# Weight Assignment Method and Application of Key Parameters in Shale Gas Resource Evaluation
## 5 Case Applications
\begin{bmatrix}1&2.75&47&2.65&613\\ 1&1.74&28.9&2.67&550\\ 1&1.84&32.2&2.68&513.5\\ 1&1.6&31.6&2.67&415.5\\ 1&0.48&35.2&2.6&414.4\\ \end{bmatrix} \]  
Calculate the resource amount of each group of data using the volume method:  
\[Y= \begin{bmatrix}209960.16\\ 73844.99\\ 81535.91\\ 56090.51\\ 18204.43\\ \end{bmatrix} \]  
Invert the matrix:  
\[\left[X^{\prime}\right]= \begin{bmatrix}1&1&1&1&1\\ 2.75&1.74&1.84&1.6&0.48\\ 47&28.9&32.2&31.6&35.2\\ 2.65&2.67&2.68&2.67&2.6\\ 613&550&513.5&415.5&414.4\\ \end{bmatrix} \]  
Invert the matrix:  
\[\left[X^{\prime}\right]= \begin{bmatrix}1&1&1&1&1\\ 2.75&1.74&1.84&1.6&0.48\\ 47&28.9&32.2&31.6&35.2\\ 2.65&2.67&2.68&2.67&2.6\\ 613&550&513.5&415.5&414.4\\ \end{bmatrix} \]  
Using the least squares method to calculate the regression vector parameters, the following matrix is the calculation process:  
\[\left[X\right]^{-1}= \begin{bmatrix}28.874&121.597&-229.347&82.747&-2.871\\ 0.717&2.054&-4.274&2.318&-0.815\\ 0.005&-0.222&0.295&-0.118&0.041\\ -11.204&-44.28&83.361&-28.701&0.824\\ -0.001&0.001&0.01&-0.012&0.002\\ \end{bmatrix} \]  
\[\left[XX^{\prime}\right]= \begin{bmatrix}377993.59&338521.16&316302.06&256199.18&255690.81\\ 338521.16&30346.37&283366.94&229449.15&228946.06\\ 316302.06&283366.94&264730.66&214387.87&213936.69\\ 256199.18&229449.15&214387.87&173649.50&173304.23\\ 255690.81&228946.06&213936.69&173304.23&172974.39\\ \end{bmatrix} \]  
\[\left[XX^{\prime}\right]^{-1}= \begin{bmatrix}377993.59&338521.16&316302.06&25619.18&255690.81\\ 338521.16&303346.37&283366.94&229449.15&228946.06\\ 316302.06&283366.94&264730.66&214387.87&213936.69\\ 256199.18&229449.15&214387.87&173649.50&173304.23\\ 255690.81&228946.06&213936.69&173304.23&172974.39\\ \end{bmatrix} \] \begin{table}
\begin{tabular}{c c c c c} \hline
**Influence** & **q (Gas Content)** & **h (Effective Shale Thickness)** & \(\rho\) **(Shale Density)** & **A (Gas-Bearing Shale Area)** \\
**Factor** & **m\textgreater{}/t** & **m** & **g/cm\textgreater{}3** & **km\textgreater{}2** \\ \hline \multirow{5}{*}{Gas content} & 2.75 & 47 & 2.65 & 613 \\  & 1.74 & 28.9 & 2.67 & 550 \\  & 1.84 & 32.2 & 2.68 & 513.5 \\  & 1.6 & 31.6 & 2.67 & 415.5 \\  & 0.48 & 35.2 & 2.6 & 414.4 \\ \hline \end{tabular}
\end{table} 
Table 8: Raw data. \[\left[X^{\prime}Y\right]=\left[ \begin{array}{c}439636.00\\ 954389.74\\ 17040960.05\\ 1169169.96\\ 242038535.11\\ \end{array} \right]\]  
\[\hat{B}=\left[ \begin{array}{c}68065920985784.10\\ 60949214766047.00\\ 56952344039223.30\\ 46134321154515.40\\ 46045691424644.80\\ \end{array} \right]\]  
Calculate \(\hat{y}\) based on the multiple linear regression equation model, where \(\hat{y}\) is the predicted value of the regression equation for the resource quantity Y:  
\[\hat{y}=\left[ \begin{array}{c}31260701225802700.00\\ 1616270170863590.00\\ 1624510758837100.00\\ 119807427433947.00\\ 46045691424644.80\\ \end{array} \right]\]  
Calculate the complex correlation coefficient:  
\[R=\frac{\sum(y-\overline{y})(\hat{y}-\overline{y})}{\sqrt{\sum(y-\overline{y})^ {2}\cdot\sum(\hat{y}-\overline{y})^{2}}} \tag{7}\]  
Through the above formula calculations and matrix operations, the complex correlation coefficient R can be determined to be 0.831, indicating that these four factors are the main influencing factors for calculating shale gas resources using the volume method, namely the gas shale area, effective thickness, and density. The positive gas content influencing factors are directly related to the local resource quantity, and their changes will directly affect the value of the resource quantity in the region. Therefore, the next step of analysis is carried out. Perform significance testing on the data using the F-distribution:  
\[F=\frac{R_{y\cdot 1\cdot 2\cdot\cdot k}^{2}}{1-R_{y\cdot 1\cdot 2\cdot\cdot k}^{2 }}\times\frac{n-k-1}{k} \tag{8}\]  
Among these, the sample size, n, is 20, and the number of independent variables, k, is 4. Calculate F = 8.37 according to the formula.