Based on the mechanical tests of shale, the reduction rate of peak strength and elastic modulus is relatively similar within 30 days of immersion (Li et al.
2020a,
b). The ratio of strength and modulus of immersed to dried shale can be used to indicate the water weakening effect on mechanical parameters (Chen et al.
2013):
$$\text{K}=\frac{{\sigma }_{w}}{{\sigma }_{c}}=\frac{{E}_{w}}{{E}_{c}}$$
(1)
where
K is softening coefficient,
\({\sigma }_{w}\) and
\({E}_{w}\) are the peak strength and elastic modulus of water immersed shale, respectively.
\({\sigma }_{c}\) and
\({E}_{c}\) are the peak strength and elastic modulus of dried shale, respectively. As the peak strength is commonly utilized to represent the shear strength, the elastic modulus of the immersed rock can be determined by multiplying the shear strength ratio of the immersed and dried shales with the elastic modulus of the dried rock:
$${E}_{w}=\frac{{\tau }_{w}}{{\tau }_{c}}\times {E}_{c}$$
(2)
where
\({\tau }_{w}\) and the
\({\tau }_{c}\) is the shear strength of the immersed and dried shale, respectively. The shear strength parameters are based on the previous inversion results,
\({c}^{w}\)=15 kPa,
\({\varphi }^{w}\) =15°. The geological prospecting data provides the elastic modulus and shear strength parameters for the dried shale,
\({E}_{c}\)=6.1 × 10
^{5} kPa,
c = 37 kPa and
\(\varphi\)=24°. Based on the Mohr–Coulomb strength theory, a soil element was considered in the hydrated carbonaceous shale as an object. It is assumed that the maximum principal stress
\({\sigma }_{1}\) on the element is vertical while the minimum principal stress
\({\sigma }_{3}\) is horizontal. As the argillaceous sandstone and carbonaceous shale above the underground water level are dry, the pore water can be disregarded under the weight of the thick overlying masses. When the unit shear failure occurs, the shear strength, maximum, and minimum principal stress can be expressed as follows:
$$\left\{\begin{array}{c}{\sigma }_{1}=\gamma z\\ {\sigma }_{3}={\sigma }_{1}{\text{tan}}^{2}({45}^{\circ }-\frac{\sigma }{2})\\ \tau =\frac{1}{2}({\sigma }_{1}-{\sigma }_{3})sin2\alpha \end{array}-2c\text{tan}({45}^{\circ }-\frac{\varphi }{2})\right.$$
(3)
where
α is the included angle between the fracture surface and the direction of the maximum principal stress,
\(\alpha ={45}^{\circ }+\frac{\varphi }{2}\). The average thickness of the argillaceous sandstone and carbonaceous shale above the water level was initially estimated to calculate the maximum principal stress (477.17 kPa), using formula (
3). Then, the strength parameters of the immersed and dried shale were introduced separately into formula (
3), and the shear strength
\({\tau }_{w}\) and
\({\tau }_{c}\) were calculated in combination with the maximum principal stress. The shear strength
\({\tau }_{w}\)=106.02 kPa,
\({\tau }_{c}\)=147.98 kPa. Finally, the shear strengths and the elastic modulus of dried shale
\({E}_{c}\) were inserted into formula (
2) to derive the elastic modulus of the immersed shale
\({E}_{w}\) = 4.4 × 10
^{5} kPa. Compared with the elastic modulus of dried carbonaceous shale, the modulus of shale after 30 days of immersion decreased by 28.4%, which is similar to the modulus loss rate of 25.05% and 31.5% measured respectively by uniaxial and triaxial tests (Bian et al.
2019; Zhao et al.
2022a,
b). Compared to short-term saturated carbonaceous shale, its elastic modulus decreases by 22.8%.