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Well log analysis is crucial 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, focusing on different sections of 5, 10, and 50 data points. Experimental results demonstrate that this approach achieves superior accuracy in filling data gaps compared to other deep learning models for spatial series analysis. The method yielded R^2 values of 0.921, 0.899, and 0.594, with corresponding mean absolute percentage error (MAPE) values of 8.320, 0.005, and 151.154, and mean absolute error (MAE) values of 0.012, 0.005, and 0.032, respectively. These results set a new benchmark for data integrity and utility in geosciences, particularly in well log data analysis.