# Study of the Migrating Mine Gas Piston Effect during Reactivation of Tectonic Faults
## 3 Computer-Generated Model and Its Parameters
#### 3.1.1 Estimation of Void Space Volume Change  
We offer an estimate of the change in the volume of the void space adjacent to the fault plane based on the value of a single-step displacement along the fault plane. The magnitude of such shift can be estimated from the results of in-mine observations and on the basis of theoretical concepts describing pulsating mode of fault wall displacement. When fault walls are displaced, seismic energy is released in a dynamic form, and the magnitude of such displacement in the source zone can be estimated by the amount of seismic energy. [PERSON] explained the relationship between the earthquake energy \(E\) and volume of the source zone \(V_{sz}\)[49]:  
\[E(erg)=1000~{}Vsz~{}(\mathrm{cm}^{3}) \tag{3}\]  
From this and similar empirical equations [50; 51], it can be calculated that the size of the fault source zone during an earthquake with up to K = 5 energy class does not exceed 10 m. It is proposed to estimate the lower limit of the displacement amplitude \(A\) (m) for brittle-plastic deformations under triaxial compression in the source zone \(L_{sz}\) (km) by the following formula [52]:  
\[lgA=0.5 lgL_{sz} \tag{4}\]  
Based on Equation (4), it can be found that the displacement amplitude for K = 5 energy class earthquake is just a few millimeters. In-field observations of the rock burst manifestations with the fault-slip mechanism also show that the amplitude of displacements along fault planes during seismic events of class K < 5 does not exceed 1 cm [37].  
To estimate the volume of the void space \(V_{vs}\) and its change \(\Delta V\) during the fault reactivation, we use the simplest model of the fault walls' contact shown in Figure 1. The \(ABCD\) figure with the indicated parameters \(h\), \(\theta\) depicts the void space (cavity) here, and the segment \(DE=Sc\) depicts the contact patch of the walls of the fault, \(A_{aV}=EF\). From geometric considerations, we can conclude that the volume of voids \(V_{vs}\) shown in Figure 1 in a fault plane section with length \(L\) and width \(W\) will be:  
\[V_{vs}=A_{aV}\cdot L\cdot W\cdot(1-n) \tag{5}\]  
where \(n\)--is a parameter characterizing the total wall contact area and \(n=\mathrm{DE/AE}\), Figure 1.  
When the walls are displaced along the arrow (Figure 1), the volume of the void space decreases by \(\Delta V\) and parameter n changes from \(n_{1}\) to \(n_{2}\).  
\[\frac{\Delta V}{V}=\frac{n_{2}-n_{1}}{1-n_{1}} \tag{6}\]  
Assuming that \(A_{aV}\) = 4 cm and \(\theta\) = 10-15\({}^{\circ}\), then we can obtain that, with a wall displacement of 1 cm, \(\Delta V\) equals fractions of a percent.  
This certainly is a very rough estimate. The volume between asperities can be filled with fault plane contact disintegration products (debris) or secondary minerals, and the single-step displacement amplitude can be an order of magnitude less. Nevertheless, the estimates made allow us to assume that, in case of a single-step fault wall displacement, the volume of free space along the fault plane will decrease by hundredths or tenths of a percent. Taking into account Equation (2), the pressure increment during displacement along the fault should be estimated as 250 Pa and more.