A FFT-based method to compute the permeability induced by a Stokes slip flow through a porous medium

Monchiet, Vincent; Bonnet, Guy; Lauriat, Guy

doi:10.1016/j.crme.2009.04.003

Cited by 36 publications

(40 citation statements)

References 12 publications

(13 reference statements)

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“…The FFT methods have been well considered for various applications in the field of linear elastic homogenization [16][17][18] and has also been adapted to other class of linear problems, among which are the static and dynamic Darcy problem [19,20] or piezoelectricity [21,22]. FFT based iterative schemes have been also successfully applied to non linear constitutive relations including elastoplastic and elastoviscoplastic materials [4,9,[23][24][25][26][27][28] and more recently non local damage models [29,30].…”

Section: Introductionmentioning

confidence: 98%

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

Monchiet¹

2015

Computer Methods in Applied Mechanics and Engineering

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Highlights• FFT based iterative schemes are reformulated in a unified variational approach.• The approach uses classic and Hashin-Shtrikman variational principles. • The method delivers rigorous bounds for the effective elastic properties of composites. • A hierarchy between the different FFT solutions is established. AbstractIn this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin-Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin-Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin-Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.

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Section: Introductionmentioning

confidence: 98%

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

Monchiet¹

2015

Computer Methods in Applied Mechanics and Engineering

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“…A direct way to upscale the permeability would be to resort to numerical homogenisation on explicit representations of the microstructure, either obtained by 3D imaging techniques or computer synthesised [21][22][23][24][25][26][27][28]. We instead turn to analytical modeling with an implicit description of the microstructure.…”

Section: Description Of the Microscopic Scalementioning

confidence: 99%

Experimental Measurements and Multi-Scale Modeling of the Relative Gas Permeability of a Caprock

Bignonnet

Duan

Egermann³

et al. 2016

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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-An experimental characterisation of a low permeability caprock is first reported. Two different methods are compared to measure the porosity: water imbibition and weighting versus gas (Argon) pycnometry. Water measured porosities appear to systematically underestimate gas measured porosities. A capillary pressure curve is derived from successive equilibrium of samples at imposed relative humidities. The relative gas permeability is thoroughly measured on the same samples. Gas porosity and effective gas permeability measurements have been carried out under different confining pressures. All measured relative gas permeabilities lie on a master curve and present a low critical water saturation beyond which the effective gas permeability nearly vanishes.In an attempt to model this last feature, the salient properties of the microstructure are extracted from electron microscopy images to propose a micro-macro model of the relative gas permeability. The multi-scale modeling is performed within the framework of random media or so-called continuum micromechanics, including localised flow effects within interfaces. The self-consistent homogenisation scheme allows retrieval of a critical water saturation, whose value is governed by the shape of the elementary particles of the material and the localisation of water within the pore space.Résumé -Caractérisation expérimentale et modélisation multi-échelles de la perméabilité elative au gaz d'une roche de couverture -Une caractérisation expérimentale d'une roche carbonatée contenant de l'argile, peu perméable, est présentée dans un premier temps. Deux méthodes différentes de mesure de la porosité sont comparées : gravitaire par imbibition d'eau et pycnométrie à l'Argon. Les mesures de porosité à l'eau sous-estiment systématiquement celles au gaz. Une courbe de pression capillaire est construite à partir d'équilibres successifs d'échantillons avec des humidités relatives imposées. La perméabilité relative au gaz est mesurée sur les mêmes échantillons. Les mesures de perméabilité effective au gaz et de porosité au gaz sont conduites sous diverses pressions de confinement. Toutes les perméabilités relatives au gaz mesurées appartiennent à une même courbe maîtresse. Cette courbe présente une faible saturation en eau critique au delà de laquelle la perméabilité effective au gaz s'annule. Dans un second temps, une tentative de modélisation de cette propriété est proposée. Un modèle micro-macro est proposé à partir d'observation de la microstructure au microscope électronique. La modélisation multi-échelles est conduite dans le contexte de l'homogénéisation des milieux aléatoires ou micromécanique des

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“…denotes the n − th application of the operator L. From (14) an (15) an iterative scheme (fixed point) can be defined to solve (13) as…”

Section: Thermal Steady-state Problemmentioning

confidence: 99%

“…Improvements of Fast Fourier Transform method can be found in [11,13,18] for dealing with nonlinear problems and arbitrary phase contrasts. See [2,15,24,27] for recent improvement of the convergence in the iterative schemes and [14,1,26,21] for other applications and recent extensions.…”

Section: Introductionmentioning

confidence: 99%

A fast method for solving microstructural problems defined by digital images: a space Lippmann–Schwinger scheme

Yvonnet

2012

Numerical Meth Engineering

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SUMMARYA fast numerical method is proposed to solve thermomechanical problems over periodic microstructures whose geometries are provided by experimental techniques, like X-ray micro tomography images. In such configuration, the phase properties are defined over regular grids of voxels. To overcome the limitations of calculations on such fine models, an iterative scheme is proposed, avoiding the construction and storage of finite element matrices. Equilibrium equations are written in the form of a Lippmann-Schwinger integral equation, which can be solved iteratively. Unlike previous algorithms based on the Fourier transform, the present scheme strictly operates in the real space domain and removes the numerical Fourier and inverse Fourier transforms at each iteration. For this purpose, the linear operator related to the Lippmann-Schwinger equation is constructed numerically by means of transformation tensors in the real space domain. The convergence and accuracy of the method are evaluated through examples in both steady-state thermics and linear elasticity problems. Computational times are found to scale linearly with the number of degrees of freedom and parallel computations can be carried out straightforwardly. The method is also illustrated on many examples involving complex microstructures, including a problem defined by a micro tomography image.

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A FFT-based method to compute the permeability induced by a Stokes slip flow through a porous medium

Cited by 36 publications

References 12 publications

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

Experimental Measurements and Multi-Scale Modeling of the Relative Gas Permeability of a Caprock

A fast method for solving microstructural problems defined by digital images: a space Lippmann–Schwinger scheme

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