  We consider the numerical behavior of the fixed-stress splitting method for
coupled poromechanics as undrained regimes are approached. We explain that
pressure stability is related to the splitting error of the scheme, not the
fact that the discrete saddle point matrix never appears in the fixed-stress
approach. This observation reconciles previous results regarding the pressure
stability of the splitting method. Using examples of compositional
poromechanics with application to geological CO$_2$ sequestration, we see that
solutions obtained using the fixed-stress scheme with a low order finite
element-finite volume discretization which is not inherently inf-sup stable can
exhibit the same pressure oscillations obtained with the corresponding fully
implicit scheme. Moreover, pressure jump stabilization can effectively remove
these spurious oscillations in the fixed-stress setting, while also improving
the efficiency of the scheme in terms of the number of iterations required at
every time step to reach convergence.
