  The classical Biot's theory provides the foundation of a fully dynamic
poroelasticity model describing the propagation of elastic waves in
fluid-saturated media. Multiple network poroelastic theory (MPET) takes into
account that the elastic matrix (solid) can be permeated by one or several
($n\ge1$) superimposed interacting single fluid networks of possibly different
characteristics; hence the single network (classical Biot) model can be
considered as a special case of the MPET model. We analyze the stability
properties of the time-discrete systems arising from second-order implicit time
stepping schemes applied to the variational formulation of the MPET model and
prove an inf-sup condition with a constant that is independent of all model
parameters. Moreover, we show that the fully discrete models obtained for a
family of strongly conservative space discretizations are also uniformly stable
with respect to the spatial discretization parameter. The norms in which these
results hold are the basis for parameter-robust preconditioners.
