# Experimental Study of Imbibition Characteristics of Silica Sol in Coal-Measure Mudstone Matrix
## 4 Test Analysis of Key Factors Affecting Imbibition Effect
### Driving Force  
The one-dimensional flow in fracture system is described by the following Equation (6) [13; 14]:  
\[v_{p}=\frac{k}{\mu_{g}}i \tag{6}\]  
where \(v_{\text{p}}\) is the pressure effect seepage velocity; \(k\) is the permeability; \(i\) is the hydraulic gradient of grout. Meanwhile, some researches [13] have shown that flow velocity in a matrix system can be expressed by:  
\[v_{i}=\sqrt{Kn}\sigma_{g}\cos\theta\left[\frac{K_{r,g}K_{r,a}}{\mu_{g}K_{r,a}+ \mu_{w}K_{r,a}}\times\frac{\partial J\big{(}S_{g}\big{)}}{\partial S_{g}} \times\frac{\partial S_{g}}{\partial x}\right] \tag{7}\]  
where \(v_{i}\) is the imbibition effect seepage velocity; \(i\) is the hydraulic gradient of grout; \(K\) is the hydraulic conductivity; \(n\) is the porosity; \(\sigma_{g}\) is the surface tension of grout; \(\theta\) is the contact angle between mudstone and grout; \(K_{r}\) is the relative hydraulic conductivity. \(J\big{(}S_{g}\big{)}\) is the Leverett equation, which can be described as:  
\[J\big{(}S_{g}\big{)}=\frac{p_{c}\big{(}S_{g}\big{)}}{\sigma_{gw}\cos\theta} \sqrt{\frac{K}{n}} \tag{8}\]  
where \(p_{c}\big{(}S_{g}\big{)}\) is the capillary force; \(J\big{(}S_{g}\big{)}\) is determined by experiment; \(S_{g}\) is the effective saturation of grout. Through Equations (6)-(8), we can see that \(v_{p}\infty k\), but \(v_{i}\infty\sqrt{K}\). For silica sol (unit weight 10.78 kN/m\({}^{3}\), dynamic viscosity 10 mPa\(\cdot\)s), \(K\approx k\times 1\times 10^{6}\)/(m\(\cdot\)s). The mudstone matrix system is extremely difficult to grout under certain confined pressures on account of its low permeability of \(10^{-6}\)-\(10^{-8}\) cm/s. However, it is easier to imbibition grout with a wondrous equivalent permeability [14; 15; 16] of \(10^{-5}\)-\(10^{-6.5}\) cm/s. These are of great significance for grouting in low-permeability mudstone. Therefore, it is considered that the driving force in a fracture system is grouting pressure, but the permeability of a matrix system is so low that grouting pressure is negligible [15] and is dominantly controlled by the imbibition effect. As Equations (7) and (8) show, we can see that the driving force of spontaneous imbibition is the wetting ability and affinity of grout on rock. The smaller the contact angle, the stronger the wetting ability, and affinity between the solid and liquid can be defined as \(F\):  
\[F=\sigma_{g}\cos\theta \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**IndexGrout**} & \multirow{2}{*}{**Silica Sol**} & \multicolumn{2}{c}{**MP364**} & \multirow{2}{*}{**P.O 425R**} \\ \cline{3-4}  & & **Part A** & & \\ \hline \(3\times d_{\text{max}}\) & 24–36 nm & 2865 nm & 24–36 nm & 747–1056 \(\upmu\)m \\ \(3\times d_{95}\) & 186 nm & 3205 nm & 10,740 nm & 1150 \(\upmu\)m \\ \(R^{3\times d_{\text{max}}}\) & 49.31\% & 34.24\% & 49.31\% & 0 \\ \(R^{3\times d_{95}}\) & 41.10\% & 32.87\% & 21.92\% & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Theoretical groutability of different grouts. The contact angle and the surface tension were tested at a room temperature of 15 \({}^{\circ}\)C using a DSA100 optical contact-angle measuring device and a K100 surface tension meter, respectively. The contact angle between water and rock sample is 39.15 \(\pm\) 0.04\({}^{\circ}\), and between silica sol and rock sample is 49.26 \(\pm\) 0.06\({}^{\circ}\) (Figure 5). The surface tension of silica sol is 73.2 mN/m. Thus, the affinity between the sample and the silica sol is 47.77 mN/m. Additionally, as shown in Table 2, the wetting ability of silica sol is slightly larger than that of water, and the affinity of silica sol is slightly smaller than that of water. Hence, the spontaneous imbibition driving force of dry mudstone is strong, but the force is restricted for mudstone saturated with water. Natural mudstone is mostly in the unsaturated state; thus, the imbibition potential in the process of grouting is very significant. The wetting ability and affinity in the mercury injection experiment behave as a capillary force.