Poro-viscoelasticity modelling based on upscaling quasistatic fluid-saturated solids

Abstract

In this paper, quasistatic models are developed for the slow flow of compressible fluids through porous solids, where the solid exhibits fading memory viscoelasticity. Problems of this type are important in practical geomechanics contexts, for example, in the context of fluid flow through unconsolidated reservoir sands and of wellbore deformation behaviour in gas and oil shale reservoirs, all of which have been studied extensively. For slow viscous fluid flow in the poro-viscoelastic media we are able to neglect the dynamic effects related to inertia forces, as well as the dissipation associated with the viscous flows. This is in contrast to the vast body of work in the poro-elastic context, where much faster flow of the viscous fluids may give rise to memory effects associated with microflows in pores of the solid media. Such problems have been treated extensively in both the dynamic and quasistatic cases. We are taking a specific type of the porous medium subject to slow deformation processes possibly inducing moderate pressure gradients and flow rates characterised by negligible inertia effects. As the result of homogenisation of such a two-phase medium, we observe the fading memory behaviour in the Biot modulus which controls the pressure increase due to skeleton macroscopic deformation and pore fluid content. Although our derivation leads to a result which is consistent with the formal phenomenological approach proposed by Biot (J Appl Phys 23:1482-1498, 1962, we offer the reader more insight into the structure of the poro-viscoelastic constitutive relations obtained; in particular, we can show that the Biot compressibility evolves in time according to the creep function while the skeleton stiffness is driven by the relaxation function.

Optimization of the porous material described by the Biot model

“…Assuming slow flows through a slightly deforming porous structure undergoing quasistatic loading of the solid phase by external forces, whereby the inertia forces can be neglected, the upscaled medium is described by the Biot-Darcy coupled system of equations which was derived by the homogenization of two decoupled problems: 1) deformation of a porous solid saturated by a slightly compressible static fluid and 2) Stokes flow through the rigid porous structure, cf. [19,24,21]. In a more general setting, homogenization of the viscous flow in deforming media was considered in [17].…”

Homogenization of the fluid-saturated piezoelectric porous media

“…As far as the quasistatic problems are considered, cf. [23], the permeability can be obtained for a specific geometry by the standard homogenization of the Stokes flow, see e.g. [25].…”

Computational homogenization and nonlinear effects in heterogeneous fluid saturated porous media

“…The homogenization methods can be applied to describe the limit models arising from asymptotic analyses of the problems for deformed elastic porous solids and steady flows, as reported in [4], cf. [5]. In this way, the Biot-Darcy model of poroelastic media for quasistatic problems is derived; it is constituted by the following equations involving the displacement and pressure fields, denoted by u and p, respectively (note e(u) = 1 2 (∇u + (∇u) T ) is the linear strain),…”