first test point and then stopped when forming a stable difference in seepage pressures across the specimen. In this case, the changes in pore pressure difference over time were recorded. The other test points were measured (in increasing order of stress) until the onset of damage.

Failure Morphology
Typical failure morphologies of specimens under different confining pressures and a pore pressure of 4 MPa are shown in Figure 2 [33]. Mainly shear failure, shear and tensile failure, and a combination of local shear failure and compactive separately occur in such specimens under confining pressures of 10, 20, and 30 MPa. A combination of local shear failure and compactive the sample exhibits the combined characteristics of shear and compactive failure. By combining these with the data linking permeability, different local structures formed by triaxial compression and fracturing into fragments of size and sphericity may be seen under deformation as causing the different changes in permeability.  Figure 3 shows the deformation of sandstone specimens under conventional triaxial compression. As the confining pressure increases, the peak strength and corresponding peak strain increase, indicative of significant plasticity. Plastic flow occurs in specimens at a confining pressure of 30 MPa. Under confining pressures of 10, 20, and 30 MPa and seepage pressures of 1, 4, and 7 MPa, stressstrain curves of sandstone during deformation are as illustrated in Figure 4. In Table 1, initial permeabilities and peak strengths of sandstone specimens under different confining pressures and seepage pressures are listed. Under the same seepage pressure, the peak strength increases with confining pressure. When the confining pressure is fixed, the peak strength decreases with increasing seepage pressure. The greater the seepage pressure, the greater the decrease in peak strength. When the confining pressure increases, the inhibitory effects of pore pressure on peak strength diminishes and the difference between peak strengths of rock specimens under different seepage pressures decrease.

Stress-Strain Relationship
At a confining pressure of 30 MPa and seepage pressures of 1 and 4 MPa, the stress drops seen at confining pressures of 10 and 20 MPa do not occur; instead, the stress remains unchanged and plastic flow occurs. Strain softening does begin until the pore pressure is increased to 7 MPa. This indicates that, due to compaction under a high confining pressure, the low pore pressure exerts little influence on the strength of the rock, while only a high pore pressure affects the strength of the rock.  Figure 3 shows the deformation of sandstone specimens under conventional triaxial compression. As the confining pressure increases, the peak strength and corresponding peak strain increase, indicative of significant plasticity. Plastic flow occurs in specimens at a confining pressure of 30 MPa.  Figure 3 shows the deformation of sandstone specimens under conventional triaxial compression. As the confining pressure increases, the peak strength and corresponding peak strain increase, indicative of significant plasticity. Plastic flow occurs in specimens at a confining pressure of 30 MPa. Under confining pressures of 10, 20, and 30 MPa and seepage pressures of 1, 4, and 7 MPa, stressstrain curves of sandstone during deformation are as illustrated in Figure 4. In Table 1, initial permeabilities and peak strengths of sandstone specimens under different confining pressures and seepage pressures are listed. Under the same seepage pressure, the peak strength increases with confining pressure. When the confining pressure is fixed, the peak strength decreases with increasing seepage pressure. The greater the seepage pressure, the greater the decrease in peak strength. When the confining pressure increases, the inhibitory effects of pore pressure on peak strength diminishes and the difference between peak strengths of rock specimens under different seepage pressures decrease.

Stress-Strain Relationship
At a confining pressure of 30 MPa and seepage pressures of 1 and 4 MPa, the stress drops seen at confining pressures of 10 and 20 MPa do not occur; instead, the stress remains unchanged and plastic flow occurs. Strain softening does begin until the pore pressure is increased to 7 MPa. This indicates that, due to compaction under a high confining pressure, the low pore pressure exerts little influence on the strength of the rock, while only a high pore pressure affects the strength of the rock. Under confining pressures of 10, 20, and 30 MPa and seepage pressures of 1, 4, and 7 MPa, stress-strain curves of sandstone during deformation are as illustrated in Figure 4. In Table 1, initial permeabilities and peak strengths of sandstone specimens under different confining pressures and seepage pressures are listed. Under the same seepage pressure, the peak strength increases with confining pressure. When the confining pressure is fixed, the peak strength decreases with increasing seepage pressure. The greater the seepage pressure, the greater the decrease in peak strength. When the confining pressure increases, the inhibitory effects of pore pressure on peak strength diminishes and the difference between peak strengths of rock specimens under different seepage pressures decrease.    At a confining pressure of 30 MPa and seepage pressures of 1 and 4 MPa, the stress drops seen at confining pressures of 10 and 20 MPa do not occur; instead, the stress remains unchanged and plastic flow occurs. Strain softening does begin until the pore pressure is increased to 7 MPa. This indicates that, due to compaction under a high confining pressure, the low pore pressure exerts little influence on the strength of the rock, while only a high pore pressure affects the strength of the rock.

Relationships of Characteristic Stress with Confining Pressure and Seepage Pressure
Characteristic stress is a parameter reflecting the stress state and triaxial strength of rock and includes compaction stress, crack-initiation stress, dilatancy stress, and peak strength. In the process of rock deformation, after the elastic phase is over, the new crack begins to expand, and the corresponding stress is defined as the crack initiation stress. Dilatancy stress refers to the stress under which the volumetric strain changes from contraction to expansion [34,35]. Based on the crack volumetric strain method [34,35], the crack initiation stress σ ci and dilatancy stress σ cd of sandstone were determined. Confining pressure borne by the rock skeleton under the influence of a positive pore water pressure is described in the form of effective confining pressure as follows: where, σ 3 and ∆P represent the confining pressure in the test and the seepage pressure, respectively. The characteristic stresses on sandstone under different effective confining pressures are listed in Table 2 and the relationship between characteristic stress and effective confining pressure is demonstrated in Figures 5 and 6.

Failure Criterion
The Mogi-Coulomb strength criterion takes the effect of intermediate principal stress on rock strength into account [36,37]: where, a and b are constants, and their relationships with parameters of the Mohr-Coulomb criterion are as follows: According to Formula (2), the data in Table 2 are fitted, as shown in Figure 7, thus obtaining the angle of internal friction of 31.2° and a cohesion of 16.4 MPa.

Failure Criterion
The Mogi-Coulomb strength criterion takes the effect of intermediate principal stress on rock strength into account [36,37]: where, a and b are constants, and their relationships with parameters of the Mohr-Coulomb criterion are as follows: According to Formula (2), the data in Table 2 are fitted, as shown in Figure 7, thus obtaining the angle of internal friction of 31.2° and a cohesion of 16.4 MPa.

Failure Criterion
The Mogi-Coulomb strength criterion takes the effect of intermediate principal stress on rock strength into account [36,37]: where, a and b are constants, and their relationships with parameters of the Mohr-Coulomb criterion are as follows: According to Formula (2), the data in Table 2 are fitted, as shown in Figure 7, thus obtaining the angle of internal friction of 31.

Relationship between Permeability and Axial Strain
The relationship between permeability, volumetric strain and axial strain of sandstone under different confining pressures and seepage pressures are shown in Figures 8 and 9. The formula for calculating the permeability under Darcy flows is [11]: where k (m 2 ) is the permeability of samples (1 darcy = 10 −12 m 2 ); Q (m 3 s −1 ) is the flow rate through the samples per unit time; μ is the dynamic viscosity coefficient of water; ΔP tands for the pressure difference between the two end-faces of the sample; L (m) and A (m 2 ) denote the height and crosssection of the sample, respectively.
In the initial deformation stage, micropores and meso-throats in sandstone are compacted and the permeability decreases. In the non-linear deformation stage, microcracks propagate in the rock and new microfractures are initiated: moreover, the permeability follows two trends: one is V-shaped, that is, it first reduces, then increases rapidly, while the other is S-shaped, such that it slowly decreases and then remains unchanged. Under a confining pressure of 10 MPa, the permeability exhibits a V-shaped trend. At a confining pressure of 20 MPa, the V-shaped trend weakens and magnitude of changes in permeability decreases. At a confining pressure of 30 MPa, the permeability follows an S-shaped trend. Under the same confining pressure, the greater the seepage pressure, the greater the initial permeability and that under the same deformation. Moreover, the V-shaped trend becomes more obvious. Under the same seepage pressure, the permeability decreases with increasing confining pressure. Furthermore, the permeability decreases when the rock suffers damage and the overall magnitude of any change in permeability decreases and the V-shaped trend gradually changes to an S-shaped one.
In comparison with the case at a high confining pressure, under a low confining pressure, the decrease in permeability is faster and its subsequent increase occurs earlier. In addition, the rate of change of permeability is greater; however, the permeability gradually changes under higher confining pressures.

Relationship between Permeability and Axial Strain
The relationship between permeability, volumetric strain and axial strain of sandstone under different confining pressures and seepage pressures are shown in Figures 8 and 9. The formula for calculating the permeability under Darcy flows is [11]: where k (m 2 ) is the permeability of samples (1 darcy = 10 −12 m 2 ); Q (m 3 s −1 ) is the flow rate through the samples per unit time; µ is the dynamic viscosity coefficient of water; ∆P tands for the pressure difference between the two end-faces of the sample; L (m) and A (m 2 ) denote the height and cross-section of the sample, respectively. In the initial deformation stage, micropores and meso-throats in sandstone are compacted and the permeability decreases. In the non-linear deformation stage, microcracks propagate in the rock and new microfractures are initiated: moreover, the permeability follows two trends: one is V-shaped, that is, it first reduces, then increases rapidly, while the other is S-shaped, such that it slowly decreases and then remains unchanged. Under a confining pressure of 10 MPa, the permeability exhibits a V-shaped trend. At a confining pressure of 20 MPa, the V-shaped trend weakens and magnitude of changes in permeability decreases. At a confining pressure of 30 MPa, the permeability follows an S-shaped trend. Under the same confining pressure, the greater the seepage pressure, the greater the initial permeability and that under the same deformation. Moreover, the V-shaped trend becomes more obvious. Under the same seepage pressure, the permeability decreases with increasing confining pressure. Furthermore, the permeability decreases when the rock suffers damage and the overall magnitude of any change in permeability decreases and the V-shaped trend gradually changes to an S-shaped one.
In comparison with the case at a high confining pressure, under a low confining pressure, the decrease in permeability is faster and its subsequent increase occurs earlier. In addition, the rate of change of permeability is greater; however, the permeability gradually changes under higher confining pressures.

Relationship between Permeability and Volumetric Strain
The relationship between permeability and volumetric strain of sandstone under different confining pressures and seepage pressures are shown in Figures 10 and 11. Volumetric strain is the sum of the axial strain ε 1 and circumferential strain ε 3 , expressed as: Compaction and dilation processes in sandstone are related to closure and propagation of internal microcracks and volumetric strain can directly reflect these two characteristics. In the compaction stage, the specimen volume and permeability decrease. After axial force reaches a certain value, many microcracks are generated in the samples. In that case, compaction-induced deformation becomes dilatant and the specimen volume increases. In this case, if the confining pressure is low and the rock is brittle, microfractures constantly coalesce and expand. Under positive pore water pressure, the internal fractures propagate and coalesce faster and the volume expands rapidly, so that the permeability increases. If the confining pressure on, and ductility of, the rock are high, skeleton particles therein are crushed and pores while microfractures coalesce, which slows the expansion and inhibits any increase of permeability. Therefore, volumetric strain can better reveal changes in permeability during compaction and dilation of the samples. In comparison with Figures 8-11, volumetric strain can more clearly reflect the effects of confining pressure and pore pressure on permeability compared with axial strain.
Under different confining pressures and seepage pressures, the permeability changes near the inflection point in the volumetric strain. In the compaction stage, the permeability first decreases, then stabilizes with reducing volumetric strain. Adjacent to the inflection point, the permeability undergoes a slow increase. Upon dilation, the permeability increases rapidly with decreasing volumetric strain. Compared with the compaction stage, the permeability during dilation is more sensitive to the decrease in volumetric strain. That is, the permeability decreases in the volume-compaction stage while increases in the volume-dilation stage.
As the confining pressure increases, the permeability tends to decrease. When the pore pressure increases, or when the confining pressure decreases, the inflection point in the volumetric strain appears earlier. With increasing seepage pressure, the permeability differs slightly from that pertaining before the inflection point of volumetric strain, but differs to a greater extent thereafter.
increases. If the confining pressure on, and ductility of, the rock are high, skeleton particles therein are crushed and pores while microfractures coalesce, which slows the expansion and inhibits any increase of permeability. Therefore, volumetric strain can better reveal changes in permeability during compaction and dilation of the samples. In comparison with Figures 8-11, volumetric strain can more clearly reflect the effects of confining pressure and pore pressure on permeability compared with axial strain.  Under different confining pressures and seepage pressures, the permeability changes near the inflection point in the volumetric strain. In the compaction stage, the permeability first decreases, then stabilizes with reducing volumetric strain. Adjacent to the inflection point, the permeability undergoes

Calculation of Non-Darcy Seepage
Based on Forchheimer's equation for non-Darcy seepage, a one-dimensional equation may be obtained by neglecting effects of volumetric body forces [38]: where, p, C α , V, and t represent the density of liquid, the acceleration coefficient of fluid, the flow rate of fluid, and time, respectively; ∂p ∂x , µ, and β denote the pressure gradient, coefficient of viscosity, and the seepage factor, respectively.
In the test process, the axial deformation stabilizes for a certain period after reaching a certain value, until the seepage reaches a steady state where: Assuming a uniform pressure gradient in the fluid under steady seepage, their values were determined by differences in seepage pressures and the height ratio of the specimens.
The data were collected at a frequency of 2 Hz. While keeping the piston displacement constant, the time series pore pressure data were collected, thus obtaining a group of stable values of pressure gradients from plotted data. The seepage velocity can be expressed as follows: where, V, V 1 , D, and d denote the seepage velocity, the relative velocity of the axial piston to the cylinder block, the diameter of the piston, and the diameter of the cylinder, respectively. The seepage velocities V 1 , V 2 , . . . ,·V n at different times can be acquired based on piston displacement. By fitting seepage velocities and pressure gradients, the following formulae were obtained [38]: Through use of the above formulae, permeability and seepage factor under non-Darcy flow were acquired, where i and m indicate the points on the time series and n represents the total number of series points.

Changes in Permeability during Non-Darcy Seepage
At a confining pressure of 10 MPa and different seepage pressures, the relationship between pore water pressure and time during non-Darcy seepage in sandstone is as shown in Figure 12 and the relationship between permeability and axial strain is demonstrated in Figure 13.  At a pore pressure of 1 MPa, the fitting equation for the relationship between pore pressure and time is given by: P = 2 × 10 −7 t 2 − 9 × 10 −4 t + 0.9981 (11) At an interval of 10 s, the pore pressure gradient is expressed as follows: According to Formula (7), the seepage velocity is given by: Formula (13) indicates that the pore pressure gradient under non-Darcy seepage shows a second-order non-linear relationship with seepage velocity and the seepage is unstable. Under Darcy seepage, the pore pressure gradient has a linear relationship to the seepage velocity, indicating that permeability evolution under non-Darcy flow is completely different from that under Darcy flow.
Under different seepage pressures, permeabilities were calculated based on Darcy and non-Darcy flow change in the same interval, with an order of magnitude ranging from 10 −15 m 2 to 10 −14 m 2 , while the change itself differs greatly. The permeability calculated based on Darcy flow shows a non-linear increasing trend, while that calculated on the basis of non-Darcy flow develops unsteadily, with a more obvious non-linear trend and multiple increase and decreases therein.

Explanation of Certain Behavioral Phenomena
The reason for the V-shaped trend in permeability is that microcracks in the rock develop and coalesce into macrocracks and open fissures are thus developed so that water can rapidly flow through the fissures, thus rapidly increasing the permeability.
The S-shaped change in permeability is mainly attributed to the fact that a high confining pressure leads to plastic flow in sandstone and inhibits initiation and coalescence of new cracks, so that the rock samples tend to undergo dilation. In addition, the local rock skeleton can be crushed into many fine particles, some of these are moved during formation of the shear zone to block the original seepage channels, resulting in a quasi-constant permeability.
Within the test range, confining pressure and pore pressure have different degrees of influence on permeability in each stage of deformation of sandstone and the effects of confining pressure on permeability are greater than those of seepage pressure, which can be explained by Biot's principle of effective stress. In the initial stage of deformation, Biot's coefficient is small in sandstone specimens. When the confining pressure is fixed, as the pore pressure increases, a small pore pressure exerts a limited influence on the effective stress and correspondingly the change in permeability is small. At a constant seepage pressure, when increasing the confining pressure, the initial effective confining pressure increases. The degree of compaction of microcracks and open pores is greater, and both the number and width of seepage channels decrease. Moreover, Biot's coefficient and the permeability decrease. During non-linear deformation, many microcracks are generated and some closed pores are connected to form open pores. Biot's coefficient increases, and the high pore pressure reduces the effective confining pressure and accelerates crack generation and propagation, thus increasing the permeability. In this stage, confining pressure and pore pressure significantly affect the permeability, and as the confining pressure is much greater than the seepage pressure, the confining pressure exerts the greater effect on the permeability.

Existing Problems
At a confining pressure of 30 MPa, sandstone undergoes a brittle-ductile transition and the permeability remains unchanged with increasing volumetric strain, which is contrary to the conventional understanding of the positive correlation between permeability and volumetric strain. It remains unclear whether this is caused by the calculation error in volumetric strain. The circumferential deformation in different parts of the samples is extremely uneven. In the test, the chain for testing the circumferential strain is placed in the middle of the samples, so the measured circumferential strain is relatively large. According to experience, the final value of the circumferential strain is taken as half of the measured value, which possibly leads to error in the calculation of volumetric strain. The effect of calculation error in the volumetric strain on the final processing of the test results warrants further investigation.
At present, there is no effective method with which to characterize the dissipation of pore pressure gradient, therefore, by referring to Forchheimer's equation for non-Darcy seepage, the seepage characteristics of sandstone were studied and permeabilities of sandstone specimens calculated based on Darcy flow and non-Darcy flow differ significantly. The determination of whether the fluid is in a Darcy or non-Darcy flow state during seepage is a key factor affecting the calculation accuracy of seepage in rock, because different seepage states need to be described using different equations, but there is no uniform criterion yet available to judge the state.
The characteristics of non-Darcy flow can be described using the Forchheimer equation; however, it is very difficult to consider all influencing factors in the equation, so the Forchheimer equation is generally fitted using test data, followed by determination of the coefficient [39,40].

1.
The permeability first decreases and exhibits two distinct trends: a V-shaped increase and a S-shaped plateau in the non-linear stage. The greater the seepage pressure, the more obvious the V-shaped trend in permeability. With increasing confining pressure, the variation in the permeability gradually changes from V-shaped to S-shaped.

2.
Compared with the case at a high confining pressure, under low confining pressure, the increase in permeability appears earlier, and the permeability rises faster. Moreover, the decrease in permeability happens faster and the time at which it increases again is earlier. The confining pressure exerts a greater effect on the permeability than the seepage pressure.

3.
In comparison with the axial strain, volumetric strain better reflects any changes in permeability during compaction and dilation of the specimens. In the compaction stage, the permeability first decreases, then stabilizes with decreasing volumetric strain. Upon dilation of the specimen, the permeability increases rapidly with increasing volumetric strain.

4.
The crack initiation stress, dilation stress, and peak stress all increase with the effective confining pressure, while decreasing with increasing seepage pressure. Failure of sandstone specimens under hydro-mechanical coupling is governed by the Mogi-Coulomb strength criterion.

5.
Permeabilities calculated based on Darcy flow and non-Darcy flow vary within the same range, while the changes therein differ significantly in terms of process: under non-Darcy flow, dissipation of pore pressure gradient and changes in permeability are significantly non-linear.