## 2 Materials and Methods  
### The Porosity and Permeability of Goaves with Caving  
One of the most important properties of roof rocks determining their ability to transform into caving is the tensile strength. This strength is the natural ability of the rock mass to resist stratification and caving of the roof rocks into the space (void) left after the mined coal as a result of vertical forces [39; 40].  
The value of the tensile strength of the roof rocks is determined by means of a down-hole penetrometer or the direct method--by stretching the sections of the vertical core of the borehole in the direction of the longitudinal axis of the borehole, and then it is determined from the following relationship:  
\[R_{tri}=0.8\frac{F}{d^{2}} \tag{1}\]  
where \(R_{tri}\) is tensile strength of the rocks (Pa), \(F\) is the applied axial load (N) and \(d\) is core diameter (m\({}^{2}\)).  
This value depends on the type of roof rocks forming the caving. The maximum value of the tensile strength of rocks in Polish mines amounts to approximately 8 MPa [1; 2]. In practice, however, such value is rare. Table 1 presents the types of roof rocks and the values of their tensile strength, as well as the characteristics of the roofs formed by these rocks.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tensile Strength of Roof Rock, Rrs, MPa** & **Description of Roof** & **Example of Rock** \\ \hline
0–0.5 & & Roof falling immediately after unveiling & clay and sandy slates, coals \\
0.5–1.5 & & Falling roof (clay and sandy slates) and weakly & clay and sandy slates, coals \\  & & self-supporting (coal) & \\
1.5–3.0 & & Crack roof, partially self-supporting and & \\  & & bearing, easily passing into a caving state & \\
3.0–4.5 & & Self-supporting roof, it goes automatically into & \\  & & a caving state without sagging into govaves & \\
4.5–6.0 & & Supporting roof, without roof falls, hardly & \\  & & passing into caving state, sagging into govaves & \\  & & Strongly compacted roof, very difficult for & \\  & & passing into caving state & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The types of roof rocks and the values of their tensile strength (own study based on [1; 2]).  
After such calculation of the tensile strength of roof rocks, it is possible to determine the permeability coefficient of goaves with caving, using the following equation [41]:  
\[k(x)=\frac{\mu_{g}}{r_{0}+ax^{2}}\text{ for }0\;\leq\;x\;\leq\;2/3\cdot l \tag{2}\]  
as well as the equation:  
\[k(x)=\frac{\mu_{g}}{r_{0}+a\left(\frac{3}{3}l-x\right)^{2}}\text{ for }2/3\cdot l \;\leq\;x\;\leq\;l \tag{3}\]  
where \(k(x)\) is permeability (m\({}^{2}\)), \(\mu_{g}\) is the coefficient of dynamic viscosity of air (Nsm\({}^{-2}\)), \(l\) the total length of the longitudinal longwalls (m), \(r_{0}\) we determine from dependence \(r_{0}=\frac{\mu}{k_{0}}\) and \(a\) we determine from dependence \(a=6\cdot 10^{9}R_{rms}{}^{-1.74}\).  
The value of the permeability coefficient of caving goaves \(k_{0}\) behind the front of the longwall is determined from the following equation [41]:  
\[k_{0}=\frac{\mu_{g}}{6}\cdot 10^{-10}R_{rms}{}^{1,44} \tag{4}\]  
The porosity of goaves varies on the basis of \"O-zone theory\" [42]. The porosity distribution along the strike direction in the middle of the working face goaves is determined from the following equation [41]:  
\[n_{x}=0.2e^{-0.0223x}+0.1 \tag{5}\]  
where \(n_{x}\) is the porosity distribution along the middle line of working face of longwall in goaves (%); \(x\) is the x position of the goaves (m).  
The porosity of goaves in dip distribution can be determined from the following equation [43]:  
\[n_{y}=e^{-0.015y}+1\text{ for }0<y<\frac{L}{2} \tag{6}\]  
\[n_{y}=e^{-0.015(L-y)}+1\text{ for }\frac{L}{2}<y<L \tag{7}\]  
where \(n_{y}\) is the porosity distribution along dip direction (%), \(y\) is the y position of the goaves (m); \(L\) is the length of the working face of longwall (m).