A continuum model for deformable, second gradient porous media partially saturated with compressible fluids

Abstract

In this paper a general set of equations of motion and duality conditions to be imposed at macroscopic surfaces of discontinuity in partially saturated, solid-second gradient porous media are derived by means of the Least Action Principle. The need of using a second gradient (of solid displacement) theory is shown to be necessary to include in the model effects related to gradients of porosity. The proposed governing equations include, in addition to balance of linear momentum for a second gradient porous continuum and to balance of water and air chemical potentials, the equations describing the evolution of solid and fluid volume fractions as supplementary independent kinematical fields. The presented equations are general in the sense that they are all written in terms of a macroscopic potential Psi which depends on the introduced kinematical fields and on their space and time derivatives. These equations are suitable to describe the motion of a partially saturated, second gradient porous medium in the elastic and hyper-elastic regime. In the second part of the paper an additive decomposition for the potential Psi is proposed which allows for describing some particular constitutive behaviors of the considered medium. While the potential associated to the solid matrix deformation is chosen in the form proposed by Cowin and Nunziato (1981) and Nunziato and Cowin (1979) and the potentials associated to water and air compressibility are chosen to assume a simple quadratic form, the macroscopic potentials associated to capillarity phenomena between water and air have to be derived with some additional considerations. In particular, two simple examples of microscopic distributions of water and air are considered: that of spherical bubbles and that of coalesced tubes of bubbles. Both these cases are suitable to describe capillarity phenomena in porous media which are close to the saturation state. Finally, an example of a simple microscopic distribution of water and air giving rise to a macroscopic capillary potential depending on the second gradient of fluid displacement is presented, showing the need of a further generalization of the proposed theoretical framework accounting for fluid second gradient effects. (C) 2013 Elsevier Ltd. All rights reserved

Flow and Transport in Tight and Shale Formations: A Review

“…A nonnegligible change in the porosity may happen due to chemical reactions such as precipitation and dissolution [170]. The transport properties are typically considered as a function of porosity ( [171][172][173][174][175] and many others) but in an evolving pore skeleton, these properties may change considerably as they depend on the details of the pore-scale geometry. To consider the effects due to the evolving microstructure is of crucial importance for obtaining reliable upscaled models, because otherwise features like pore clogging or damaging of the structure will not be captured at all.…”

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

“…These procedures may also produce continuous models (see, e.g., [87,88]) where the microscopic microstructure is accounted for through constitutive anisotropy or constrained kinematics. Another relevant example is given by solids with interconnected pores, saturated, or partially saturated by compressible fluids (see, e.g., [89][90][91][92]). …”

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“…Naively speaking, this corresponds to the approach taken by the strain-gradient theory where the second gradient of displacement is included in the expression of the strain energy. and Madeo et al [2013] apply the Lagrangian formalism to deduce the evolution equations. This procedure seems analogous to the present theory whereby the field equations are derived through the application of the Lagrangian formalism to the gauge potential.…”