  This work is concerned with the iterative solution of systems of quasi-static
multiple-network poroelasticity (MPET) equations describing flow in elastic
porous media that is permeated by single or multiple fluid networks. Here, the
focus is on a three-field formulation of the problem in which the displacement
field of the elastic matrix and, additionally, one velocity field and one
pressure field for each of the $n \ge 1$ fluid networks are the unknown
physical quantities. Generalizing Biot's model of consolidation, which is
obtained for $n=1$, the MPET equations for $n\ge1$ exhibit a double saddle
point structure.
  The proposed approach is based on a framework of augmenting and splitting
this three-by-three block system in such a way that the resulting block
Gauss-Seidel preconditioner defines a fully decoupled iterative scheme for the
flux-, pressure-, and displacement fields. In this manner, one obtains an
augmented Lagrangian Uzawa-type method, the analysis of which is the main
contribution of this work. The parameter-robust uniform linear convergence of
this fixed-point iteration is proved by showing that its rate of contraction is
strictly less than one independent of all physical and discretization
parameters.
  The theoretical results are confirmed by a series of numerical tests that
compare the new fully decoupled scheme to the very popular partially decoupled
fixed-stress split iterative method, which decouples only flow--the flux and
pressure fields remain coupled in this case--from the mechanics problem. We
further test the performance of the block triangular preconditioner defining
the new scheme when used to accelerate the GMRES algorithm.
