Well log analysis is significant for hydrocarbon exploration, providing detailed insights into subsurface geological formations. However, gaps and inaccuracies in well log data, often due to equipment limitations, operational challenges, and harsh subsurface conditions, can introduce significant uncertainties in reservoir evaluation. Addressing these challenges requires effective methods for both synthetic data generation and precise imputation of missing data, ensuring data completeness and reliability. This study introduces a novel framework utilizing sequence-based generative adversarial networks (GANs) specifically designed for well log data generation and imputation. The framework integrates two distinct sequence-based GAN models: time series GAN (TSGAN) for generating synthetic well log data and sequence GAN (SeqGAN) for imputing missing data. Both models were tested on a dataset from the North Sea, Netherlands region. For the imputation task, the input comprises logs with missing values and the output is the corresponding imputed logs; for the synthetic data generation task, the input is complete real logs and the output is synthetic logs that mimic the statistical properties of the original data. All log measurements are normalized to a 0-1 range using min-max scaling, and error metrics are reported in these normalized units. Different sections of 5, 10, and 50 data points were used. Experimental results demonstrate that this approach achieves superior accuracy in filling data gaps compared to other deep learning models for spatial series analysis. The imputation method yielded \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {R}^{2}$$\end{document} values of 0.92, 0.86, and 0.57, with corresponding mean absolute percentage error (MAPE) values of 8.320, 0.005, and 166.6, and mean absolute error (MAE) values of 0.012, 0.002, and 0.03, respectively. The synthetic generation yielded \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {R}^{2}$$\end{document} of 0.92, MAE, of 0.35, and MRLE of 0.01. These results set a new benchmark for data integrity and utility in geosciences, particularly in well log data analysis.