To investigate the effects of hydration on the mechanical properties of tight sandstones, specimens were divided into three groups: 1) dry sample, 2) fully water-saturated, and 3) incomplete water-saturated. The tight sandstone samples used in this paper are all subsurface rocks, which were taken from tight sandstone gas reservoirs. These tight sandstone samples have an average porosity of 12% and an average permeability of 4.8mD. The mineral composition is introduced in the following sections of this paper. These tight sandstone samples do not contain oil.

The preparation steps of specimens at saturation states are presented below.

I. Fully water-saturated samples:

The dry specimens were weighed and soaked in a potassium sulfate solution until they became completely saturated (Figure 1A). The dried specimens were weighed and saturated in a drying vessel (Figure 1B).

To calculate the water saturation of the specimens, the water content of the specimens was divided by the water content of the fully saturated specimens. Noted that drilled cores or rock blocks were used in the specimens, and artificial fractures were not allowed during the preparation process.

The mechanical properties of rock are controlled by the mineral composition and microstructure of rock. The changes in rock mechanical properties are due to the changes in rock microstructure and mineral composition caused by hydration. In order to investigate the mechanism of the influence of hydration on rock mechanical properties, the microstructure and mineral composition of rocks under different saturation states were tested and analyzed. The samples with different saturation states were analyzed by triaxial compression tests, tensile strength tests, field-emission scanning electron microscopy (FESEM), and X-ray diffraction (XRD) analysis.

Triaxial compression tests were performed on an MTS-816 true triaxial compression test system. The sample diameter and height were 24.7 mm and 51.02 mm, respectively. The diameter error did not exceed 0.3 mm, and the maximum parallelism at both ends of each sample did not exceed 0.05 mm. Experimental specifications complied with the laboratory test standards recommended by the International Society for Rock Mechanics (ISRM). The mechanical parameters of rocks, such as compressive strength, Young’s modulus, Poisson’s ratio, were experimentally determined.

The Brazilian disk method recommended by the International Committee for Rock Mechanics Experiments was adopted to measure the tensile strength of rocks. It must be noted that Brazilian disc test is an indirect measure of tensile strength of a rock. The sample diameter and thickness were 24.6 mm and 11.6 mm, respectively. Tensile stress was calculated by Eq. (1). σ t = 2 P π D t (1) where P means the peak load during Brazilian disc test, MPa; Dt represents the diameter of the rock sample, mm.

The WIF propagation length was analyzed by the stress intensity factor around the fracture tip. To calculate the WIF propagation length, the stress around the fracture during water flooding should be known. This kind of stress could be calculated by flow-geomechanics-coupled simulations. The analysis steps of WIF propagation are depicted below.

(1) According to flow-geomechanics-coupled numerical simulation results, the in-situ stress on the reservoir after water injection was calculated.

To analyze the stress during water flooding, a flow-geomechanics-coupled numerical simulation model was used (Wang and Peng, 2014; Lei et al., 2020). This model included two governing equations: i) flow equation of two-phase fluids through porous media and ii) rock deformation equation.

The propagation process of WIF was mainly affected by the reservoir stress and the fluid pressure in the fracture. The in-situ stress on the fracture was symmetrically distributed along the water injection well (Figure 4). It was assumed that the fracture lengths before and after WIF propagation were 2H ₂ and 2H, respectively. The half-length of fracture propagation was h = H − H ₂ .

According to the fracture mechanics theory, the stress intensity factor acting on the fracture wall at the crack tip could be expressed as K I A = 1 π H ∫ − H H p x H + x H − x d x (17) where H is the half-length of the fracture m), p(x) is the net pressure in the fracture (MPa), and K _IA is the stress intensity factor at the crack tip (MPa•m^0.5) (Qiangbo, 2017).

Owing to the stress concentration phenomenon at the fracture tip, stress distributions on the fracture wall and at the fracture tip could be simplified as σ c 2 = B 2 + a 2 x , H 2 < x ≤ H σ b 2 = B 1 + a 1 x , H 1 < x ≤ H 2 σ a = σ 1 , − H 1 < x ≤ H 1 σ b 1 = B 1 − a 1 x , − H 2 < x ≤ − H 1 σ c 1 = B 2 − a 2 x , − H ≤ x ≤ − H 2 (18)

The initial WIF length was 2H₂. To calculate the stress intensity factor at the fracture tip, the uniform fluid pressure in the fracture (P _f ) and the effects of the ground stresses σ _a , σ _b1 , and σ _b2 were considered. Therefore, the pressure acting on the fracture wall could be expressed as p x = p f − σ b 2 , H 1 < x ≤ H 2 p f − σ a , − H 1 < x ≤ H 1 p f − σ b 1 , − H 2 ≤ x ≤ − H 1 (19)

Now, substituting Eq. (19) into Eq. (17), K I A = 1 π H ( ∫ − H 1 H 1 p f − σ 1 H + x H − x d x + ∫ − H 2 − H 1 p f − B 1 + a 1 x H + x H − x d x + ∫ H 1 H 2 p f − B 1 − a 1 x H + x H − x d x (20)

Further, the integration of the above equation leads to K I A = 2 P f − σ a Asin H 1 H 2 H / π H 2 + 2 P f − B 1 H 2 Asin H 2 H 2 − A ⁡ sin H 1 H 2 / π H − 2 a 1 H 2 H 2 2 − H 1 2 / π H 2 (21)

If the K _IA value obtained from Eq. (21) is higher than the fracture toughness (K _IC), WIF propagation will occur. It was assumed that the new WIF length was H. To calculate the pressure acting on the fracture wall (Eq. (22)), the impacts of σ _a , σ _b1 , σ _b2 , σ _c1 , σ _c2 on WIF and the fluid pressure in the fracture (P _f ) were considered. p x = p f − σ c 2 , H 2 < x ≤ H p f − σ b 2 , H 1 < x ≤ H 2 p f − σ a , − H 1 < x ≤ H 1 p f − σ b 1 , − H 2 ≤ x ≤ − H 1 p f − σ c 1 , − H ≤ x ≤ − H 2 (22)

Now, substituting Eq. (22) into Eq. (17), K I A = 1 π H ∫ − H − H 2 ( p f − B 2 + a 2 x H + x H − x d x + ∫ − H 2 − H 1 p f − B 1 + a 1 x H + x H − x d x + ∫ − H 1 H 1 p f − σ 1 H + x H − x d x + ∫ H 1 H 2 p f − B 1 − a 1 x H + x H − x d x + ∫ H 2 H p f − B 2 − a 2 x H + x H − x d x (23)

The integration of the above equation leads to Eq. (24). It is noticeable that when H increases, K _IA decreases continuously. When K _IA decreases to K _IC, the propagation of WIF stops. Therefore, when K _IA is equal to K _IC in Eq. (24), the updated WIF length H) is obtained. K I A = 2 P f − B 2 H Asin 1 − A ⁡ sin H 2 H / π H − 2 a 2 H H 2 − H 2 2 / π H + 2 P f − σ 1 Asin H 1 H H / π H + 2 P f − B 1 H Asin H 2 H − A ⁡ sin H 1 H / π H − 2 a 1 H H 2 − H 1 2 − H 2 − H 2 2 / π H (24)

Numerical simulations were conducted to calculate the K _IA at the fracture tip during water flooding. To simplify the calculation, a two-dimensional geometric model of 350 m × 200 m size was developed to represent the reservoir. A total of 4,800 grids with a size of 3.5 m × 2.08 m was divided into 100 grids horizontally and 48 grids vertically. The basic parameters of the model are presented in Table 6. The ranges of the parameters were consistent with experimental results. Some parameters were not tested and obtained from previous reports. The model was composed of two wells: one is the injection well located at the left end of the model and the other one is the production well located at the right end of the model (Figure 5A). Both wells were connected to one hydraulic fracture located in the middle of the model. At time t = 0, the pressure at each point in the reservoir was 10 MPa and the initial oil saturation was 0.8. The left and right sides of the model represent constant pressure boundaries. The injection pressure on the injection well was 15.1 MPa, and the production pressure on the production well was 6.5 MPa. No flow boundaries were applied on the upper and lower sides of the model. Referring to the horizontal principal level of tight sandstone reservoir, the minimum and maximum horizontal principal stresses for rock deformation were 20 MPa and 45 MPa, respectively. Roller support boundary conditions were applied to the left and bottom of the model (Figure 5B).

It is noticeable from Figure 6B that during waterflooding, the water saturation of the reservoir changed sharply, especially around the hydraulic fracture at the left side of the reservoir. The maximum water saturation reached 0.9, and the maximum change was 0.7. The difference in water saturation between areas around the hydraulic fracture and other areas in the reservoir was extremely large. After 240 days of water injection, the water saturation around the hydraulic fracture at the left side of the reservoir was 0.7–0.9, whereas the water saturation in other areas was around 0.2. Therefore, a heterogeneous distribution of water saturation occurred in the reservoir. According to experimental results, the mechanical properties of rocks changed with the increase of water saturation. Figure 6C reveals that the Young’s modulus of the reservoir changed significantly after 240 days of water injection. The Young’s modulus around the hydraulic fracture decreased from 14 GPa to 7 GPa; therefore, a heterogeneous distribution of Young’s modulus also occurred.

The heterogeneous distribution of Young’s modulus played an important role in stress variation. Figure 7 displays the stress evolution in the y-direction (vertical to the hydraulic fracture) around the hydraulic fracture. In most areas around the fracture (less than 130 m in Figure 7, length of WIF = 140 m), stress values were almost equal to the minimum horizontal stress of 20 MPa. Owing to WIF, stress concentration occurred at the fracture tip (near 140 m in Figure 7) and turned into tensile stress in some areas. After 150 m, the stress started to increase. After 240 days of water injection, stress values in the areas less than 130 m or greater than 170 m changed slightly; however, stress values in the areas between 130 m and 150 m changed sharply. In most areas around the fracture tip, the stress became tensile (minus stress value in Figure 7), causing WIF propagation.

According to Eq. (22), stress distributions on the fracture surface and in front of the fracture tip at 0 days and 240 days of water injection were calculated.

The σ _y distribution at 0 days is of water injection could be expressed as σ c 2 = − − 181.85 + 1.3111 x , 140 < x ≤ H σ b 2 = 67.09 + − 0.4365 x , 110 < x ≤ 140 σ a = 18.82 , − 110 < x ≤ 110 σ b 1 = 67.09 − − 0.4365 x , − 140 < x ≤ − 110 σ c 1 = − 181.85 − 1.3111 x , − H ≤ x ≤ − 140 (25)

The σ _y distribution after 240 days of water injection could be described as σ c 2 = − 216.248 + 1.4976 x , 140 < x ≤ H σ b 2 = 87.14 + − 0.6166 x , 110 < x ≤ 140 σ a = 18.87 , − 110 < x ≤ 110 σ b 1 = 87.14 − − 0.6166 x , − 140 < x ≤ − 110 σ c 1 = − 216.248 − 1.4976 x , − H ≤ x ≤ − 140 (26)

Now, substituting Eqs. (25) and (26) into Eq. (21), it could be inferred that the K _IA values at the fracture tip after 0 days and 240 days of water injection were −3.18693.1869 MPa m^0.5 and 25.86919 MPa m^0.5, respectively. According to the experiment, the K _IC of the sandstone sample was 0.4249 MPa m^0.5; therefore, WIF could not propagate at 0 days of water injection. Now, substituting Eq. (26) into Eq. 24 and assuming K _IA = K _IC , the fracture length H was calculated as 159.49 m. The initial fracture half-length was 140 m; hence, the half-length of WIF propagation was 19.49 m.

The mechanical parameters of rocks, such as Young’s modulus and Poisson’s ratio, varied with water saturation. Therefore, eight groups of damage variables were set to illustrate the effects of damage variables on WIF propagation. Parameters relating to stress distributions on the fracture surface and at the fracture front under different damage variables are presented in Tables 7, 8, and the corresponding results are displayed in Figure 8 and 9. When the injection pressure and the fracture length were constant, K _IA increased quadratically with the damage variable. When the damage variable of Young’s modulus was less than 0.5, K _IA was less than K _IC ; thus, WIF could not propagate. When the damage variable was greater than 0.5, the WIF propagation length increased exponentially with the damage variable. Therefore, the damage variable reduced the difficulty of fracture propagation and increased the WIF propagation length, implying that in the later stage of waterflooding, the damage variable increased, WIF easily propagated, and the WIF propagation length was larger than that in the early stage of waterflooding, leading to the formation of a sudden water influx.

Generally, fracture propagation is controlled by the injection pressure. To investigate the influences of injection pressure on fracture propagation, water injection pressures were set to 14 MPa, 15 MPa, 15.5 MPa, 16 MPa, 17 MPa, and 19 MPa.

The variations of K _IA at the fracture tip and the half propagation length of WIF are displayed in Figure 10, respectively. All these results were obtained from the same damage variable. A linear relationship was detected between K _IA and the injection pressure. Moreover, the half propagation length of WIF also varied linearly with the injection pressure. When the damage variable of Young’s modulus was close to 1 and the injection pressure was 14 MPa, WIF propagated. When the damage variable was taken into account, the critical pressure for WIF propagation was reduced. However, in practical applications, the injection pressure is set as a constant value; thus, sudden WIF propagation and sudden water influx occur.

Generally, the injection pressure is set close to the minimum horizontal stress. Therefore, in this analysis, the water injection pressure (P _f ) was set as 19 MPa. The initial WIF lengths for different cases were 30 m, 60 m, 90 m, 120 m, and 140 m. The K _IA values and half propagation lengths of WIF for different cases were calculated. The damage variable was set as 0.6.

Figure 11 and 12 indicate that the K_IA and propagation length of WIF increased exponentially with the initial WIF length, implying that WIF could propagate when the initial WIF length was long enough. According to the fitting function in Figure 12, when the initial WIF length was greater than 114 m, the power index was larger than 1; thus, the propagation length of WIF was almost 40 m, indicating that WIF propagation became significant in the later stage of waterflooding. To avoid sudden water influx, the injection pressure should be reduced in the later stage of waterflooding.