## 5 Surface Subsidence Prediction Method for UCG
### Basic Roof Subsidence Space Calculate
#### 5.1.2 Roof Subsidence in the Combustion Cavity  
After the completion of gasification, several combustion cavities with roof exposure exist between isolated coal pillars. In general, the length of each mining combustion cavity is much larger than its width. Thus, it can be regarded as a simply supported beam system under uniform load. In this case, the differential deflection equation of the basic roof rock beam above the combustion area is as follows:  
\[E_{r}I_{r}\frac{d^{4}w_{b}(x)}{dx^{4}}=q(n(cl^{\prime}+gl^{\prime})<x<n(cl^{ \prime}+gl^{\prime})+gl^{\prime},n=1,2,3,4\ldots) \tag{2}\]  
\(E_{r}\) is the average elastic modulus of the basic roof, due to the roof burnt \(E_{r}=\sum\limits_{i=1}^{k}E_{ri}/k\), \(k\) is the number of layers within the influence range of the temperature field inside the roof, \(I_{r}\) is the moment of inertia of the roof, \(I_{r}={h_{r}}^{3}/12\), \(w_{b}\) is the relative deflection of the roof in the combustion cavity, \(q\) is the load of the overburdened strata \(q=rH\), \(r\) is the average bulk density of the overlying strata, \(H\) is the thickness of the roof to the surface, and \(n\) is the number of combustion cavity. The boundary equation under the simply supported beam condition is as follows:  
\[x=n(cl^{\prime}+gl^{\prime}),\ wb=0,\ \frac{\partial^{2}w_{b}}{\partial x^{2}}=0 \tag{3}\]  
\[x=n(cl^{\prime}+gl^{\prime})+gl^{\prime},\ wb=0,\ \frac{\partial^{2}w_{b}}{ \partial x^{2}}=0 \tag{4}\]  
The roof deflection of the combustion cavity can be solved as follows:  
\[ \begin{array}{c}w_{b}(x)=\frac{q(x-n(cl^{\prime}+gl^{\prime}))}{24E_{r}I_{r }}[g{l^{\prime}}^{3}-2 gl^{\prime}[x-n(cl^{\prime}+gl^{\prime})]^{2}+[x-n(cl^ {\prime}+gl^{\prime})]^{3}\}\\ (n(a^{\prime}+b^{\prime})<x<n(a^{\prime}+b^{\prime})+a^{\prime},n=1,2,3,4 \ldots)\end{array}  \tag{5}\]  
compression of coal pillar caused by stress concentration, the isolated coal pillar can be regarded as a continuously distributed Winkler elastic foundation (Figure 7). In this case, its equivalent elastic coefficient is set as \(k\), and then, the calculation method of \(k\) is as follows:  
\[\frac{nE_{c}cl^{\prime}}{h}=(ncl^{\prime}+(n+1)gl^{\prime})k \tag{6}\]  
\(E_{c}\) is the elastic modulus of coal, and \(h\) is the thickness of the coal seam. In this case,  
\[k=\frac{NE_{c}gl^{\prime}}{(Ngl^{\prime}+(N+1)cl^{\prime})h} \tag{7}\]  
Assuming that the load acting on the coal pillar is a uniform load, the additional load \(q\) acting on the elastic foundation resulting in the compression of the coal pillar can be calculated by the following formula:  
\[q^{\prime}=\frac{q(cl^{\prime}+gl^{\prime})/cl^{\prime}-q}{(cl^{\prime}+gl^{ \prime})/cl^{\prime}} \tag{8}\]  
According to [PERSON]'s elastic foundation theory, the compression of the coal pillar is directly proportional to the pressure at the point. \[\sigma=kw_{c}(x) \tag{9}\]  
Thus, the differential equation of roof deflection continuously can be derived as follows:  
\[E_{r}I_{r}\frac{d^{4}w_{c}(x)}{dx^{4}}+kw_{c}(x)=q^{\prime} \tag{10}\]  
The general solution of the homogeneous differential equation above is as follows:  
\[w_{c}(x)=e^{-\lambda x}(C_{1}\sin\lambda x+C_{2}\cos\lambda x)+e^{\lambda x}( C_{3}\sin\lambda x+C_{4}\cos\lambda x) \tag{11}\]  
Among them, \(\lambda=\sqrt[4]{\frac{k}{4E_{r}I_{r}}}C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are undetermined coefficients. The particular solution of the non-homogeneous differential equation is as follows:  
\[w_{c}*(x)=\frac{q^{\prime}}{k} \tag{12}\]  
Then, the solution of the differential equation is as follows:  
\[w_{c}(x)=e^{-\lambda x}(C_{1}\sin\lambda x+C_{2}\cos\lambda x)+e^{\lambda x}( C_{3}\sin\lambda x+C_{4}\cos\lambda x)+\frac{q^{\prime}}{k} \tag{13}\]  
Figure 7: Schematic diagram of equivalent elastic foundation. Assuming that the two ends of the beam on the elastic foundation are simply supported, there are boundary conditions.