# Effect of Water Content and Cementation on the Shear Characteristics of Remolded Fault Gouge
## 4 Discussion
### Shear Strength Model for High-Clay-Content Fault Gouge  
The Mohr-Coulomb criterion is a widely accepted theoretical model for evaluating the shear strength of geomaterials, expressing shear strength \(\tau\) as a function of cohesion \(c\), internal friction angle \(\varphi\), and normal stress \(\sigma\), in the following form:  
\[\tau=c+\sigma\cdot\tan\varphi \tag{1}\]  
However, the classical Mohr-Coulomb model cannot be directly applied to fault gouge with variable water contents and cementation degrees because its parameters do not explicitly reflect the influence of these evolving hydro-mechanical conditions. In recent years, several studies have attempted to incorporate such factors--particularly water content, suction, and stress--into empirical or extended formulations of the Mohr-Coulomb model, improving its applicability to unsaturated soils and soft rock materials [51; 52; 53; 54; 55]. Inspired by these developments, we introduce the effects of water content and cementation degree into the strength model for fault gouge, and the criterion is rewritten as  
\[\tau(W,D)=c(W,D)+\sigma\tan[\varphi(W,D)] \tag{2}\]  
where \(W\) (%) is water content and \(D\) (%) is degree of cementation. The experimental results presented in Sections 3 and 4.2 demonstrate that cohesion exhibits a nonlinear response to water content, increasing initially and then decreasing, while increasing cementation degree leads to a monotonic enhancement of cohesion. In contrast, the internal friction angle tends to decrease as water content rises, but this softening effect becomes less pronounced under higher cementation conditions. These tendencies are captured by  
\[c(W,D)=AW^{k_{1}}e^{-\lambda_{1}W}+\alpha D^{m} \tag{3}\]  
\[\varphi(W,D)=Ce^{-k_{2}Wc^{-\lambda_{2}D}}\ \approx\ \tan[\varphi(W,D)] \tag{4}\]  
where A, \(\alpha\), C, k\({}_{1}\), \(\lambda_{1}\), m, k\({}_{2}\), and \(\lambda_{2}\) are empirical constants. Because all measured friction angles lie between 10\({}^{\circ}\) and 40\({}^{\circ}\), tan \(\varphi\) differs from \(\varphi\) (in radians) by \(\mathcal{<}\)7%; the single function \(\varphi(W,D)\) therefore replaces tan \(\varphi(W,D)\) in Equation (2) without an appreciable loss of accuracy and keeps the parameter set compact. Substituting Equations (3) and (4) into Equation (2) gives the semi-empirical shear strength model  
\[\tau=\left(AW^{k_{1}}e^{-\lambda_{1}W}+\alpha D^{m}\right)+\sigma\cdot\left( C\cdot e^{-k_{2}Wc^{-\lambda_{2}D}}\right) \tag{5}\]where \(\tau\) (kPa) is the shear strength, and \(\sigma\) (kPa) is the applied normal stress. This model maintains the fundamental structure of the Mohr-Coulomb criterion while effectively capturing the nonlinear and coupled effects of water and cementation on shear strength. To calibrate and verify the model, a total of 60 fault gouge samples obtained from the experiments described in Sections 2 and 3 are used. The dataset is divided using a random seed in Python 3.11.7, with 40 samples used for training and parameter fitting, and the remaining 20 samples used for model testing. The empirical parameters in the model were obtained by nonlinear least-squares fitting using the curve_fit function in Python (SciPy library). The fitting process was conducted by minimizing the squared difference between the predicted and measured shear strength on the training set. Based on the training data, the fitted shear strength prediction model is expressed as follows:  
\[\tau=\left(10^{-6}\cdot W^{9.496}\cdot e^{-0.524W}+0.00540\cdot D^{8.840}\right) +\sigma\cdot\left(0.603\cdot e^{-0.0319W\cdot e^{-0.740D}}\right) \tag{6}\]  
The comparison between predicted and measured shear strength is shown in Figure 11.  
As illustrated in the figure, there is a strong correlation between predicted and actual values. It is generally accepted that a coefficient of determination \(R^{2}>0.8\) indicates good predictive reliability [56].