Abstract
We perform direct numerical simulations of the flow through a model of deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus G, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux (q) is a nonlinear function - steeper than linear - of the pressure-difference (ΔP) across the medium. Furthermore, the flux is larger for a softer medium (smaller G). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is an universal function of ΔP/G, which is confirmed by the present simulations.
Highlights
Percolation of water through soil is one of the oldest problems in hydrodynamics
The simplest poroelastic problem is that of linear poroelasticity where we assume that the flow of the liquid is governed by the Darcy’s law and the solid skeleton has linear constitutive relation and undergoes small deformation
Different elastic models will result in different expression for the function f (b) in eqn (9)
Summary
Percolation of water through soil is one of the oldest problems in hydrodynamics. The fluid passes through a network of irregularly arranged interstices between solid objects. Poroelasticity play an important role in understanding the transport through a wide range of materials ranging from individual cells,[2,3] to biological tissues, e.g., soft-tissues,[4,5,6] bones,[7] even to hydraulic fracture.[8,9] The simplest poroelastic problem is that of linear poroelasticity where we assume that the flow of the liquid is governed by the Darcy’s law and the solid skeleton has linear constitutive relation and undergoes small deformation. As our model we choose a bed, a two-dimensional hexagonal lattice with defects, of soft elastic cylinders immersed in a liquid Using both direct numerical simulations – a set of fully coupled equations for a viscoelastic solid in contact with a Newtonian fluid – and theory, we show that at scales that are large compared to the diameter of a cylinder the flux versus pressure-difference relationship in the system is a Darcy-like equation.
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