In numerical modeling of two-phase flows, a longstanding focus has been the treatment of contact lines, defined as the intersection of a fluid-fluid interface with a solid wall boundary. Contact lines play an important role in the fluid mechanics of a variety of applications, including flow in porous media, electrochemical systems, microfluidics, and industrial processes such as printing and coating. Various treatments have been proposed for various two-phase flow methods that attempt to model the physics of contact lines. In this work, we present an analysis of contact line models for a two phase flow method known as the Conservative Diffuse Interface (CDI) model, which models two phase flow by coupling the Navier-Stokes equations to an equation for a phase indicator function known as the phase field that distinguishes between the two phases. As a diffuse interface method, CDI employs a regularization term that maintains the interface as a computationally tractable diffuse region of finite thickness. For CDI, the regularization is based on second-order spatial derivatives, in contrast to the popular Cahn-Hilliard model, for which the regularization is based on fourth-order spatial derivatives. The second-order nature of CDI enables simpler and more efficient numerical implementations compared to the Cahn-Hilliard model but also presents challenges when it comes to the dynamics of contact lines and maintaining physical behavior of contact line speed and contact angle. Because the Cahn-Hilliard model employs a fourth-order spatial operator, it admits two wall boundary conditions for the phase field, and well-established contact line models for the Cahn-Hilliard model often rely on both boundary conditions to simultaneously maintain mass conservation and incorporate contact line physics. CDI is second-order and therefore admits only one wall boundary condition for the phase field, so a different strategy is required. In this presentation, we present a contact line treatment for CDI. The phase field boundary condition, along with the near-wall discretizations for regularization and surface tension, is designed to maintain mass conservation while having no effect on contact line physics. Discretization error in computing regularization and surface tension can lead to spurious contact line motion, but accuracy improvements are offered that minimize this effect. Contact line physics is then incorporated using a slip boundary condition based on the generalized