A Review on Capillary Condensation in Nanoporous Media: Implications for Hydrocarbon Recovery from Tight Reservoirs ⇑ Elizabeth Barsotti A, Sugata P

Fuel 184 (2016) 344–361

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Review article A review on capillary condensation in nanoporous media: Implications for hydrocarbon recovery from tight reservoirs ⇑ Elizabeth Barsotti a, Sugata P. Tan a, Soheil Saraji a, , Mohammad Piri a, Jin-Hong Chen b

a Department of Petroleum Engineering, University of Wyoming, Laramie, WY 82071, USA b Aramco Services Company: Aramco Research Center – Houston, TX 77084, USA

highlights

Insight into capillary condensation may improve gas recovery from tight reservoirs. Insight into capillary condensation is limited by the scarcity of experimental data. A review on the experimental data available in the literature is presented. A review on theories for modeling capillary condensation is presented. The extension of experimentally veriﬁed models to the reservoir scale is promoted.

article info abstract

Article history: The key to understanding capillary condensation phenomena and employing that knowledge in a wide Received 11 May 2016 range of engineering applications lies in the synergy of theoretical and experimental studies. Of particular Received in revised form 24 June 2016 interest are modeling works for the development of reliable tools with which to predict capillary Accepted 27 June 2016 condensation in a variety of porous materials. Such predictions could prove invaluable to the petroleum Available online 15 July 2016 industry where an understanding of capillary condensation could have signiﬁcant implications for gas in place calculations and production estimations for shale and tight reservoirs. On the other hand, Keywords: experimental data is required to validate the theories and simulation models as well as to provide possible Capillary condensation insight into new physics that has not been predicted by the existing theories. In this paper, we provide a Conﬁnement Hydrocarbon brief review of the theoretical and experimental work on capillary condensation with emphasis on the Nanopores production and interpretation of adsorption isotherms in hydrocarbon systems. We also discuss the Tight reservoirs implications of the available data on production from shale and tight gas reservoirs and provide recommendations on relevant future work. Ó 2016 Elsevier Ltd. All rights reserved.

Contents

1. Introduction ...... 345 2. Capillary condensation of hydrocarbon gases in tight formations ...... 345 3. Capillary condensation: theoretical perspective...... 347 3.1. Density and phases ...... 347 3.2. Mechanism of condensation ...... 347 3.3. Ideal case: the Kelvin equation ...... 350 3.4. Advanced models ...... 351 4. Capillary condensation: experimental perspective...... 353 4.1. Nanoporous media: materials and characterization ...... 353 4.2. Measurement techniques ...... 354 4.3. Experimental data on hydrocarbon systems ...... 355

⇑ Corresponding author. E-mail address: [email protected] (S. Saraji).

5. Conclusions and final remarks ...... 358 Funding...... 358 Appendix A. Supplementary material...... 358 References ...... 358

1. Introduction hydrocarbons in shale and tight formations, impediments to production from them remain and are manifested in nanoscopic An improved understanding of the physical behavior of properties such as fine grain sizes [18,22], nanopores confined fluid is important to a multitude of disciplines and will [18,22–24,27,61], low porosities (2–10%) [19], and nanodarcy allow for the development of better insights into catalysis [1–3], permeabilities [18,19,22,24,33], as well as complex mineral chemistry [4], geochemistry [5], geophysics [1], nanomaterials [1] compositions [62]. These characteristics limit conventional and improved methods of battery design [2], carbon dioxide methods of reservoir evaluation [33], complicating estimations of sequestration [6,7], drug delivery [2], enhanced coalbed methane original hydrocarbons in place and ultimate recovery [18,20] and recovery [8], lubrication and adhesion [1], materials characteriza- culminating in an inability to accurately predict the profitability tion [9–13], micro/nano electromechanical system design [14], of a reservoir. Case in point, a good history match for oil production pollution control [1,7,15–17], and separation [2], as well as hydro- from wells in the middle Bakken formation is obtained only after carbon production from shale and other tight formations [18–35]. considering the fluid phase behavior in small pores [24]. In oil production, for example, full advantage of enhanced oil In estimating hydrocarbon recovery, the physicochemical recovery by carbon dioxide injection into shale formations can only properties of the reservoir fluids are combined with information be taken once a better understanding of confined fluid behavior, about the petrophysical properties of the matrix in order to including the phase equilibria, is gained [36,37]. interpret well logs [20,21,61], compute original hydrocarbons in It is well known that the physical behavior of fluids in confined place [20,24], determine drainage areas, calculate well spacing spaces differs from that in the bulk [1,2,4,5,7,14,15,18,20–22,24, [27], evaluate various production scenarios, and predict ultimate 26–28,38–55]. In nanoporous media with pore diameters less than recovery. For shale and tight reservoirs, uncertainties in the 100 nm [56] and greater than 2 nm, molecular size and mean free determinations of water saturation [32], capillary pressure, and path cannot be ignored compared to pore size [1,23,57]. At this absolute and relative permeabilities [31,32] along with scale, due to confinement, distances are decreased among mole- non-Darcy flow [33], delayed capillary equilibrium, and confined cules, so intermolecular forces are large, and consequently, phase phase behavior necessitate comprehensive theoretical and experi- behavior becomes not only a function of fluid-fluid interactions, mental studies of these nanoscale phenomena and the develop- as it is in the bulk, but also a function of fluid-pore-wall interac- ment of specialized methods for estimating hydrocarbon recovery. tions. Capillary and adsorptive forces [1,15,18,23,26,27,57] alter In shale gas reservoirs (i.e., at reservoir conditions), strong phase boundaries [1,2,7,15,18,21–24,26,27,42,45,49,50,52,55], affiliation of reservoir fluids to pore walls is often present. Because phase compositions [1,27,52,58], interfacial tensions [22], fluid hydrocarbon gases are predominately stored in the organic-matter densities [1,5,23,24,49,51], fluid viscosities [18,22], and saturation nanopores [24] of the shale in which they are the wetting fluid pressures [20,24,42,43,46]. The extent to which the phase behavior [20,21], capillary condensation is highly probable, although more is altered by confinement depends on the interplay of the information is needed to understand how and when it occurs. fluid-fluid and the fluid-pore-wall interactions. Although pore size, Typical compositions of petroleum gases can be found in Table 1 shape, and interconnectivity; pore wall roughness, composition, for conventional geological formations and shale formations. and wettability; and fluid composition and molecular size are Capillary condensation has major implications for estimating qualitatively known to influence the physical behavior of confined hydrocarbons in place in shale and tight gas reservoirs. This is in fluids [1,5,27,45,52], a quantitative understanding of the relative strict contrast to conventional gas reservoirs where nanopores effects of each characteristic is presently lacking. represent an inconsequential percentage of the total porosity in

2. Capillary condensation of hydrocarbon gases in tight Table 1 formations Typical compositions of conventional and unconventional petroleum gases. Component Mole fraction Tight formations and their analogs, shale gas and shale oil Conventionala Shaleb reservoirs, are unconventional resources, which are defined as rock Methane 0.9500 0.6192 formations bearing large quantities of hydrocarbons in place that, Ethane 0.0320 0.1408 as a result of reservoir rock and fluid properties, cannot be Propane 0.0020 0.0835 economically produced by conventional methods. Only in the past n-Butane 0.0003 0.0341 decade have the depletion of conventional reservoirs and the Isobutene 0.0003 0.0097 increasing worldwide demand for hydrocarbons generated enough n-Pentane 0.0001 0.0148 Isopentane 0.0001 0.0084 interest in shale and tight reservoirs to establish technological Hexane 0.0000 0.0179 innovations that make the production from these resources Heptane 0.0000 0.0158 profitable. Octane 0.0000 0.0122 Shale is ‘‘a laminated, indurated rock with [more than] 67% Nonane 0.0000 0.0094 Decane+ 0.0000 0.0311 clay-sized minerals” [59]. The U.S. Energy Information Nitrogen 0.0100 0.0013 Administration estimates that 345 billion barrels of recoverable Carbon Dioxide 0.0050 0.0018 oil and 7,299 trillion cubic feet of recoverable gas are stored in shale Oxygen 0.0002 0.0000 formations worldwide, making shale oil accountable for 9% of total a Typical composition of conventional natural gas composition taken from (proven and unproven) oil reserves and shale gas accountable for Driscoll and Maclachlan [63]. 32% of total gas reserves [60]. Despite the abundance of b Composition of Eagle Ford Shale gas taken from Deo and Anderson [64]. 346 E. Barsotti et al. / Fuel 184 (2016) 344–361 comparison to macropores, and thus, no significant changes to the pairings. It must be noted that capillary condensation data, not overall phase behavior is observed [54]. In spite of the implications adsorption data, is the missing prerequisite for EOS parameteriza- of capillary condensation for unconventional gas reservoirs, cur- tion. Adsorption without phase transition is much better understood rent methods of calculating reserves in shale only consider than capillary condensation, and since it is associated with surface adsorbed gas on the pore walls and free gas in the pore bodies forces, i.e., the disjoining pressure, which appears to be negligible [20,21,30,32,33,35,60] with the largely immobile adsorbed phase in comparison to capillary condensation forces [65], little can be accounting for up to half of the total gas in place [19]. In one study, gained from adsorption-only experiments, in this respect. For this Chen et al. estimated in principle that accounting for capillary reason, we exclude adsorption-only data from Section 3.3 on avail- condensation could increase reserve estimations by up to three- able experimental results. to six-times [20]. Because the densities of phases are different in The current reviews on capillary condensation in the literature nanopores, however, more reliable models are needed in order to either are composed for general applications across a wide variety confirm such estimations. On the other hand, the presence of a of industries with no mention of oil or gas [66] or introduce capil- condensed phase would reduce the permeability in the reservoir lary condensation as it pertains to the much broader studies of as it blocks the gas flow near the wellbore. Nevertheless, confine- phase behavior [1,67], adsorption, materials characterization ment by nanopores introduces different conditions at which two [2,51,68], and fluid dynamics [69]. This paper, however, focuses phases may coexist. Therefore, the net effect of accounting for specifically on the implications of capillary condensation phenom- capillary condensation on the estimates of recoverable reserves is ena for the petroleum industry. In this endeavor, an overview of still unknown. Again, as mentioned earlier, reliable models are the relevant theoretical and experimental methods for the study needed to obtain more accurate estimates. of the capillary condensation of hydrocarbon gases will be pre- Prerequisite to the parameterization of the equations of state sented to reflect the current state of knowledge and to suggest (EOS) used in such models and their validation for applications in the direction for future research. Such future research will provide real systems is comprehensive experimental data on capillary a fundamental understanding of capillary condensation, without condensation. At present, however, experimental data is hardly which applications of it in shale and tight formations cannot be available even for many of the simplest hydrocarbon-adsorbent reliably effective.

Fig. 1. Confined phases and densities: (a) local densities in a pore, where the adsorbed phase contains molecules layered on the wall, while the capillary condensation occurs when the low-density phase (vapor-like phase) abruptly changes into a higher-density phase (condensed phase) in the pore space far from the wall. Adapted with permission from grand-canonical Monte Carlo (GCMC) simulation by Walton and Quirke [70]. Copyright 1989 Taylor & Francis, Ltd. (b) Cartoon showing confined density defined as the number of moles of a given confined fluid phase filling a pore divided by the total volume of the pore. E. Barsotti et al. / Fuel 184 (2016) 344–361 347

3. Capillary condensation: theoretical perspective bulk vapor phase (i.e., adsorbate) are physically bound to the surface of pore-walls (i.e., adsorbent) to form a monomolecular The key to understanding the confinement phenomena, from layer of the adsorbed phase. If the pores are small enough and which shale gas and/or tight gas recovery may benefit, lies in the vapor phase sufficiently wets the pore surface, systematic studies of capillary condensation. The current, albeit inter-molecular forces can build multiple molecular layers of the sparse, body of knowledge regarding capillary condensation is adsorbed phase until, at some threshold temperature and pressure the result of studies from many disciplines, primarily adsorption below the bulk phase boundary of the fluid, a new condensed experiments and molecular simulation studies. phase nucleates and fills the pore [1,2,49,52,55,68,72–75]. This phase is separated from the gas or bulk vapor phase at the pore 3.1. Density and phases throat by a curved meniscus [76]. This phenomenon is called capillary condensation, and is generally categorized either as a second Prior to further discussion, there is a need to define a common order phase transition for one-dimensional (cylindrical) pores [49] ground in using the terms density and phases to describe the behav- or as a first order [2,49,68,76] or nearly-first-order phase transition ior of confined fluids. As facilitated by the established theoretical [69] for two- (slit-like) and three-dimensional pores. and experimental approaches for fluids in the bulk, we use our Capillary condensation data is commonly derived from adsorp- understanding from the bulk properties for the case of confined tion isotherms that relate the amount of fluid adsorbed on a solid fluids even though we know it may not apply at the nanoscale of surface (i.e., in the pore space) to the operating bulk pressure at the confinement. constant temperature [76]. In ordered nanoporous materials, As known from many studies, the fluid molecules are unevenly adsorption isotherms displaying a steep vertical or near-vertical distributed inside each pore as is shown in Fig. 1a, which is step are indicative of the rapid pore filling associated with capillary adapted from a molecular simulation [70], due to fluid-pore-wall condensation [38]. Of the variety of adsorption isotherms recog- interactions as manifest in pore wettability [16,71]. Therefore, nized by the International Union of Pure and Applied Chemistry the density is localized in the confined space; there are more mole- (IUPAC) [47,56], the Type IVa, IVb, and V isotherms (see Fig. 2) cules near the pore wall. To still use density in the usual meaning, are most consistent with this observation and, thus, are the most the confined density is taken to be the moles of the fluid in the pore relevant to the study of capillary condensation in ordered nanopor- divided by the total pore volume as shown in Fig. 1b. ous materials [17,45–47,56]. As in the bulk, densities may be used to infer states of matter or Although ordered nanoporous materials – porous materials phases. Therefore, based on Fig. 1, two different phases may exist containing a regular array of nanopores with the same geometry in nanopores and are of primary interest to shale and tight gas – are not representative of real adsorbents such as shale rocks, they recovery. The first phase is the vapor-like phase in the pore prior are a mainstay of present research into capillary condensation. to condensation with an average density of qA, while the second Their well-defined pores allow for studies into even the most basic, is the liquid-like condensed phase with an average density of qL yet still not understood, phenomena related to fluid-pore-wall that forms at capillary condensation [1,2,42,49,52,68,69,72]. The interactions that are not observable in adsorbents with highly first phase consists of molecules that are mostly adsorbed on the irregular arrays of pores and poorly defined surfaces. pore walls, so it has a density, as newly defined, between that of Adsorption isotherms exhibiting capillary condensation can the bulk vapor phase and that of the condensed phase [49,72,73]. either be reversible, as in Type IVb isotherms, or irreversible (i.e., From this point on, we call it the adsorbed phase. The bulk vapor hysteretic), as in Type IVa and Type V isotherms. The hysteresis phase outside the pore and this adsorbed phase are in thermody- loop in the Type IVa isotherm indicates capillary condensation namic equilibrium with each other prior to condensation. At [76,77] when the temperature of the confined fluid is above the condensation, the equilibrium also involves the condensed phase. triple-point of the bulk fluid [1]. The different paths of adsorption After the phase transition, the condensed phase replaces the and desorption in hysteretic isotherms are dependent on pore adsorbed phase in the equilibrium. chemistry, geometry and temperature [47,73] and may arise either from the formation of metastable states (Type H1 hysteresis) 3.2. Mechanism of condensation [1,2,38,56,66,74], or from pore blocking or networking effects due to heterogeneity of pore geometries, such as is found in Exposing gas to a clean, outgassed porous material introduces a inkbottle shaped pores or interconnected pores (Type H2 hystere- phenomenon well known as adsorption, where molecules of the sis) [47,56,66,73,74,76].

Fig. 2. IUPAC Type IV(a), IV(b), and V isotherms. The shaded regions in the figure indicate the condensation/evaporation steps. Types IV(a) and IV(b) isotherms are typical of nanoporous adsorbents in which the adsorbate is the wetting phase. Type IV(a) isotherms may exhibit hysteresis caused either by the formation of metastable states or from pore blocking or networking. The first is exhibited in the IUPAC H1 hysteresis loop as indicated by the solid line [1,2,38,56,66,74], while the second is exhibited in the IUPAC H2 hysteresis loop as indicated by the dotted line [47,56,66,73,74,76]. While Type V isotherms are characteristic of adsorbents in which the adsorbates are non-wetting [56]. The wettability of the adsorbent to its adsorbate is demonstrated by the shape of the isotherm prior to condensation. Wetting fluids almost immediately begin to form layers on the sides of the pores as evidenced by a convex curve, while non-wetting fluids do not share this behavior, as shown by their concave curve. Adapted with permission from Thommes et al. [56]. Copyright 2015 IUPAC. 348 E. Barsotti et al. / Fuel 184 (2016) 344–361

no hysteresis occurs; this is denoted as the hysteresis critical

temperature (Th) [55,66,73,78]. Th is less than TCp, while decreases in temperature from Th result in expansion of the hysteresis loop [13,38,55,73,77,78], as illustrated in Fig. 3. This is supported by the work of Morishige and coworkers, who experimentally observed capillary condensation at a wide range of temperatures

from below Th to TCp, including the disappearance of the differences between confined fluid phases at TCp, for argon, nitrogen, oxygen, ethylene, and carbon dioxide in MCM-41 [38,73] and SBA-15 [78]. These results were further corroborated for many fluids in wide variety of adsorbents by Qiao et al. [40], Russo et al. [15], Yun

et al. [7] and Tanchoux et al., who also noted that Th decreases with pore size [13]. However, relatively recent studies from Morishige et al. and Horikawa et al., which detailed the capillary condensation of water in ordered mesoporous carbons (OMC’s), are in disagreement with these ﬁndings [72,79]. Unlike the capillary condensation of gases in mesoporous silicas where condensation pressures rose with

Fig. 4. Isotherms for water in OMC (7.0 nm pore diameter) [72]. The positions of the adsorption and desorption branches of the hysteresis loop are constant regardless of temperature. Adapted with permission from Morishige et al. [72]. Copyright 2014 American Chemical Society.

In many senses, as previously mentioned, capillary condensation is the phase transition of the confined fluid into a condensed phase, similar to the vapor-liquid phase boundary in the bulk. For a pure fluid in the bulk, this phase transition can occur up to the critical point (TC), above which only a homogenous supercritical phase can exist. For a confined fluid in a given adsorbent, the condensed phase and the lighter adsorbed phase are distinguishable only up to the so-called pore critical temperature (T ), which is normally Fig. 5. Experimental indication that the adsorption branch of the isotherm (open Cp circles) is closer to equilibrium than the desorption branch (filled circles). Data is for lower than T [1,49,73,78]. For fluids that show hysteresis in their C O2 in 4.4-nm MCM-41 [73]. Adapted with permission from Morishige et al. [73]. adsorption isotherms, there also exists a temperature above which Copyright 2004 American Chemical Society. E. Barsotti et al. / Fuel 184 (2016) 344–361 349 increasing temperature [38,78], the condensation pressures of the scattering (SAXS). Based on discrepancies between theoretical and water in OMC’s remained constant with increasing temperature experimental isotherms, Gor et al. concluded that capillary conden- (Fig. 4) [72,79]. Morishige et al. attributed this inconsistency to sation changes the elastic properties of SBA-15 but not of MCM-41. differences in the wettabilities of the adsorbents to their respective They attributed this difference to the presence of micropores in the fluids [72]. SBA-15, alone, even though their SBA-15 sample (8.1 nm) had more Regardless of the relationships among the confined critical than twice the pore diameter of their MCM-41 sample (3.4 nm). temperatures and condensation pressures, for all fluids confined Thus, it cannot be known from their work how pore size affects in porous media, TCp and Th depend on the fluid chemistry, changes in pore wall elasticity during capillary condensation [8]. pore-wall chemistry, pore size, and/or pore geometry [1,66]. The In a complimentary work by Günther et al., small angle X-ray diffrac- complex dependency of Th on these properties has given rise to tion was used to show that increasing adsorption before capillary controversy as to which branch of the hysteresis loop represents condensation causes pores to expand, while capillary condensation the equilibrium phase transition [12,15,55,66,74]. causes the pores to contract [50]. Therefore, changes in pore Neimark et al., who matched data from their model to diameter due to adsorption, as shown in the work of Gor et al. and experimental isotherms and Derjaguin-Broekhoff-de Boer Günther et al., could affect the onset of capillary condensation [8,50]. isotherms for argon and nitrogen in MCM-41 type pores [80], Because the strain of the adsorbent is directly related to the proposed the desorption branch as the equilibrium branch. This is pressure of its occupying fluid, the strain isotherms produced by supported by the theory, where minimization of the grand potential Gor et al. can also be interpreted as showing the pressures of the energy results in a wider range of available fluid configurations fluid within the pores. Through this interpretation, it is evident (i.e., densities) for the adsorption branch than the desorption that adsorbed fluid layers before capillary condensation possesses branch, thus indicating that more metastable states develop during a positive pressure (i.e., cause positive strain, or expansion, of the adsorption so that desorption, rather than adsorption, occurs at the adsorbent), while the condensed phase during capillary condensa- true phase equilibrium [42]. This designation is further supported by tion has a negative pressure (i.e., causes negative strain, or contrac- Pellenq et al., who have compared isotherms from their mean field tion, of the adsorbent), often referred to as tension or being model to those of argon in MCM-41-type materials [81], and stretched. The condition under tension is supported by simulation has been adopted by the IUPAC [56]. However, it is in strict contrast observations, such as those by Long et al. [4], who found that for to the experimental work of others [38,72–74,78]. In their pores with widths greater than 5 molecular diameters of the experimental studies, Morishige and coworkers have investigated confined fluid, the pressure in the condensed phase was always the temperature progression of the chemical potential difference negative. Both the work of Gor et al. and Long et al. are consistent (with respect to that of bulk liquid) of adsorption and desorption with evidence from capillarities, such as water plugs in nanochan- during capillary condensation in hysteretic isotherms [73,78],as nels [82], peculiar behavior of soil water [83], centrifuge capillary- shown in Fig. 5. Based on the continuity of the slope along the pressure experiments [84], and sap transport in trees [85]. Though adsorption branch across Th in Fig. 5, Morishige and coworkers pre- liquids under negative pressure are theoretically in a metastable sent a compelling argument for the existence of thermodynamic state [86], they show stable behavior in confinement [84]; and thus equilibrium only during adsorption. In other words, their work can exist even for geological times [135]. They may also exceed the shows that the adsorption branch is the true phase transition and, stability limit where cavitation should occur to stabilize the system thus, should be used for identifying capillary condensation from [87]. This fact may offer an alternative explanation to the delayed adsorption isotherms [14,69]. desorption phenomenon if adsorption is considered to be the true Based on observations from the adsorption isotherms, capillary phase equilibrium. condensation in nanopores is attributed to strong intermolecular Likewise, evidence as to the fluid-wall interactions during forces [15,18,23,26,27,46,57,69,76], although the nature of the capillary condensation has been produced by Naumov et al. [46]. pore-fluid interactions and the ways in which pore geometry, Using Pulsed Field Gradient Nuclear Magnetic Resonance (PFG pressure, and temperature quantitatively affect these interactions NMR), they showed that for cyclohexane in Vycor glass and porous is unknown [1]. However, some progress has been made in the anal- silicon, the hysteresis of the adsorption isotherm is accompanied ysis of the forces exerted by capillary condensed fluids on the walls by hysteresis of the self-diffusivities [46]. Self-diffusivity is the ran- of their confining pores, and vice versa. For example, Gor et al. have dom, microscopic movement of the fluid molecules due only to compared capillary condensation adsorption isotherms of their own thermal energy [46]. Naumov et al. attribute hysteresis n-pentane in MCM-41 and SBA-15 to both experimental and of the self-diffusivities to the differences in the densities of the theoretical capillary condensation strain isotherms [8]. Strain iso- fluid filling the pores during adsorption and desorption [46]. The therms, as shown in Fig. 6, are plots of relative pressure versus strain self-diffusivity was lower during desorption due to the at constant temperature, typically produced via small angle X-ray pore-blocking effects that occur when evaporation proceeds via

Fig. 6. Typical strain isotherms for adsorbents during the capillary condensation of (a) a wetting ﬂuid and (b) a non-wetting ﬂuid [8]. These correspond to the IUPAC Type IV (b) and Type V adsorption isotherms, respectively [8]. Adapted with permission from Gor et al. [8]. Copyright 2013 American Chemical Society. 350 E. Barsotti et al. / Fuel 184 (2016) 344–361

In capillary condensation, the Young-Laplace equation is used in the vapor-liquid equilibria (VLE) to account for the effect of the nanopores. For bigger pores, where the vapor phase may be considered ideal and the liquid phase is incompressible, the use of the Young-Laplace equation in the VLE leads to the Kelvin equation [12,14,44,46,88,90,91]: PV 1 2c ln ð2Þ sat q P LRT rp

where PV is the operating pressure of the vapor phase at which capillary condensation occurs, Psat is the saturated vapor pressure q of the fluid in the bulk, L is the molar density of the liquid phase in the bulk, R is the universal gas constant, and T is the absolute temperature. If both sides of Eq. (2) are multiplied by temperature, then the expression on the left-hand side is the same as the vertical axis in Fig. 5, while the right-hand side is a continuous function of temperature. Therefore, Morishige and coworkers have practically shown that the adsorption branch behaves according to Eq. (2), which accounts for the phase equilibrium. Fig. 7. Hysteresis of self-diffusivities of cyclohexane in Vycor glass at 297 K as The Kelvin equation is used extensively in both experiments compared to that in the corresponding adsorption isotherm. Adapted with and models to mathematically describe capillary condensation permission from Naumov et al. [46]. Copyright 2008 American Chemical Society. [81,90]. It is frequently employed in the prediction of the occur- rence of capillary condensation under different conditions [14,42,46] and the evaluation of the forces that are exerted on cavitation, which is at least partially dependent on pore geometry the adsorbent by condensates [14,75], the thickness of adsorbed [46]. A self-diffusivity hysteresis loop is shown in comparison to layers [75,88], pore size [75], and pore size distribution [44,88,92]. hysteresis of an adsorption isotherm in Fig. 7. Despite its frequent use, the Kelvin equation is based on many assumptions that may not be valid for scenarios associated with capillary condensation. For example, the Kelvin equation assumes 3.3. Ideal case: the Kelvin equation that the liquid is incompressible, the vapor is ideal, and both the surface tension and the molar density are independent of the pore To account for fluid-pore interactions, many theoretical meth- radius [55,89,91], all of which are not necessarily true for fluids con- ods take into consideration the pressure difference between the fined in nanopores. Furthermore, it does not account for adsorbed confined and the bulk phases in the form of the disjoining pressure, phases or the fluid-pore wall forces that cause them [2,46,55]. the capillary pressure, or both. Before capillary condensation Likewise, because it is based on macroscopic thermodynamics for occurs, disjoining pressure is the only form of interfacial pressure a liquid and vapor in equilibrium, its accuracy when used in and exists as the pressure difference between the adsorbed layers hysteretic isotherms is largely dependent on whether or not the and the vapor phase filling the pore body [8,12,14]. Once the true equilibrium branch is selected for the calculations [2,46,55]. adsorbed film reaches its limit of stability, the layers of the adsorbed phase converge upon each other in the process of capillary condensation [12], and the center of the pore is filled with the condensed phase which is separated from the bulk vapor at the pore throat junction by a meniscus. At this point, equilibrium is reached between the bulk vapor and the condensed phase in the pore, and the capillary pressure can be defined as the difference in the pressures on either side of the meniscus [8]. It is important to note, however, that even after capillary condensation occurs there are still some layers of fluid adsorbed on the pore wall, meaning that a disjoining pressure is still present, although it now represents the pressure difference between the adsorbed layers and the condensed phase. In some studies, the disjoining pressure appears to be negligible in comparison with the capillary pressure [65].

Capillary pressure, Pcap, is deﬁned by the Young-Laplace equation [14,18,22–24,46,88,89]. The equation for the case of a cylindrical pore is:

2c Pcap ¼ PNW PW ¼ cos h ð1Þ rp where PNW is the pressure of the non-wetting phase, which is commonly the bulk vapor phase; PW is the pressure of the wetting Fig. 8. The difference between capillary condensation pressure for nitrogen in a slit phase, which is commonly the liquid-like condensed phase in the pore predicted by the Kelvin equation (the dashed line) and GCMC simulation (the solid line). The experimental data for a similar system is shown by the filled circles. pore; c is the surface tension between these two fluid phases; r p The figure shows significant deviations between predictions by the Kelvin equation h is the radius of the pore; and is the contact angle of the meniscus and experimental data for pore diameters below 7 nm. Taken from Walton and with the pore wall. Quirke [70], in the format adapted by Aukett et al. [95]. Copyright 1992 Elsevier. E. Barsotti et al. / Fuel 184 (2016) 344–361 351

Although the Kelvin equation and its variants have been shown [1,2,4,44,66,68,69,101,102] and Molecular Dynamics (MD) to be valid in pores with radii as small as 4 nm for some confined [1,66,68,69]. Because molecular simulations can account for the systems [43,93,94], the validity of its assumptions decreases with individual interactions of atoms and molecules [69], they allow pore size [91]. A study by Aukett et al. showed that the Kelvin for easier description of complicated molecular structures [66] equation deviates from simulation results and experimental data and provide for more invasive investigation into the underlying in the prediction of capillary condensation for pores smaller than mechanisms of capillary condensation compared to DFT 7nm(Fig. 8) [95]. Nevertheless, using the gas composition from [2,44,53,66]. Likewise, molecular simulations allow for easy a Marcellus well and the pore size distribution of kerogen pores adjustment of properties that are difficult, if not impossible, to in a hypothetical shale rock, Chen et al. incorporated the multicom- control in experiments [1,66,69]. However, oversimplifications of ponent version of Kelvin equation [96] into estimations of gas in input functions, inadequate algorithms, and the extensive amount place [20]. The resulting theoretical adsorption isotherms of computational time required to perform realistic simulations exhibited condensation steps, which, when incorporated into Chen [58], render current molecular simulations as inaccurate as DFT et al.’s estimations, lead to increased gas in place estimates for the [2,66]. The uncertainties as to the influence of the various chemical hypothetical rock of up to six times [20]. and physical parameters persist and, along with the present Some efforts have been made to extend the accuracy of the inability to exactly model both real and synthesized nanoporous Kelvin equation to nanoscale systems, among which are the inclu- materials, generate inaccuracies that prevent further progress into sion of the effect of the meniscus on the surface tension [97] and the development and evaluation of theories [52]. Even so, simula- direct adjustments to the pore radius to account for the thickness tions are useful for qualitative studies of capillary condensation of the adsorbed phase [92,98]. and are particularly valuable for their insight into properties that Regardless, when the Kelvin equation is applied in the are experimentally inaccessible. A sample density profile of characterization of porous media, it has been shown to underesti- confined fluid available through GCMC simulations is shown in mate pore size by approximately 25% in pores with radii below Fig. 1a, which is in excellent agreement with that available through 10 nm [99]. Because many studies involving capillary condensation DFT [103]. include porous materials with radii below 10 nm [8,13,15,72,74] Long et al. used GCMC simulations to evaluate the effect of pore and below 4 nm [6–8,10,13,15,38–40,48,72,73,78], more accurate geometry on the pressure of argon [4]. Based on their simulations, approaches are needed to better describe the fluid phase behavior they showed that, for a given temperature, the capillary in nanopores. condensation of argon occurred first (at the lowest pressure) for spherical pores, followed by cylindrical pores and then slit pores 3.4. Advanced models [4]. They attributed this to the direct proportionality of the pore wall curvature to pore-fluid interactions as manifest in a pressure Efforts to account for large departures from the ideal tensor for the argon that, over the given pressure range, exhibited a assumptions of the Kelvin equation include the utilization of rapidly increasing tangential pressure [4]. various versions of density functional theory (DFT), e.g., non-local Singh et al. used grand-canonical transition-matrix Monte Carlo density functional theory (NLDFT) and quench solid density (GC-TMMC) simulations to study the critical properties, surface functional theory (QSDFT). DFT relies on the minimization of the tensions, phase coexistence, density profiles, and orientation grand potential energy, which accounts for all the thermodynamic profiles of methane, ethane, propane, butane, and octane in gra- energies of the system at the same chemical potentials throughout phite and mica slit pores with different widths [53]. Though their all phases, to achieve the most stable energy state for the fluid- findings are in general agreement with experimental data for the pore system [1,2,12,66]. DFT deals with energy functionals [100], given adsorbate-adsorbent systems, their study suffers from a lack which also account for pore geometry [2] and fluid-pore wall inter- of comprehensiveness; for their simulations included only a lim- actions [12], and results in the fluid density profile within the pores ited selection of fluids in just two pore types [53]. In another work, at the most stable energy state (Fig. 1a) [1,2,12,66]. In studies of Yun et al. were able to use their own experimental isotherms to capillary condensation, this density can correspond either to the validate GCMC simulations and ideal adsorbed solution theory condensed or the vapor phase in the pore [12]. Using DFT, meta- (IAST) for the prediction of the adsorption of methane, ethane, stable states can be modeled for the investigation of hysteresis, and a methane-ethane mixture in MCM-41 [7]. Likewise, He and although many calculations for identification of the sorption branch Seaton used their own experimental isotherms to validate GCMC (adsorption or desorption) that corresponds to phase equilibrium simulations for the prediction of the adsorption of carbon dioxide, are contradictory to the aforementioned experimental findings of ethane, and a carbon dioxide-ethane mixture in MCM-41 [104]. Morishige et al. [2,12,66]. Indeed, comparisons of DFT isotherms Although neither Yun et al. nor He and Seaton specifically studied with experimental isotherms have shown DFT to be incapable of capillary condensation in their systems, their data showed precise quantitative predictions of hysteresis in even the simplest condensation at some experimental conditions. cylindrical pores [12,66]. The inaccuracies of DFT have been While the above rigorous computational approaches do not attributed to oversimplifications in the functionals that include provide accurate descriptions of macroscopic behavior, simpler the disregard of pore wall surface roughness [2], models of infinite methods with equations of state (EOS) have been used to model pores that have little relation to their finite experimental counter- the confined phase equilibria. In such modeling, the equilibrium parts, and improper treatments of pore-fluid potentials [2,12,100]. of the condensed phase in pores (L) and the vapor phase in the bulk Despite its shortcomings, however, NLDFT, in particular, has gained (V) can be generalized in terms of chemical potential, or more popularity for its ability to estimate pore size distribution more commonly, in terms of fugacity, where the equifugacity equations accurately than approaches involving the Kelvin equation [2,12]. for a mixture with N components are expressed as: With the current lack of experimental data and the difficulty in obtaining them, molecular simulations have been carried out for ^ ^ f Lðx; T; PLÞ¼f Vðy; T; PVÞ i ¼ 1; ...; N ð3Þ adsorbents of varying wettability [52,53] and pore geometry [4] i i for theory validation purposes. Similar to DFT, molecular simulations offer microscopic treatments of capillary condensation that Note that the pressures of the phases are different. The pressure take into account fluid-fluid and fluid-pore wall interactions [2], difference can be expressed in terms of capillary pressure as in most commonly grand canonical Monte Carlo simulation (GCMC) Eq. (1) where L is the condensed phase, which is wetting; and V is 352 E. Barsotti et al. / Fuel 184 (2016) 344–361

Fig. 10. Qualitative shift of a phase envelope when shifted-phase parameters are Fig. 9. Qualitative shift of a phase envelope when bulk-phase parameters are applied to a cubic EOS [18]: dew points and bubble points can be higher than those applied to a cubic EOS [106]: the critical point (C.P.) does not change; it remains the in the bulk. same as in the bulk. Du r r 2 the bulk vapor phase, which is non-wetting; x and y are the fluid C ¼ 0:9409 0:2415 ð4Þ u r r composition in phases L and V, respectively. C p p In addition to Eq. (3), the system has to satisfy the material where u can be temperature or pressure, and DuC is the shift of the balance. Solving these equations is analogous to the vapor-liquid critical property in the pores from its value in the bulk. phase equilibrium calculation in the bulk, but at different pres- Zarragoicoechea and Kuz studied the critical temperature and sures across the phase boundary. This approach is free from pressure shift of argon, xenon, carbon dioxide, oxygen, nitrogen, assumptions imposed on the Kelvin equation while still dependent and ethylene in MCM-41 [109]. In view of their ability to match on the validity of the Young-Laplace equation for extremely small experimental data relatively well [109], it would be beneficial to pores. see further validations of their model with the hydrocarbon Though a simple cubic EOS may also be used here, some constituents of shale and/or tight reservoirs, as presented in Table 1. difficulties arise in using this class of EOS, as it requires critical Using another approach, Alharthy et al. correlated shifted properties for parameterization. In nanopores, where the critical critical pressures and temperatures to results from GCMC points of pure substances are shifted from their bulk values, a cubic simulations by Singh et al. for n-alkanes in graphite slit pores, EOS must be applied either by using the shifted critical properties where d is the pore diameter [18,53]: as the new EOS parameters [18] or by ignoring the confinement- DP induced shift altogether, thus, using the bulk-phase parameters C ¼0:4097d þ 1:2142 ð5aÞ [57,105] while the Laplace equation, Eq. (1), is added to account PC DT for the effects of confinement [24,106]. ln C ¼ 0:093764 0:929d ð5bÞ For multicomponent fluids in confinement, it is generally TC agreed upon that the phase envelope in the P-T phase diagram is The main drawback of the approach with shifted critical also shifted from that of the bulk, even though the shift of the crit- properties, however, is that, as in Eq. (5a), there is no way to reduce ical point has not yet been experimentally verified. An analysis of the systems to the bulk phase if the porous medium is removed, the envelope shift has been recently made [106] using a theoretical where the new parameters in cubic EOS refer to different chemical approach inspired by the multicomponent version of the Kelvin substances in the bulk. Furthermore, as seen in Fig. 10, the bubble equation [96,107]. In the analysis, which applied a cubic EOS with points can be higher or lower than those in the bulk phase, while the Laplace equation, the critical point of a multicomponent fluid the dew points are even higher than their bulk-phase counterparts, was found to be invariant, while the cricondentherm (the which both contradict the notion introduced by experimental data maximum temperature of the envelope) was found to shift to a [89]. higher temperature, as shown in Fig. 9. The bubble points were also In view of the present lack of experimental data necessary to found to be suppressed for temperatures up to the critical point, as validate the above approaches, some studies [26,110] have even were the dew points up to the bulk cricondentherm. However, the combined the approaches that utilize both the shifted critical portion of the envelope between the critical-point temperature properties as the EOS parameters and the Laplace equation to and the bulk cricondentherm was found to shift to higher pres- account for the effects due to the surface tension. sures. These findings, except for the shift of the cricondentherm, In an effort to better account for the effects of nano- are similar to those of some other studies including that of confinement, additional terms can be supplied to a cubic EOS. Nojabaei et al. [24] and Jin and Firoozabadi [108], where the For example, Travalloni’s term [57], written here as f, can be added cricondentherm was found to be invariant. to the Peng-Robinson EOS [111] in the following form: On the other hand, when using shifted critical properties as the new EOS parameters, the resulting phase envelopes dramatically ¼ RT a ap ð ; d ; e ÞðÞ P 2 f rp p p 6 change, as shown in Fig. 10 [18]. The shifted critical properties v bp v2 þ v 2bp bp can be numerically derived from theory, as done by

Zarragoicoechea and Kuz with regard to the ratio of molecular where dp ={dp,i} and ep ={ep,i} are the parameters that represent the diameter (r) to pore radius (rp) [109]: potential width and potential energy between the pore wall (p) and E. Barsotti et al. / Fuel 184 (2016) 344–361 353 the fluid molecules (i), respectively, while the pore-modified energy discrepancy between the calculated and experimental dew-point and size parameters of the EOS are: pressures is currently of unknown origin that needs further XX investigations. bi þ bj ap ¼ xixjaij 1 ð7Þ The main drawback impeding this approach is the scarcity of hðrPÞ Xi j experimental phase-equilibrium data in the literature for confined systems of both pure components and their mixtures from which bp ¼ xibiCiðbi; rpÞð8Þ i to derive the parameter k in Eq. (11). Therefore, the promising results of this approach warrant experimental measurements on where h and C are functions of the pore radius r and must reduce to p the phase behavior of confined fluids for its further development. h ? 1 and C ? 1 in the bulk, i.e., when r ? 1, so that the modified p For unconventional oil and gas recovery applications, those parameters also reduce to the bulk parameters. measurements must be eventually made both with hydrocarbons The original Travalloni’s work [57] also presents the term f and other compounds generally present in the entrapped oil or added to van der Waals EOS, which is used to construct isotherms gas and with adsorbents representative of reservoir rock. for methane, ethane, toluene, 1-propanol, nitrogen, and hydrogen adsorbed on MCM-41, MSC-5A molecular sieve, DAY-13 zeolite, and JX-101 activated carbon. Though their work closely matched 4. Capillary condensation: experimental perspective experimental data, the majority of their experimentally verified work did not include capillary condensation [57]. Nevertheless, 4.1. Nanoporous media: materials and characterization its good prediction of the condensation of ethane in MCM-41 makes it a promising candidate for future studies [57]. In order to improve the fundamental understanding of the Likewise, contributions from surface forces can be added to the physics involved in capillary condensation, reliable experimental Laplace equation within the Peng-Robinson EOS environment [65]: data in nanoporous media with disconnected pores, uniform pore geometry, and known homogeneous chemistry is required. The a b 2c A P ¼ P 123 ð9Þ interpretation of experimental data in such porous media is mostly rp rp p 3 rp 6 z0 straightforward. The most commonly used adsorbents are the ordered nanoporous silicas, MCM-41 [6–8,10–13,15,38,40,50,73, where A123 is the Hamaker constant between the bulk phase b and 77,78,115] and SBA-15 [8,9,12,15,74,78,115]. MCM-41 and the pore material in the presence of condensed phase a, and z0 is SBA-15 both contain hexagonally ordered cylindrical pores con- the distance to the pore wall. The Hamaker constant can be derived structed of silicon dioxide, which can be easily synthesized in a as [112]: variety of pore sizes [116,117]. A comprehensive comparison of 3 the characteristics of MCM-41 to those of SBA-15 can be found in A123 ¼ ðn ðe1; e2; e3ÞkT þ n ðn1; n2; n3ÞhmaÞð10Þ 4 1 2 the work of Galarneau et al. [117]. The simple geometry of MCM-41 and SBA-15 and their oil-wet where ei is the dielectric constant, ni is the refractive index of mate- counterparts, i.e., ordered mesoporous carbons (OMC’s) [72], are rial i, k is the Boltzmann constant, h is the Planck constant, and ma is the adsorption frequency. However, because the additional contri- bution of the surface forces turns out to be small in comparison to that of the surface tension in the Laplace equation, it can usually be safely neglected [65]. Recently, a robust EOS based on statistical mechanics, i.e., the perturbed-chain statistical associating fluid theory (PC-SAFT EOS) [113], has been used to calculate fugacity coefficients coupled with the modified Laplace equation [89]. The modified Laplace equation is written as: c a ¼ b 2 ð Þ Prp Prp 11 rpð1 kÞ where k is a new parameter that depends on the fluid and the pore material and is readily derived from capillary condensation experimental data. With the use of experimental data for the derivation of the parameter k, this particular approach differs from its predeces- sors in that the actual effects due to the fluid-wall interactions are well represented. Consequently, the EOS has strong predictability in modeling confined fluid mixtures when the bulk behavior of the system is known. This approach has been applied to study the capillary condensation of non-associating fluids in MCM-41 (nitrogen, argon, oxygen, carbon dioxide, n-pentane, n-hexane), SBA-15 (nitrogen), and Vycor (nitrogen, nitrogen-argon mixture, krypton-argon mixture) [89]; and associating fluids in MCM-41 (water, ethanol), stainless Fig. 11. Differences in pore diameter derived from different characterization steel plates (ethanol, acetone, acetone-ethanol mixture, ethanol- methods for two different MCM-41 samples using nitrogen isotherms at 77 K. water mixture), and porous carbon plates (ethanol, acetone, Sample 2 was synthesized to have a smaller pore diameter than Sample 1. Adapted acetone-ethanol mixture, ethanol–water mixture) [114]. All results from Kruk et al. [11] (BJH(a) and BJH(d): BJH method applied to the adsorption proved consistent with the available data from experiments and branch and desorption branch of a hysteretic isotherm, respectively; NLDFT: non-local density functional theory; BET: pore sizes derived from the BET specific simulations except in the case of dew-point pressure predictions surface area and the BET pore volume; and Geometric: pore size calculated by for an ethanol-water mixture in stainless steel, i.e., a strongly relating the volume of the pore space to the volume of the solid adsorbent for a associating fluid in a highly polar porous medium [114]. The highly uniform array of cylindrical pores [i.e., MCM-41 and SBA-15]) [11]. 354 E. Barsotti et al. / Fuel 184 (2016) 344–361 over-simplified compared to naturally-occurring adsorbents and both commercial [7,10,11,16,46,48,72,74] and homemade porous media encountered in industry. More realistic models for [7,13,38,46,48,49,73,78] apparatuses are available in the literature. these applications are adsorbents with interconnected pores, Gravimetric apparatuses also come in a variety of homemade including controlled pore glass (CPG) [46,49], Vycor glass [39,46], [8,15,16,126] and commercial [6,9,17,40,72,77,115] designs. One SBA-16 [15], LPC [15], mesocellular foam [15], MCM-48 [16], silica of the most popular gravimetric apparatuses used in the literature glass [46], and shale core plugs [21]. Although it is challenging to is the magnetic suspension balance [6,17,51,72,127]. In extending interpret data from these more complex adsorbents, they are more measurements to multicomponent adsorbates for both volumetry relevant to practical applications of capillary condensation and gravimetry, a gas chromatograph is required to derive the phenomena. This is particularly true to the petroleum industry concentrations of the individual components that are condensed where capillary condensation happens in the random pore in the pore space, as will be later described [7,51]. networks, geometries, and chemistries of shale and tight reservoir The popularity of gravimetry and volumetry has led to compar- rocks. Nevertheless, systematic studies need to start with very ison of the two [127]. While volumetry is known for its economy simple cases before dealing with complex ones. and simplicity [127], and gravimetry is known for its comprehen- Regardless of the adsorbent, appropriate characterization of the siveness; a comparison of the accuracy of the two is difficult, since material is needed, including information about pore wall such a comparison is partially dependent upon the applicability of wettability [51], pore size, pore volume, pore size distribution, pore each specific apparatus. Application limitations include tempera- connectivity, and pore surface area [9–13,41,48], in order to help sys- ture, pressure, chemical resistance, and flow potential. For example, tematize the interpretation of capillary condensation data many of the microbalances (excluding magnetic suspension bal- [2,9–12,41,51,55,76,88,92,117–121]. Commonly used characteriza- ances [5]) used in gravimetric measurements cannot withstand tion methods include the Clausius-Clapeyron equation [122,123] for temperatures or pressures far removed from standard conditions analysis of pore surface homogeneity [6,7,15,16],the [2,76], while the accuracy of volumetric measurements are highly Barrett-Joyner-Halenda (BJH) method [55,66,92] for determination dependent on the equations of state used [127]. Furthermore, of pore size distribution [9,11,16,17,48,72,77],andthe volumetry, which is often used for the characterization of adsor- Brunauer-Emmett-Teller (BET) method [2,51,56,76,118,121,124] for bents [2,76], can easily withstand extreme temperatures such as quantitative determination of the specific surface area of the adsor- the boiling point of nitrogen (77 K) [2,76], while gravimetry gives bent and qualitative determination of the fluid-solid interaction direct, real-time recordings of the amount adsorbed or desorbed, energy [7–9,11,13,16,38,48,72–74,77]. A critical examination of these providing insight into the kinetics of capillary condensation. methods can be found elsewhere in the literature [11,56,88,92, By pairing a volumetric and a gravimetric apparatus with 118,120]. Scrutiny of them has led to complementary and alternative similar capabilities, Belambkhout et al. carried out a comparison characterization methods including X-ray diffraction (XRD) [6,7,9–11, of the two for high-pressure adsorption measurements [127].In 13,15,16,38,40,41,48,72,77,78,125], transmission electron micro- their study, Belambkhout et al. used a homemade volumetric scopy (TEM) [48,77,115], scanning electron microscopy (SEM) apparatus and a RubothermÒ magnetic suspension balance for high [21,45,48,77], geometrical considerations [11], the alpha-plot method pressure adsorption measurements [127]. Their results showed [40,74], and density functional theory [2]. A comparison among pore volumetric data to differ from gravimetric data by 3%, with the sizes determined using different characterization methods can be gravimetric method considered to be more accurate, especially at found in Fig. 11. high pressures where extreme deviations between the volumetric In spite of the obvious differences in the results obtained from and gravimetric data became predominate [127]. In view of this, various characterization methods, as shown in Fig. 11, it is difficult Belambkhout et al. concluded that, in terms of accuracy, the two to order the characterization methods hierarchically with respect measurement techniques can be interchangeably used at low pres- to accuracy. Of primary importance are limitations in accuracy sures (less than 360 psi) as long as care is taken to accurately due to the assumptions used in the different mathematical measure the reference volumes of volumetric apparatuses, while interpretations of the experimental data. Other factors include appropriate equations of state are used when necessary, and gas analytical and experimental limitations such as small sample mass, leaks in the apparatuses are prevented [127]. small sample cell volume, condensation among adsorbent Like volumetry and gravimetry, some alternative methods particles, the appropriate identification of the equilibrium branch produce adsorption isotherms, such as NMR [46] and optical (in hysteretic isotherms), and a proper curve fit between interferometry [45], which measure condensation by detecting experimental data points [120]. the intensity of a nuclear spin echo or of the spectra of reflected light, respectively. Others are used in systems where a surface ten- 4.2. Measurement techniques sion is defined to infer the growth of condensate from the change in height of a meniscus (multiple beam interferometry) Measurement techniques used to study capillary condensation [90,93,94,128] or the change in force between a probe and the in nanoporous media mostly involve adsorption experiments. adsorbed or capillary condensed fluid (AFM) [128,129]. While Although various apparatuses have been used, as described later, optical interferometry and multiple beam interferometry are both the most commonly employed techniques are volumetry [7,9–11, dependent on changes in the wavelength of light for their 13,38,46,48,49,72–74,78] and gravimetry [6,8,15–17,21,40,49,72, measurements, pulse-echo ultrasonic wave transit measurements 77,115] because of their applicability to the broadest range of [75], as the name suggests, uses the transit time of ultrasonic conditions pertaining to nanopore size, temperature, and pressure waves to determine pore filling and the longitudinal modulus [2]. Volumetry uses an equation of state to relate the known pres- and effective shear modulus of the adsorbent [75]. For instance, sures, temperatures, and volumes of two holding vessels to the in the work of Schappert et al., pulse-echo ultrasonic wave transit amount of gas adsorbed and/or condensed in the pores of the measurements were coupled with capacitative distance sensor adsorbent; while gravimetry utilizes a balance to directly weigh measurements to directly evaluate changes in adsorbent sample changes in the mass of the adsorbent, which result from adsorption size (i.e., pore diameter) due to strain generated during adsorption [51]. or condensation within the pore space [75]. Some measurement Volumetric apparatuses are most commonly used for the methods also make use of X-ray diffraction peaks to identify characterization of porous materials through nitrogen adsorption capillary condensation (SAXD) [50,130] or use X-ray scattering to at 77 K [2,6,9–11,16,38,48,72,74]. The detailed descriptions of determine how the lattice parameter of an adsorbent changes as E. Barsotti et al. / Fuel 184 (2016) 344–361 355

Table 2 Alternative measurement techniques used in the literature for studies of capillary condensation.

Adsorbate Adsorbent Method Reference Deuterium MCM-41 Neutron Diffraction Floquet et al. [134] Deionized Water Mica-Cytop Dielectric Stack Electro-wetting on Dielectric Surface Force Apparatus Gupta et al. [90] Water SBA-15 SAXS Erko et al. [131] Water Vycor Glass Spectrophotometry Ogwa and Nakamura [135] Deuterated Water MCM-41 Neutron Spin Echo Measurements Yoshida et al. [136] Deuterated Methane SBA-15 SANS Chiang et al. [137] Argon Vycor Glass Positron Annihilation Spectroscopy Alam et al. [39], Jones and Fretwell [133] Carbon Dioxide Packed Silica Spheres Visual Observation Ally et al. [138] Propane Packed Silica Spheres Visual Observation Ally et al. [138] Isopropanol Nanoporous Alumina Optical Interferometry Casanova et al. [45] Pentane MCM-41 SAXD Günther et al. [50] Benzene Silicon Wafers Fourier-Transform IR Microscope with Focal Plane Array Lauerer et al. [139] Detector and a Single Element Detector Krypton MCM-41 Neutron Diffraction Floquet et al. [134] Krypton Vycor Glass Positron Annihilation Spectroscopy Jones and Fretwell [133] Cyclohexane Mica Surface Force Apparatus Maeda and Israelachvili [128] Cyclohexane Vycor 7930 NMR Naumov [46] n-Hexane Controlled Pore glass, Woodford Shale Core Weight Gain, NMR Chen et al. [21] Plugs, Sandstone Core Plugs Toluene Nanoporous Alumina Optical Interferometry Casanova et al. [45] Octane Controlled Pore Glass DSC Luo et al. [140] Decane Controlled Pore Glass DSC Luo et al. [140] Dodecaﬂuoropentane MCM-41 SAXD Günther et al. [50] Dodecaﬂuoropentane SBA-15 SAXD Zickler et al. [130] Octane + Decane Controlled Pore Glass DSC Luo et al. [140] Nitrogen + Argon Vycor Glass Positron Annihilation Spectroscopy Alam et al. [39] Krypton + Argon Vycor Glass Positron Annihilation Spectroscopy Jones and Fretwell [132]

a result of capillary condensation (SAXS) [8,131]. Alternatively, The second is a static method, in which the overall composition unlike any of the aforementioned methods which all involve the of fluid is kept constant throughout the entire experiment. In this measurements of various properties at constant temperature, method, pressure increases result in depletion of the more differential scanning calorimetry (DSC) and positron annihilation selectively adsorbed component from the bulk fluid until the spectroscopy [39,132,133] are employed at constant pressure. In dew-point curve of the confined fluid is hit. Therefore, the situation the latter case, this results in the creation of adsorption isobars. is different from that experienced for the phase behavior of a bulk Similar to isotherms, isobars may exhibit a step or a hysteresis loop fluid, where the composition of the bulk vapor phase is the same as indicative of capillary condensation. A compilation of data the overall composition of the unconfined fluid at the dew point. available from alternative experimental methods is presented in Because the dew-point pressure of the confined fluid and the Table 2. corresponding composition of the bulk vapor are unknown before- More recently, a preliminary work has emerged on the direct hand, it is not straightforward to compare the result from this type imaging of the fluid phases during capillary condensation [139]. of experiment with that obtained from observations of the phase Using infrared microscopy, Lauerer et al. were able to capture behavior of a fluid in the bulk with the same overall composition. images of benzene in two different silicon 5–10 nm pore slit To decide which experiment is more relevant, the circum- geometries during adsorption and desorption at different stances encountered in gas reservoirs and in their recovery must pressures, including the condensation step [139]. Refinement of be considered. The overall composition in the reservoir might not this novel approach may allow for direct, visual observation of be known, but that of the bulk vapor phase (i.e., produced hydro- the distribution of fluid molecules throughout the pore, including carbon gases from production wells) can be analyzed and, due to observations of the mechanisms by which adsorption and the enormity of its presence, may be assumed constant. Therefore, desorption occur [139]. the open-flow setup with constant vapor composition would In the case of gas mixtures, the composition of equilibrium provide data with more practical use to the petroleum industry. phases at the point of capillary condensation is another required parameter for characterization of the system. Two primary 4.3. Experimental data on hydrocarbon systems methodologies exist for extending measurements to multicomponent systems. The first is the open flow method employed by We have collected published experimental adsorption iso- Yun et al. [7], in which bulk fluid of a known composition is flowed therms, from which the data on capillary condensation may be through the adsorbent at constant temperature, until steady state derived, for components typically found in petroleum gases in flow is achieved. Therefore, the bulk vapor in a control volume Table 3. It should be noted that this table is not an exhaustive list surrounding the adsorbent is at constant temperature, pressure, of research on this topic due to limited space in this paper. and composition, necessitating constant chemical potential [7]. Likewise, a compilation of fluids not found in petroleum gases is At each pressure step, the total weight of the adsorbed phase is tabulated in the Supplementary Material. This latter data set could plotted against the bulk pressure to get the adsorption isotherm be useful for general validation purposes. of the mixture. Likewise, using a chromatograph, the number of As seen from Table 3, most of the data was measured at a single moles for each component of the mixture may be individually temperature and/or for a single pore size, while model parameters plotted versus the bulk pressure to determine the uptake of each that represent the interaction between fluid molecules and the pore compound over the course of the isotherm. walls may vary with temperature and pore size. This introduces a 356 E. Barsotti et al. / Fuel 184 (2016) 344–361

Table 3 List of available experimental adsorption isotherms, from which capillary condensation data may be derived, in the literature for petroleum gas components. (V represents volumetry, and G represents gravimetry.)

V Adsorbate Adsorbent Pore size (nm) Temperature (K) P /Psat Method Reference Methane MCM-41 2.5 77.5 0–1 V Llewellyn et al. [141] (deuterated) Ethane MCM-41 2.7–3.9 264.6–273.55 0–1 V He and Seaton [104] and Yun et al. [7] Ethylene MCM-41 1.8 144.1–148.1 0–1 V Morishige et al. [38] n-Butane MCM-41 2.1–3.6 283 0–1.2 V Ioneva et al. [48] n-Pentane MCM-41d 2.0–4.57 258–298 0–1 V, G Rathousky´ et al. [142] and Russo et al. [15,143] SBA-15 7.31–8.14 258–298 0–1 G Findenegg et al. [144], Gor et al. [8] and Russo et al. [15,143] MCM-48 3.78 298 0–1 G Russo et al. [143] SBA-16 8.15 298 0–1 G Russo et al. [143] HSB 3.54 298 0–1 G Russo et al. [143] CMK-3 4.90 298 0–1 G Russo et al. [143] LPC 20.62 298 0–1 G Russo et al. [143] MCF 13.47–20.47 298 0–1 G Russo et al. [143] Alumina N/Ac 308 0–1 G Tzevelekos et al. [145] membrane Cyclopentane MCM-41 3.7–4.57 253–293 0–1 V Rathousky´ et al. [142] Neopentane MCM-41d 1.9–9.9 258–333 0–1 V, G Carrott et al. [146], Long et al. [147], Qiao et al. [40], Russo et al. [15,143] and Tanchoux et al. [13] SBA-15 7.31–7.87 258–273 0–1 G Russo et al. [15,143] MCM-48 3.78 273 0–1 G Russo et al. [143] SBA-16 8.15 273 0–0.95 G Russo et al. [143] HSB 3.54 273 0–0.9 G Russo et al. [143] CMK-3 4.90 298 0–0.9 G Russo et al. [143] LPC 20.62 273 0–1 G Russo et al. [143] MCF 13.47–20.47 273 0–1 G Russo et al. [143] n-Hexane MCM-41 1.9–4.2 293–323 0–1 G Carrott et al. [146], Jänchen et al. [148] and Qiao et al. [40] MCM-48 3.2 303.15 0–0.8 G Shim et al. [16] Benzene MCM-41 1.9–4.4 273–303 0–1 V, G Carrott et al. [146], Choma et al. [149], Jänchen, et al. [148] and Nguyen et al. [150] MCM-48 3.2 303.15 0–0.75 G Shim et al. [16] Cyclohexane MCM-41 2.6–3.9 298 0–0.8 G Long et al. [147] Vycor 7930 6 297 0–0.9 V, NMR Naumov, Sergej [46] n-Heptane MCM-41 2.7 298 0–0.97 G Zandavi and Ward [71,115] SBA-15 6.4–6.6 293–298 0–0.95 V, G Kierys et al. [151] and Zandavi and Ward [115] Porous Silicon 6.5 291 0–0.95 V Grosman et al. [152] Shale 3.8 298 0–0.95 G Zandavi and Ward [153] Methylcyclohexane MCM-41d 2.39–4.57 298 0–1 G Russo et al. [143] SBA-15 7.31–7.59 298 0–0.95 G Russo et al. [143] MCM-48 3.78 298 0–1 G Russo et al. [143] SBA-16 8.15 268–298 0–0.95 G Russo et al. [15,143] HSB 3.54 298 0–0.95 G Russo et al. [143] CMK-3 4.90 298 0–0.97 G Russo et al. [143] LPC 20.62 268–298 0–1 G Russo et al. [15,143] MCF 13.47–30.47 278–298 0–1 G Russo et al. [15,143] n-Octane MCM-41 2.6 298 0–0.95 G Zandavi and Ward [71] CPGh 4.3–38.1 N/A N/A DSC Luo et al. [140] Shale 3.8 298 0–1 G Zandavi and Ward [153] n-Butylbenzeneb Anodized Aluminaf 6.9–12.2 273 0–0.8 V Nonaka [154] Porous Silicon 1.7–9.7 273 0–0.8 V Nonaka [154] n-Nonane MCM-41 2.88 313 0–0.6 G Berenguer-Murcia et al. [155] Decaneh CPG 4.3–38.1 N/A N/A DSC Luo et al. [140] Water Mesoporous Silica 3.8–10.5 298 0–1 G Hwang et al. [156] MCM-41e 2.1–4.0 293–323 0–1 G Branton et al. [157,158], Jänchen et al. [148], Kittaka et al. [159], Russo et al. [160], Saliba et al. ,[161] and Zandavi and Ward [115] SBA-15 6.6–8.9 278–298 0–1 G, SAXD Erko et al. [131], Hwang et al. ,[156] and Zandavi and Ward [115] OMC 6.2–9.2 264.8–298 0–1 V, G Morishige et al. [72] and Horkawa et al. [79] PSM 2.0 323 0–0.9 G Saliba et al. [161] RD-silica 2.1 323 0–0.9 G Saliba et al. [161] Nitrogena MCM-41 2.4–4.4 63.3–119.2 0–1 V Morishige et al. [38] and Morishige and Ito [78] SBA-15 6.0–11 69–121.8 0–1 V Morishige and Ito [78] OMC 6.2–9.2 77 0–1 V Morishige et al. [72] Oxygen MCM-41 1.8–4.4 77–139.4 0–1 V, G Branton et al. [162], Inoue et al. [163], Morishige et al. [38], E. Barsotti et al. / Fuel 184 (2016) 344–361 357

Table 3 (continued)

V Adsorbate Adsorbent Pore size (nm) Temperature (K) P /Psat Method Reference Morishige and Nakamura [73] and Sonwane et al. [164] FSM1 2.5 77 0–1 G Inoue et al. [163] FSM2 3.4 77 0–1 G Inoue et al. [163] GB1 3.0 77 0–1 G Inoue et al. [163] Carbon Dioxide MCM-41 1.8–4.4 185.5–273 0–1 V, G Berenguer-Murcia et al. [155], He and Seaton [104], Sonwane et al. [164], Morishige and Nakamura [73] and Morishige et al. [38] Carbon Monoxide MCM-41 2.5 77 0–1 V Llewellyn et al. [141] Methane + Ethane MCM-41 3.9 264.75 0–435g V Yun et al. [7]

CO2 + Ethane MCM-41 2.70 264.6 7.25– V He and Seaton [104] 261g Octane + Decaneh CPG 4.3–38.1 N/A N/A DSC Luo et al. [140]

a The capillary condensation of nitrogen has been widely observed for the purpose of materials characterization. As such, only a few sample references are given here [38,73,78]. b Although the adsorption isotherms clearly indicate capillary condensation in the form of hysteresis loops, some isotherms are incomplete [154]. c The adsorbent was a pellet of compacted alumina powder. The powder had an average particle diameter of 20 nm [145]. d Some MCM-41 samples included the surface group trimethylsilane [165], were grafted with aluminum [160], or had been modiﬁed with chloromethyltriethoxysilane [143]. e Some MCM-41 samples were grafted with aluminum [160]. f Experiments were carried out in anodized aluminum pores with and without silicate coatings [154]. g Pressures are given in psi for mixtures. h DSC measurements were carried out at atmospheric pressure for temperatures from 320 K to 540 K. The CPG adsorbent was treated to include the surface group hexamethyldisilazane [140].

serious barrier to developing robust models that have sound small isosteric heats of adsorption to characterize weak predictability. Most of the experimental data is also limited to ideal fluid-pore interactions, and vice versa [38]. adsorbents and single component fluids, which are far removed In the investigation of fluid-pore-wall interactions, as manifest from the conditions encountered in practical applications. Despite in wettability, Zandavi et al. have shown the contact angle of the these limitations, the current body of experimental work on condensed phase with the pore wall to be zero once capillary con- hydrocarbon systems presents many breakthroughs in the effort densation has occurred. This was done by calculating pore size to understand confined phase phenomena. Such breakthroughs based on isotherms for water, heptane, octane, and toluene in include progress made in evaluating fluid-fluid and fluid-pore MCM-41 and SBA-15 while assuming a contact angle of zero and interactions in confined spaces. then comparing the results to pore size values taken from trans- For example, to show how fluid properties affect capillary con- mission electron microscopy [115]. However, the wettability of densation, Ioneva et al. produced separate capillary condensation nanopores does differ from one adsorbate to another, as shown isotherms for ethane, propane, and butane in MCM-41 samples by Shim et al. using thermogravimetric analysis [16]. They found [48]. They found the onset of capillary condensation to occur at a the wetting order of several fluids in MCM-48, which they relative vapor pressure of 0.8 for ethane, 0.62 for propane, and described as ‘‘energetically heterogeneous” based on isosteric heat 0.38 for n-butane in 3.6 nm diameter pores at 283.15 K [48]. This calculations, in order of decreasing wettability to be acetone, provides evidence that, in single-component isothermal systems, methanol, n-hexane, benzene, cyclohexane, and toluene [16]. capillary condensation generally occurs at lower relative pressures Furthermore, in their observations of the capillary condensation for the fluids with larger molecules. The same hierarchy of of toluene, methylcyclohexane, neopentane, and n-pentane, Russo capillary condensation pressure with regard to bulk vapor pres- et al. used isosteric heats to study the effects of pore geometry and sures has been observed by Russo et al. in their study of toluene, interconnectivity [15]. Their capillary condensation isotherms for methylcyclohexane, n-pentane, and neopentane [15]; Casanova MCM-41, SBA-15, SBA-16, LPC, and mesocellular foam (MCF), led et al. in their study of isopropanol and toluene [45]; and Shim to heat calculations, where significantly higher heats of desorption et al. in their study of benzene, toluene, n-hexane, cyclohexane, (rather than heats of adsorption) for fluids in SBA-16 and LPC were acetone, methanol, methyl ethyl ketone, and trichloroethane [16]. observed, indicating that both adsorbents have narrow pore These findings are consistent with the Kelvin equation, i.e. Eq. mouths and facilitate a desorption mechanism different than that (2), where the molar density of the liquid, which depends on the of MCM-41 or SBA-15, whose fluids exhibit heats of desorption molecular size, is inversely related to the capillary condensation that are only slightly larger than their heats of adsorption [15]. pressure. Similarly, the identical heats of desorption and adsorption for With further regard to fluid-fluid interactions, Shim et al. found MCF were attributed to larger pore mouths [15]. The effect of pore differences among the heats of capillary condensed benzene, cyclo- geometry is echoed by a more fundamental study by Ioneva et al., hexane, hexane, toluene, trichloroethylene, acetone, and methanol who showed that for n-butane at 283 K in pore diameters of in MCM-48, despite the fact that all adsorbates wet the adsorbent 2.1 nm, 2.6 nm, 2.8 nm, and 3.6 nm, capillary condensation [16]. This was taken as a qualitative indication that fluid-fluid occurred at relative vapor pressures of 0.20, 0.23, 0.34, and 0.38, interactions do play a role in capillary condensation. In particular, respectively [48]. This supports theoretical predictions that Qiao et al. calculated the isosteric heat of capillary condensation capillary condensation occurs first in the smallest pores and was for hexane in MCM-41 and found it to be higher than the heat of confirmed in the works of Tanchoux et al. and Qiao et al., who condensation for hexane in the bulk and to increase with decreas- both studied the capillary condensation of hexane in MCM-41 ing pore size [40]. On the other hand, Morishige et al., have used [13,40]. 358 E. Barsotti et al. / Fuel 184 (2016) 344–361

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