Machine learning applications in reservoir modeling are hindered by the limited availability of well log data, a common challenge in the oil and gas industry. We propose VAEc-tMC, a domain-informed Conditional Variational Autoencoder that generates synthetic well-log data conditioned on rock type. Addressing a critical gap by existing generative models that rely solely on statistical reconstruction, our model embeds geological domain knowledge into the latent space, and optimizes a modified objective with an adaptive Student-t reconstruction loss and a beta-weighted KL regularizer, improving stability under heavy-tailed data. When used for data augmentation, the synthetic samples preserve inter-log dependencies and substantially enhance downstream classification, accuracy 39→63%, F1-score 36→68%, AUC 0.46→0.80 on a held-out well. Beyond the geological context, the proposed approach illustrates a generalizable strategy where domain-aware generative models with adaptive loss functions provide a robust solution for data-efficient learning in scientific domains facing data scarcity, noise, and heavy-tailed distributions.