## 2 Materials and Methods
### Method  
Capillary water absorption tests of cement-based materials usually apply the weighing method, which includes the upwards, side, and downwards suction methods. The most commonly used method is the downwards suction method, as shown in Figure 1. In this experiment, before a specimen of cement-based material was exposed to water, the specimen was first dried in an oven at 105 \({}^{\circ}\)C to a constant weight and weighed with an accuracy of 0.1 g, and then all surfaces, except the test surface, were sealed with epoxy resin to ensure one-directional transport of water. After the epoxy cured, the specimen was placed with the test surface facing down on a support frame in a water tank at a temperature of 20 \(\pm\) 1 \({}^{\circ}\)C. During the test, the water level was maintained at 5 mm above the bottom of the specimen, and the specimen was removed every 15 min. The surface moisture was removed with a damp cloth, the specimen was weighed immediately, and the condition of the specimen was observed. The total test time was at least 8 h.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Code**} & **Water/** & \multicolumn{6}{c}{**Mix Proportion (kg/m\({}^{3}\))**} & \multicolumn{6}{c}{**Steel Fibre**} & **Compressive Strength (MPa)** \\  & **Binder** & **Cement** & **Fly** & **Silica** & **Quartz** & **Coarse** & **Steel** & **Superplasticizer** & **Water** & **Volume V/** & **Strength (MPa)** \\  & **Ratio** & & **Ash** & **Fume** & **Sand** & **Aggregate** & **Fibre** & **Supplasticizer** & **Water** & **(\%)** & **(\%)** \\ \hline PC-1 & 0.13 & 2344 & 0 & 0 & 0 & 0 & 0 & 117.2 & 234.4 & 0.0 & 108.8 \\ PC-2 & 0.17 & 2344 & 0 & 0 & 0 & 0 & 58.6 & 363.3 & 0.0 & 135.1 \\ PC-3  
Since only one side of a specimen absorbed water during the test, a one-dimensional model was generally used to analyze the test results, and the extended version of [PERSON]'s law was adopted. We assumed that the pore structure of the concrete did not change during the test, and the Boltzmann variable [29,30]\(\lambda(\theta)=\kappa/t^{1/2}\) was introduced. The Richards equation was obtained, as shown in Equation (1), where \(x\) is the depth of capillary water absorption. The water absorption per unit area of cement-based materials tended to be linearly proportional to the square root of time. This law is also known as the square root law:  
\[V_{\rm w}(t)=\int_{\theta_{\rm init}}^{\theta_{\rm cap}}xd\theta=\sqrt{t}\int _{\theta_{\rm init}}^{\theta_{\rm cap}}\lambda(\theta)d\theta=S\sqrt{t}, \tag{1}\]  
where \(V_{\rm w}(t)\) is the water absorption of the test piece at \(t\), \(\theta\) is the water content, \(t\) is the elapsed time, \(\theta_{\rm cap}\) is the boundary water content when the capillary is saturated during the absorption process, \(\theta_{\rm init}\) is the initial water content, and \(S\) is the capillary water absorption coefficient. However, the test results showed that the curve fitted to the water absorption vs. square root of time did not pass through the origin. Therefore, Formula (2) was typically used to analyze the test results [31], and a was the intercept:  
\[V_{\rm w}=St^{1/2}+a, \tag{2}\]  
Although Equation (2) was used to analyze the test results, many studies have shown that the initial cumulative water absorption by cement-based materials obeys the square root law. As the water absorption time increased, a deviation was observed. Therefore, many scholars have divided the absorption process into sections and analyzed it in two stages to obtain the first-stage capillary water absorption coefficient and the second-stage capillary water absorption coefficient (represented as \(S_{1}\) and \(S_{2}\) below, respectively). Generally, the change in the capillary water absorption coefficient is presumed to be related to the two behaviors described below.