Argument mining has garnered increasing attention over the years, with the recent advancement of Large Language Models (LLMs) further propelling this trend. However, current argument relations remain relatively simplistic and foundational, struggling to capture the full scope of argument information. To address this limitation, we propose a systematic framework comprising 14 fine-grained relation types from the perspectives of vertical argument relations and horizontal discourse relations, thereby capturing the intricate interplay between argument components for a thorough understanding of argument structure. On this basis, we conducted extensive experiments on three tasks: argument component prediction, relation prediction, and automated essay grading. Additionally, we explored the impact of writing quality on argument component prediction and relation prediction, as well as the connections between discourse relations and argumentative features. The findings highlight the importance of fine-grained argumentative annotations for argumentative writing assessment and encourage multi-dimensional argument analysis.