## 2 The Equations of Coupled THC Model
### The Hydrochemical and Hydraulic Erosion Equation
#### 2.1.1 Basic Equations  
The porosity and the volume fraction of the fluidized solids are defined as  
\[\phi=\frac{V_{p}}{V}=\frac{V_{f}+V_{hs}+V_{cs}}{V} \tag{1}\]  
\[c_{h}=\frac{V_{hs}}{V_{p}}=\frac{V_{hs}}{V_{f}+V_{hs}+V_{cs}} \tag{2}\]  
\[c_{c}=\frac{V_{cs}}{V_{p}}=\frac{V_{cs}}{V_{f}+V_{hs}+V_{cs}} \tag{3}\]  
where \(\phi\) is the porosity of the porous rock; \(V\), \(V_{p}\), \(V_{f}\), \(V_{hs}\) and \(V_{cs}\) are the volume of the porous rock, pore, fluid, the fluidized solids from hydraulic erosion, and the fluidized solids from chemical erosion, respectively, m\({}^{3}\), in which the volume of the fluidized solids from hydraulic erosion and chemical erosion are actually the reduced volume of the solid skeleton due to hydraulic erosion and chemical erosion; \(c_{h}\) and \(c_{c}\) are the volume fraction of the fluidized solids from hydraulic erosion and the fluidized solids from chemical erosion, respectively.  
Therefore, the equivalent densities of the four phases are defined as  
\[\rho_{1}=\frac{m_{s}}{V}=\rho_{s}\frac{V_{s}}{V}=\rho_{s}\frac{V-V_{p}}{V}=(1- \phi)\rho_{s} \tag{4}\]  
\[\rho_{2}=\frac{m_{f}}{V}=\rho_{f}\frac{V_{f}}{V}=\rho_{f}\frac{\phi V-V_{hs}-V _{cs}}{V}=(1-c_{h}-c_{c})\phi\rho_{f} \tag{5}\]\[\rho_{3}=\frac{m_{ls}}{V}=\rho_{s}\frac{V_{hs}}{V}=c_{t}\phi\rho_{s} \tag{6}\] \[\rho_{4}=\frac{m_{cs}}{V}=\rho_{s}\frac{V_{cs}}{V}=c_{c}\phi\rho_{s} \tag{7}\]  
where \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\) and \(\rho_{4}\) are the equivalent density of the solid skeleton, fluid, the fluidized solids from hydraulic erosion, and the fluidized solids from chemical erosion, respectively, kg/m\({}^{3}\); \(m_{s}\), \(m_{f}\), \(m_{ls}\) and \(m_{cs}\) are the mass of the solid skeleton, fluid, the fluidized solids from hydraulic erosion, and the fluidized solids from chemical erosion, respectively, kg; \(\rho_{s}\) and \(\rho_{f}\) are the solid density and fluid density, respectively, kg/m\({}^{3}\); \(V_{s}\) is the volume of the solid skeleton, m\({}^{3}\).  
The mass balance for the solid skeleton can be expressed as  
\[\frac{\partial\rho_{1}}{\partial t}+\
abla\cdot\left(\rho_{1}v_{sk}\right)=I_ {hr}-R \tag{8}\]  
where \(v_{sk}\) is the fluid velocity of the solid skeleton, m/s; \(I_{hr}\) and \(R\) are the mass sources and sinks, kg/(m\({}^{3}\cdot\)s), \(I_{hr}\) represents the removed mass due to the hydraulic erosion here, and \(R\) represents the removed mass due to the chemical erosion here. \(R\) also represents the chemical reaction rate.  
The mass balance for the fluidized solids from hydraulic erosion can be expressed as  
\[\frac{\partial\rho_{3}}{\partial t}+\
abla\cdot\left(\rho_{3}v_{fs}\right)=I_ {hp} \tag{9}\]  
where \(v_{fs}\) is the fluid velocity of the fluidized solids from hydraulic erosion, m/s; \(I_{hp}\) is the mass sources and sinks, kg/(m\({}^{3}\cdot\)s), which represent the produced mass from the solid skeleton. Therefore,  
\[I_{hr}=-I_{hp} \tag{10}\]  
According to the previous research [26],  
\[v_{sk}\cong 0 \tag{11}\]  
\[c_{t}\phi v_{fs}=\lambda\phi_{,i} \tag{12}\]  
\[c_{h}\ll 1 \tag{13}\]  
where \(\lambda\) is the porosity diffusivity coefficient, m\({}^{2}\)/s.