## 2 Construction of Productivity Prediction Method of Reservoir  
### Construction of Fracture Effectiveness Index  
The effectiveness of the reservoir is provide an objective measure of the productivity of an individual well. The study area is characterized by reservoirs with buried hills that have developed fractures serving as storage space and effective seepage channels for oil and gas.  
The effectiveness of fractures is a direct indicator of the reservoir's performance. Two main criteria are used to evaluate the effectiveness of fractured reservoirs: firstly, the degree of fracture development [18; 19], which encompasses the number, scale and distribution of fractures. The existence of fractures greatly improves the permeability of the reservoir and increases the storage space. The better developed the fractures, the higher the quality of the reservoir and the greater the degree of enrichment. The fracture density determined by imaging logging typically reflects the extent of fracture development [20; 21]. Conversely, array acoustic logging enables the characterization of fracture development by measuring changes in shear wave amplitude and attenuation [22; 23]. Fracture seepage capacity, or the conductivity of fractures during fluid migration, is also an important factor [24; 25]. When the degree of fracture development is identical, higher permeability leads to faster oil and gas migration and better supply capacity, which indicates the higher productivity potential of the oil and gas reservoir. Imaging logging is commonly employed to calculate fracture opening and porosity for the characterization of a single fracture's percolation properties. Meanwhile, array acoustic logging can demonstrate the reservoir's permeability based on the amplitude and reflection coefficient of Stoneley wave [26; 27].  
By thoroughly assessing the criteria for evaluating the effectiveness of fractured reservoirs, in conjunction with wall core data and imaging logging technology, we can determine effective fractures and calculate fracture parameters to quantitatively evaluate the longitudinal distribution characteristics of fractures. Using measurements of fracture density to describe the degree of fracture development, the width of fractures, and the porosity of fractures to reflect their Seepage capacity, we have constructed a fracture effectiveness index, Fi.  
\[F_{i}=\frac{\text{Log}(Z_{i}\times\varphi_{i}W_{i})-\text{Log}(Z_{i}\times \varphi_{i}W_{i})_{\text{min}}}{\text{Log}(Z_{i}\times\varphi_{i}W_{i})_{ \text{max}}-\text{Log}(Z_{i}\times\varphi_{i}W_{i})_{\text{min}}} \tag{1}\]  
where \(Fi\) is the fracture effectiveness index, dimensionless; \(Z\) is the fracture density, \(1/\text{m}\); \(W\) is the fracture opening, um; \(\varphi_{i}\) is the fracture porosity, %. The subscript min is the calculated minimum and the subscript max is the calculated maximum.  
Array acoustic logging data has an advantage in detecting fractures at a greater radial depth by employing acoustic parameters such as shear wave amplitude and attenuation parameters. This compensatory method addresses the shallow depth detection limitation of imaging logging, and results in a more comprehensive evaluation of fracture effectiveness.  
\[F_{j}=\frac{\text{Log}(\frac{1}{ATTU}\times AMP)-\text{Log}(\frac{1}{ATTU} \times AMP)_{\text{min}}}{\text{Log}(\frac{1}{ATTU}\times AMP)_{\text{max}}- \text{Log}(\frac{1}{ATTU}\times AMP)_{\text{min}}} \tag{2}\]  
where \(Fj\) is the acoustic effectiveness index, dimensionless; \(ATTU\) is the attenuation of the waveform in the array sound wave, \(1/\text{m}\); \(AMP\) is the amplitude of the sound wave, mv. The subscript min is the calculated minimum value, and the subscript max is the calculated maximum value.