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This paper is devoted to the analytical calculation of the contribution and opening displacement tensors of an arbitrarily oriented crack in a transformed transversely isotropic (TraTI) matrix. It generalizes a recent work considering elliptically orthotropic (EO) symmetry, as EO is a particular case of TraTI. The latter allows to explore a wider range of symmetries, including non-orthotropic symmetries. The approach is based on the linear transformation between boundary value problems with TraTI and transversely isotropic (TI) bodies. A detailed analysis shows that TraTI fourth-order tensors are described by a set of 11 parameters (5 material parameters and 6 angles defining the transformation). This is a significant enrichment beyond the set of EO tensors which depends on 7 parameters (4 material parameters and 3 angles). New analytical results are obtained for elliptical and circular cracks embedded, respectively, in TraTI, orthotropic TraTI and monoclinic TraTI matrices. It is shown that the most general case leading to analytical derivations of crack contribution and opening displacement tensors is that of a TraTI matrix with a specific restriction: the initial crack must be aligned along the isotropy plane of the TI matrix. To the best of our knowledge, this corresponds to the current largest space of matrix anisotropy allowing analytical derivation of the compliance of a single elliptical crack embedded in an infinite matrix as well as an extension of analytical 3D results showing a coupling between opening and shear modes. Numerical results are presented to illustrate potential applications of the method for the case of an arbitrarily oriented elliptical crack in an infinite uniform matrix of stiffness of arbitrary anisotropy. The best-fit problem investigated in previous papers is revisited and a new algorithm providing the closest stiffness tensor for which there exists an analytical solution to the crack opening displacement tensor is developed. Numerical applications to real TI materials are finally presented.

In situ pull-out tests were performed in claystones at the Andra Underground Research Laboratory of Meuse/Haute-Marne on fully grouted bolts. The bolts were instrumented with strain gauges along th...

International audience; In this paper, we derive the second-order crack opening displacement tensor for an arbi- trarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is ob- tained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have nei- ther the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a cor- rection factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in de- tail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.

The aim of this paper is to extend recent works devoted to the study of the effect of 3 D pores of concave shape embedded in isotropic matrix to the case of transversely isotropic (TI) matrix. In the first part of the paper, approximate relations for the compliance contribution tensor of pores of two reference shapes, supersphere and axisymmetrical superspheroid, are developed on the basis of 3 D Finite Element Modelling, recently presented in a companion paper, and known exact solutions for the limiting cases of spherical pores and circular crack. In the second part application to effective elastic coefficients of transversely isotropic materials such as clay rocks, in the frame of homogenization theory is presented to illustrate the impact of concavity parameter on overall properties.

This paper presents a reduced-scale physical model of a mine stope used to reproduce the underground stope backfilling practiced in some Canadian mines. The objective is to study the geomechanical behavior of the waste rock barricades in interaction with the mine backfill. The instrumentations, along with visual observations and preliminary results, are presented. The main results demonstrated that: (i) the stability of the barricade depends on its physical properties (e.g., size, location, particles gradation and compaction) and the frictional behavior at the barricade/drift walls interface, and (ii) for two backfill formulations, cemented and uncemented, prepared with 70% of solid mass concentration, the volumetric strain due to self-weight consolidation of the backfilled room was higher for uncemented backfill (16%) than for the cemented one (4.5%). In addition, the results highlighted the importance of using shotcrete around the downstream face of the barricade, mostly at the top, to close the gaps and bind the barricade particles, which improves its stability.

The main objective of this work is to estimate the compliance contribution tensor of the concave pore inhomogeneity surrounded by a transversely isotropic matrix. In this light, we make use of a recently developed adapted boundary conditions based Finite Elements Method to incorporate the matrix anisotropy and the correction of the bias induced by the bounded character of the mesh domain, which allows to accelerate the computation convergence without sacrificing its accuracy. The correction of the boundary conditions is given as functions of the Green tensor and its gradient as dependent on the anisotropic elasticity of the matrix material, which are rigorously calculated by means of the Fourier transform based integral method in particular for regularizing the singularities on the symmetric axis of the transverse isotropy. Simultaneously by complying with the numerical homogenization technique, the compliance contribution tensor is computed for different forms of pores (e.g. superspheroidal and superspherical ones, etc) embedded in an transversely isotropic matrix. The proposed numerical method is shown to be efficient and accurate after several appropriate assessment and validation by comparing its predictions, in some particular cases, with analytical results and some available numerical ones. Finally, the effect of the pore concavity on the compliance contribution tensor is quantitatively illustrated.

This paper focuses on effective elastic properties, and thermal conductivity of materials constituted by an isotropic homogeneous solid matrix containing both ellipsoidal and non-ellipsoidal pores....

The present paper focuses on materials containing superellipsoidal pores described by equation | x 1 | 2 p + | x 2 | 2 p + | x 3 /γ | 2 p = 1. This shape is concave when 0 p p > 0.5. They are oblate when γ 1 Combined effect of two shape factors - parameter p describing the concavity and aspect ratio γ describing extend of oblateness/prolateness - on compliance contribution tensor is analyzed numerically and approximated analytically for most important case of p < 1. It is shown that, in this case, the combined effect is equal to the product of two effects considered separately. The obtained approximate expressions are used to calculate the effective elastic properties of a heterogeneous material containing pores of superellipsoidal shape.

We have characterized the microstructure of a heterogeneous oolitic rock (limestone from Lavoux, France) with X-ray nano-computed tomography. This rock comprises an assemblage of porous grains (oolites), irregularly shaped three-dimensional pores, and inter-oolitic crystals (cement). To model the effect of this microstructure on the macroscopic behavior of the rock, we approximate the porous oolites by spheres, and the irregularly shaped pores by ellipsoids. This approximation is performed based on the principal component analysis PCA, which provides the geometrical properties such as length of semi-axes and orientation of resulting ellipsoids. The sphericity of the approximated oolites was calculated and the value close to 1 allows us to consider oolites as spheres. To verify the approximation in the case of pores, we evaluated the contribution of these irregularly shaped three-dimensional pores to the overall elastic properties. Thus, compliance contribution tensors for 3D irregular pores and their ellipsoidal approximations are calculated using the finite element method. These tensors were compared to the compliance tensors for ellipsoids obtained using analytical solutions based on Eshelby's theory and a relative error is estimated to evaluate the accuracy of the approximation. This error produces a maximum discrepancy of 4.5% between the two solutions respectively of pores and ellipsoids which verifies the proposed approximation procedure based on principal component analysis. The FEA numerical method is verified by comparing the numerical solution of compliance contribution tensors of the ellipsoids to the known analytical solution of these same shapes based on Eshelby's theory. The difference between these two solutions does not exceed 3%. Compliance contribution tensors are finally used to compare effective elastic parameters of a material containing irregular pores via the Maxwell homogenization scheme. These elastic parameters (bulk modulus and shear coefficient) coincide with a maximum deviation of 5% with ones for a material with ellipsoidal pores.

We focus on derivation and experimental verification of the replacement relations that link the overall thermal conductivities of heterogeneous materials with the same matrix and microstructure but having inhomogeneities with different properties. First, we derive replacement inequalities based on Hashin-Shtrikman bounds that relate overall thermal conductivity of a composite to thermal conductivity of a.porous material. Hashin-Shtrikman bounds are also used to derive analogy of Gassmann equation for thermal conductivity. Then, we use formalism of property contribution tensors to obtain replacement relations for anisotropic materials containing ellipsoidal inhomogeneities. These relations coincide for different homogenization schemes – non-interaction approximation, Mori-Tanaka scheme, and Maxwell scheme, provided, that both effective conductivity of a composite and porous material are calculated in the framework of the same method. In the case of the overall isotropy, all of these relations coincide with Gassmann equation derived from Hashin-Shtrikman bounds. We check the possibility to apply these relations to 3-D non-ellipsoidal inhomogeneities on example of a supersphere using numerical simulations. The replacement relations are approximate with satisfactory accuracy for convex superspheres, while the error is significant for concave shapes. For this case, we suggest a modification that involves an extra shape factor that can be determined, for example, from comparison of the average Eshelby tensor for conductivity and conductivity contribution tensor of a pore. To verify the approach, thermal conductivity of 85 quartz sandstone specimens – dry and saturated with water and kerosene – of porosity varying from 0.14 to 0.29 is measured using optical scanning technique. The average pore shape and thermal conductivity of the dense quartz matrix are determined from best fitting of the conductivity-porosity curve for dry sandstone. Then these parameters are used in the replacement relation for sandstone saturated with water and kerosene. The comparison of the experimental data with theoretical predictions shows a good accuracy of the proposed approach.

International audience; The paper focuses on the construction of one particle homogenization schemes for an ageing linear viscoelastic composite. The composite consists of an isotropic host matrix and randomly located spheroidal inhomogeneities. Both phases are assumed to be ageing linear viscoelastic. Expressions for the effective viscoelastic properties in the frameworks of Non Interaction Approximation as well as Maxwell and Mori-Tanaka-Benveniste homogenization schemes are derived. A special attention is paid to the far-field solution to the one-particle problem since it plays a major role in the Maxwell scheme.

This paper provides new estimates for the Representative Elementary Area (REA) sizes of the following two shales that are actively being studied in the framework of deep disposal of radioactive waste: Callovo-Oxfordian (COx) claystone from the Meuse/Haute-Marne underground research laboratory (Eastern France) and Toarcian argillite from the experimental station of Tournemire (Southern France). The REA sizes, named LREA, were obtained from two mineral maps following the classic “counting box” (CB) method and a statistical approach that introduces the concept of a “statistical” REA. Following this approach, a “statistical” REA is related not only to the microstructure and the properties of each of the components but, above all, to a given precision in the estimation of the effective property depending on the number of realizations that one is ready to generate. The probabilistic concept of realization is here, from a practical viewpoint, a subdomain of a mineral map in which the apparent morphological or mechanical properties have to be calculated. In this study, the apparent elastic moduli of the subdomain have been estimated using two micromechanical models. The first micromechanical model consisted of an inclusion-based model for which spherical nonclay grain is embedded in a clay matrix in which the values of its transverse isotropic stiffness tensor have been taken from literature. The second micromechanical model was an isotropic inclusion-based model for which a spherical nonclay grain is embedded in a clay matrix; the elastic moduli values have been inverted by a Monte-Carlo approach from the engineering moduli of both shales under study. Our calculations have shown the following results: (i) the statistical morphological LREA considering the surface clay fraction are of the same order of magnitude as those measured in other shales and those obtained by the simple CB method, with relative error values between 5 and 10%, (ii) the mechanical LREA values associated with the bulk modulus and shear modulus are significantly greater than that of the morphological LREA, and (iii) the mechanical LREA estimates of the shear modulus are greater than that of the bulk modulus. Moreover, our study highlights that these qualitative results do not depend on the chosen micromechanical models and, thus, would be independent of the underlying anisotropic nature of shale.

The present work is devoted to the determination of linear effective thermal conductivity of porous rocks characterized by an assemblage of grains (oolites) coated by a matrix. Two distinct classes of pores, i.e. micropores or intra oolitic pores (oolite porosity) and mesopores or inter oolitic pores (inter oolite porosity), are taken into account. The overall porosity is supposed to be connected and decomposed into oolite porosity and matrix porosity. Within the framework of Hashin composite sphere assemblage (CSA) models, a two-step homogenization method is developed. At the first homogenization step, pores are assembled into two layers by using self-consistent scheme (SCS). At the second step, the two porous layers constituting the oolites and the matrix are assembled by using generalized self-consistent scheme (GSCS) and referred to as three-phase model. Numerical results are presented for data representative of a porous oolitic limestone. It is shown that the influence of porosity on the overall thermal conductivity of such materials may be significant.

International audience; Significant progresses have been realized during the last decades on both macroscopic and micro-mechanical modeling of induced damage in brittle materials. Most damage models developed so far were devoted to initially isotropic materials. This work is devoted to modeling of induced damage in an initially anisotropic material. A numerical micro-mechanical damage model is proposed, using an Eshelby inclusion solution based homogenization method. Based on the numerical integration of the exact Green’s function and using an appropriate coordinate frame rotation method, an efficient numerical algorithm is proposed to determine the Hill tensor for an arbitrarily oriented family of cracks embedded in a transversely isotropic elastic matrix. Based on this, the effective elastic properties of cracked materials are determined through a rigorous up-scaling procedure using three different homogenization schemes, and taking into account interactions between the initial material anisotropy and induced cracks. A specific damage criterion is then defined in the framework of irreversible thermodynamics to describe the progressive growth of damage. The proposed model is finally implemented in a computer code and applied to study mechanical responses of cracked materials in different loading paths. Again, effects of the initial anisotropy and homogenization schemes are investigated.

This paper addresses the problem of the electrical conductivity tensor calculation for a transversely isotropic material that contains inhomogeneities of arbitrary orientation. For this goal, we first construct the electrical conductivity contribution tensor for an arbitrarily oriented isolated ellipsoidal anisotropic inhomogeneity embedded in a transversely isotropic matrix. The general case of an orthotropic ellipsoidal inhomogeneity unaligned in an anisotropic matrix with different classes of symmetry can be considered. This solution is used as the basic building block of various homogenization techniques: the Mori–Tanaka–Benveniste scheme, Maxwell scheme, and differential scheme. The approach is illustrated by an application to a transversely isotropic mudstone rock, composed of a clay matrix containing inhomogeneities of calcite and quartz. We analyse the origins of the extent of anisotropy of the effective conductivity tensor, distinguishing among the shape, orientation distribution, and anisotropy of the inhomogeneities on the one hand and the anisotropy of the matrix on the other hand. Numerical results show that the orientation distribution of the inhomogeneities significantly affects the overall anisotropy in the case of inhomogeneities with low aspect ratio(s). Limiting cases of aligned and randomly oriented inhomogeneities provide bounds of the extent of anisotropy for the overall conductivity tensor.

International audience; We calculate effective properties of a porous material with non-ellipsoidal concave pores. The pore shape is described by equation of a supersphere x(2p) + y(2p) + Z(2p) = 1 that is convex when p > 0.5 and concave whenp < 0.5. Compliance and resistivity contribution tensors for a superspherical pore are calculated using finite element method and approximated by analytical expressions for p < 1. These tensors are used to evaluate effective elastic and conductive properties of a material with superspherical pores via non-interaction approximation, Mod-Tanaka scheme and Maxwell scheme. We show that the geometrical parameters entering expressions for the elastic moduli and conductivity are the same and establish cross-property connection for such materials. These connections coincide with ones for a material with spherical pores.

International audience; The present work focuses on effective thermal conductivity of oolitic lime-stones, characterized by an assemblage of porous grains (oolites), mesopores and solid grains. Two distinct scales of pores, micropores or intra oolitic pores and mesopores or inter oolitic pores are taken into account. At the first step, micropores are homogenized inside the oolites by using self consistent homogenization scheme. The second homogenization step describing transition from the mesoscale to the macroscale, is performed by using a recent reformulation of the Maxwell homogenization scheme (see [1]). At the mesoscale, porous oolitic inclusions are quasi spherical whereas two families of mesopores are considered according to analysis of photomicrographs: (1) randomly oriented oblate spheroidal pores and (2) concave pores. The proposed model is compared to a simplified one when all the pores are of ellipsoidal shape. The relevancy of the ellipsoidal approximation is then evaluated. In particular, the influence of the shape of the mesopores on the overall thermal conductivity is discussed. Comparisons between multi-scale model based on Maxwell homogenization method and experimental data show that effects of porosity and saturating fluids on overall conductivity are correctly predicted when concave pores are taken into account.

International audience; We calculate Eshelby tensor for inclusions of non-ellipsoidal shape. We focus on the superspherical shape described by equation X-2P + y(2p) + Z(2p) 0.5 and concave when p < 0.5. We propose a numerical approach to perform: integration on the surface of the superspherical inclusion necessary to compute the average Eshelby tensor. Validation of the method is done by comparison of the results with analytical solutions for a spherical inclusion (p = 1) and with numerical results of Onaka (2001) (p > 1).

In this paper, we used a theoretical model for the variation of Eulerian porosity, which takes into account the adsorption process known to be the main mechanism of production or sequestration of gas in many reservoir of coal. This process is classically modeled using Langmuir's isotherm. After implementation in Code_Aster, a fully coupled thermo-hydro-mechanical analysis code for structures calculations, we used numerical simulations to investigate the influence of coal's hydro-mechanical properties (Biot's coefficient, bulk modulus), Langmuir's adsorption parameters, and the initial liquid pressure in rock mass during CO2 injection in coal. These simulations showed that the increase in the values of Langmuir's parameters and Biot's coefficient promotes a reduction in porosity because of the adsorption process when the gas pressure increases. Low values of bulk modulus increase the positive effect (i.e., increase) of hydro-mechanical coupling on the porosity evolution. The presence of high initial liquid pressure in the rock mass prevents the progression of injected gas pressure when CO2 dissolution in water is taken into account.

International audience; Characterization of time-dependent brittle deformation is important for estimating the long-term stability of rock structures in shallow environments such as underground mines which are exposed to water or in a water vapor environment. The poro-mechanical (pore pressure) and chemical (stress corrosion reactions) effects of fluids at crack tips are responsible for the subcritical propagation of microcracks. This brittle creep results in the weakening of porous rocks over time (i.e., static fatigue). The focus of this work was to characterize using very long duration uniaxial multi-step creep tests, both the fluids effects on subcritical stress corrosion cracking in a porous rock (oolithic iron ore). Different fluids with increasing levels of chemical influence (oil, ethanol and water) were tested and both acoustic and mechanical properties were investigated. The evolution of the cumulative number of AE (Acoustic Emission) events, which reproduces the shape of the creep curves very efficiently, and static elastic properties indicate that micro-cracking plays the main role in the creep process. The poro-mechanical effect, which is predominant under partially saturated (water or ethanol) conditions, decreases the rate of subcritical cracking through capillary attraction forces. These forces harden porous rocks by modifying the effectiVe stress state, increasing the activation energy barrier of the stress corrosion process and the fracture toughness and decreasing the stress intensity factor. The chemical effect of fluids is related to minerals developing stress corrosion reactions at crack tips which enhance subcritical cracking. Immersion in water/ethanol annihilates capillary forces and decreases the activation energy of the chemical reactions, thus increasing dilatancy, the rate of stress corrosion cracking, AE activity and rock weakening. Under saturated conditions, the timedependent strength and time to failure increase as the chemical influence of the interstitial fluid decreases but the cumulative number and energy of AE at failure do not seem to be influenced by the chemistry of fluids. The short-term strength and deformability are also influenced by the level of chemical influence of the interstitial fluid but to a less remarkable extent because of the slow rate of chemical reactions. Water is therefore the most effective chemical agent promoting stress corrosion of iron ore among the fluids tested in our study.

International audience; The pulse test is usually considered to be an efficient method for measuring the permeability of weakly permeable porous rocks. Classically, the test consists of imposing a pressure drop on the base of a cylindrical sample and measuring the pressure variations in the upstream and downstream reservoirs. In the present work, we propose a new apparatus and procedure for hollow cylindrical samples in which the pressure drop is imposed in an axial hole and the pressure is measured both in the hole and on the circumference of the sample. Unlike the classical axial pulse test, this configuration results in a flow in both the axial and transversal directions rather than only in the axial direction. For transverse isotropic rocks, this configuration allows the assessment of the permeability in the isotropy planes and normal to the isotropy planes in a single sample when the samples are appropriately oriented. The test is fully hydro-mechanically coupled; therefore, no analytical solution exists. The test is then interpreted through fully coupled numerical modeling in 3D, considering the anisotropy of the samples. In previous works, we developed and implemented a transverse isotropic poroelastic constitutive law in the finite element code Code_Aster (Edf), as well as an inverse method coupled to the numerical modeling for parameter identification. The method is adapted to the radial pulse test and then applied to hollow cylindrical samples of Meuse/Haute-Marne argillite with the axis either parallel or perpendicular to the isotropy planes. Although this method requires 2 samples for the assessment of permeability in the isotropy planes and normal to the isotropy planes in the axial pulse test, the method applied to the radial pulse test allows the assessment of intrinsic permeability in both directions on a single sample, which allows freeing ourselves from a heterogeneity effect.

International audience; This article concerns the problem of collapses of abandoned underground (room-and-pillar) iron mines in Lorraine (North-Eastern France). A study on the ageing of iron ore was carried out using different analytical techniques (SEM-EDS, EMPA, X-ray diffraction, Mössbauer and ICP spectroscopy). As iron ore is a reactive environment in an oxidizing atmosphere created by anthropic activity, this kind of weathering is triggered or activated by chemical processes, and differs from diagenetic evolution under reducing conditions. The weathering is characterized by the chemical alteration/oxidation of reduced minerals (siderite and berthierine) in the inter-oolithic cement, mechanical disaggregation of oolithes, recrystallization of berthierine and neo-crystallization of a honeycombed hematite crust (i.e., rust) on all mineral surfaces. This corrosion of iron ore was found to occur at a large scale in pillars, especially in very old pillars which have been exposed to the mine atmosphere for 70–100 years. The alteration of the inter-oolithic cement which ensures material cohesion was not found to have been compensated by the crystallization of new minerals, thus inducing the decrease in the micro/macro-mechanical properties of iron ore (micro-indentation tests at millimetric scale, micro-compression triaxial tests at centimetric scale and macro-compression triaxial tests at decimetric scale). This chemical weathering of anthropic-origin plays a large-scale and major role in the long-term behavior of pillars, and is probably one of the main causes of mine collapses that occured in Lorraine. Therefore, the decision to stop pumping of mine drainage, thus inducing flooding of iron mines (by groundwater), was the correct decision, since oxidation reactions in iron ore are necessarily less important under anaerobic condition than under aerobic condition. In addition, the knowledge of iron ore ageing allows for improved assessment of the long-term stability of iron mines which have yet to collapse (i.e., flooded iron mines and also iron mines which are still above the ground water-table).

In this chapter, numerical methods for modeling of fluid-driven cracks in rocks are presented. The modeling of the coupled fluid flow in continuous media is first reminded, with presentation of several applications. Then the modeling of the fractured rock is considered. A review of the most common methods for coupled fluid flow in fractured porous rocks is presented and the manuscript focuses on the extended finite element method (XFEM) method combined with hydromechanical couplings. Several problems in relation with fluid-driven crack in porous rocks are foreseen, such as partial saturation, anisotropy, or crack initiation. Some details about numerical issues are also given.

This paper is devoted to a micro-mechanical study of induced anisotropic micro-cracks in initially anisotropic materials. The effective elastic properties of cracked materials are determined by a rigorous up-scaling homogenization procedure based on an efficient numerical method to calculate Hill polarization tensor. A linear damage criterion is defined in the framework of irreversible thermodynamics to describe the growth of damage. Multiple families of micro-cracks in different orientations are taken into account by using the Gauss-type numerical integration method. Numerical assessments are proposed by using a PCW estimation and a set of 33 families of micro-cracks for uniaxial tension and compression tests. Induced crack density distributions are investigated for both isotropic and anisotropic materials. Effects of the initial anisotropy on crack propagation process are clearly demonstrated.

International audience; The present work focuses on the characterization of the geometry of the microstructure of porous oolitic rocks. These rocks are constituted by an assemblage of porous grains (oolites), pores and inter-granular crystals. X ray 3D Computed Tomography is used to identify the different components of these rocks by applying an algorithm based on grayscale values. This analytical method allows the characterization of the porous network (size, spatial distribution, and volume fraction), oolites and inter-oolitic crystals. The microstructure of these porous rocks has a significant effect on their macroscopic behavior. This micro-macroscopic relationship is taken into account in micromechanical models developed within the framework of the homogenization theory (e.g., Maxwell scheme) of random heterogeneous media. X ray tomography images showed that pores have irregular shapes, so the micromechanical modeling based on analytical solution is not relevant. Then, pores are approximated by ellipsoids using principal components analysis (PCA) method, which allows us to obtain the geometrical properties such as length of semi-axes and orientation of ellipsoids. To validate mechanically this approximation, we compared the contribution of irregularly shaped 3D pores and ellipsoidal pores to the effective elastic properties. The relative error due to this ellipsoidal approximation is then estimated. The compliance contribution tensors of irregular 3D pores are evaluated numerically using finite element method while those of approximated ellipsoidal pores are obtained analytically and numerically. The same procedure of approximation is applied on oolites. Shape and spatial parameters, such as the volume, radius and center of each oolite are also determined. The sphericity of the approximated oolites is calculated. The obtained values are close to 1, so oolites can be reasonably approximated by spheres.

A three dimensional micro-mechanical model is developed for modeling micro-crack growth and plastic frictional sliding in initially anisotropic quasi brittle materials under compressive loading. Macroscopic strains are attributed to elastic deformation of matrix and displacement discontinuity on micro-cracks. Effective elastic properties of cracked materials are determined using a Eshelby's solution based linear homogenization technique by considering micro-cracks as spheroidal inclusions. An efficient numerical method is used to calculate Hill polarization tensor for spheroidal micro-cracks arbitrarily embedded in transversely isotropic solid matrix. Based on this, the plastic strain related to frictional sliding on closed micro-cracks is determined by combining irreversible thermodynamics and homogenization method. A specific plastic friction criterion is formulated in terms of the local stress field on crack surfaces. The presence of a back stress tensor allows description of material hardening and softening without any additional functions. A specific damage evolution law is finally proposed. The evolutions of the friction-related plastic strain and crack-propagation induced damage are inherently coupled. A series of numerical assessments are presented for various loading paths such as uniaxial compression, triaxial compression and shear. The obtained numerical results clearly reveal that the macroscopic behaviors of cracked materials are strongly affected by the initial anisotropy. Finally, the performance of the proposed micro-mechanical model is verified by comparing numerical results and experimental data for a typical rock-like materials, shale.

The present work is devoted to the estimate of linear effective poroelastic properties of isotropic porous rocks characterized by an assemblage of porous grains ( oolites ), mesopores and solid grains. Examples of such type of microstructure may be found in oolitic limestones. Two distinct classes of pores are taken into account, micropores or intra oolitic pores and mesopores or inter oolitic pores . In the framework of Hashin Composite Sphere Assemblage ( CSA , Hashin, 1962 ) and self-consistent models, a two step homogenization model is developed. At the first step, micropores are homogenized inside the oolites. At the second step, oolites, mesopores and solid grains are assembled by using the self consistent scheme. Levin’s theorem and transformation field analysis method are used to derive poroelastic coefficients (Biot coefficient and solid Biot modulus). Simple explicit formula are presented for effective poroelastic coefficients in the general case of double porosity media, under the assumption of spherical shape for all constituent particles and isotropic distribution. Finally comparisons between model and experimental data are presented for an oolitic limestone which includes micro-indentation results obtained at mesoscale and poromechanical results obtained at macroscale. Comparisons show that in the case of low macro stiffness, a degraded interfacial transition zone ( ITZ ) surrounding oolites, need to be introduced to fit the data

This paper focuses on the effect of concavity of pores on elastic properties of porous materials. We consider a pore having a shape of a supersphere of unit radius (x)2p + (y)2p + (z)2p = 1 focusing mostly on the case p ≤ 1. Using FEM analysis of Sevostianov et al (2008), we propose simple approximate formulae for components of the compliance contribution tensor of the supersphere. From these formulae, we identify the microstructural parameter describing its contribution into effective elastic properties. The derivation is illustrated by comparison of the results with the known ones for a spheroidal pore of the aspect ratio γ.

The French National Agency for Nuclear Waste Management is currently studying the Callovo-Oxfordian claystone as a potential repository for nuclear waste. This study focuses on the hydromechanical (HM) unsaturated response of the rock subjected to hydraulic loading. Static atmosphere drying tests highlighted the HM coupling and mechanical anisotropy of the samples. A coupled poroelastic model was proposed, involving highly nonlinear equations. Numerical simulations were performed using the finite element code Code_Aster. Through a sensitivity analysis, the most suitable parameters for estimation were found to be the Biot coefficient and 3 of the 4 independent parameters of the Van Genuchten–Mualem relations. The parameters were estimated according to an inverse procedure that minimizes a cost functional. The estimates could not outline any hysteretic behavior, but they appeared to be divided into two groups (depending on their fitted values). The convergence and stability of the inverse algorithm were studied, and the results showed that direct computations must be run prior to the inversion to ensure its convergence. No plasticity occurred in the samples (according to the model), whereas irreversible phenomena do take place in situ.

The present work is devoted to the determination of linear effective poroelastic properties of porous rocks characterized by an assemblage of grains ( oolites ) coated by a matrix. In the framework of Hashin composite sphere assemblage (CSA [1] ) models, a two-step homogenization method is developed. Three porous materials are considered and assembled into a CSA : oolite ( o ) coated by an interfacial transition zone ITZ ( i ) and a matrix ( m ). The overall porosity is supposed connected and decomposed into oolite porosity, interphase porosity and matrix porosity. The three constitutive porous materials ( o , i , m ) are homogenized at the first step and a sensitive study is performed on the homogenization scheme by comparing self consistent SC , differential self consistent DSC , generalized self consistent GSC , iterative self consistent ISC and Mori Tanaka schemes. At the second step, the simple three-phase model [2] is firstly recalled and general relations for effective linear poroelastic coefficients are presented in the four-phase model case [3] . Mathematical analogy between linear thermoelasticity and linear poroelasticity is used to establish theoretical results in the context of CSA .

The present paper addresses an approximate analytical model for contribution of an arbitrarily oriented circular crack into effective elastic compliance of a transversely isotropic material. We numerically examine the bounds of applicability of the hypothesis that change in elastic potential due to an arbitrarily oriented circular crack in a transversely- isotropic material can be approximated by the change calculated for a certain isotropic environment. In particular, we obtained that the error of such an approximation is less than 20% if the extent of anisotropy is moderate - the ratio of Young's moduli in transverse and in-plane directions is less than 1.87. The obtained result can be used for development of a simple model for microcracked transversely-isotropic materials with mild-to-moderate extent of anisotropy.

International audience

International audience; The present paper focuses on the Eshelby inclusion problem which is revisited in the framework of ageing linear viscoelasticity. All known results established in linear viscoelasticity thanks to the correspondence principle are recovered as particular cases of a general solution extended to ageing. A closed form solution is presented for the Hill and Eshelby tensors related to an ellipsoidal inclusion embedded in an infinite anisotropic medium. The case of the isotropic medium is investigated in detail and related solutions are presented for spherical and ellipsoidal inclusions. A numerical procedure which operates in time domain and based on the trapezoidal rule to determine Volterra integral operators allows to efficiently calculate Hill and Eshelby tensors for a wide range of behaviours including in particular time-dependent Poisson ratios. Validation and verification of the new developed solution are presented for the ageing spherical inclusion on the basis of recent published results and it is completed by comparisons with finite element simulations for the general and novel case of an oblate spheroid embedded in an ageing linear viscoelastic matrix.

Transversely-isotropic material with an arbitrarily oriented penny-shaped crack is considered. We calculate fourth-rank compliance contribution tensor of the crack and second-rank crack opening displacement tensor and examine their dependence on crack orientation. It is shown that this dependence for the crack opening displacement tensor is negligible if transverse isotropy has elliptic character, i.e. if material symmetry can be described in terms of a certain second rank tensor.

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.

Poromechanical behaviour modelling of the Callovo–Oxfordian argillite under saturated and partially saturated conditions is proposed using the equivalent stress concept. In comparison with the previous works on this rock, the particular form of the yield criterion and the plastic flow potential proposed here help to better describe the rock behaviour in tension–stress paths. The evolution of the poroelastic parameters due to the induced cracks is also considered in a simple way. Due to its physical nature, different from classical soils, the partially saturated behaviour of this rock could not be correctly described by any of the pre-existing theories usually used for the partially saturated porous media (soils). Based upon experimental results on this rock and developing some ideas proposed by other authors, an extension of the saturated elastic–plastic model in unsaturated field is proposed. The key hypothesis of this extension is the evolution of the Biot’s coefficient as a function of the suction, justified by laboratory results and micromechanical analyses. The predictions of the model in saturated and partially saturated conditions are compared with laboratory results and a good general agreement is found.

International audience; This paper revisits the ``Local Porosity Theory'' developed by Hilfer to propose a ``local clay theory'' (LCT) that establishes a quantitative relationship between the effective elastic properties and clay distribution of shales. This approach is primarily based on a ``local simplicity'' assumption; under this assumption, the complexity of spatial clay distribution can be captured by two local functions, namely, the local clay fraction distribution mu and the local percolation probability lambda which are calculated by partitioning a mineral map. The local clay fraction distribution provides information about spatial fluctuations of clay fraction and the local percolation probability describes the spatial fluctuations in the clay connectivity. This LCT was applied to a mineral map made from a Callovo-Oxfordian mudstone sample for which petrophysical data were available. This application demonstrated that the local functions mu and lambda can be reasonably approximated by a Gaussian and simple sigmoid function, respectively. LCT was also used in a sensitivity analysis to evaluate the impact of the spatial fluctuations of the clay phase and clay connectivity on the effective elastic properties of shales. This impact has been investigated by comparing the effective properties obtained from LCT with those provided by a percolating inclusion-based model (a non-clay spherical grain embedded in a transverse isotropic clay matrix with neither spatial clay fluctuations nor spatial fluctuations in the clay connectivity). This inclusion-based model was built from a differential effective medium scheme. The results of this sensitivity analysis indicated that spatial clay fluctuations have only, a slight impact on the calculated effective properties. However, when the spatial fluctuations of clay connectivity were considered, the quantitative deviations between the effective properties inferred from LCT and those from the percolating inclusion-based model were large exceeding 50% in some cases.

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.

This paper deals with a new strategy for initial stress identification by stress relaxation methods, coupled with finite element calculation, and applied to the overcoring test. The back-analysis of such a test uses an inversion method which consists in the minimisation, with a gradient-based algorithm, of a cost functional of least-squares type, which quantizes the difference between measured and computed strains. The computed strains are assessed by three-dimensional finite element modellings of the overcoring test. The inversion methodology is applied to a recent in situ overcoring test performed at Mont Terri laboratory, Switzerland. The inversion gave good results and allows us to validate the inversion methodology. The constitutive law considered for this application is transverse isotropic elasticity but the inversion method developed is applicable to most constitutive laws and every kind of in situ test.

Mining collapses currently occur in Lorraine (France) above abandoned room-and-pillar iron mines during the transient stage of mine flooding. The mining rocks are exposed in situ to various hydrous conditions: full saturation–partial saturation (ventilation)–full saturation. In order to explain the mechanisms involved in mining collapses, we have performed one- and two-dimensional numerical simulations based on a poroplastic model and within the framework of partially saturated and continuous porous media. These numerical simulations model the hydrous cycle imposed to the rocks in these underground deep mines. It results from this hydromechanical modelling that the final stage of resaturation induces transient tensile stresses that lead to failures at the roof and at the floor of the galleries. These instabilities appear during the transient stage of resaturation only if the variation of the equivalent interstitial pressure π is important. The studied key points are the complex geometry of the model (rectangular rooms and dissymmetry of the upper and lower boundaries conditions) and the kinetics of the fluids diffusion processes that makes the reversibility very long and induces an important hysteresis in the hydromechanical cycle. The convergence measurements in an in situ experimental site, in which the flooding of the mine has been simulated, are well modelled.

This paper deals with the modelling of the long-term behaviour of a natural gypsum rock, that manifests a volumetric strain variation during the creep, a significant influence of the relative humidity on the creep strain rate and an inverse creep during unloading. Based upon laboratory results a creep model describing primary and inverse creep is firstly constructed. The creep strains are seen as a sum of the recoverable and the viscoplastic strains. The firsts are results of two competitive creep mechanisms: the forward reversible creep strain, which is a function of the active stress and the backward creep itself a function of the cumulated deformation energy. According with laboratory tests, the viscoplastic strains have been considered to be mean-stress independent, which yields to an Odqvist-like creep model with some modifications in order to describe the primary creep and the volumetric strain evolution. The role of the relative humidity in the long-term behaviour of this rock is taken into account by the evolution of the creep activation energy. The model has a limited number of parameters, all accessible from standard laboratory tests. Its description was completed by the procedure of the parameters identification. Then, the predictions of the model have been compared with laboratory results and a good concordance has been found with laboratory data during the first stages of creep. Some discrepancies at the very late stages of the creep are explained by a damage of the rock during the creep. Accordingly, an extension of the model has been proposed that takes into account this damage. Consequently, the extended model fits better laboratory results and allows describing of the accelerated creep.

In the underground waste isolation projects such as the ANDRA'one in the site of Bure, the transport properties of the surrounding rock mass is of fundamental importance. To measure very low permeability, we use the modified version of the pulse test proposed by Hsieh et al. [Hsieh, P.A., Tracy, J.V., Neuzil, C.E., Bredehoeft, J.D., Silliman, S.E., 1981. A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks — I. Theory. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. Vol. 18, pp. 245–252] which enables the intrinsic permeability, k , and the specific storage coefficient, S s , of rocks such as mudstone to be characterized. In this paper, the special effort performed on the laboratory apparatus design, to ensure a good sensitivity of the rock response with respect to both parameters, k and S s , is presented. In addition, two parameters identification procedures are proposed: the graphical method given by Hsieh et al. [Hsieh, P.A., Tracy, J.V., Neuzil, C.E., Bredehoeft, J.D., Silliman, S.E., 1981. A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks — I. Theory. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. Vol. 18, pp. 245–252] and a parameter identification based on the solution of an inverse problem. The efficiency of the apparatus design and the parameters identification procedures is then demonstrated though some pulse tests performed on the Meuse/Haute-Marne mudstone.

The effect of suction on the behaviour of iron ore has been studied from both physical and mechanical points of view. The porosity and the suction phenomena have been analysed using different experimental techniques. Uniaxial compressive tests on partially saturated samples have shown that the suction is responsible for strength and cohesion improvement. Considering the theory of partially saturated porous soils of Coussy and Dangla (Mecanique des sols non satures (2002 edn). Hermes Science: 2002; 390), we have proposed a constitutive law for partially saturated iron ore. The real increase in the apparent cohesion due to the capillary attraction forces is overestimated if the yield function is written in terms of effective stresses. The effect of the capillary cohesion has been modelled with a function in the expression of the apparent cohesion of the yield function. The effect of suction on the mechanical behaviour has been represented in the effective stresses space and in the total stresses space like the Alonso model (Geotechnique 1990; 40:405–430). Copyright © 2005 John Wiley & Sons, Ltd.

This paper deals with the methods of determination of permeability in saturated and partially saturated conditions for low permeable porous rocks such as argillites. The modified version of the pulse test proposed by Hsieh et al. [Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 18 (1981) 245] has been used to characterize permeability in the saturated case. It enables the hydraulic diffusivity and then the intrinsic permeability and the specific storage coefficient to be characterized. In partially saturated conditions the method of saline solution to impose relative humidity and then capillary pressure has been used. The permeability in the partially saturated range can be deduced form measurements of transient weight loss and deformations of a sample submitted to a decrease of relative humidity in an hermetic chamber. In the two cases, pulse and drying tests, original experimental devices have been developed. The parameter-identification procedure based on the solution of corresponding inverse problems is presented. First approach explicit or semi-explicit solutions for the direct problems are used. It shows that the simplified linear approach is useful to obtain correct order of magnitude of unknowns parameters.

A plastic deviatoric model with hardening is developed on the basis of geomechanical tests performed in the saturated case on low permeable porous material such as argillite. This model is a generalized Mohr–Coulomb plastic criterion combined with a Drucker–Prager plastic potential and the hardening parameter is the plastic distortion. Three different hardening functions have been introduced on the basis of triaxial tests: an increase of friction angle, a decrease of cohesion after a threshold and a contractancy to dilatancy transition for volumetric plastic strain. This plastic model has been adapted to the partially saturated case. The effective stress is expressed thanks to the equivalent interstitial pressure π. Numerical results are presented for the excavation and monotonous ventilation of a deep cylindrical cavity. A first plastification due to excavation is followed by a second one due to desaturation. The extent of the non-saturated zone provokes an extent of a plastic zone in the rock mass. Analysis shows that the origin of the plastification can be found in the deviatoric stresses because mean effective stresses are compressive during drying. Copyright © 2004 John Wiley & Sons, Ltd.