It looks like the $project_type variable is not defined anywhere in the scope of your artwork page. You have to look into how to get the artwork’s project type on the artwork page.
Is it a custom taxonomy? If yes, you would use functions like get_the_terms. I don’t know what the project type taxonomy is called (how it was registered), so you will have to the the research on your own.
Maybe it is defined as post meta? In this case you would use get_post_meta function.
Whatever it is, once you get the term (or post meta) you can print its value using print_r( get_the_terms( $post->ID, 'taxonomy_name' ) ); to learn how the data is structured there and how to extract what you need from it (the taxonomy name and taxonomy link).
What you said in your last comment – it might be useful, yes, but it’s hard to say without seeing the entire class. By the name of the function – it looks like it might get just the link to portfolio for you. I’m not sure if (and where) the class is instantiated so can’t tell you for sure if you can use its functions on your artwork page. Again, can’t help without seeing the code.