Consolidation around a borehole embedded in media with double porosity under release of geostatic stresses

The paper deals with the stress, displacement, pore and fissure pressures fields induced by the drilling and/or the pressurization of a vertical borehole in a formation of water-saturated porous media with double porosity. The solution includes the boundary condition of non-hydrostatic in situ state of stress. The solid skeleton is assumed to behave as a linearly poroelastic material with compressible constituents. The analytical solution is derived in Laplace’s space and transformed to the time domain using a numerical inversion technique. The histories of pore and fissure pressures are illustrated to show the influence of permeabilities of the pore and fissure systems.     

COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.     

Explicit Effective Constants for an Inhomogeneous Porothermoelastic Medium

An arbitrary anisotropic micro-inhomogeneous (composite) poroelastic medium is considered, containing a random set of ellipsoidal inhomogeneities with different poroelastic characteristics. The properties of these constituents are described by the linear porothermoelastic theory of Biot. One of the self-consistent schemes named effective field method is used to develop explicit expressions for the effective porothermoelastic constants (tensor of the frame elastic compliances , tensor of the generalized Skempton’s coefficients , tensor of thermal expansion coefficients , Biot’s constants , and the heat capacity at constant stress for the static porothermoelastic theory. It is shown that for two components composite porous material these expressions are interconnected and can be expressed only via the components of tensor . Some special cases are considered for the isotropic main material (matrix).     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

Three-dimensional finite/infinite elements analysis of fluid flow in porous media

Fluid flow in naturally fractured porous media can always be regarded as an unbounded domain problem and be better solved by finite/infinite elements. In this paper, a three-dimensional two-direction mapped infinite element is generated and combined with conventional finite elements and one direction infinite element to simulate poroelasticity. Therefore, the entire semi-infinite domain can be included in the numerical analysis. Both single- and dual-porosity porous media are considered. For the purpose of validation, we compare the results of finite/infinite elements with those of finite elements under two extreme boundary conditions. The comparison indicated that mapped infinite element is an appropriate approach to model fluid flow in porous media and provides an intermediate solution.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Cylindre transverse isotrope poroélastique chargé cycliquement : application à un ostéonCyclic loading of a transverse isotropic poroelastic cylinder: a model for the osteon

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. This cylinder is considered as a model for an osteon, the basic unit of cortical bone. The fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed, as this could influence bone remodeling. To cite this article: A. Rémond, S. Naili, C. R. Mecanique 332 (2004).     

Compressible and incompressible constituents in anisotropic poroelasticity: The problem of unconfined compression of a disk

The governing equations for the theory of anisotropic poroelastic materials with incompressible constituents undergoing small deformations are developed from the theory of anisotropic poroelastic materials without the constituent incompressibility constraint. The development of the constituent specific incompressibility constraint is accomplished by restricting the elastic constants rather than by introducing Lagrange multipliers, the conventional method. The advantage of the approach is insight into the nature of the elastic response that characterizes incompressible poroelasticity. An application of the theory to the unconfined compression of a circular porous disk is presented to illustrate the effects of compressibility vs. incompressibility and transverse isotropy vs. isotropy.     

Transverse isotropic poroelastic osteon model under cyclic loading

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. This cylinder models an osteon, basic unit of cortical bone. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. Fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed as it could influence bone remodeling. Strain rate of loading is shown to play a significant role in mass flux in the porous material.