Coupling Stokes Flow with Inhomogeneous Poroelasticity

We investigate the behaviour of flux-driven flow through a single-phase fluid domain coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linearly elastic solid filled with the same viscous fluid. The material properties of the poroelastic domain, that is permeability and elastic parameters, depend on the inhomogeneous initial porosity field. We identify the dimensionless parameters governing the behaviour of the coupled problem: The ratio between the magnitudes of the driving velocity and the Darcy flows in the poroelastic domain, and the ratio between the viscous pressure scale and the size of the elastic stresses in the poroelastic domain. We consider a perfusion system, where flow is forced to pass from the single-phase fluid to the biphasic poroelastic domain. We focus on a simplified two-dimensional geometry with small aspect ratio and perform an asymptotic analysis to derive analytical solutions. The slender geometry is divided in four regions, two outer domains that describe the regions away from the interface and two inner domains that are the regions across the interface. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanical response when coupled with a fluid domain. The analysis reveals that, in the interfacial zone, the fluid and the elastic behaviours of this coupled Stokes-poroelastic problem can be treated separately via (i) a Stokes-Darcy coupling and (ii) the solid skeleton being stress free. This latter finding is crucial to derive the coupling condition across the outer domains for both the elastic part of the poroelastic domain and the fluid flow. Via specification of heterogeneous material properties distribution, we reveal the effects of heterogeneity and deformability on the mechanics of the poroelastic domain.