  We consider the interaction between a poroelastic structure, described using
the Biot model in primal form, and a free-flowing fluid, modelled with the
time-dependent incompressible Stokes equations. We propose a diffuse interface
model in which a phase field function is used to write each integral in the
weak formulation of the coupled problem on the entire domain containing both
the Stokes and Biot regions. The phase field function continuously transitions
from one to zero over a diffuse region of width $\mathcal{O}(\varepsilon)$
around the interface; this allows the equations to be posed uniformly across
the domain, and obviates tracking the subdomains or the interface between them.
We prove convergence in weighted norms of a finite element discretisation of
the diffuse interface model to the continuous diffuse model; here the weight is
a power of the distance to the diffuse interface. We in turn prove convergence
of the continuous diffuse model to the standard, sharp interface, model.
Numerical examples verify the proven error estimates, and illustrate
application of the method to fluid flow through a complex network, describing
blood circulation in the circle of Willis.
