Computing Gröbner bases is known to have a very high upper bound on computation time with respect to input length. Due to the connection between polyhedral geometry and Gröbner bases through the Gröbner fan, one can attempt an incremental approach to compute Gröbner bases. First computing a Gröbner basis with respect to an `easy' term order and transforming that result to a Gröbner basis with respect to the desired term order by using information about this polyhedral fan is done by a family of algorithms termed as Gröbner walk. We implemented two variants of the Gröbner walk in the computer algebra system OSCAR and compared their performance with classical Gröbner basis methods already found in OSCAR.