The current Text-based Speech Editing (TSE) techniques have focused on reducing the difference between the generated speech segment and the reference target in the editing region, ignoring its local and global fluency in the context and original utterance. To maintain the speech fluency, we propose a fluency speech editing model, termed FluentEditor, by considering fluency-aware training criterion in the TSE training. Specifically, the acoustic consistency constraint aims to smooth the transition between the edited region and its neighboring acoustic segments consistent with the ground truth, while the prosody consistency constraint seeks to ensure that the prosody attributes within the edited regions remain consistent with the overall style of the original utterance. The subjective and objective experimental results on VCTK demonstrate that our FluentEditor outperforms all advanced baselines in terms of naturalness and fluency. The audio samples and code are available at https://github.com/AI-S2-Lab/FluentEditor.