## 2 Materials and Methods
### Analytical Models  
Analytical models were carried out to explore the theoretical expectations comparing GPR results before and after filling the fractures. Rock fractures can be generally considered thin beds with a distance between the two interfaces less than the resolution limit determined by [PERSON]'s criterion. When a fracture in a homogeneous rock is considered a layer, it gives two signals of opposite polarity, which are reflected by the two interfaces of the fracture. If the two distinct signals are clearly identifiable and separated in time, the fracture thickness can be calculated by the time difference between the two reflections, knowing the signal velocity in the filling layer. The difference between a thin layer and a thick layer depends on the resolution of the radar system. Conventionally, the vertical resolution is estimated as a quarter of the wavelength calculated with the nominal frequency of the antenna. At 3 GHz, the theoretical expected resolution is 25 mm, 13.4 mm, and 14.4 mm for the air, resin, and PVC filling material, respectively. Thus, 14 mm is about the maximum thickness that we can consider as a thin bed either before or after filling the fracture. When contemplating the fracture as a thin layer, the two reflections can overlap in such a way that just a single composite wavelet is identifiable, and this interference may be constructive or destructive. The response of the thin layer is the sum of the primary reflections and multiple reflections that occur within the layer [46; 54; 55]. Therefore, the reflection coefficient is affected by the properties of the thin bed (i.e., thickness and filling material) because it is related to the permittivity contrast between the rock and the filling  
Figure 6: Data processing results for measurements on the air-filled and PVC-filled fractures with the thickness value varying from 0.25 mm to 16 mm (see Table 1). Note that the horizontal axis is the stacked trace number: 1–22 for the air-filled fractures and 23–44 for the PVC-filled fractures. material as well as to the ratio of the bed thickness to the wavelength within the thin bed. The value of the reflection coefficient changes linearly with the thin bed thickness for layers much thinner than the wavelength. The magnitude of the reflection coefficient oscillates between zero and a maximal value as the layer becomes thicker, depending on the destructive or constructive interference between the primary and multiple reflections [46]. This behavior can be analytically predicted using the following equation [46]:  
\[|\Gamma|=\left|\frac{2R_{12}\sin(2\pi d/\lambda)}{\sqrt{\left(1-R_{12}^{2} \right)^{2}+\left(2R_{12}\sin(2\pi d/\lambda)\right)^{2}}}\right| \tag{1}\]  
where \(\Gamma\) is the thin bed reflection coefficient, \(R_{12}\) is the normal incidence electromagnetic reflection coefficient between mediums 1 and 2 (rock and filling material, respectively), and \(\lambda\) is the pulse wavelength inside the thin bed. The following equation is used to calculate \(R_{12}\):  
\[R_{12}=\frac{\sqrt{\epsilon_{1}}-\sqrt{\epsilon_{2}}}{\sqrt{\epsilon_{1}}+ \sqrt{\epsilon_{2}}} \tag{2}\]  
where \(\epsilon_{1}\) and \(\epsilon_{2}\) are, respectively, the permittivity of medium 1 and medium 2. We used Equation (2) to predict the theoretical behavior of the air-filled, resin-filled, and PVC-filled thin beds. Figure 7 indicates the predicted theoretical difference between the air-filled and the resin-filled fractures. Although only two thicknesses of 1 mm and 2 mm were tested in the laboratory measurements with resin layers, a wide thickness range (0-40 mm) is plotted here in order to explore the trend of the reflection coefficient variations. The results illustrate that a 33% difference between the air-filled and the resin-filled fractures is expected for the smallest fracture apertures. The percentage difference theoretically increases to 100% when the fracture thickness is equal to half the wavelength calculated within the resin.