## 3 Methods and Results
### D Volume Strain
#### 3.2.3 Results of 3D Volume Strain in the Study Area  
The 3D volume strain permits the simultaneous assessment of strain and volumetric changes resulting from the creation of fissure structures. The purpose of this section is to investigate and describe the 3D volume strain changes with distance from the fault in a specific part of the study area.  
Generally, a complete strain analysis is performed in three dimensions. These values can be found in the Flinn diagram or similar diagrams that illustrate the strain's elliptical shape, also referred to as the strain geometry. Furthermore, stereographic net tools can be used to visualize the direction of principal strains.  
Field investigations of 3D strain are frequently based on combining the two-dimensional values of several oriented sections. It will be simple to calculate the 3D strain if each section contains two axes of the principal strain. Using the two-dimensional strain values calculated in the previous sections and according to Equation (7), the three-dimensional strain was obtained in the study area (Table 4).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Section** & **\(\alpha\)** & **\(\gamma\)** & **B** & **R\({}_{\text{XY}}\)** & **R\({}_{\text{YZ}}\)** & **D** \\ \hline
1 & 60 & 80 & 40 & 1.32 & 2.04 & 0.32 \\
2 & 60 & 80 & 40 & 1.08 & 1.67 & 0.45 \\
3 & 60 & 80 & 40 & 0.92 & 1.43 & 0.43 \\
4 & 60 & 80 & 40 & 1.105 & 0.714 & 0.3 \\
5 & 60 & 80 & 40 & 0.95 & 1.47 & 0.68 \\
6 & 60 & 80 & 40 & 1.18 & 1.83 & 0.84 \\
7 & 30 & 75 & 75 & 1.05 & 1.63 & 0.39 \\
8 & 60 & 80 & 40 & 0.71 & 1.1 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical values of volume strain for the desired sections. Abbreviations: \(\alpha\), B, and \(\gamma\) = angles formed by the normal of the plane and each of the principal stress axes (x, y, z); R\({}_{\text{XY}}\), R\({}_{\text{YZ}}\) = strain ratio on planes XY and YZ; D = Eigenvalues of the matrix.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Section** & **The Initial Area of the Rock Mass, Along Section (A0)** & **Changes in Rock Mass Area Area Area of Fissure Structures,** & **Volumetric Strain (V=\(\frac{\text{AA}}{\text{A}_{0}}\))** \\ \hline
1 & 74.7 & 24 & 0.321 \\
2 & 344 & 79 & 0.229 \\
3 & 325 & 81 & 0.249 \\
4 & 358 & 96.5 & 0.269 \\
5 & 249.4 & 92.4 & 0.37 \\
6 & 220 & 85.5 & 0.388 \\
7 & 256 & 69.5 & 0.271 \\
8 & 91 & 30 & 0.329 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of the surface volume strain on the plane XZ.  
Figure 7: Topography cross-sections obtained along the desired directions on the XZ plane for eight sections which are shown in Figure 5. Part (**a**) is related to section 1, part (**b**) is related to section 2, and so on. Using integration methods, we obtained the surface strain values on the XZ plane. The vertical and horizontal axes are the elevation data and the length of the cross-sections, respectively.  
The average volumetric strain in the study area is 14.68 percent. Figure 8 illustrates the volume strain changes in various sections of the study area, indicating that the greatest volumetric strain occurs along Sections 5 and 6, which contain the North Birjand fault's youngest splay.