Our paper focuses on Interactive Visual Narratives (IVNs), a digital storytelling mode where users can interact, manipulate and co-create narratives. We propose a unified IVN Model that facilitates discourse production in IVNs, providing users control while generating narratives. We aim to bring a design perspective to IVN studies by reconsidering narrative discourse production techniques. Our study aims to identify the elements and structure of an IVN and how they integrate to form the structure. We conducted a morphological study to answer questions about narrating and visually presenting a story. The IVN Model, which explains the structure and production of discourse, comprises Story (What) and Discourse (How). The IVN Construction Method and IVN Discourse Mode visualize the discourse and its manifestation. There is an Order of presentation and an Order of telling. The elements of IVN include the Key Elements, Formation Devices and Presentation Devices. Our study found a set of consistent elements present across all IVNs. The Key Elements are the foundational elements without which an IVN cannot exist. The Formation Devices help decipher the discourse mode used, while the Presentation Devices facilitate the stylistic presentation of an IVN.