  The incompressible $\mu(I)$-rheology has been used to study subaerial
granular flows with remarkable success. For subaquatic granular flows, drag
between grains and the pore fluid is substantially higher and the physical
behaviour is more complex. High drag forces constrain the rearrangement of
grains and dilatancy, leading to a considerable build-up of pore pressure. Its
transient and dynamic description is the key to modelling subaquatic granular
flows but out of the scope of incompressible models. In this work, we advance
from the incompressible $\mu(I)$-rheology to the compressible
$\mu(J)$-$\phi(J)$-rheology to account for pore pressure, dilatancy, and the
scaling laws under subaquatic conditions. The model is supplemented with
critical state theory to yield the correct properties in the quasi-static
limit. The pore fluid is described by an additional set of conservation
equations and the interaction with grains is described by a drag model. This
new implementation enables us to include most of the physical processes
relevant for submerged granular flows in a highly transparent manner. Both, the
incompressible and compressible rheologies are implemented into OpenFOAM and
various simulations at low and high Stokes numbers are conducted with both
frameworks. We found a good agreement of the $\mu(J)$-$\phi(J)$-rheology with
low Stokes number experiments, that incompressible models fail to describe. The
combination of granular rheology, pore pressure, and drag model leads to
complex phenomena such as apparent cohesion, remoulding, hydroplaning, and
turbidity currents. The simulations give remarkable insights into these
phenomena and increase our understanding of subaquatic mass transports.
