From Macromolecular Thermodynamics to Reservoir Simulation

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Energy & Fuels 2007, 21, 1646-1654

Coal Chemistry for Mechanical Engineers: From Macromolecular Thermodynamics to Reservoir Simulation Vyacheslav Romanov* U.S. Department of Energy, National Energy Technology Laboratory, P.O. Box 10940, M/S 58-207, Pittsburgh, PennsylVania 15236-0940 ReceiVed September 25, 2006. ReVised Manuscript ReceiVed February 15, 2007

In pilot trials and commercial scale field demonstrations of CO2 storage in coal seams, quite often unexpected problems with coal swelling around injector and reducing injection efficiency (e.g., Allison unit in the San Juan Basin, RECOPOL in Poland, Hokkaido project in Japan, etc.) can stall or even terminate the site development. To avoid the costly mistakes with the prospective site evaluation, the state of the art in reservoir modeling needs to be improved by taking into account coal properties at the macromolecular level. The current models are based on the rock mechanics, which ignores decades of experimental and theoretical studies of interaction between coal and injected fluids. A pseudopolymer approach is introduced to the modelers as a viable alternative, especially, at medium to high fluid pressures. Further, it is discussed how the thermodynamics of CO2 dissolution in the macromolecular network of the coal matrix can be incorporated into geomechanical models.

Introduction In conventional reservoir simulators, coal permeability is directly derived from the pore pressure and temperature changes via the medium compressibility and dilatability coefficients. However, these parameters are not sufficient to deduce pore volume changes when the injected fluid changes the medium properties.1-4 More accurate modeling involves coupling of a reservoir model that describes the fluid flow in the porous medium with a geomechanical model accounting for the medium deformation. The fully coupled algorithm solves the entire problem in one simulator,4,5 whereas the simulators coupling method1,6,7 uses two simulators to solve the two sets of equations, for mechanical and flow problems, with the data exchange via the Gauss-Seidel staggered coupling algorithm or a preconditioned (by solving the mechanical problem) conjugate gradient method.8 The majority of reservoir simulators employ the mechanical engineering approach that treats coal as a porous elastic medium with an instantaneous deformation response. Typically, the simulation of compressible Darcy flow through the cleats and * Telephone: (412) 386-5476. Fax: (412) 386-4806. E-mail: [email protected]. (1) Tortike, W. S.; Farouq, A. J. Can. Pet. Technol. 1993, 32, 28-37. (2) Settari, A.; Mourits, F. M. Coupling of Geomechanics and Reservoir Simulation Models. In Computer Methods and AdVances in Geomechanics; Siriwardane, H. J., Zeman, M. M., Eds.; A. A. Balkema: Rotterdam, 1994; Vol, 3, pp 2151-2158. (3) Ruisten, H.; Teufel, L. W.; Rhett, D. W. Influence of Reservoir Stress Path on Deformation and Permeability of Weakly Cemented Sandstone Reservoirs. Presented at the SPE Annual Technical Conference, Denver, CO, October 6-9, 1996. (4) Gutierrez, M.; Lewis, M. The Role of Geomechanics in Reservoir Simulation. Presented at SPR/IRSM Eurock’98, Trondheim, Norway, July 8-10, 1998; pp 439-448. (5) Chin, L. Y.; Raghavan, R.; Thomas, L. K. Fully-Coupled Geomechanics and Fluid-Flow Analysis of Wells with Stress-Dependent Permeability. Presented at the SPE Interntional Conference and Exhibition in China, Beijing, China, November 2-6, 1998; pp 269-248. (6) Settari, A.; Mourits, F. M. A Coupled Reservoir and Geomechanical Simulation System. SPE J. 1998, 219-226; SPE paper 50939.

Fickian diffusion through fractal micropores is coupled to geomechanical models with direct relations between the porosity changes due to strain variation, the total stress changes associated with the pressure variation, and the medium properties. In such models, the coal porosity is assumed to be a function of the state variables, e.g., via the Langmuir adsorption isotherm as in the Palmer-Mansoori equation,9,10 assuming that the compressibility of the solid matrix is limited to the viscoelastic response to macroscopic forces and that changes in the pore volume are mostly due to so-called “absolute” adsorption of the fluid on the coal surface. However, the Langmuir model is questioned in supercritical fluid sorption due to coal swelling10,11 and the transformation strains can be 2 or 3 orders of magnitude larger than elastic strains.12 The rigorous mass balance condition should include not only adsorption on the pore walls but also capillary condensation in small pores and dissolution of the fluid in the coal matrix. Recent field demonstrations of carbon dioxide sequestration in unminable coal seams and enhanced coalbed methane (ECBM) recovery revealed large discrepancies between field results, simulation predictions, and laboratory data.13-18 One of the main concerns is that the current models fail to reliably (7) Settari, A.; Walters, D. A. Advances in Coupled Geomechanical and Reservoir Modeling with Applications to Reservoir Compaction. Presented at the SPE Reservoir Simulation Symposium, Houston, TX, February 1417, 1999; pp 1-13. (8) Daı¨m, F.; Eymard, R.; Hilhorst, D.; Mainguy, M.; Masson, R. Oil Gas Sci. Technol.-ReV. IFP 2002, 57, 515-523. (9) Palmer, I.; Mansoori, J. How permeability depends on stress and pore pressure in coalbeds-a new model. In Proceedings of the Annual Technical Conference and Exhibition, V. sigma, ReserVoir Engineering, Society of Petroleum Engineers: Richardson, TX, 1996; pp 557-564, SPE paper 36737. (10) Mazumder, S.; Karnik, A.; Wolf, K.-H. A. A. Swelling of Coal in Response to CO2 Sequestration for ECBM and its Effect on Fracture Permeability. SPE J. 2006, 390-398; SPE paper 97754-PA. (11) Clarkson, C. R.; Bustin, R. M. Fuel 1999, 78, 1333-1344. (12) Levitas, V. I. Continuum mechanical fundamentals of mechanochemistry. In High Pressure Surface Science and Engineering, Section 3; Gogotsi, Y., Domnich, V., Eds.; Institute of Physics: Bristol, 2004; pp 159-292.

10.1021/ef060476p CCC: $37.00 © 2007 American Chemical Society Published on Web 03/30/2007

Coal Chemistry for Mechanical Engineers

predict the dramatic changes in permeabilitysup to a complete loss of flow due to closed cleatssupon CO2 injection. These problems escalate in the region of supercritical CO2.19,20 This is related to uncertainties in the nonequilibrium equation-ofstate (EOS) but also to volumetric effects of sorption and structural relaxation. The main reason for the failure, a gap between the rock mechanics and pseudopolymer chemistry approaches, can be bridged by a dynamic relation between macroscopic stress and strain tensors stemming from microscopic changes that occur in coal mixing with the permeating fluid (e.g., structural relaxation upon dissolution of CO2 in the macromolecular network of coal). In this work, the reference mixture is defined as an ideal solution in the sense of Raoult’s law.21 In order to adapt this relationship to CO2 sequestration in unminable coal seams and the enhanced coalbed methane (ECBM) recovery modeling, the definition of the stress, σ, should be generalized to account for thermodynamically induced effects. It requires a formulation of theoretical framework tying the thermodynamics of dissolution and the mechanical properties of the mixture within a broad range of pressures. Additionally, the affinity of the macroscopic and microscopic deformations needs to be addressed. These problems will be discussed in the following sections. Pseudopolymer Approach Coal is a composite material that consists of a variety of coalified vegetative remnants (macerals, such as vitrinite, fusinite, and liptinite) mixed with various inorganic minerals, which all differ in swellability, hardness, and shape; even within a given maceral class differences in swellability were observed.22 Appreciable regions of the coals may not be swollen even by strongly interacting liquids that swell coal by more than 100%.23,24 Chemically, coal structure, excluding anthracites and the high rank coals that are graphitic in their nature, is generally viewed as a macromolecular network of cross-linked (chemically bonded and physically entangled) chains or clusters of aromatic rings,25 in which is dissolved a mix of soluble and physically (13) Palmer, I. Presentation on Geomechanics and Volumetric Changes in Coal. Presented at the Enhanced Coalbed Methane Recovery and CO2 Sequestration, Applied Technology Workshop, Denver, CO, October 2729, 2004. (14) Reeves, S.; Davis, D.; Oudinot, A. A Technical and Economic SensitiVity Study of Enhanced Coalbed Methane RecoVery and Carbon Sequestration in Coal; DOE Topical Report, Washington, D.C., March, 2004. (15) Reeves, S.; Taillefert, A.; Pekot, L.; Clarkson, C. The Allison Unit CO2-ECBM Pilot: A ReserVoir Modeling Study; DOE Topical Report, Washington, D.C., February, 2003. (16) Law, D. H.-S.; van der Meer, L. H. G.; Gunter, W. D. Comparison of Numerical Simulators for Greenhouse Gas Sequestration in Coalbeds, Part IV: More Complex Problems. Presented at the 2nd Annual DOE Conference on Carbon Sequestration, Alexandria, VA, May 5-8, 2003. (17) van Bergen, F.; Pagnier, H. IEA GHG Programme Greenhouse Issues 2005, 79, 6. (18) Gale, J. CO2 Storage in Coal Seams-An IEA GHG PerspectiVe; IEA Greenhouse Gas R&D Programme: Brisbane, Australia, 2005. (19) Romanov, V. N.; Goodman, A. L.; Larsen, J. W. Energy Fuels 2006, 20, 415-416. (20) Romanov, V.; Soong, Y.; Schroeder, K. Chem. Eng. Technol. 2006, 29, 368-374. (21) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomez de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: New York, 1986. (22) Brenner, D. Fuel 1983, 62, 1347-1350. (23) Norinaga, K.; Iino, M.; Cody, G. D.; Thiyagarajan, P. Energy Fuels 2000, 14, 1245-1251. (24) Larsen, J. W.; Green, T. K.; Kovac, J. J. Org. Chem. 1985, 50, 4729-4735.

Energy & Fuels, Vol. 21, No. 3, 2007 1647

Figure 1. Simplified representation of the cross-linked macromolecular structure of coal:25 (C) carbon atoms; (O) clusters of aromatic rings; (b) molecules of the penetrant (CO2); (E- -) extractable matter.

trapped “guest” molecules (Figure 1).24 Such structure permits coal solubilization26 and (anisotropic) swelling in appropriate solvents,27,28 while unswollen coal is in a glassy state (not in the lowest energy state) under typical reservoir conditions.29 The temperature required to cause a transition from glass to rubber (with liquidlike internal viscosity) is the glass transition temperature. It was observed that the early stages of solubilization and swelling (small deformations) are highly reversible,22 which provides a close analogy between the coal properties and those of polymeric materials. However, rapid differential swelling and drying of the more accessible regions may result in severe deformations (fracturing or dislocations) of even homogeneous vitrinite, caused by stresses due to expansion constrained by the underlying rigid material. The similarity of the macromolecular structure of coal and that of cross-linked polymers prompted the application of polymer sorption theories to coal.30 Alternative approaches31,32 will not be considered in the following analysis. Diffusion of fluid into a polymer network is a combination of two analytically treatable cases: (A) The Fickian diffusion (case I) mechanism is controlled by the concentration gradient. Extensions of the Fickian diffusion may include a contribution due to viscoelastic stress.33,34 (B) The case II (anomalous) transport mechanism is characterized by a well-defined diffusion front, preceded by a region of low permeate concentration, which results from the Fickian (25) Vahrman, M. Fuel 1970, 49, 5-16. (26) Dryden, I. G. C. Fuel 1951, 30, 145-158. (27) Cody, G. D., Jr.; Larsen, J. W.; Siskin, M. Energy Fuels 1988, 2, 340-344. (28) Suuberg, E. M.; Otake, Y.; Langner, M. J.; Leung, K. T.; Milosavljevic, I. Energy Fuels 1994, 8, 1247-1262. (29) Suuberg, E. M. Energy Fuels 1997, 11, 1103-1104. (30) Milewska-Duda, J. Fuel 1993, 72, 419-425. (31) Takanohashi, T.; Iino, M.; Nishioka, M. Energy Fuels 1995, 9, 788793. (32) Berkowitz, N. On Some Inconsistencies in Current Concepts of Coal Chemistry. In Technology and Use of Lignite, Proceedings of the 11th Biennial Lignite Symposium, Washington, DC, June 15-17, 1981; Kube, W. R., Sondreal, E. A. White, D. M., Eds.; U.S. DOE: Washington, D.C., 1982; Vol. 1, pp 414-422. (33) Brochard, F.; De Gennes, P. G. PhysicoChem. Hydrodyn. 1983, 4, 313-322. (34) Cohen, D. S.; White, A. B. J. Polym. Sci. B: Polym. Phys. 1989, 27, 1731-1747.

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diffusion into the glassy network, with concentration dependent diffusivity. The permeate dissolution in the polymer induces the glass-to-rubber transition, which leads to mechanical deformation of the network in response to the reversible35 thermodynamic swelling stress (not the total stress) in the transition region.36 There is no discontinuity in the penetrant concentration at the phase transition as can be found in standard chemical systems.37 Under conditions of enhanced solubility, geopolymers such as bituminous coal exhibit case II transport behavior.38,39 At low to moderate temperatures, as the penetrating fluid dissolves in the macromolecular network of coal, the network density decreases and the large molecular chain motions increase.40 At the pressures necessary for carbon dioxide sequestration, dissolution in coal can exceed adsorption. More CO2 is dissolved as the coal rank decreases.41 Structural changes induced by this dissolution process include swelling, microcavity formation, and the phase transition from the glassy to the rubbery state requiring rearrangement of the chain segments. (Plasticizers enable the rearrangement by increasing the available free volume.) Thus, an increase of the penetrant (plasticizer) concentration in the network causes a similar effect as a decrease of the glass transition temperature.42,43 Despite the similarities, plasticization induced by CO2 absorption is not equivalent to plasticization induced thermally. Both the mechanism (anomalous as the plasticization is induced by CO2, while still Fickian after the thermal plasticization44) and time scale of mass transport can be different.45 The only experimental study of the combined thermal and the CO2-induced bituminous coal softening was conducted in the 1980s using a high-pressure microdilatometer.43 The reported results showed a dramatic drop in the coal softening temperature from 430 to about 60 °C at CO2 pressures above 4 MPa (Figure 2). This finding certainly needs to be verified, because it raised serious concerns over the practicality of carbon dioxide storage in coal, if it can flow and block the cleats at these temperatures. Assuming that Khan’s data actually reflect the miscibility properties of the CO2-coal system, one can reasonably argue that the shape of the coal softening line (the locus of points corresponding to the glass-to-rubber transition of the system), which apparently intersects the pure CO2 gas-liquid coexistence curve at its critical point (C1), indicates the importance of the long-range correlations due to a critical behavior of CO2, which may result in significant free volume effects in the vicinity of C1. Such effects may also result from the local density augmentation (“molecular charisma”sa dynamic exchange

between small solvent molecules and large solute molecules, whose characteristic energy exceeds the solvent-solvent interaction energy)46 due to specific short-range effects. In the standard excess sorption isotherms that assume constant void volume, V0, the free volume effects may reveal themselves in the form of the local maxima and minima.19,20,47-50 Typical positions of the experimentally observed maxima (max1 and max2) and the local minimum (min) of “sorption” at 55 °C are marked on the pressure-temperature diagram (Figure 2). It appears that the “max1” anomaly on some sorption isotherms of high rank coals correlates with Khan’s phase transition for CO2-coal system. This is yet another argument supporting the theory of CO2 dissolution in coals. Plasticized coals having less than about 86% C (dmmf) rearrange to a more associated form in which CO2 is less soluble.41 The coals having more that 86% C (dmmf) may rearrange to a more or to a less highly associated structure.51,52 Another peak (max2 in Figures 2 and 3) appears at the pressures above the CO2 cluster fluctuation ridge,53,54 which suggests that CO2 solubility changes due to density fluctuations near the critical isochore could play a role in either sorption or excess volume changes. Thermodynamics of the free volume effects and compressibility will be discussed in the next section. In favor of the volumetric nature of the anomalies speaks the fact that manometric and gravimetric methods yield very

(35) Rodrigues, C. F.; de Sousa, M. J. L. Int. J. Coal Geol. 2002, 48, 245-251. (36) Thomas, N.; Windle, A. H. Polymer 1982, 23, 529-542. (37) Edwards, D. A.; Cohen, D. S. AIChE J. 1995, 41, 2345-2355. (38) Alfrey, T. E.; Gurnee, E. F.; Lloyd, W. G. J. Polym. Sci. C 1966, 12, 249-261. (39) Mazumder, S.; Bruining, J.; Wolf, K.-H. A. A. Swelling and Anomalous diffusion mechanisms of CO2 in coal. Proceedings of the International Coalbed Methane Symposium, Tuscaloosa, AL, May 22-26, 2006; paper 0601. (40) Peppas, N. A. Pharm. Acta HelV. 1985, 60, 110-111. (41) Larsen, J. W. Int. J. Coal Geol. 2004, 57, 67-70. (42) Hsieh, S. T.; Duda, J. L. ACS Polym. Mater. Sci. Eng. 1984, 51, 703-706. (43) Khan, M. R. Thermoplastic Properties of Coal Pyrolyzed at Elevated Pressures: Role of Experimental Variables, Inorganic Additives and Preoxidation. Ph.D. dissertation, The Pennsylvania State University, State College, PA, 1985. (44) Barrie, J. A. Water in Polymers. In Diffusion in Polymers; Crank, J., Park, G. S., Eds.; Academic Press: New York, 1968. (45) Vincent, M. F.; Kazarian, S. G.; Eckert, C. A. AIChE J. 1997, 43, 1838-1848.

(46) Eckert, C. A.; Knuston, B. L. Fluid Phase Equilib. 1993, 83, 93100. (47) Busch, A. Thermodynamic and kinetic processes associated with CO2 sequestration and CO2-enhanced coalbed methane production from unminable coal seams. Ph.D. dissertation, RWTH Aachen University of Technology, Netherlands, 2005. (48) Ozdemir, E. Chemistry of the adsorption of carbon dioxide by Argonne Premium Coals and a model to simulate CO2 sequestration in coal seams. Ph.D. dissertation, University of Pittsburgh, Pittsburgh, PA, 2004. (49) Krooss, B.; van Bergen, F.; Gensterblum, Y.; Siemons, N.; Pagnier, H. J. M.; David, P. Int. J. Coal Geol. 2002, 51, 69-92. (50) Toribio, M. M.; Oshima, Y.; Shimada, S. Coal Adsorption Capacity Measurements Using Carbon Dioxide at Sub-Critical and Supercritical Conditions. Proceedings of the 7th International Conference on Greenhouse Gas Control Technologies, Vancouver, Canada, September 5-9, 2004. (51) Yun, Y.; Suuberg, E. M. Prep. Pap.sAm. Chem. Soc., DiV. Fuel Chem. 1992, 37, 856-865. (52) Yun, Y.; Suuberg, E. M. Prep. Pap.sAm. Chem. Soc., DiV. Fuel Chem. 1993, 38, 489-494. (53) Nishikawa, K.; Takematsu, M. Chem. Phys. Lett. 1994, 226, 359363. (54) Nishikawa, K.; Takematsu, M. Chem. Phys. Lett. 1997, 271, 188.

Figure 2. Phase diagram for CO2 (liquid-gas, C1 ) critical point of the pure fluid) and CO2-coal (coal softening) for low volatile bituminous coals with more than 86% C (dmmf). Anomalies (local maxima, max1 and max2, and a local minimum, min) in the CO2 sorption isotherms (55 °C)20 correlate with the phase transitions (free volume effects).43,53-54



Figure 3. Comparison of the manometric (for high, (1 and low (0.1 K temperature oscillations) and the gravimetric experimental data for the Pocahontas no. 3 powder from Argonne Premium Coal Bank, reduced to the excess (Gibbs) sorption at 328 K (55 °C).19,20

similar data after adjustment for the void volume changes (Figure 3), although the nonequilibrium EOS effects obviously play the role as demonstrated by the enhanced anomalies in the experiments with larger ((1 °C) temperature oscillations. The experimental aspects of the sample volume variability during the sorption experiments are discussed elsewhere.19,20 The stress relaxation process accompanying the penetrant dissolution is controlled by rupture and disentanglement of the polymer chains;55 the relaxation time constant is associated with the length of the polymer in relation to the entanglement network of the glassy state. For the limiting case II transport, the macroscopic strain relaxation rates and the permeate front’s velocities can be derived from the network relaxation rates.56 The strain rate is related to the osmotic stress or the chemical potential of diluent in quasi-equilibrium with the polymer, which is the main subject of a discussion in the next section. Thermodynamics of the Problem In general form, the permeate transport across the mushy precursor layer (“membrane”) can be described by the following linear approximations:

∆p - ∆π1 ∆p - ∆π1 ) L2 η1R2L2

J1 ) P12

(1)

where P12 is the permeability coefficient, L2 is the thickness of the membrane, η1 is the penetrant viscosity, R2 is the resistance of the membrane per unit of thickness, ∆p is the pore pressure drop, and ∆π1 is the osmotic pressure across the layer. Permeability P, which is the steady-state, pressure- and thickness-normalized flux through a polymer membrane, can be expressed as the product of the penetrant solubility and diffusivity in the polymer.57 When it is dominated by diffusivity, materials are typically more permeable to smaller molecules than to larger ones. Recently, however, engineers announced the development of highly branched, cross-linked poly(ethylene oxide) molecular sieve that is virtually impermeable to small hydrogen molecules but displays excellent permeability to much larger carbon dioxide molecules, after the initial increase of the CO2 concentration sorbed in the polymer.58 Such a phenomenon is a trademark of case II diffusion. In the above example, highly (55) Peterlin, A. J. Polym. Sci. B: Polym. Lett. 1965, 3, 1083-1087. (56) Cody, G. D.; Botto, R. E. Macromolecules 1994, 27, 2607-2614. (57) Baker, R. W. Membrane Technology and Applications, 2nd ed.; Wiley: New York, 2004.

flexible polymer branches lead to weak size-sieving properties and high diffusion coefficients, which resulted in domination of the solubility factor. Similarly, a comparison of the sorption of carbon dioxide and ethane, molecules of similar size, by coal shows much greater CO2 uptake (permeability) than ethane, because CO2 readily dissolves in coals and ethane does not.59 (Only part of the higher CO2 sorption can be attributed to the size difference and the linear geometry of the carbon dioxide molecule, which results in more efficient “slipped parallel” packing at high densities.60) Reasonably assuming that the primary CO2 interaction with coal is via dispersion forces due to the molecule’s quadrupolar moment, the solubility can be roughly estimated via an activity coefficient of the low-density permeating fluid, γ1 (defined using symmetric convention for normalization, γL1 ) 1 for the rubbery state of equilibrium), in a regular (nonzero enthalpy of mixing but no free volume effects) solution by using ScatchardHildebrand solubility parameters, δi:21

RT ln γ1 ≈ ν1(δ1 - δ h )2

(2)

where T is the temperature, δ h is a volume-fraction average of the solubility parameters of all the components in the solution, and ν1 is the molar volume of the penetrant. The experimental data for the solubility of CO2 in some heavy hydrocarbons are described by the relatively simple Krichevsky-Kasarnovsky equation:61

ln(f1/y1) ) ln(f

L 1/

yL1 )

νj∞1 + (p - pL) RT

(3)

where p is the pressure; f1, y1, and νj∞1 are the fugacity, molar fraction, and partial molar volume (at hypothetical infinite dilution) of the gas; L stands for the reference state; and when the reference state corresponds to ideal dilute solution, f L1 /yL1 ) H1,2 which is known as Henry’s constant. This equation62 was developed for gases which are slightly soluble in low volatile solvents. Its linearity is expected to break down when the solubility becomes very large or the pressure is too high.63 Moreover, the high solubility case should be treated in conjunction with the relaxation phenomena resulting in the anomalous (case II) transport mechanism. In order to derive the stress-strain relationship for the limiting case II transport, it is reasonable to assume that the global elastic field relaxes much faster than the local strain and the local relaxation is isothermal. Under these conditions, the swelling of the coal is governed by the viscoelastic response rather than penetrant diffusion. The elastic constants that characterize the driving force of the phase transition can be defined as partial second derivatives of the Helmholtz free energy (second derivatives of the internal energy give the adiabatic constants) with respect to material deformation coordinates.64 (58) Lin, H.; Van Wagner, E.; Freeman, B. D.; Toy, L. G.; Gupta, R. P. Science 2006, 311, 639. (59) Larsen, J. W.; Hall, P.; Wernett, P. C. Energy Fuels 1995, 9, 324330. (60) Skoulidas, A. I.; Sholl, D. S.; Jhonson, J. K. J. Chem. Phys. 2006, 124, 1-7. (61) Gasem, K. A. M.; Robinson, R. L., Jr. J. Chem. Eng. Data 1985, 30, 53-56. (62) Krichevsky, I. R.; Kasarnovsky, J. S. J. Am. Chem. Soc. 1935, 57, 2168-2171. (63) Dodge, B. F.; Newton, R. H. Ind. Eng. Chem. 1937, 29, 718724. (64) Steinle-Neumann, G.; Cohen, R. E. J. Phys.: Condens. Matter 2004, 16, 8783-8786.


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The relationship between stress, σ, and strain, , is usually oversimplified. In classical linear elasticity theory, this relationship is provided by Hooke’s law (σ ) ^). Viscoelastic materials on the other hand exhibit behavior characteristic of both classical Hookean solids and Newtonian fluids, which are modeled by a spring (σ ) E) and a dashpot (σ ) η˘ ), respectively, be it due to deformation under an applied load or internal deformation caused by a diffusing penetrant. Such materials are said to possess “memory”,65 which is reflected in the constitutive relationship between the stress, σ, and strain, , tensors:66

σ(x,t) ) ^(0)(u(x,t)) -

∂^(t - s) (u(x,s)) ds (4) ∂s

∫0t

The tensor of elastic coefficients, ^(t) is expected to be a smooth monotonic decreasing function of time and u(x,t) is a displacement from the equilibrium position as a function of space and time coordinates. The discrete rheological representation of materials in connection to their finite 3D generalization, and numerical implementation allows an internal variable formulation, which establishes the thermodynamic feasibility of the models.67 For example, using the same strain for combination of the spring and dashpot yields the Voigt model. (For simplicity, we do not take into account the gravity forces.) This gives the constitutive law in hereditary form as68

(τ) ) e-t/τ(0) +

1 η

∫0te-(t-s)/τσ(s) ds

(5)

When the local stress is purely osmotic, σ ) ∆π, and the mushy region is an isotropic Newtonian fluid with effective bulk viscosity η2, the strain rate related to activity a1, and the chemical potential, µ1, of the fluid dissolved in the network (inertia is considered negligible relative to friction69):

-η2

∂ ) ∂t

∫µµ

/ 1 L 1

dµ1 ≈ c1RT ln a1 νj1

(6)

where c1 ) 1/ν1 (regular solution approximation) and νj1 is the partial molar volume of the penetrant; * here corresponds to the activity of the fluid that induces the glass-to-rubber transition. Then, the initial (φ2 f φ/2) solution for the Kelvin-Voigt viscoelastic element (eq 5) transforms into

∆π2 (1 - e-(K*/η2)t) + (φ/2)e-(K*/η2)t )K*

(7)

where φ2 is the volume fraction of the network and ∆π2 is the osmotically induced viscoelastic swelling pressure; K* is the osmotic bulk modulus of the swelling region, which can be estimated using the partial second derivative of the Gibbs free (65) Vieth, W. R. Diffusion in and through Polymers: Principles and Applications; Oxford University Press: New York, 1991. (66) Ferry, J. D. Viscoelastic properties of polymers; Wiley: New York, 1970. (67) Papoulia, K. D.; Asce, M.; Panoscaltsis, V. P.; Korovajchuk, I. A Class of Models of 3D Finite Viscoelasticity. Proceedings of the 14th Engineering Mechanics Conferemce, The University of Texas at Austin, Austin, TX, May 21-24, 2000. (68) Shaw, S.; Whiteman, J. R. Some Partial Differential Volterra Equation Problems Arising in Viscoelasticity. Proceedings of the CzechosloVak Conference on Differential Equations and Their Applications (EQUADIFF 9), Masaryk University, Brno, Czech Republic, August 2529, 1997;pp 183-200. (69) Tanaka, T.; Hocker, L. O.; Benedek, G. B. J. Chem. Phys. 1973, 59, 5151-5159.

energy (free enthalpy) with respect to the relative network volume on assumptions of the decorated lattice model70,71 and incompressibility of the rubbery state of the mixture, thus yielding K* ≈ -φ2(∂π2/∂φ2). (An important correction for mixtures with strong interaction between the molecules is derived in Appendix A.) Generally, the elastic coefficients defined in such fashion are not constants,68,72 yet the osmotic bulk modulus is ostensibly dominated by dilution free energy and does not depend on the changes of cross-link density during the phase transformation to the extent that the shear modulus does.71 It was found (theoretically and experimentally) that osmotic pressure is proportional to (c2)9/4 and elastic (shear) moduli are proportional to (c/2)9/4, where c2 is the polymer concentration (c/2 is the equilibrium concentration) in the semidilute regime.73 Note that we made an implicit transition to microscopic stress and strain variables, which assumes a subsequent volume averaging over specific coal macerals and accounting for a local pore structure. The real problem should consider a superposition of macroscopic stress and anisotropic effects of osmotic swelling, as well as the surface tension74 caused by adsorbed molecules. These terms can be treated as global parameters with respect to the osmotic swelling or shrinkage. A contribution of shear strains to the equation of motion for the local plastic relaxation can be derived within the framework of Landau theory of the free enthalpy expansion,75 a phenomenological framework that originated as a description of secondorder (continuous) phase transformations,76 which will not be treated in this work. If intermolecular forces within the adsorbed layer are negligible (one of the conventional Langmuir model assumptions), the surface tension term vanishes and the major problem is to estimate the parameters in eq 7. Similarly, the contribution of the purely mechanical stresses can be neglected in the first-order approximation because of the limited elastic compressibility of the rubbery medium. A comprehensive mechanochemical approach to interpretation of geophysical experiments, involving the carbon phase transitions under conditions of very high plastic shear and normal stresses, is discussed elsewhere.12 One can estimate ∆π2 (φ2 f φ/2) in eq 7 (also ∆π2 ≈ ∆π1 in eq 1 for binary mixtures) by defining the activity of a solvent, a1, in a regular polymer solution near the equilibrium (rubbery state)77

∆µR2 νj1∆π2 1 = ln a1 ) ln(1 - φ2) + 1 - φ2 + χ12φ22 + RT r RT (8)

[

(

)

]

where r is the number of statistical segments between crosslinks (junctions) and ∆µR2 is the residual contribution due to differences in intermolecular forces and in free volumes between the components, as well as the elastic properties of the network: (70) Bastide, J.; Candau, S.; Leibler, L. Macromolecules 1981, 14, 719726. (71) Bastide, J.; Duplessix, R.; Picot, C.; Candau, S. Macromolecules 1984, 17, 83-93. (72) Buchel, A.; Sethna, J. P. Phys. ReV. Lett. 1996, 77, 1520-1523. (73) Horkay, F.; Zrinyi, M. Macromolecules 1982, 15, 1306-1310. (74) Bullard, J. W.; Garboczi, E. J.; Carter, W. C. J. Appl. Phys. 1998, 83, 4477-4486. (75) Mu¨ller, J.; Grant, M. Phys. ReV. Lett. 1999, 82, 1736-1739. (76) Landau, L. D. Zh. Eksp. Teor. Fiz. 1937, 7, 19, 627; translated in Collected Papers of L. D. Landau; Ter Haar, D., Ed.; Pergamon Press: Oxford, 1965; pp 193, 216. (77) Flory, P. J. Principles of polymer chemistry; Cornell University Press: Ithaca, NY, 1953.


{ [( )

∆µel2 1 φ2 ) RT r r1-R

1/3

-


]}

φ2 2

(9)

where the affinity factor, R, introduced here ranges from 0 (excluded-volume blob, de Gennes c* model)78 to 1 (small deformations of elementary chains affine with the macroscopic deformations of the sample, Flory-Rehner theory).77 Here, r ) NA /(Fν1); NA is Avogadro’s number, F is the dry density of the network, and is the number averaged molecular weight between cross-links (Gaussian distribution, r > 3)79 that can be obtained experimentally utilizing the FloryRehner equilibrium equation.42,80 A simple substitution of eq 9 into eq 8 allows us to transform eq 7 into a direct relationship between the strain and the relative concentration of the fluid dissolved in coal, if not for the uncertainty of χ12. According to the lattice theory,81 the FloryHuggins polymer-solvent interaction parameter, χ12, is determined by the interchange potential energies that characterize interactions between the polymer segment and a solvent molecule, Γ12, between pairs of solvent segments, Γ22, and pairs of solvent molecules, Γ11, and is independent of solvent concentration (νj1 ≈ ν1):

1 z Γ12 - (Γ11 + Γ22) 2 χ12 = kT

[

]

(10)

where z is the effective coordination number (typically, z ∼ 10). Although the lattice theory is certainly not well suited for coal,82 ∆π1 should not be significantly affected by χ12 variation with fluid concentration well below the supercritical limit. At low concentrations, the interaction energies can be expressed via polarizability and the molar enthalpy of vaporization, based on London’s theory of dispersion forces,83 which is a fair approximation for the CO2-coal interaction due to the typically nonpolar (quadrupolar) structure of the CO2 molecule, except for the supercritical phase where the resonant structures may contribute to specific interactions.28,84 Chemical similarity between the sorbent molecule and the coal is the main criterion of applicability of the above approach at low permeate pressures. On the other hand, for high (liquidlike) fluid densities, Prigogine’s theory21,85,86 provides a good approximation for the enthalpy of mixing and entropic contributions due to free volume differences of the mixing molecules (equation-of-state effects). Another variable in eq 8 that becomes very important at high fluid concentrations is the partial molar volume of the penetrant, νj1. For the near critical permeating fluid, thermodynamic equations for solubility are more conveniently expressed in form of the complementary parameters of the solid solute (n1νj1 + n2νj2 ) V, where n1 and n2 are the molar amounts of the solvent and the solute and V is the total volume of the mixture without phase separation). Solubility effects are enhanced near the (78) Painter, P.; Shenoy, S. Energy Fuels 1995, 9, 364-371. (79) Veytsman, B.; Painter, P. Energy Fuels 1997, 11, 1250-1256. (80) Peppas, N. A.; Bures, P.; Leobandung, W.; Ichikawa, H. Eur. J. Pharm. Biopharm. 2000, 50, 27-46. (81) Eyring, H.; John, M. S. Significant Liquid Structures; Wiley: New York, 1969. (82) Hsieh, S. T.; Duda, J. L. Fuel 1987, 66, 170-178. (83) Kohler, F. Monatsh. Chem. 1957, 88, 857-877. (84) Kazarian, S. G.; Vincent, M. F.; Bright, F. V.; Liotta, C. L.; Eckert, C. A. J. Am. Chem. Soc. 1996, 118, 1729-1736. (85) Prigogine, I. The Molecular Theory of Solutions; North-Holland Publishing: Amsterdam, 1957. (86) Patterson, D.; Delmas, G. Discuss. Faraday Soc. 1970, 49, 98105.

solvent’s transition into the supercritical phase. It follows from standard thermodynamic relations that the minimum of νj2 corresponds to the point at which the rate of increase in solubility of a solid, y2 (molar fraction under saturation conditions), is maximized:87

( ) ∂ ln y2 ∂p

T

)

ν0S j2 2 -ν ∂ν p j2 RT + y2 0 ∂y2

[

∫

( ) ]

(11)

dp

T,p

The solute partial molar volumes are very large and negative in the highly compressible near-critical region.88 The extrema in the solute concentration occur at νj2 ) ν0S 2 (molar volume of the pure solid). For example, the partial molar volume, νj2 of naphthalene at infinite dilution (negligible solute-solute interaction) in supercritical CO2 takes a minimum near the critical isochor extension curve due to attractive dispersion forces, which is also true for the other studied supercritical solution systems.88 The rate of the solute concentration change is high and positive in this region (νj2 < 0), which results in two closely spaced solutions (minimum and maximum) of eq 11 for y2. In the above example, the solubility of naphthalene increased by a factor of ∼17, between 7 and 9 MPa at 35 °C, which corresponds to the density fluctuation ridge, where the correlation length and the numberdensity fluctuation take the maximum values and the solvent molecules are known to form clusters around the solute molecules92 or aggregate themselves (typical CO2 cluster L 2628 Å).53,54 Such dramatic free volume effects and the related “inaccessible” (or alternatively void) volume changes are apparently manifested in the local minima and maxima of the excess sorption data (Figure 3).46 For the second class (type II) binary mixtures of low volatile hydrocarbons such as naphthalene, biphenyl, and octacosane, which are similar to the ring and chain segments of the coal network, the critical mixture curve is broken due to incipient demixing (instabilities and phase separations) between the lower critical end point (LCEP) and the upper critical end point (UCEP).90 The similarity of the higher temperature portion of the three-phase (Solid-Liquid-Gas) coexistence curve for such mixtures (Figure 4) and the Khan-Jenkins coal softening (glassto-rubber transition) curve (Figure 2) allows one to expect that solubility behavior of coals near the critical point of CO2 will be very unstable and difficult to predict theoretically. Cubic equations-of-state and perturbation models enjoyed some success in correlation of solubilities, but they lack predictive power.88 Thus, some vital experimental data are required for accurate simulations. A chemical-type model of cluster formation may be superior to purely physical EOS treatment, but there has been no conclusive evidence of specific interaction between CO2 and coal reported so far. Experimental Support for Modeling In building the framework for the streamlined carbon sequestration research stemming from the first principles, it (87) Paulaitis, M. E.; Johnston, K. P.; Eckert, C. A. J. Phys. Chem. 1981, 85, 1770-1771. (88) Brennecke, J. F.; Eckert, C. A. AIChE J. 1989, 35, 1409-1427. (89) Debenedetti, P. G. Chem. Eng. Sci. 1987, 42, 2203-2212. (90) Lu, B. C.-Y.; Zhang, D.; Sheng, W. Pure Appl. Chem. 1990, 62, 2277-2285. (91) Skerget, M.; Novak-Pintaric, Z.; Knez, Z.; Kravnja, Z. Fluid Phase Equilib. 2002, 203, 111-132. (92) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1991, 30, 19942005.


RomanoV

〈n2〉 )

m2

(14)

where m2 is the mass of the mixed (in equilibrium) solid. Alternatively, eq 11 can be rewritten for molar fraction of the solute segments (or “isomers”) in equilibrium with the solvent

y˜ 2 ) Figure 4. Pressure-temperature (P-T) phase diagram for binary mixtures of low volatile hydrocarbon solutes (such as naphthalene, biphenyl, and octacosane,90 chemically representative of the coal network components) with a supercritical CO2 solvent. C1 and C2 are critical points of the pure solvent and pure solute, respectively; The 1 symbol designates the triple point of the pure solute. The three-phase (SLG) coexistence curve has a temperature minimum between the LCEP and the UCEP.

would be unfair to put full responsibility for the accuracy of the final predictions entirely on the modelers. Half of the load belongs to the experimental studies. The ultimate goal is to replace most if not all of the fudge factors by experimentally determined values of the valid physical parameters. This will require significant changes in experimental methods as well. Specifically, the CO2 sorption studies should take into account the free volume effects.19,20 If the (low partial pressure) helium volume is monitored during the sorption studies,20 the amount of dissolved fluid (e.g., supercritical CO2) can be separated from the adsorbed amount, when condensation in micropores is negligible:

mf ) FfVHe + ma + md ma md V0 ) VHe + + +V h E(md, t) Fa Ff

(

)}

(12)

where VHe is the helium volume, mf is the total injected amount of fluid, Ff is the measured density of the free phase, ma and Fa are the adsorbed layer mass and density, md is the amount of dissolved fluid (filling micro- and mesopores should not be confused with coal matrix permeation), and V h E is the free volume correction due to dissolution effects. Thus, it appears that the osmotic bulk modulus (defined in the previous section) can be determined experimentally. The key issue here is a possibility of gradual dissolution of the initially adsorbed layer, which would decrease the volume fraction of the network as it expands, still in accordance with eq 7. The volumetric effects of dissolution can be estimated using the decorated lattice model:

(1/φ2) ) 1 +

∫0

ν2 d ≈ 1 + rνj1

(13)

Equation 11 cannot be applied to coal directly, because the coal matter for the most part remains cross-linked even after the quasi-equilibrium mixing with dissolved fluid. Conventionally, it should be reflected in the equation-of-state84-86,91,92 corrections to χ12 in eq 8, of which the perturbation model (statistical associating fluid theory) is suitable for branched networks and requires only one adjustable binary interaction parameter to fit the experimental data.93 However, the molar amount of cross-linked chains can be interpreted in terms of the pseudo-molecular weight , which can be estimated experimentally:42,80

φ2 r - (r - 1)φ2

(15)

A procedure for experimental evaluation of the elastic coefficients at the various stages of dilution is described elsewhere.69,94 However, care should be taken to control the specimen confinement, porosity, and the rates of dilution and recompression. Strictly speaking, the coal compressibility in response to osmotic pressure is different from compressibility in response to elastic stress, because the initial glassy state is far from the equilibrium,29 which results in thermodynamic irreversibility of the osmotically induced relaxation. This factor should be taken into account for correct interpretation of the experimental data. Once the bulk modulus K is known, the viscosity coefficients can be estimated by fitting the relaxation time constant to eq 7. Discussion At this point, the modelers may still be looking for some magic to turn the entire analysis essentially into a single equation that can be readily incorporated into popular (invalid) coal reservoir models and breathe life into the rock-mechanics approach. Such a possibility will be evaluated for several limiting cases of CO2-coal interaction. First, let us analyze the fast and slow CO2-induced relaxation cases for the readily plasticizable coals. The quintessential finding of this work is that there cannot be any direct functional relationship between porosity and strain. In fact, the strain increase induced by elastic stresses causes the global reduction in porosity, while the osmotically induced strain decrease does the same job locally. A natural approach to solving such problems is to estimate the global porosity ΦG within the quasielastic framework (e.g., via the Palmer-Mansoori equation) and then to compute the heterogeneous local porosity ΦL changes caused by the pseudo-polymer (macromolecular) thermodynamic relaxation of specific coal macerals. For an isotropic case of slow relaxation, after elementary algebraic transformations of eq 7 (for purely osmotic stress), the local porosity changes can be reduced to a simple formula:

(

ΦL ) ΦG - (φ/2) -

)

∆π2 (1 - e-t/τ)(1 - ΦG) (16) K*

where the time constant τ ) η2/K*. (Negative values of ΦL in eq 16 would indicate an outflow of the mixture into the neighboring regions.) If there are any additional (global) stresses, the equation of motion similar to the one for the fluid (eq 1) should be written for a ductile phase of coal as well. Since the rubbery equilibrium state of plasticized coals does not retain “spatial memory” of the original stresses, the coal anisotropy is permanently removed after the phase transition95,96 (93) Shine, A. D. Polymers Supercritical Fluids. In AIP Properties of Polymers Handbook; Mark, J. E., Ed.; AIP Press, 1996; pp 249-256. (94) Tanaka, T.; Fillmore, D. J. J. Chem. Phys. 1979, 70, 1214-1218. (95) Larsen, J. W.; Flowers, R. A., II; Hall, P.; Carlson, G. Energy Fuels 1997, 11, 998-1002. (96) Yun, Y.; Suuberg, E. M. Energy Fuels 1992, 6, 328-330.



and will have no effect on the global stress-strain relationship under confining conditions of the coal seam. However, residual stress anisotropy can affect the directionality of the initial coal swelling locally, wherever the coal matter, during the phase transition, expands into neighboring pores and fractures, and primarily face cleats, thus reducing the widths of the fluid transport pathways or by compressing the macerals that do not interact with the permeating fluid. This can be accounted for by introducing a two-directional (Appendix A) tensor of osmotic pseudo-elastic coefficients; shear components are neglected in a first order approximation, in accordance with the experimental findings.94 The fast swelling of the rubbery phase is only different by the rate of swelling that is high enough to justify the instant equilibrium approximation for the porosity correction:

δΦ ≈ ((φ2) - (φ/2))(1 - Φ)

KOS ) -V

Summary Although not all of the geomechanical processes related to carbon sequestration and CBM projects are well understood, it is recommended that the efforts of the modelers focus on systematic approaches based on the sound physical principles. Traditional models are correlative in nature and strictly speaking cannot be used to predict the performance of new projects. The overwhelming amount of fudge factors used in the state-of-theart computational models complicates the history matching for the old projects as well. Without direct ties to experimental data for fundamental molecular scale characteristics of the coal, it becomes increasingly more difficult to justify the acceptable ranges of the fitting parameter values. The proposed here strainvolume fraction relationship for osmotically driven network relaxation (eqs 7 and 16), on the other hand, offers a direct way of taking into account the molecular level interactions and incorporating them into the geomechanical models. The above analysis suggests that no single equation can be used to amend the Palmer-Mansoori theory, because there cannot be any functional relationship between the strain and porosity advocated by the rock-mechanics approach. This is the reason why enormous state-of-the-art computational efforts directed toward finding the right fudge factors within the pure rock-mechanics approach should inevitably continue to fail. While building a universal geomechanical model is beyond the scope of this work, it serves to provide a new direction for those modelers who are interested in the predictive rather than correlative power of their models.

(A1)

T

where partial differentiation implicitly is applied to the special case of n2 ) constant. Thus, still using the decorated lattice approach,

( ) ∂ν1 ∂n1

dφ2 dV dν2 dV dν1 dV + + + )))φ2 V ν2 V ν1 V ∂V ∂n1

n2

( )

n2

V dV ν1 V (A2)

On the other hand, by definition, νj1 ) (∂V/∂n1)n2 and hence

νj1 ) ν1 +

(17)

where (1/φ2) f 1 - ln a1/(K*νj1RT) (see eqs 8 and 13); this can be readily incorporated into the current simulators. What may happen though if the swelling potential (strain) is very high but yet the fluid penetrating into the coal matrix cannot transform it to the rubbery state? In that case, the above approach is not applicable. As the initial glassy state is far from equilibrium, it may result in nonhomogeneous fracturing caused by weakening at the more accessible regions constrained by the underlying brittle material, which is described by the Weibull statistics. Such a process is irreversible and its effect on permeability is very difficult to predict.

(∂V∂π)

( )

V ∂ν1 ν1 ∂n1

(A3)

n2

Combining eqs A2 and A3 yields

and

dV νj1 dφ2 ) V ν1 φ2

(A4)

()( )

KOS ) -

νj1 ∂π φ ν1 2 ∂φ2

(A5)

T

A.2. Two-Directional Anisotropy. Under the natural reservoir conditions, the most practicable approach is to separate the coordinates of the viscoelastic properties into a direction of the principal (overburden) compression, with strain II, and the directions perpendicular to it, with strain ⊥:

dV ) dII + 2d⊥ V

(A6)

Equation A5 will still hold as the osmotic pressure is isotropic and not subject to partition, while the dynamic relationship between the strain tensor and the osmotic pressure will split into two linear differential equations:

KII + ηII˘ II ) -π 2K⊥ + η⊥˘ ⊥ ) -π

}

(A7)

where directionality of the viscosity coefficients is rooted in higher cross-link density along the principal axis due to the closer proximity of the neighboring chains caused by the preferential compression in the vertical direction (Figure A1).

Acknowledgment. The author thanks John W. Larsen for fruitful discussions and acknowledges the ORAU research grant.

Appendix A: Remarks on the Osmotic Bulk Modulus A.1. Isotropic Case. It is more natural to define the osmotic bulk modulus similar to the corresponding elastic property:

Figure A1. Simplified representation of anisotropic swelling of a coal matrix (black, initial volume; gray, equilibrium volume) and its effect on transport of the fluid injected into coal seam. Arrows show the directions of the fluid flow and the primary direction of coal swelling (II is the initial principal strain).


These two factors (lower initial strain and lower viscosity) will result in much shorter time to reach the equilibrium in the horizontal direction, after which the osmotically induced (unobstructed) expansion will continue only along the primary axis. Nomenclature Variables and Parameters a ) activity, c ) concentration, mol/m3 ^ ) tensor of elastic coefficients, Pa E ) Young’s modulus, Pa f ) fugacity, Pa J ) flux per unit surface area, kg/(s‚m2) K ) bulk modulus, Pa L ) length, m ) number averaged molecular weight between cross-links, kg m ) mass, kg NA ) Avogadro’s number, n ) molar amount, P ) permeability, kg/(Pa‚s‚m) p ) pressure, Pa R ) universal gas constant, 8.314472 J/(K‚mol) R ) specific resistance of membrane, m/kg r ) number of statistical segments between cross-links (junctions), s ) dummy variable, s t ) time, s T ) temperature, K

RomanoV u(x,t) ) displacement from equilibrium position at a position x[m] and time t, m V ) total volume of the mixture without phase separation, m3 V0 ) void volume, m3 y ) molar fraction, Greek Letters Γ ) potential energy of molecular interaction, J γ ) activity coefficient, δ ) Scatchard-Hildebrand solubility parameter, J0.5/m1.5 ) strain, η ) viscosity, Pa‚s µ ) chemical potential, J/mol ν ) molar volume, m3/mol π ) osmotic pressure, Pa F ) density, kg/m3 σ ) stress tensor, Pa τ ) time constant, s Φ ) porosity, φ ) volume fraction, χ ) Flory-Huggins interaction parameter, Other Notation ) used to indicate differentiation with respect to t xj ) used to indicate partial property L ) used to indicate a standard state 1 ) as a subscript, used to indicate the property of the fluid 2 ) as a subscript, used to indicate the property of the coal or polymer •

EF060476P

From Macromolecular Thermodynamics to Reservoir Simulation

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