## 2 Research Methods for Reservoir Fluid PVT High-Pressure Physical Property Analysis
### Prediction and Analysis of Formation Crude Oil Viscosity
#### 2.3.3 Construction of Graph Convolution Prediction Model
Data integration: With the assistance of the Laplacian matrix L, the model combines all kinds of information within the graph. At each graph layer, Chebyshev polynomials are employed to approximate the graph signal for the purpose of constructing a filter. Essentially, this method considers the average weight of surrounding nodes as a crucial feature and effectively extracts and combines such information from different data sub-graphs. 2. Feature decomposition: This model adopts a universal approximation strategy that is based on the graph structure pattern. Specifically, the operation of approximating the Laplacian matrix is accomplished through Chebyshev polynomial operations. This operation enables efficient convolution operations on graph-structured data in the spectral domain, along with good scalability. During the operation, the Laplacian matrix can be decomposed into its eigenvectors and eigenvalues. The specific calculation formula is presented in Equation (6). \[L=U\Lambda U^{-1}=U \begin{bmatrix}\lambda_{1}&\cdots&\ldots\\ \vdots&\ddots&\vdots\\ \cdots&\cdots&\lambda_{n}\end{bmatrix} U^{-1}\] (6) In Equation (6), U is represented as a column matrix in the form of \(\text{U}=\left(\overrightarrow{\mu_{1}},\overrightarrow{\mu_{2}},\ldots \overrightarrow{\mu_{n}}\right)\), which is the Chebyshev matrix. Meanwhile, \(\Lambda\) is a diagonal matrix that is composed of the eigenvalues of L, and \(U^{-1}\) refers to the inverse matrix of U. 3. Normalization processing: In this paper, the eigenvectors of the Laplacian matrix were utilized to carry out the convolution operation. To enhance the computational performance as well as the information expression ability, the eigenvectors were normalized in this paper. The specific approach was to divide each eigenvector by the square root of its corresponding eigenvalue. By doing so, each eigenvector would have a unit length, which is beneficial for the subsequent spectral convolution operations. This normalization step guarantees the orthogonality among the eigenvectors. As a result, it realizes the effective conversion from spatial-domain information to frequency-domain information. Consequently, this paper made use of spectral convolution to handle image data and improve its accuracy by means of eigenvector normalization processing. The specific processing formula is presented in Equation (7). \[\widetilde{\mu_{i}}=\frac{\mu_{i}}{\sqrt{\lambda_{i}}},\;\;i=1,2,\ldots,N\] (7) In Equation (7), \(\mu_{i}\) represents the eigenvector that corresponds to node \(i\) within the graph. \(\lambda_{l}\) is the eigenvalue of the Laplacian matrix that corresponds to node i; in other words, it is the eigenvalue associated with that particular eigenvector. \(\widetilde{\mu_{i}}\) stands for the normalized eigenvector, and N indicates the maximum number of nodes present in the graph. Graph convolution: In this step, the graph convolution operation is achieved through the Chebyshev polynomial of the Laplacian matrix. Specifically, the new feature vector of the node is produced by linearly combining the input feature of each node with the features of its adjacent nodes. From a mathematical perspective, graph convolution fundamentally involves the linear weighted sum of a node's feature and the features of its neighboring nodes to create a new feature vector. The calculation formula for graph convolution is presented in Equation (8). This formula encompasses the node feature vector f, the weight parameter \(g_{\theta}\), which is obtained through training, and the learnable parameter \(\theta\) within the model. Additionally, the Hadamard product \(\odot\) was utilized to depict the relationship between the node feature vectors. \[f^{*}g_{\theta}=U\Big{(}\left(U^{T}g_{\theta}\right)\odot\left(U^{T}f\right) \Big{)}\] (8)
5. Matrix product: In the task of predicting correlations within time-series data, the feature vector of the previous time point can play a significant role in effectively grasping the correlation that exists between adjacent time points.