Particulate Processes in Porous Media 157 subject to (8-32) i cr is the critical shear stress and k e is an erosion rate constant given by (Khilar and Fogler, 1987): 0 when > l and = 0 otherwise (8-33) (8-34) There are several alternative ways of expressing the hydraulic force. The Rabinowitsch-Mooney equation for non-Newtonian fluid wall shear- stress, i w , in pipes is given by (Metzner and Reed, 1955): d* D (8-35) The non-Darcy equation can be modified by applying the capillary tubes analogy and the procedure by Ikoku and Ramey, Jr. (1979): V-e (8-36) where \L e is the effective viscosity given by (8-37) k' and n are some empirical parameters, which assume k' = \JL and n' = 1 for Newtonian fluids. The critical shear stress, i cr , is a function of the particle stickiness to the surface characterized by the k x constant and the particle concentration at pore surface (Civan, 1990, 1996): T =^c« (8-38) where a is an empirical constant. 158 Reservoir Formation Damage Based on Eqs. 8-35 and 8-36, the excess shear stress can be correlated for one-dimensional horizontal flow: •-T, (8-39) The previous studies are mostly limited to one-dimensional Newtonian fluid flow and they typically used (Civan et al., 1989; Khilar and Folger, 1987; Gruesbeck and Collins, 1982; Cernansky and Siroky, 1985; Ohen and Civan, 1989, 1990): (8-40) In general, for multi-dimensional flow (Civan, 1996) (8-41) where if/ is the flow potential and D is the hydraulic tube diameter tensor for anisotropic media. 5 is a unit vector. Particle Transfer Across Fluid-Fluid Interfaces The driving force for particle transfer between two fluid phases is the wettability of the fluid phases relative to the wettability of the particles. Particles prefer to be in the phase that wets them (Muecke, 1979) (see Figure 8-9 by Civan, 1994). But, mixed-wet particles tend to remain on the interface where they are most stable (Ivanov et al., 1986) (see Figure 8-10). In the region involving the interface between wetting and non- wetting phases, it can be postulated that particles A in a weaker wettability phase-1 first move to the interface and then migrate from the interface to a stronger wettability phase-2 according to the following consecutive processes (Civan, 1996): Nonwetting phase - 1 —» Interface —> Wetting phase - 2 (8-42) Therefore, the following power-law rate expressions can be proposed: Paniculate Processes in Porous Media 159 SOLID-FLUID INTERFACE VELOCITY PROFILE WETTING FLUID (PHASE 1) NON-WETTING FLUID (PHASE 2) SOLID MATRIX (PHASES) FLUID-FLUID INTERFACE Figure 8-9. Particle retention at solid-fluid and fluid-fluid interfaces and the velocity profiles in multi-phase systems (after Civan, 1994; reprinted by permission of the U.S. Department of Energy). i y FLUID > •'; •Phase'2-"-•-•••;' • SOLID . . v PARTICLE •••.••': ••' ' . • •; •; \/Phase 1 Figure 8-10. A particle stabilized at a fluid-fluid interface (modified after Ivanov et al., 1986; reprinted by permission of the author and Academic Press; after Civan, 1994; reprinted by permission of the U.S. Department of Energy). 160 Reservoir Formation Damage dR Ml = •A\ _ (8-43) p _ *^M12 _ T, poA _ T, pP^ ^12— ./^ , x^^Al^Al / M2 y M12 (8-44) \ -* A2 _ . «4 — " 4 9 **• 4 1 "7 (8-45) ?i A1 and A- A2 are some rate constants; a A and {3 A are some empirical exponents of intensities; and ? A1 , t An , and t A2 are the time delays due to the inertia of the respective transfer processes. The initial conditions are: = 0 (8-46) The rate of particle transfer can be expressed per unit volume of the I th phase according to the following expression: (8-47) The particles captured at the interface are assumed to migrate at a speed determined by the relative speed of the fluid phases. Exercise 1. Using Einstein's equation, Eq. 8-4, estimate the Brownian diffu- sivity of a l|0jn diameter fine particle in water at 20°C temperature. (Answer: D = 4.3xl(T 9 cm 2 /s; McDowell-Boyer et al., 1986) References Amaefule, J. O., Kersey, D. G., Norman, D. L., & Shannon, P. M., "Advances in Formation Damage Assessment and Control Strategies," CIM Paper No. 88-39-65, Proceedings of the 39th Annual Technical Meeting of Petroleum Society of CIM and Canadian Gas Processors Association, June 12-16, 1988, Calgary, Alberta, 16 p. Cernansky, A., & Siroky, R., "Deep-bed Filtration on Filament Layers on Particle Polydispersed in Liquids," Int. Chem. Eng., Vol. 25, No. 2, 1985, pp. 364-375. Paniculate Processes in Porous Media 161 Chang, F. F., & Civan, F., "Modeling of Formation Damage due to Physical and Chemical Interactions between Fluids and Reservoir Rocks," SPE 22856 paper, Proceedings of the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, October 6-9, 1991, Dallas, Texas. Chang, F. E, & Civan, E, "Predictability of Formation Damage by Modeling Chemical and Mechanical Processes," SPE 23793 paper, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 1992, Lafayette, Louisiana, pp. 293-312. Chang, F. E, & Civan, E, "Practical Model for Chemically Induced Formation Damage," J. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 123-137. Civan, E, "A Generalized Model for Formation Damage by Rock-Fluid Interactions and Particulate Processes," SPE Paper 21183, Proceedings of the SPE 1990 Latin American Petroleum Engineering Conference, October 14-19, 1990, Rio de Janeiro, Brazil, 11 p. Civan, E, "Evaluation and Comparison of the Formation Damage Models," SPE Paper 23787, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 1992, Lafayette, Louisiana, pp. 219-236. Civan, E, "Predictability of Formation Damage: An Assessment Study and Generalized Models," Final Report, U.S. DOE Contract No. DE-AC22- 90BC14658, April 1994. Civan, E, "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE 28709 paper, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412. Civan, E, "Modeling and Simulation of Formation Damage by Organic Deposition," Proceedings of the First International Symposium on Colloid Chemistry in Oil Production: Asphaltenes and Wax Deposition, ISCOP'95, Rio de Janeiro, Brazil, November 26-29, 1995, pp. 102-107. Civan, E, "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Symposium, Lafayette, Louisiana, February 14-15, 1996, pp. 311-326. Civan, E, Knapp, R. M., & Ohen, H. A., "Alteration of Permeability by Fine Particle Processes," J. Petroleum Science and Engineering, Vol. 3, Nos. 1/2, October. 1989, pp. 65-79. Dullien, F. A. L., Porous Media Fluid Transport and Pore Structure, Academic Press, London (1979), 396 p. Gruesbeck, C, & Collins, R. E., "Particle Transport Through Perforations," SPEJ, December 1982, pp. 857-865. Gruesbeck, C., R. E. Collins, "Entrainment and Deposition of Fine Particles in Porous Media," SPEJ, December 1982, pp. 847-856. 162 Reservoir Formation Damage Hallow, J. S., "Incipient Rolling, Sliding, and Suspension of Particles in Horizontal and Inclined Turbulent Flow," Chem. Eng. Sci., Vol. 28, 1973, pp. 1-12. Himes, R. E., Vinson, E. E, & Simon, D. E., "Clay Stabilization in Low- Permeability Formations," SPE Production Engineering, August 1991, pp. 252-258. Ikoku, C. U., & Ramey, Jr., H. J., "Transient Flow of Non-Newtonian Power-Law Fluids in Porous Media," Supri-TR-9, Report No. E(04- 3)1265, U.S. Department of Energy (February 1979) 257. Ivanov, I. B., Kralchevsky, P. A., & Nikolov, A. D., "Film and Line Tension Effects on the Attachment of Particles to an Interface," J. Colloid and Interface ScL, Vol. 112, No. 1, 1986, pp. 97-107. Ives, K. J., "Deep Bed Filters," in Rushton, A. (Ed.) Mathematical Models and Design Methods in Solid-Liquid Separation, 1985 Martinus Nijhoff Publishers, pp. 90-332. Khilar, K. C., & Fogler, H. S., "Colloidally Induced Fines Migration in Porous Media," in Amundson, N. R. and Luss, D. (Eds.), Reviews in Chemical Engineering, Freund Publishing House LTD., London, England, January-June 1987, Vol. 4, Nos. 1 and 2, pp. 41-108. Khilar, K. C., & Fogler, H. S., "Water Sensitivity of Sandstones," SPEJ, February 1983, pp. 55-64. Kia, S. E, Fogler, H. S., & Reed, M. G., "Effect of Salt Composition on Clay Release in Berea Sandstones," SPE 16254, February 1987. King, R. W., and Adegbesan, K. O., "Resolution of the Principal For- mation Damage Mechanisms Causing Injectivity and Productivity Impairment in the Pembina Cardium Reservoir," SPE Paper 38870, Proceedings of the 1997 Annual Technical Conference and Exhibition held in San Antonio, Texas, October 5-8, 1997, pp. 277-288. Lichtner, Water Resources Research, Vol. 28, No. 12, December 1992, pp. 3135-3155. McDowell-Boyer, L. M., Hunt, J. R., & Sitar, N., "Particle Transport Through Porous Media," Water Resources Research, Vol. 22, No. 13, December 1986, pp. 1901-1921. Metzner, A. B., & Reed, J. C., "Flow of Non-Newtonian Fluids—Corre- lation of the Laminar, Transition, and Turbulent Flow Regions," AIChE J., Vol. 1, No. 4, 1955, pp. 434-440. Millan-Arcia, E., Civan, F. "Characterization of Formation Damage by Particulate Processes," J. Canadian Petroleum Technology, Vol. 31, No. 3, March 1992, pp. 27-33. Muecke, T. W., "Formation Fines and Factors Controlling their Movement in Porous Media," JPT, April 1979. Ochi, J., & Vernoux, J F., "Permeability Decrease in Sandstone Reser- voirs by Fluid Injection-Hydrodynamic and Chemical Effects," /. of Hydrology, Vol. 208, 1998, pp. 237-248. Paniculate Processes in Porous Media 163 Ohen, H. A., & Civan, F., "Predicting Fines Generation, Migration and Deposition Near Injection and Production Wells," Proceedings of the First Regional Meeting, American Filtration Society, Houston, Texas, October 30-November 1, 1989, pp. 161-164. Ohen, H. A., & Civan, F, "Simulation of Formation Damage in Petroleum Reservoirs," SPE 19420 paper, Proceedings of the 1990 SPE Symposium on Formation Damage Control, Lafayette, Louisiana, February 22-23, 1990, pp. 185-200. Ohen, H. A., & Civan, F, "Simulation of Formation Damage in Petroleum Reservoirs," SPE Advanced Technology Series, Vol. 1, No. 1, April 1993, pp. 27-35. Pautz, J. F, Crocker, M. E., & Walton, C. G., "Relating Water Quality and Formation Permeability to Loss of Injectivity," SPE 18888 paper, Proceedings of the SPE Production Operations, Oklahoma City, Oklahoma, March 13-14, 1989, pp. 565-576. Rushton, A., "Mathematical Models and Design Methods in Solid-Liquid Separation," NATO ASI, 1985, No. 88, Ed. A. Rushton, Martinus Nijhoff. Wojtanowicz, A. K., Krilov, Z. and Langlinais, J. P., "Study on the Effect of Pore Blocking Mechanisms on Formation Damage," Paper SPE 16233, presented at Society of Petroleum Engineers Production Operations Symposium, Oklahoma City, Oklahoma, March 8-10, 1987, pp. 449-463. Wojtanowicz, A. K., Krilov, Z. and Langlinais, J. P., "Experimental Determination of Formation Damage Pore Blocking Mechanisms," Trans, of the ASME, Journal of Energy Resources Technology, Vol. 110, 1988, pp. 34-42. Chapter 9 Crystal Growth and Scale Formation in Porous Media* Summary In this chapter, the inorganic and organic precipitation/dissolution phenomena, and their effect on the size of the suspended particles and porosity variation are discussed and formulated. Introduction Civan (1996) describes that: Injection of fluids and chemicals for improved recovery, and libera- tion of dissolved gases, such as CO 2 and light hydrocarbons from the reservoir fluids approaching the wellbore during produc- tion, and variation of fluid saturations can alter the temperature, pressure, and composition of the fluids in the near wellbore region and tubing. Consequently, the thermodynamic and chemical balance may change in favor of precipitate separation, aggregation of preci- pitates, crystal growth, and scale formation. Precipitates cause formation damage by changing the wettability and permeability of petroleum bearing rock and cause scale formation and clogging in tubing and pore throats. Inorganic Precipitation Typical inorganic precipitates include anhydrate (CaCO 3 ), gypsum (CaSO 4 '2H 2 O), hemihydrate (CaSO4' ] / 2 H 2 O\ barite (BaSO 4 ), celestite * Parts reprinted by permission of the Society of Petroleum Engineers from Civan, ©1996 SPE, SPE 31101 paper. 164 Crystal Growth and Scale Formation in Porous Media 165 (SrSO 4 ), magnesium sulfide (MgSO 4 ) originating from mixing sea water with brine, and rock and brine interactions (Oddo and Tomson, 1994; Atkinson and Mecik, 1997); ironhydroxide gel (Fe(OH)^ originating from the acid dissolution and precipitation of iron minerals such as pyrhotite (FeS), pyrite (FeS 2 ), hematite (Fe 2 O 3 ), magnetite (Fe 3 O 4 ), and siderite (FeCO 3 ) (Rege and Fogler, 1989); silicium tetra hydroxide gel (Si(OH) 4 ] originating from the alkaline dissolution and precipitation of minerals in shaly sandstones such as quartz and kaolinite (Labrid, 1990); and polymeric substances produced by in-situ gelation (Todd et al., 1993), alcohol induced crystallization (Zhu and Tiab, 1993), separation of elemental sulfur (Roberts, 1997); and surfactant precipitation (Arshad and Harwell, 1985). Following Oddo and Tomson (1994), precipitation/dissolution reactions can be symbolically represented by: An v 3 Pr (9-1) where Me represents a cation or metal ion such as 5V +2 , Ca +2 , Mg +2 , An represents an anion such as CO 3 2 , SO 4 2 , and Pr represents a solid precipitate such as CaCO 3 , MgCO 3 , BaSO 4 , Fe(OH) 3 , Si(OH) 4 . v p v 2 , and v 3 are some stoichiometric coefficients. Oddo and Tomson (1994) correlated the saturation solubility product, K sp , empirically as a function of temperature, T, pressure, p, and ionic strength, 5"., for typical systems. Hence, the saturation ratio given by the following equation can be used to determine whether the conditions are favorable for precipitation (Oddo and Tomson, 1994): F s =[MeV(AnY/K sp (9-2) F s < 1 indicates an undersaturated solution, condition unfavorable for scaling, if F s = 1, the solution is at equilibrium with the solid scale, and F s > 1 indicates a supersaturated solution, condition favorable for scaling. Organic Precipitation Typical organic precipitates encountered in petroleum production are paraffins and asphaltenes. Paraffins are inert and asphaltenes are reactive substances. They are both sticky, thick, and deformable precipitates (Chung, 1992; Ring et al., 1994). Therefore, they can seal the pore throats and reduce the permeability to zero without needing to reduce the porosity to zero and their deposition at the pore surface and tubing wall is irreversible unless a solvent treatment is applied (Leontaritis et al., 1992): The saturation ratio is given by: 166 Reservoir Formation Damage (9-3) where F s < 1 for undersaturated solution, F s = 1 for saturated solution, and F s > 1 for supersaturated solution. X A is the mole fraction of the dissolved organic in oil and (X A ) S is the organic solubility at saturation conditions. (X A ) S is predicted using the thermodynamic model by Chung (1992). Crystallization Majors (1999) explains that "Crystallization is the arrangement of atoms from a solution into an orderly solid phase." and "Growth is simply the deposition of material at growth sites on an existing crystal face." The process is called primary nucleation if there are no crystals present in the solution to start with and crystallization is occurring for the first time. Primary nucleation can be homogeneous or heterogeneous (Majors, 1999). Homogeneous nucleation occurs inside the solution without contact with any surface. Heterogeneous nucleation occurs over a solid surface. The process is called secondary nucleation if there are already some crystals present in the system over which further deposition can occur. The schematic chart given in Figure 9-1 by Majors (1999) describes the concentration-temperature relationship for nucleation. As can be seen, the primary nucleation process requires a sufficiently high concentration of c 1 § o B u Undersaturated Saturation line Temperature Figure 9-1. Concentration vs. temperature diagram for crystal formation (after Majors, 1999; reprinted by permission of the Chemical Processing Magazine). [...]... Precipitation," SPE 2 48 51, Proceedings of the 67th Annual technical Conference and Exhibition of the SPE held in Washington, D.C., October 4-7, 19 92, pp 86 9 -87 8 Civan, F., "Correlation of the Pit Depth in Crystal Etching by Dissolution," J of Colloid and Interface Science, Vol 222, No 1, pp 15 615 8, 2000 Civan, F., "A Multi-Purpose Formation Damage Model," SPE 311 01 paper, SPE Formation Damage Symposium,... Pte Ltd., 19 95, pp 4 18 -4 28 Hunkeler, F and Bohni, H., "Determination of Pit Growth Rates on Aluminum Using a Metal Foil Technique," Corrosion, Vol 37 (11 ), 19 81 , pp 645-650 Labrid, J., "Modeling of High pH Sandstone Dissolution," Proceedings of the International Technical Meeting held jointly by the Petroleum Society of CIM and the SPE in Calgary, June 10 -13 , 19 90, pp 81 / 1- 21 Leetaru, H E., "Reservoir. .. 9-3) 2U (9 -8) (9-9) A = (9 -10 ) By combining the various efforts, Eq 9-6 can be expressed as (Walton, 19 69; Putnis and McConnell, 19 80 ; Richardson and McSween, 19 89 ; Schneider, 19 97; and Stumm and Morgan, 19 96): 17 0 Reservoir Formation Damage (9 -11 ) The depositing substance and the substrate surface match well when a cs . and the particle concentration at pore surface (Civan, 19 90, 19 96): T =^c« (8- 38) where a is an empirical constant. 15 8 Reservoir Formation Damage Based on Eqs. 8- 35 and 8- 36, the . Gruesbeck and Collins, 19 82 ; Cernansky and Siroky, 19 85 ; Ohen and Civan, 19 89 , 19 90): (8- 40) In general, for multi-dimensional flow (Civan, 19 96) (8- 41) where if/ is the flow . Vol. 17 , No. 1/ 2, February 19 97, pp. 12 3 -13 7. Civan, E, "A Generalized Model for Formation Damage by Rock-Fluid Interactions and Particulate Processes," SPE Paper 211 83 ,