Extended Poromechanics For Adsorption-induced Swelling .
Extended poromechanics for adsorption-induced swellingprediction in double porosity media: modeling andexperimental validation on activated carbonLaurent Perrier, Gilles Pijaudier-Cabot, David GrégoireTo cite this version:Laurent Perrier, Gilles Pijaudier-Cabot, David Grégoire. Extended poromechanics for adsorptioninduced swelling prediction in double porosity media: modeling and experimental validation onactivated carbon. International Journal of Solids and Structures, Elsevier, 2018, 146, pp.192-202. 10.1016/j.ijsolstr.2018.03.029 . hal-01756957 HAL Id: 1756957Submitted on 9 Apr 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
2Extended poromechanics for adsorption-induced swelling prediction in doubleporosity media: modeling and experimental validation on activated carbon3Laurent Perrier, Gilles Pijaudier-Cabot1 , David Grégoire1, 1University Pau & Pays Adour,Laboratoire des Fluides Complexes et leurs Réservoirs, LFCR-IPRA, UMR5150,Campus Montaury, F-64600 Anglet, France4567891011121314151617AbstractNatural and synthesised porous media are generally composed of a double porosity: a microporosity where thefluid is trapped as an adsorbed phase and a meso or a macro porosity required to ensure the transport of fluids to andfrom the smaller pores. Zeolites, activated carbon, tight rocks, coal rocks, source rocks, cement paste or constructionmaterials are among these materials.In nanometer-scale pores, the molecules of fluid are confined. This effect, denoted as molecular packing, inducesthat fluid-fluid and fluid-solid interactions sum at the pore scale and have significant consequences at the macroscale,such as instantaneous deformation, which are not predicted by classical poromechanics. If adsorption in nanoporesinduces instantaneous deformation at a higher scale, the matrix swelling may close the transport porosity, reducingthe global permeability of the porous system. This is important for applications in petroleum oil and gas recovery, gasstorage, separation, catalysis or drug delivery.This study aims at characterizing the influence of an adsorbed phase on the instantaneous deformation of micro-tomacro porous media presenting distinct and well-separated porosities. A new incremental poromechanical frameworkwith varying porosity is proposed allowing the prediction of the swelling induced by adsorption without any fittingparameters. This model is validated by experimental comparison performed on a high micro and macro porousactivated carbon. It is shown also that a single porosity model cannot predict the adsorption-induced strain evolutionobserved during the experiment. After validation, the double porosity model is used to discuss the evolution of theporomechanical properties under free and constraint swelling.18Keywords: Adsorption, swelling, double porosity media, poromechanical modelling19Introduction2021222324252627Following the IUPAC recommendation (Sing et al., 1985; Thommes et al., 2015), the pore space in porous materials is divided into three groups according to the pore size diameters: macropores of widths greater than 50 nm,mesopores of widths between 2 and 50 nm and micropores (or nanopores) of widths less than 2 nm. Zeolites, activatedcarbon, tight rocks, coal rocks, source rocks, cement paste or construction materials are among these materials. Inrecent years, a major attention has been paid on these microporous materials because the surface-to-volume ratio (i.e.,the specific pore surface) increases with decreasing characteristic pore size. Consequently, these materials can trap animportant quantity of fluid molecules as an adsorbed phase. This is important for applications in petroleum and oilrecovery, gas storage, separation, catalysis or drug delivery. Corresponding1 D.author: david.gregoire@univ-pau.frGrégoire and G. Pijaudier-Cabot are fellows of the Institut Universitaire de France.Accepted manuscript in International Journal of Solids and Structures (DOI: 10.1016/j.ijsolstr.2018.03.029)The final publication is available at: https://doi.org/10.1016/j.ijsolstr.2018.03.029 .2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ .
7879For these microporous materials, a deviation from standard poromechanics (Biot, 1941; Coussy, 2004), is expected. In nanometer-scale pores, the molecules of fluid are confined. This effect, denoted as molecular packing,induces that fluid-fluid and fluid-solid interactions sum at the pore scale and have significant consequences at themacroscale, such as instantaneous deformation. A lot of natural and synthesised porous media are composed of adouble porosity: the microporosity where the fluid is trapped as an adsorbed phase and a meso or a macro porosity required to ensure the transport of fluids to and from the smaller pores. If adsorption in nanopores induces instantaneousdeformation at a higher scale, the matrix swelling may close the transport porosity, reducing the global permeabilityof the porous system or annihilating the functionality of synthesised materials. In different contexts, this deformationmay be critical. For instance, in situ adsorption-induced coal swelling has been identified (Larsen, 2004; Pan andConnell, 2007; Sampath et al., 2017) as the principal factor leading to a rapid decrease in CO2 injectivity during coalbed methane production enhanced by CO2 injection. Conversely, gas desorption can lead to matrix shrinkage andmicrocracking, which may help oil and gas recovery in the context of unconventional petroleum engineering (Levine,1996). The effects of adsorbent deformation on physical adsorption has also been identified by Thommes and Cychosz (2014) as one of the next major challenges concerning gas porosimetry in nano-porous non-rigid materials (e.g.metal organic framework). In conclusion, there is now a consensus in the research community that major attentionhas to be focused on the coupled effects appearing at the nanoscale within microporous media because they may havesignificant consequences at the macroscale.Experimentally, different authors tried to combine gas adsorption results and volumetric swelling data (see e.g. Goret al. (2017) for a review). The pioneering work of Meehan (1927) showed the effect of carbon dioxyde sorption onthe expansion of charcoal but only mechanical deformation was reported and adsorption quantities were not measured. Later on, different authors (Briggs and Sinha, 1933; Levine, 1996; Day et al., 2008; Ottiger et al., 2008; Piniet al., 2009; Hol and Spiers, 2012; Espinoza et al., 2014) performed tests on bituminous coal, because it is of utmostimportance in the context of CO2 geological sequestration and coal bed reservoirs exploitation. However, most resultswere not complete in a sense that adsorption and swelling experiments were not measured simultaneously (Meehan,1927; Robertson and Christiansen, 2005) or performed on exactly the same coal samples (Ottiger et al., 2008). Otherauthors presented simultaneous in situ adsorption and swelling results but the volumetric strain was extrapolated froma local measurement – using strain gauges (Levine, 1996; Harpalani and Schraufnagel, 1990; Battistutta et al., 2010)or LVDT sensors (Chen et al., 2012; Espinoza et al., 2014) – or by monitoring the silhouette expansion (Day et al.,2008). Perrier et al. (2017) presented an experimental setup providing simultaneous in situ measurements of bothadsorption and deformation for the same sample in the exact same conditions, which can be directly used for modelvalidation. Gas adsorption measurements are performed using a custom-built manometric apparatus and deformation measurements are performed using a digital image correlation set-up. This set-up allows full-field displacementmeasurements, which may be crucial for heterogeneous, anisotropic or cracked samples.As far as modeling is concerned, molecular simulations are the classical tools at the nanoscale. Important effortshave been involved in molecular simulations in order to characterise adsorption-induced deformation in nanoporousmaterials (Vandamme et al., 2010; Brochard et al., 2012a; Hoang and Galliero, 2015) and these investigations showedon few configurations that pressures applied on the pore surfaces may be very high (few hundred of MPa), depending on the thermodynamic conditions and on the pore sizes. Note that an alternative approach based on a non-localdensity functional theory can be used to obtain highly resolved evolutions of pore pressure versus pore widths andbulk pressure in slit-shaped pores for a large spectrum of thermodynamic conditions on the whole range of microporewidths, even for complex fluids (Grégoire et al., 2018). However, if macroscopic adsorption isotherms may be reconstructed in a consistent way from molecular simulations through the material pore size distribution (Khaddour et al.,2014), molecular simulation tools are not tractable to predict resulting deformation at a macroscale due to the fluidconfinement in nanopores (pore sizes below 2 nm). Note that Kulasinski et al. (2017) proposed a molecular dynamicstudy where macroscopic swelling may be reconstructed from water adsorption in mesoporous wood (pore sizes in[4 10] nm). If adsorption is essentially controlled by the amount and size of the pores, the mechanical effect of thepressure build up inside the pores due to fluid confinement requires some additional description about the topologyand spatial organization of the porous network which is not easy to characterize, for sub-nanometric pores especially.Such a result motivates the fact that swelling is usually related to the adsorption isotherms instead of the pore pressuredirectly, the mechanical effect of the pore pressure being hidden in the poromechanical description.In this context, different enhanced thermodynamical or poromechanical frameworks have been proposed withinthe last ten years to link adsorption, induced deformation and permeability changes (e.g Pan and Connell (2007);2
PbTransport porosity φM saturatedDouble porosity mediasubmited to isotropicsurrounding bulk pressure Pbwith intersttal fuid at confnedpressure PM.Adsorpton porosity φμ saturatedwith intersttal fuid at confnedpressure Pμ.Figure 1: Schematic of a double porosity media.97Pijaudier-Cabot et al. (2011); Brochard et al. (2012b); Vermorel and Pijaudier-Cabot (2014); Perrier et al. (2015);Nikoosokhan et al. (2014)). For instance, Brochard et al. (2012b) (resp. Vermorel and Pijaudier-Cabot (2014)) proposed enhanced poromechanical frameworks where swelling volumetric deformation may be estimated as a functionof the bulk pressure and a coupling (resp. a confinement) coefficient, which may be deduced from adsorption measurements. However, if these models are consistent with experimental results from the literature, they cannot be consideredas truly predictive because the model parameters have to be identified to recover the experimental loading path. Anincremental poromechanical framework with varying porosity has been proposed by Perrier et al. (2015) allowing thefull prediction of the swelling induced by adsorption for isotropic nano-porous solids saturated with a single phasefluid, in reversible and isothermal conditions. This single porosity model has been compared with experimental dataobtained by Ottiger et al. (2008) on bituminous coal samples filled with pure CH4 and pure CO2 at T 45o C and afair agreement was observed for these low porosity coals but these types of models have to be enhanced to take intoaccount the intrinsic double porosity features of such materials.This study aims at characterizing the influence of an adsorbed phase on the instantaneous deformation of microto-macro porous media presenting distinct and well separated porosities in term of pore size distribution. A modelaccounting for double porosity is proposed and validated by in-situ and simultaneous experimental comparisons.The novelty of the approach is to propose an extended poromechanical framework taking into account the intrinsicdouble porosity features of such materials capable of predicting adsorption-induced swelling for high porous materialswithout any fitting parameters.981. An incremental poromechanical framework with varying porosity for double porosity media808182838485868788899091929394959699100In this section, an incremental poromechanical framework with varying porosity proposed by Perrier et al. (2015)for single porosity media is extended to double porosity media.101102103104105106107108109We consider here a double porosity medium with distinct and separated porosities. The small porosity is calledadsorption porosity (φµ ) and the larger one transport porosity (φ M ). This medium is considered as isotropic witha linear poro-elastic behaviour and it is immersed and saturated by a surrounding fluid at bulk pressure Pb underisothermal conditions. Confinement effects may change the thermodynamic properties of the interstitial fluids in bothporosities. The adsorption porosity is saturated by an interstitial fluid of density ρµ at pressure Pµ . The transportporosity is fully saturated by an interstitial fluid (single-phase) of density ρ M at pressure P M (see Fig. 1).For saturated isotropic porous solids, in reversible and isothermal conditions and under small displacementgradient assumptions, classical poromechanics may be rewritten for double porosity media Coussy (2004) :dG̃ s dΨ s dW s(1) σi j : dεi j P M dφ M Pµ dφµ d( P M φ M Pµ φµ ) {z} {z}(2)dΨ s dW sσi j : dεi j φ M dP M φµ dPµ .3(3)
110111112113In Eq. 1–3, (G̃ s , Ψ s , W s ) are respectively the Gibbs free energy, the Helmholtz free energy and the mechanical workof the skeleton. The state variables (εi j , φ M , φµ ) are respectively the infinitesimal strain tensor, the transport porosityand the adsorption porosity. The associated thermodynamical forces (σi j , P M , Pµ ) are respectively the Cauchy stresstensor and the fluid pore pressures in both porosities.114115For an isotropic linear poro-elastic medium, the state equations are then given by: σi j φM φµ G̃ s εi j G̃ s, PMand then: PG̃µs(4) dσ dφ M dφµ 116117118K(φ M , φµ )dε b M (φ M , φµ )dP M bµ (φ M , φµ )dPµb M (φ M , φµ )dε bµ (φ M , φµ )dε dP MN MM (φ M ,φµ )dP MNµM (φ M ,φµ ) dPµN Mµ (φ M ,φµ ).dPµNµµ (φ M ,φµ )In Eq. 4, (σ σkk /3) and (ε εkk ) are respectively the total mean stress and the volumetric strain. (K, b M , bµ , N MM ,N Mµ , NµM , Nµµ ) are respectively the apparent modulus of incompressibility and six poromechanical properties, whichdepends on the two evolving porosities φ M and φµ and on the constant skeleton matrix modulus.119120Considering a single cylindrical porosity2 , homogenization models Halpin and Kardos (1976) yield to: s (1 φ) K(φ) KGs Gs K sφ b(φ) 1 K(φ)Ks121, Gs ,3K s (1 2ν s )2(1 ν s )N(φ) .(5)Ksb(φ) φIn Eq. 5, φ is the porosity and (G s , ν s ) are respectively the shear modulus and the Poisson ratio of the skeleton matrix.122123124125126Practically and for high porosity media, an iterative process of homogenization is chosen to avoid discrepanciesin the apparent properties estimation as noticed by Barboura (2007). The iterative process of homogenization for acylindrical porosity is detailed in Appendix A. Full details on the iterative processes for both spherical and cylindricalporosities are presented in Perrier et al. (2015).127128129130131132Considering that the two porosities are distinct, well separated and both cylindrical, the iterative process canbe used in two successive steps to determine the different modulus of incompressibility. Note that this two stephomogenization process may be reversed as well to estimate the skeleton properties knowing the apparent ones: (K, G) Fn Fn (K s , G s , φµ ), φ M(6)and (K s , G s ) Rn (Rn (K, G, φ M ), φµ ) .In Eq. 6, (Fn , Rn ) stand as the standard and the reverse iterative processes of homogenization defined in Eq. A.1 and A.2respectively. K s and G s (resp. K and G) are the skeleton (resp. apparent) incompressible and shear modulii.133134Based on stress/strain partitions (Coussy, 2004) and on the response of the medium saturated by a non-adsorbable2 Thisassumption is discussed in Perrier et al. (2015) where both spherical and cylindrical porosities are considered.4
135fluid (Nikoosokhan et al., 2013), the six poromechanical properties (b M , bµ , N MM , N Mµ , NµM , Nµµ ) may be identified: b M 1 KKµ , bµ K( K1µ K1s ) 1 bM φM ,1111 N MMKµN Mµ NµM (b M φ M )( Kµ K s ) , (7) (bµ φµ ) 111 (b M φ M )( Kµ Ks )Nµµ Kswith136137Kµ Fn (K s , G s , φµ ) .For a porous medium saturated by a fluid under isothermal conditions (isotropic surrounding/bulk pressure: Pb ,density: ρb ), dσ dPb and Eq. 4 yield to: b dε dP Ks Mdφ M φ.(8) K s dPb dφ φµ dPµ138139140KsbTherefore, classical poromechanics predicts a shrinkage of the porous matrix and a decrease of the porosity underbulk pressure. This has been confirmed by experimental measurements (e.g. Reucroft and Patel (1986) on a naturalcoal with a non-adsorbable 156157158159Considering that the fluid is confined within both porosities, the thermodynamic properties (pressures: P M , Pµ ,densities: ρ M , ρµ ) of the interstitial fluid in the two porosities (φ M , φµ ) differ from the surrounding ones (Pb , ρb ) but thethermodynamic equilibrium imposes that the three fluids are chemically balanced (equality of the chemical potentialsµb , µ M and µµ ). Assuming that the Gibbs-Duhem equation (dP ρdµ) still applies for both the surrounding fluid andthe interstitial ones, a macroscopic relation between the interstitial pore pressures and the surrounding one may bederived similarly to the relation initially proposed by Vermorel and Pijaudier-Cabot (2014) and used in Perrier et al.(2015) for single porosity media: dPbb dP M ρ M dP 1 χ ρbM .(9) dPµ ρµ dPb dPbρb1 χµIn Eq. 9, (χ M 1 ρρMb ) and (χµ 1 ρρµb ) are the confinement degrees in the transport and in the adsorption porositiesexrespectively, which characterize how confined is the interstitial fluid due to the number of adsorbate moles nexM and nµthat exceeds the number of fluid moles at bulk conditions in porosities φ M and φµ respectively: ρ b Vφ MnexM χ with ntot nextotM MM Mn M .(10) ex ρb Vφµ ex χµ ntotµ with ntot nµ MµnµIn Eq. 10, (VφM , Vφµ ) are the connected porous volume corresponding to the transport porosity φ M and to the adsorptionexporosities φµ respectively, nexM and nµ are the number of adsorbate moles that exceeds the number of fluid moles attottotbulk conditions and n M and nµ are the total number of moles of interstitial fluid in porosities φ M and φµ respectively.Generally, there is no way to link separately the two confinement degrees χ M and χµ to quantities that can be measuredexperimentally because the partition of the excess number of adsorbate moles nex , which can be measured experimenextally, within the two porosities is unknown (nex nexM nµ ). However, assuming that the two scales of porosities arewell separated, one can consider that most of the adsorption phenomenon oc
28 For these microporous materials, a deviation from standard poromechanics (Biot, 1941; Coussy, 2004), is ex- 29 pected. In nanometer-scale pores, the molecules of fluid are confined. This e ect, denoted as molecular packing, 30 induces that fluid-fluid and fluid-solid interactions sum at the pore scale and have significant con
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7th January 2004 11:52 Elsevier/MFS mfs 4 Chapter 1 Fundamentals of Poromechanics (P–p) p p P p (a) ε 1 (P p)/Kε ε 1 ε 2 (P bp)/Kε 2 P / K s (b) (c) Figure 1.1 A porous rock submitted to an isotropic pressure P on its external surface and a pore pressure p (c). This
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