  In this paper, a parallel domain decomposition method is proposed for solving
the fully-mixed Stokes-dual-permeability fluid flow model with Beavers-Joseph
(BJ) interface conditions. Three Robin-type boundary conditions and a modified
weak formulation are constructed to completely decouple the original problem,
not only for the free flow and dual-permeability regions but also for the
matrix and microfractures in the dual-porosity media. We derive the equivalence
between the original problem and the decoupled systems with some suitable
compatibility conditions, and also demonstrate the equivalence of two weak
formulations in different Sobolev spaces. Based on the completely decoupled
modified weak formulation, the convergence of the iterative parallel algorithm
is proved rigorously. To carry out the convergence analysis of our proposed
algorithm, we propose an important but general convergence lemma for the
steady-state problems. Furthermore, with some suitable choice of parameters,
the new algorithm is proved to achieve the geometric convergence rate. Finally,
several numerical experiments are presented to illustrate and validate the
performance and exclusive features of our proposed algorithm.
