# Quantitative Characterization of Micro-Scale Pore-Throat Heterogeneity in Tight Sandstone Reservoir
## 2 Materials and Methods
### Mathematical Theory and Method  
Information entropy, first proposed by [PERSON], the father of information theory, can be used to measure the uncertainty of information and to represent the complexity of a system [12]. The higher the entropy value of a system, the more complex it is [13; 14; 15; 16; 17; 18]. The essence of information entropy is the quantitative analysis of distribution uniformity. The steps of establishing a quantitative characterization model of pore-throat micro-scale heterogeneity in a tight sandstone reservoir based on information entropy are as follows:  
(1) Pressure and volume of mercury injection in tight sandstone samples are obtained based on the constant velocity mercury injection-experiment;  
(2) Experimental data are processed to obtain the distribution data of pore radius and throat radius in a tight sandstone reservoir, and the distribution-frequency spectrum of the pore radius and throat radius in a tight sandstone reservoir are drawn;  
(3) The distribution interval of the pore radius or throat radius is intercepted, which is taken as the object, and it is assumed that it contains m pores or throats. The occurrence frequency of different pore radii or throat radii are normalized, and a sequence is generated. According to Formula (1), the information entropy of different pore radius or throat radius intervals is calculated, namely, micro-scale pore-radius entropy [16] and micro-scale throat-radius entropy. \[H=-\sum_{i=1}^{m}z_{i}\ln z_{i} \tag{1}\]  
where \(z_{i}\) is the standard value of the occurrence frequency of the \(i\)th pore radius or throat radius. In a certain pore or throat interval, when the occurrence frequency of different pore radii or throat radii are the same, namely, \(z_{1}=z_{2}=\ldots=z_{m}=1/M\), the entropy is at its maximum (In\(M\)). This means that the size distribution of the pores or throat reaches an even state within the pore or throat interval. In practical applications, due to the differences in the number of pores or throats that are contained in each pore or throat interval, the divided pores or throat intervals may be different, and the information entropy of pores' radius or throat radius is often not comparable. Therefore, a dimensionless parameter (\(J\)) is introduced to represent pore uniformity or throat uniformity. According to Formula (2), the uniformity of pores or throats in different scales can be calculated, which is used to quantitatively characterize the uniformity of pore throat distribution at different scales in a tight sandstone reservoir;  
\[J=\frac{-\sum_{i=1}^{m}z_{i}\ln z_{i}}{M} \tag{2}\](4) The overall distribution uniformity of pore radius or throat radius in a tight sandstone reservoir is calculated;  
1 In the process of step 4, it is assumed that \(N\) pore radius or throat radius intervals are divided, and the total frequency of the pore or throat in each interval is \(Q_{j}\) (\(j\) = 1, 2,..., \(N\)). According to Formula (3), the percentage of pore or throat in each interval in the total frequency of pores or throats that are obtained from the rock sample test is calculated. \[P_{j}=\frac{Q_{j}}{\sum_{j=1}^{N}Q_{j}} \tag{3}\]  
2 According to Formula (4), the overall distribution uniformity of pores or throat is calculated. \[M=\sum_{j=1}^{N}p_{j}j_{j} \tag{4}\]  
where \(J\) represents the uniformity of the \(j\)th pore or throat;  
(5) In order to use information entropy to determine the weight of the pore and throat, and to solve the joint uniformity of the pore and throat, it is necessary to analyze the dispersion degree of the pore and throat. Generally, the dispersion degree of the \(j\)th index (pore or throat) depends on \(h_{j}\), and the more dispersed the value distribution of the \(j\)th index (pore or throat) is, the greater the corresponding \(h_{j}\) value is, indicating that the significance of the \(j\)th index (pore or throat) is also higher. Among all \(n\) indexes (pore or throat), the weight of the \(j\)th index (pore or throat) can be determined by Equation (5).