In this paper, the Uniaxial and Triaxial Mechanics Instrument(ITEM:TAWA-2000), which is a hybrid loading equipment under maximum axial pressure 2 000 KN and maximum confining pressure 140 MPa for uniaxial compression test and triaxial compression test, was used to simulate the failure process. The rock samples cored at different directions with respect to the plane of initial fracture coming from the third section of Bashenjiqike Formation (K₁bs) Kela-2 area in Kuqa depression. Each sample was prepared by ISRM suggested method (ISRM, 2007) with diameter of 50 mm and length of 100 mm, with ends grounded to be flat to 0.01 mm and less than 0.2 mm, and the vertical deviation was less than 0.25° . Triaxial tests were carried out by multi-stage loading method, and the loading rate was set to 0.01 mm/s. In loading course the confining pressure was set to eleven stages with interval of 5 MPa:0 MPa, 5 MPa, 10 MPa, 15 MPa, 20 MPa, manually as the axial pressure increases where at all times axial loads exceed confining pressure by no more than one tenth of the rock UCS until peak stress reached^[46]. In this study, the mechanical parameters (e.g., σ _t, σ _c, τ _s, E, G, μ , C₀ and φ ) of low-permeability sandstone were acquired (Table 1, Table2).

Table 1

Table 1

Table 1 Experimental datasets from uniaxial compressive tests Sample No. Rock type σ c /MPa E /GPa G /GPa μ Failure mode K101-DC3 Mid-fine sandstone 34.0 3.45 0.058 Tension K101-DB3 Silty sandstone 43.55 5.03 2.32 0.082 Tension K103-DA3 Mid-fine sandstone 50.51 6.36 2.98 0.069 Tenso-shear K101-AA1 Calcareous siltstone 183.68 16.08 7.00 0.148 Tension K101-DA2 Calcareous siltstone 167.18 16.78 7.77 0.082 Tension K105 Siltstone 30.28 4.05 1.94 0.045 Tenso-shear

Table 1 Experimental datasets from uniaxial compressive tests

After a series of tests, rock strength and mechanical properties were documented in Table 1 and Table 2 by uniaxial compressive tests, triaxial compressive tests and splitting tests etc. Where σ _t is the tensile strength of intact rock, τ _s is the shear strength, σ _c is the uniaxial compressive strength, E is the Young’ s modulus or elastic modulus, G is the shear modulus, μ is the poisson’ s ratio, C₀ is the cohesive strength of rock, φ is the friction angle of rock. According to the obtained rock properties listed in Table 1, the failure mode are categorized by σ _c, σ _t, τ _s, E, G and μ as different anisotropy rocks^[47].It can be concluded that the calcareous siltstone has stronger brittleness than other sandstones, such as fine sandstone, siltstone, argillaceous sandstone, etc.Table 2 shows the variation of triaxial compressive strength and friction angle of the fine sandstone and siltstone with respect to different confining pressures. It should be noted that the maximum peak strength will be obtained when the confining load is close to real underground conditions(approximately 5 MPa).

Table 2

Table 2

Table 2 Experimental datasets from triaxial compressive tests Experiment Type Experiment Index Finestone Siltstone Density test Dry density ρ (g/cm3) 2.187 2.195 Uniaxial compression experiment Uniaxial stress σ c(MPa) 34.51 30.47 E(GPa) 2.06 4.39 μ 0.082 0.136 Triaxial compression experiment σ 3=5 MPa Axial stress at failure σ 1(MPa) 57.79 61.14 E(GPa) 5.85 3.35 σ 3=10 MPa μ 0.093 0.114 Axial stress at failure σ 1(MPa) 68.37 93.06 E(GPa) 5.94 11.76 σ 3=15 MPa μ 0.109 0.197 Axial stress at failure σ 1(MPa) 96.96 E(GPa) 8.91 μ Shear strength index calculation Regression calculating strength under zero confining pressure σ 0(MPa) 34.72 30.26 K in fitting formula 3.959 6.259 Cohesive strength C(MPa) 8.73 6.05 Friction angle φ (° ) 36.63 46.43

Table 2 Experimental datasets from triaxial compressive tests

Fig.1 compares the mechanical properties with failure mode of different sandstone in study area according to the above mentioned. Three failure modes were identied: Tension failure, shear failure and mixed failure (tenso-shear fracture). Accordingly the extensional fracture criteria and shear fracture criteria should be taken together to explain the failure behavior of intact rock masses^[33]. Generally, in numerical modeling of tectonic stress field, Griffith Criterion is utilized for the calculation of extensional fracture, and Coulumb-Mohr Criteria is preferred for orientation of shear fracture^{[48, 49]}. As a matter of fact, under condition of high confining pressure, high temperature and slow loading ratio, a kind of shear fracture with angle less than 45° oblique to the pressing direction will occur. On the contrary, under rapid loading ratio, dilated breaking will occur parallel to compression direction, which belongs to typical brittle failure mode^[50].

As depicted in Fig.3, variation of failure forms and combination show a transitional trend with various confining pressure. There are actually many reasons that explain differences between failure modes obtained for criterions such as stresses, cohesion, friction and mineralogy of rocks. Hence, cohesive strength, C₀ and friction angle, φ of siltstone and finestone were determined from linear portion of Mohr envelopes at θ =24.5° and θ =19.09° as presented in Table 2. Among the samples, there appeared a large number of fissures and fractures, consisting of types of tension, compression and shear, such as sample K101-DC3, K101-DA4 and K101-DA2 performed as tension-shear opening fractures, sample K103-DA3, K101-DB3 and K105 performed as shear fracture, with some tensile characteristics. Among them, fracture trace in sample K101-DB3 is characterized by a series of tensile micro-cracks to form and propagate into macroscopic shear zone with angle of rupture less than 45° , just coinciding with the Mohr-Coulomb theory. However, Jaeger and Hoek and Brown’ s works are of importance that they thought the failure criterion should be modi-ed to take into account the anisotropy of strength properties. Rupture shapes through triaxial compression tests in Fig.3c, such as sample L12-1, L12-2 and L12-3 respectively with confining pressure 5 MPa, 10 MPa and 15 MPa, and sample L25-1, L25-2and L25-3 respectively with confining pressure 5 MPa, 10 MPa and 20 MPa, etc., are obviously different from that of unaxial samples. As a result, it can be concluded that the fracture mechanical property is mostly influenced by state of stress and certain confining pressure, i.e., along with the increasing confining pressure, the shear action becomes stronger, and the angle of rupture increases(Fig.3c).

Fig.3

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	Fig.3 Results of different rocks after mechanical experiments with different confining pressure (a) Rock failure modes of uniaxial compressive tests. (b) Rock failure modes of triaxial compression tests. (c)Standard rock samples coming from drill cores used for texts

Generally, thedifferences between mechanical parameters or properties are mainly influenced by multiple factors including rock composition, confining pressure, pore pressure, temperature, fluid media, loading ratio, etc. For this study, predecessors in this field have already done a lot of interesting researches, and achieved fruitful results. On the basis of the above experiments, we can change the next experiment scheme as follows: To select core samples coming from Kela-2 gas field in Kuqa Depression with similar lithology and mechanical properties, including some initial microcracks or weakness planes to simulate the failure process. To acquire complete stress-strain curves and σ _c(peak strength) in strength tests under uniaxial stress, the CT MPR and microscopic thin section observations were carried out to record the fracture evolution process at different loading stage, i.e.σ _c 10% σ _c 20%, σ _c 30%, ……, σ _c 80%, σ _c 95% with steady rate(0.01 mm/s). After loading, rock samples were taken out and consolidated with glue, either to be scanned on equipment— CD-600BX or observed on level thin section and vertical thin section under a high-powered microscope(Fig.4). Finally, based on a series of stress-strain curves, the strain energy density and fractures density of low-permeability sandstone were calculated and described in detail to set up certain relationship between stress and fracture parameters.

Fig.4

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Fig.4 Evolution diagram of microcracks in low-permeability sandstone under uniaxial cyclic compression tests (a) Thin slices of microcracks and macrocracks obtained parallel to the cross-section of rock samples under plane polarized light.(b)Overview of microcracks statistics by slice A, B, C in low-permeability sandstone after compression test.(c) Inner cracks are finely scanned with CT scanner (precision less than 10um) at different loading stages(0, 0.5 σ c, 0.65 σ c, 0.85 σ c, σ c and > σ c) and finally 3D-crack images are reconstructed

Based on mechanical tests of medium grain marble, granite and aplite, Brace et al.^[51]divided the brittle deformation course into four stages: Compaction, elastic deformation, dilatation and failure. According to axial and radial strain curves, at first, most microcracks in rocks propagate along weaknesses spots sequentially, when reach the region of one-third and two-thirds of uniaxial compression strength, the rocks dilate obviously and produce a large number of cracks. Therefore, the macroscopic behavior of rock is generally controlled by number of microcracks and rock properties.

Recently, a lot of arguments for cracking mode under tensile stress appeared, some thought that shear fractures only formed under deep triaxial stress conditions, and extensional fractures only form under shallow tensile stress conditions. Through many experiments, Dai et al.^[52] and Omid et al.^[46] discussed the possible conditions for shear fractures with tensile stress existing, which indicated that tension criterion should be preferred to determine whether extensional fracture occurred firstly. If no fracture took place, then Mohr-Coulomb criterion would be used to determine whether shear fracture occurred, was known as a kind of combined tension-shear failure criterion. Firstly, an enveloping curve was extracted from a series of Mohr circles on base of experimental results, then two-step Mohr-Coulomb curve is drew out^[53], as shown in Fig.5.

Fig.5

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	Fig.5 Mohr enveloping curve of low-permeability sandstone (a) and two-step Mohr-Coulomb curve in study area(b) extracted from triaxial compression tests

Here, φ ₁, φ ₂ respectively is internal frictional angle under different confining pressure, k₁, k₂ respectively is the straight slope in curve, and σ ₀=5 MPa is cut-off value of confining pressure. It can be concluded as follows:Fig.5 implies the strong relationship between confining pressure and failure modes, i.e. when the confining pressure is less than a value, φ is quite large, and θ =45° -φ /2 is relatively small, extensional fracture is popular. When confining pressure is relatively large, failure mode will transform to shear fracture or compresso-shear fracture. The detailed relative equations as follows:

If 0< σ ₃≤ σ ₀=5 MPa,

φ =φ ₁=51.83° (angle of rupture)

θ =θ ₁=45° -φ /2=19.09° (1)

If σ ₃> σ ₀=5 MPa, φ =φ ₂=41° (angle of rupture)

θ =θ ₂=45° -φ /2=24.5° (2)

Thus, it is not difficult to find that under compression tress state, the two-step Mohr-Coulomb criterion is preferred for the low-permeability sandstone^{[47, 50]}. However, under tension stress state, the modified Griffith criterion is preferred, that is why the above criterion is inapplicable. As a result, the criterions for siltstone and fine sandstone will be selected as follows:

(1)If σ ₃≥ 0, the Mohr-Coulomb criterion will be selected, that is

σ1-σ32≥C0cosφ+σ1+σ32sinφ (3)

Where, if σ ₃> σ ₀=5 MPa, φ =φ ₂=41° , angle of rupture θ =θ ₂=45° -φ /2=24.5° .

If 0< σ ₀=5 MPa, φ =φ ₁=51.83° , angle of rupture θ =θ ₁=45° -φ /2=19.09° , cohesion

C₀=6.53 MPa as actual value.

(2) If σ ₃< 0, Griffith criterion is selected firstly, next, a discussion according to special condition should be taken^[54].

If(σ ₁+3σ ₃)> 0, the relative criterion is (σ1-σ2)2+ (σ2-σ3)2+ (σ3-σ1)2

=24σ _T(σ ₁+σ ₂+σ ₃), cos2θ = σ1-σ32(σ1+σ3)(4)

If(σ ₁+3σ ₃)≤ 0, which is simplified to

σ ₃=-σ _T, θ =0 (5)

According to Fig.3 and Fig.4, sample K103-DA3 and sample K103-DA1 all belong to fine sandstone with compact structure and high density than mid-fine sandstone, which easily lead to brittle failure under rapid uniaxial compression, relatively mid-fine sandstone sample K101-DC1 and sample K101-DC4 have lower peak strength and shorter stress-strain curves, especially when contains number of original microcracks, the stage of elasticity get shorter, during which only extensional failure or shear failure will occur with the presence of pore pressure or uniaxial pressure in Fig.6a. Having described macroscopic and microscopic anisotropy and mechanical behavior of during loading, Fig.6b provides an overview of microcracks evolution law in relation to steady loading processes that at last lead to overall destruction of rock. Here we emphasize that stress field does not always cause macroscopic fractures to improve the permeability in rocks, the large number of microcracks also play a similar important role, as a result, the key value 0.85 σ _c can be defined as peak strength value, by which the evolution curve is divided into five stages:

Fig.6

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Fig.6 Uniaxial stress-strain curves of various micro-fractured low-permeability sandstone under cyclic compression loading(a) and evolutionary process from microcracks to macrocracks(b) In area OA-AB few microcracks form, in area BC the rock dilate obviously, producing a large number of microcracks could generate, in area CD series of macrocracks appear, accompanied with the stress declining, fracture density and aperture increasing.Only results up to peak stress and 85 percent of peak stress are included

(1) Inperiod of compaction by initial damage(OA segment), in Fig.5b, c, the microcracks in porous rocks began to close with compression, allowing rocks to become dense, fracture aperture to reduce significantly and permeability to die away gradually.

(2) AB segmenton stress-strain curve presents proximate straight line feature, during which rock occurs elastic deformation with few obvious extensional fractures appearing, although axial strain increases significantly, unlike studies on carbonate samples of middle Tarim area by Song^[34]and Gale^[55]. Under this condition, mutually parallel or nearly parallel extensional fractures are not developed here, as most of microcracks belong to intragranular fracture type on surface of the particles, with aperture less than 0.001 mm and extended distance less than 0.2 mm. Overall, in this stage, there develop few fractures, which present scattered in distribution and seem grid and directionless trading.

(3) The BC segment in Fig.5b is no longer a straight line, with increasing strain on radial, which is considered as a dilatation stage. As the yield limit of the rock, the stress of point B is equal to 0.85 σ _c, after this point, fracture number increases rapidly and are of strong orientation and approximately equal in distances, all of which prove that fractures tend to develop along anisotropy when the angle between pre-existing microcracks and σ ₁ is low(< 30° ).

(4) While stress reaches the peak strength (point C) (Fig.5b), the overall macro-rupture occurs, along with the serious decrease of stress and significant increase of strain.

(5) Breakdown of sandstoneas evolution result of intragranular microcracks, mainly includes development of original microcracks, aggregation of microcracks, interconnection of macrocracks. From above, it can be indicated that a large number of new fractures appeared at the compressive stress equaling to 0.85 σ _c, which can be defined as an indication or precursor for microcracks to sprout and develop during loading. Xia et al.^[56]conducted uniaxial compression experiment on MTS with sandstone samples under various stress conditions, and found that microcracks increased as stress boosted, especially when stress reached or exceeded 0.85 σ _c, the number of fractures exploded instantaneously and the number of large-scale fractures (length > 0.4 mm) increased faster than that of small-scale fractures (length≤ 0.3 mm), meanwhile the number of fractures with small angle of≤ 30° with respect to principal stress increased faster than that of fractures with large angle of> 30° (Fig.4).

According torock mechanics theory^[57], geo-stresses, important influencing factors of the stability of the deep rock mass, are the main factors controlling development of fractures in crust, which is characterized by three-dimensional state of stress including two horizontal principal stresses(σ _H and σ _h) and a vertical principal stress(σ _v) in most conditions. In the stress field, if the rock appears distorted as time passes and much elastic strain energy accumulates everywhere; the energy intensity usually can be characterized by rock’ s strain energy per unit volume, which is strain energy density^[58].

ω̅=12σ1ε1+σ2ε2+σ3ε3(6)

Where ω̅is the strain energy density of low-permeability sandstone. By the generalized Hooke’ s law, stress-strain relationship is formulated.

ε1=1Eσ1-μ(σ2+σ3) 7ε2=1Eσ2-μ(σ1+σ3)8ε3=1Eσ3-μ(σ1+σ2)9

Combing Eq.(7), Eq.(8), Eq.(9) with Eq.(6), new strain energy density is given by

ω̅=12E[σ12+σ22+σ32 -2μ(σ1σ2+σ2σ3+σ1σ3)](10)

Where ε is the strain, μ is the Poisson’ s ratio.

Theoritically, low-permeability sandstone is characterized by strong brittleness, according to brittle fracture mechanics theory and maximum tensile stress theory, when elastic strain energy release rate accumulating in brittle rocks which is equals to the energy per unit volume for generating fractures, rocks will break. Especially the fine sandstone and mid-fine sandstone in Kela-2 area, when surrounding three-dimensional stress state reaches the rock strength, macro brittle fractures will occur with strain energy releasing, part of which will offset the surface energy of new fractures, the rest of which will offset in form of elastic waves^[59]. However, for fractures, the elastic wave energy is so weak that can be completely ignored, so we assume that all of fractures in rocks are caused by the releasing energy, here law of conservation of energy must be abided.

Dvf=SfV=ω̅fJ(11)

Where D_vf is fracture volume density per unit volume (m²/m³); ω̅fis strain energy density for new fractures(J/m³); V is characterization of unit cell volume (m³); S is surface area of new fractures(m²); J is the required energy per unit area for fractures, i.e., fracture surface energy (here the energy is different from and far less than theoretical value of molecular dissociation).

In Eq.(11) if ω̅fis regarded as the residual strain energy density by current strain energy density per unit volume subtracting elastic strain energy density for new fractures( ω̅e), a modified formula will be given by:

Dvf=SfV=ω̅fJ=ω̅-ω̅eJ=1Jω̅-ω̅eJ=aω̅+b(12)

Where ω̅eis the necessarily overcoming elastic strain energy for new fractures, a= 1J, b=- ω̅eJis relative coefficient, which can be derived by two methods. One way is based on the values of J, ω̅eis calculated by physical concept, then makes conversion to obtain values of a and b. Another way is through experimental data, curve fitting and statistical approach is to obtain the coefficients, by contrasting, and the former has clearer physical meaning in practice.

In most cases, there exist three types of stress environment in earth crust, such as type Ia, type II and type III, and, different mechanical behaviors appear under different stress state, for example, shear fractures probably form in the extrusion conditions and extensional fractures probably form in the extension environment, so individual condition must be discussed respectively.

As stated above, 0.85 σ _c as peak period for the initiation of fracture swarm, its corresponding strain density energy is close to ω̅ein theory, in this case, we assume that ω̅eis strain energy density at σ =0.85 σ _c, which to be plugged into a formula to calculate value of J, values of a and b can be confirmed finally. Based on mechanical experimental results of low-permeability sandstone in Kela-2 area, the average uniaxial compressive strength of fine-median sandstone has been acquired, such as σ _c=42.69 MPa, 0.85 σ _c=36.28 MPa, accordingly calculated ω̅e=1.133× 10³ J/m³. Here Eq.(12) can be transformed to J= ω̅-ω̅eDvf, values of J are worked out as follows(Table 3).

Table 3

Table 3

Table 3 Strain energy parameters of four specimens in Kela-2 area Num. 0.85 σ c /MPa ε /× 10-4 ω̅ /(J/m3) σ i /MPa ε i /× 10-4 ω̅e /(J/m3) Dvf /(m2/m3) J /(J/m2) K101-DB3 43.55 74.37 1.619× 105 36.28 62.46 1.133× 105 44.7 1 087.25 K103-DA3 50.52 64.07 1.618× 105 36.28 62.46 1.133× 105 44.6 1 087.44 KL1-DB1 34.02 58.10 1.265× 105 28.34 50.20 0.891× 105 34.92 1 391.38 KL3-DA2 39.47 50.05 1.264× 105 28.34 50.20 0.885× 105 34.84 1 391.44 Note: σ i is initial stress, ε i is initial strain, and all stresses are in MPa.

Table 3 Strain energy parameters of four specimens in Kela-2 area

Taking average of J in Table 3 to acquire J̅=1 087.35 J/m², together with ω̅e=1.133× 10⁵ J/m³ to be substituted into the Eq.(12), quantitative relationship between strain density and fracture volume density is founded.

Dvf=9.2×10-4ω̅-104.2 (13)

In order to test and verify the reliability of ω̅e, curve fitting method should be taken to calculate a and b, by comparing with data by first method, the final fitting results are showed in Fig.7.

Dvf=8.04×10-4ω̅-86.128 (14)

Where the correlation coefficient reaches 0.995, and the relative errors for coefficients a and b by two different methods are 12.6% and 17.3% respectively, both within allowed error range. And it should bepointed out that regardless of the rock is intact or fractured, the fitting curve exhibits positive linear relationship between parameters. Even if there is a small devi-ation, the main reasons still can be caused by aniso- tropic mechanical properties, i.e. combination of pre-existing weakness planes and unruptured rocks. Whatever it takes, conforming ω̅eby strain energy density corresponding to 0.85 σ _c is an effective approach, only a small amount of rock samples are used.

Fig.7

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	Fig.7 Linearized relational graphs between strain energy density and fracture volume density at key stress point (0.85 σ c) based on stress strain curves The data from reference[60] and the experimental results showed good agreement

As has been proved by the experiments, relationship between strain energy density and fracture volume density under complex state of stress is unlike that under unaxial state of stress, without popularity in practice. In the former case, elastic strain energy ω̅eof rocks is determined not only by inherent mechanical properties but also by state of stress. Nevertheless, there still exist some similarity between curves of triaxial compression test and uniaxial compression test, for example, once axial stress reaches 0.85 σ _max, a number of precursory microcracks will appear, whose strain energy is just equal to ω̅eat this time.

Table 4

Table 4

Table 4 Stress-strain state and fracture parameters of rocks in experiments Lithology Sample Num. Sampling stress σ Q /MPa Bearing ratio σ Q/σ c Axial deformation ratio ε h/ε hc Radial deformation ratio ε r/ε rc Volume density /(m2/m3) Aperture /μ m Stain energy density /(J/m3) siltstone and fine sandstone K101-DC1 30.34 0.67 0.74 0.2 0.00 0.0000 7.938× 104 K101-DC2 19.38 0.45 0.57 0.21 0.00 0.0000 4.003× 104 K101-DC3 34.0 0.8 1.1 0.95 22.50 0.0240 1.356× 105 K101-DC4 23.18 0.54 0.78 0.35 0.00 0.0000 6.592× 104 K101-DB1 37.36 0.88 0.91 1.10 13.40 0.0080 1.242× 105 K101-DB2 28.35 0.66 0.69 0.49 0.00 0.0000 7.099× 104 K101-DB3 43.55 1.02 1.02 1.38 43.00 0.0280 1.619× 105 K103-DA1 39.05 0.91 0.74 0.45 0.00 0.0000 1.044× 105 K103-DA2 19.64 0.46 0.33 0.24 0.00 0.0000 2.326× 104 K103-DA3 50.52 1.18 0.88 0.67 40.35 0.0020 1.618× 105

Table 4 Stress-strain state and fracture parameters of rocks in experiments

If the minimum principal stress σ ₃ is given, and σ ₃≥ 0 MPa under triaxial state of stress (in contrast σ ₃< 0 MPa representing tensile state of stress), using above-mentioned failure criteria, we get the minimum value of maximum principal stress for rock failure.

σ1min=2C0cosφ+(1+sinφ)σ31-sinφ15

In fact, the calculatedσ ₁ does not represent actual maximum principal stress, but exactly equals bursting stress σ ₃, here for the later convenience, we try using σ _p to replace σ ₁.

σp=2C0cosφ+(1+sinφ)σ31-sinφ16

Under any stress state σ _ij= σ1σ2σ3, if σ ₃≥ 0 MPa is substituted in Eq.(16) to obtain σ ₁> σ _p, which meet the rupture conditions, then according to elastic theory, the value of ω̅ecorresponding to 0.85 σ _p is computed, combining Eq.(6) and Eq.(7) we get

ω̅e= 12σ _p(0.85)ε _p(0.85)= 120.85 σ _p× 0.85 ε _p= 12E0.85σp0.85σp-2μ(σ2+σ3))(17)

Combining Eq.(12) and Eq.(13) we get

ω̅f=ω̅-ω̅e=12(σ1ε1+σ2ε2+σ3ε3)-12E0.85σp0.85σp-2μ(σ2+σ3))](18)

Combining previous research results, in study area, there is no clear relationship between poisson’ s ratio and confining pressure, however, clear linear relationship is discovered between strength, elastic modulus and confining pressure through a lot of experimental data, and further analysis demonstrate that there appears a linear relationship between compressive strength and elastic modulus despite of the great change of confining pressure and difference in unaxial compression strength(Fig.8a), which can be expressed as

E=112.6×2C0cosφ+(1+sinφ)σ31-sinφ19

Where φ is the internal friction angle, C₀ is the cohesive strength of rocks.

Putting Eq.(19) and Eq.(17) into Eq.(18), Eq.(18) is simplified to

ω̅f= ω̅- ω̅e= 12(σ ₁ε ₁+σ ₂ε ₂+σ ₃ε ₃)- 0.8522C0cosφ+(1+sinφ)σ3225.2(1-sinφ)+0.85μ(σ2+σ3)112.6(20)

Undoubtedly, under confining pressure conditions, energy accumulated in the rocks not only must overcome the molecular internal cohesive forces, but also spares part of its own to resist confining pressure, as a result, the needed power for fractures per unit area increase in progress, a comprehensive superposition result is given by

J=J0+ΔJ21

Where J₀ is fracture surface energy without confining pressure or under unaxial compressive stress (J/m²), Δ J is the additional surface energy caused by confining pressure σ ₃ (J/m²), μ is Poisson’ s ratio.

Fig.8

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Fig.8 Main controlling factors for variation of fracture parameters under various confining pressure (a) Linearized relationship between compressive strength and elastic modulus under confining pressure in Kela-2 area. (b)Relation diagram between fracture surface energy and confining pressure at depth. Unlike the unaxial compression condition, there will consume a certain amount of extra energy to resist confining pressure, the higher the confining pressure, the higher the fracture surface energy

Now we assume a default fracture with aperture b, area S and confining pressure σ ₃(Fig.8b), then work of σ ₃ for the fracture is defined by

ω̅=σ ₃bS (22)

Where Δ J= ω̅/s=σ ₃b (23)

Combing Eq.(20), Eq.(21)and Eq.(23) into Eq.(12), we can get the fracture volume density.

D_vf= ω̅fJ= 12(J0+ΔJ)(σ ₁ε ₁+σ ₂ε ₂+σ ₃ε ₃)- 0.8522C0cosφ+(1+sinφ)σ3225.2(1-sinφ)(J0+ΔJ)+ 0.85μ(σ2+σ3)112.6(J0+ΔJ)(24)

To determine the value of σ _p, some conversions are needed based on Eq.(24), with experiments, there σ _p=σ ₁ and strain is given, the above equations are equivalent to

Dvf=ω̅fJ =(σ1ε1+σ2ε2+σ3ε3)2(J0+ΔJ) -0.852σ1ε12J0+ΔJ25

To verify the validity of aboveequations or mechanical models, firstly we use Eq.(25) to compute fracture volume density, then a comparison is carried out by actual data in the study area. It is necessary to emphasize that some of original data is drawn from the triaxial test results of sandstone in Fig.9.

Totally therelative errors of 94.11% of samples are less than 15%, demonstrating the accuracy and practicability of the models (Fig.8). Meanwhile through detailed statistics between computed results and measured data, several major laws have been found out. Firstly, with the increase of σ ₃ or confining pressure, ω̅fdecreases drastically, according with the actual situation, and where there exists a clear boundary with corresponding value=8 MPa, which described the curve bifurcation phenomenon. Secondly, with the increasing of differential stress σ ₁-σ ₃ or stress intensity, the relative errors decreases constantly, until downwards intersecting with axes-x at σ ₁-σ ₃=120 MPa. Thirdly, with the increasing of stress intensity, ω̅fincreases gradually along two approximate parallel curves(Fig.8d).

In order to meet the demands of the real reservoir development behind, linear fracture density in low-permeability sandstone is needed as well as fracture surface density with respect to the calculated model of fracture volume density D_vf. If stress σ ₁ is tilted horizontally by an angle θ from the fracture plane, the linear fracture density is calculated as follows

D_lf= 2DvfL1L3sinθcosθ-L1sinθ-L3cosθL12sin2θ+L32cos2θ(26)

Where D_lf is linear fracture density (1/m), L₁, L₂, L₃ are respectively the lengths in directions of σ ₁, σ ₂ and σ ₃(m) in Representative Elementary Volume(REV).

If stress σ ₁ is parallel to fracture plane, the linear fracture density is calculated as follows

D_lf=D_vf (27)

In extensional basins, there always exists type Ib of stress state which is usually with one horizontal principal stress negative(tensile) and the other positive(compressive), both are small within several MPa. In contrast, when the rocks remain in the vertical stress which is positive, till to ground surface the compressional force gets small enough to be neglected^[61]. From perspective of rock mechanics, as nearing the ground surface, the stress has become smaller, under the state dominated by one or two extensional principal stress, there must appears σ ₃< 0, at this time the appropriate applied fracture criteria is not Mohr-Coulomb criterion but Griffith criterion, and the foundation of corresponding stress-strain relationship also has a bit difference from the above.

Fig.9

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Fig.9 Comparison between computed results and actual measured data of fracture density (a)The relative errors between measured volume density of fracture by drill cores and calculated density by mechanical model range from 0.22 percent to 10.87 percent, local error up to 20%, which shows good reliability.(b)A clear boundary in confining pressure Ca. 8 MPa exists to divided relation curves into two areas, i.e. low confining pressure area and high confining area.(c)The higher the differential stress(σ 1-σ 3), the larger the relative errors. (d) The higher the differential stress(σ 1-σ 3), the bigger the strain energy density, and both relation curves the two curves parallel to each other with same slope

Certainly, under extension state, rock appears stronger brittle deformation than that of compression state, such as stress and strain which are good linear relationship before peak strength, after that, rock strength rapidly decreases to zero without any precursor microcracks. By contrast, under compression state around peak strength value, rock normally undergoes several deformation stages with noticeable precursor microcracks and residual strength, and the total accumulated strain from beginning is also bigger than that of the former.

In conclusion, when tension stress appears, rock preferred occurring tension fracture or tension-shear composite fracture approximately perpendicular to the maximum principal stress, at this time, an essential thing must be satisfied that the elastic strain energy density ω̅efor fractures is equal to that of peak strength under uniaxial tension, which is not the one corresponding to 0.85 σ _max under uniaxial compression.

Throughout theexperiments, σ _t, E and the ultimate strain ε _t are acquired now, then ω̅eis also computed out as follows

ω̅e=12σtεt=12Eσt2 (28)

Finally we can get the fracture volume density:

Dvf=1Jω̅-ω̅e=1Jω̅-12Eσt2(29)

According tothe above description, when(σ ₁+3σ ₃)> 0, by Griffith criterion, there should have

cos2θ = σ1-σ32(σ1+σ3)

θ = arccos(σ1-σ3)/2(σ1+σ3)2(30)

When(σ ₁+3σ ₃)≤ 0, θ =0, we get

D_lf=D_vf (31)

As we all know, in three-dimensional in-situ stress system of coordinates, thecalculation of fracture strike and dip depends on reasonable projection, e.g. axes-x agrees with east-west direction, axes-y agrees with north-south direction, axes-z agrees with vertical direction. Thus, the strike angle α in the x-y plane can be characterized by its normal vector of fracture surface. Here n̅={l m n} denotes cosine of normal vector, if projected in the x-y plane, whose inclination angle α _Y with respect to axes-y is obtained, and α _Y=arctan(-l/n)(Fig.10).

if 0≤ α _Y< 90° , α =90° -α _Y (32)

if 90° < α _Y< 0, α =(-90° -α _Y)+360° (33)

In three-dimensional in-situ stress system of coordinates, inclination angle α _dip between the lx+my+nz=0 plane and y=0 plane is assumed as the inclination angle between fracture surface and x-y plane, which is given by

cosα _dip= |l·0+m·1+n·0|l2+m2+n202+12+02

= ml2+m2+n2, 0° ≤ α _dip≤ 90° (34)