change GET query string

When user submits filter form the url will be example.com/search?brand=&model=foo

and then I filter products according to these values in GET field, when filtering I discard that 'brand' because it doesnt have a value.

then I load same view again with filtered data. 

But in browser stays same example.com/search?brand=&model=foo 

I can delete brand from url  using javascript before submitting filter form. But I want to know if there is way to delete brand from url in php
I mean when send example.com/search?brand=&model=foo request the responded url will be example.com/search?model=foo

if I build url string using http_build_query() It wont change the url field in browser it example.com/search?brand=&model=foo