希望用我的网站取代WP的认证系统

On my website WordPress is located in http://www.mywebsite/blog and is simply another section of the website.

I would like to integrate my own authentication system, which uses php sessions, into WordPress so that a user that registers and logs in my website has access to every sections, without having to register/login a second time when they enter the WordPress section.

In other words I am looking to completly get rid off WordPress's login and register pages, and make it so my regular users conserve their account when they go to the WordPress section and have the same privileges/functionnalities as if they were authentified into WordPress itself. As I would not be using WordPress's wp_users database.

As of now, on my website, I simply start the session, and check the user's privileges before calling functions to grant them permissions/functionnalities access.

I'm currently at step 0 of this project as I found very little useful informations on the topic (related questions like here, here, and here haven't helped me much and are 2/3+ years old). I tried http://wordpress.stackexchange.com but haven't found answers.

Has anyone ever done that ? How can I achieve this (if it is possible) ?

Any suggestions/thoughts on this would be really helpful to me. Thanks