# Research on Grouting Pressure of Earth-Rock Dams Reinforced by Polymer Based on Discrete Elements
## 3 Principle of Fluid-Solid Coupling Model  
### Mathematical Model of Polymers  
To establish the discrete element model for subsequent analysis, understanding the chemical reaction mechanism of the polymer and establishing the requisite mathematical model is necessary. Given the complexity of the actual chemical reaction, and the inherent limitations of the discrete element model, this study focuses on the primary raw materials and key reaction processes, extracting and numerically simulating the key parameters. The grouting material under investigation herein is a two-component polyurethane polymer, undergoing gel and foaming reactions concurrently, which collectively determine its final structure and properties. Initially, the foaming process yields minimal foam, with the primary body remaining liquid and exhibiting high fluidity; as the reaction progresses, viscosity increases, transforming the liquid into a foam colloid. Subsequently, the foam expands to fill the rockfill pores, followed by further gelation. In this study, the polymer foaming expansion process is simplified as fluid seepage, necessitating parameter determination to describe the flow dynamics. As per regulations, the polymer must have a density not less than 1 g/cm\({}^{3}\) and a viscosity not exceeding 1000 mPa\(\cdot\)s.  
1. Rheological property Polymer fluid is called complex fluid. Because the main performance of polymer after a chemical reaction is to produce foam and expand, when the foam size is small enough, the foam can be regarded as a diluted foam suspension. It can be regarded as a generalized Newtonian fluid [14]. The characteristic flow velocity is 0.05 m/s, the fluid density is 1000 kg/m\({}^{3}\). The corresponding Reynolds number \(Re\) is shown in Equation (1):  
\[Re=\frac{\rho vL}{\mu}=\frac{1000\times 0.05\times 1}{0.05}=1000 \tag{1}\]  
In the formula, \(\rho\) is the fluid density, \(\mu\) is the kinetic viscosity, \(v\) is the characteristic flow velocity, and \(L\) is the characteristic length. At this time, the expansion process of polymer foam can be regarded as laminar flow.  
2. Viscosity  
The viscosity of polymer is not only affected by the shear rate, but also by the elastic deformation. Such fluid is called viscoelastic fluid. In order to accurately simulate the expansion process of polymer foams, the Castro-Macosko model [15] is used to characterize the viscosity \(u_{\mathrm{f}}\), both \(\eta_{\infty}\) and \(E_{\mathrm{n}}/R_{\mathrm{g}}\) are polymer coefficients, \(C^{*}\) represents the polymer conversion rate, and \(C^{*}_{\mathrm{g}}\) denotes the polymer gel conversion rate. The variables \(a\), \(b\), and \(c\) are all coefficients of the viscosity model, see Equation (2):  
\[u_{\mathrm{f}}=\eta_{\infty}e^{(\frac{E_{\mathrm{n}}}{R_{\mathrm{g}}T})}.( \frac{C^{*}_{\mathrm{g}}}{C^{*}_{\mathrm{g}}-C^{*}})^{a+bC^{*}+cC^{*2}} \tag{2}\]  
3. Thermal conductivity.  
The thermal conductivity of polymers is often very small. According to the empirical formula [16], the thermal conductivity \(k\) of polymers is only related to the density \(\rho\). When the polymer density is greater than 48 kg/m\({}^{3}\), see Equation (3):  
\[k=8.7006\times 10^{-8}\rho^{2}+8.4674\times 10^{-5}\rho+1.16\times 10^{-2} \tag{3}\]