Scale Formation and Dissolution at the Pore Surface
The rate of scale formation at the pore surface can be expressed similarly to crystal growth. The pore volume relates to the diameter of an equivalent spherical shape pore space by (Civan, 1996):
Thus, the pore surface relates to the porosity according to:
C3 and C4 are some shape factors and C5=C4/C^3. The porosity of the solid porous matrix can be expressed as a sum of the instantaneous porosity, <|>, and the pore space occupied by the scales, <{>5, as:
Therefore, substituting Eqs. 9-32 and 9-34 into Eq. 9-23 yields (Civan, 1996):
and k'c is a scale formation rate constant incorporating the above mentioned shape factors and some constants. The minus sign in Eq. 9-35 is for the reduction of porosity by scale formation at the pore surface. Thus, assuming the rock porosity, (|)r, remains constant and substituting Eq. 9-33 into 9-35 leads to an equation similar to Ortoleva et al. (1987):
Assume that the surface area of crystal available for growth can be expressed empirically by:
in which /(({>) is the specific surface of the mineral-fluid contact area (surface area per unit mineral mass) expressed as a function of porosity. Civan (1996) approximated this function according to Eq. 9-32. Thus, substituting Eq. 9-40 into Eq. 9-23 leads to Holstad's (1995) equation:
Holstad (1995) expressed the temperature dependency of the crystallization rate constant by the Arrhenius equation:
where FM, AM, and EM denote an empirical mineral property factor, an Arrhenius pre-factor, and the activation energy. Liu et al. (1997) used a similar equation
where k° is the high-temperature (T —> °°) limit of the rate constant. The effects of various conditions on dissolution rates, including lithologic variation, hydrodynamics, ionic strength, saturation state, mixedkinetic control, and surface treatment, have been investigated by Raines and Dewers (1997, 1997), Hajash Jr. et al. (1998), and Merino and Dewers (1998).
Crystal Surface Displacement by Dissolution and Precipation
The dissolution and precipitation of a crystalline matter in contact with a solution can be studied by measuring the progress of the crystal surface as a function of time. Hunkeler and Bohni (1981) and Dunn et al. (1999) used this technique. Civan (2000) determined that the position of the progressing crystal surface could be correlated by:
for which x, x0 and xt are the instantaneous, initial, and final surface positions, respectively, k is a rate constant, and M is the amount of solute precipitated or dissolved, given by:
where t is time, c0 and c, are the solute concentrations of the solution at the beginning and equilibrium, respectively, and D is the diffusion coefficient of the solute. Civan (2000) verified this model using the Dunn et al. (1999) measurements of the pit depth during barite dissolution.
References
Arshad, A., & Harwell, J. H., "Enhanced Oil Recovery by Surfactant- Enhanced Volumetric Sweep Efficiency," SPE 14291, Annual Technical Conference and Exhibition of SPE, Las Vegas, Nevada, September 22- 25, 1985.
Atkinson, G., & Mecik, M., "The Chemistry of Scale Prediction," J. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 113-121.
Chang, F. F., & Civan, F., "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.
Chung, T.-H., "Thermodynamic Modeling for Organic Solid Precipitation," SPE 24851, Proceedings of the 67th Annual technical Conference and Exhibition of the SPE held in Washington, D.C., October 4-7, 1992, pp. 869-878.
Civan, F., "Correlation of the Pit Depth in Crystal Etching by Dissolution," J. of Colloid and Interface Science, Vol. 222, No. 1, pp. 156- 158, 2000.
Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31101 paper, SPE Formation Damage Symposium, Lafayette, Louisiana, February 14-15, 1996, pp. 311-329.
Dawe, R. A. and Zhang, Y., "Kinetics of Calcium Carbonate Scaling Using Observations from Glass Micromodels," Journal of Petroleum Science and Engineering, Vol. 18, No. 3/4, pp. 179-187, 1997.
Dunn, K., Daniel, E., Shuler, P. J., Chen, H. J., Tang, Y, and Yen, T. F, "Mechanism of Surface Precipitation and Dissolution of Barite: A Morphology Approach," /. Colloid Interface Sci. Vol. 214, 1999, pp. 427-437.
Dunning, W. J., "General and Theoretical Introduction," Nucleation, Zettlemoyer, A.C. (Ed.), M. Dekker, Inc., New York, New York, 1969, pp. 1-67.
Hajash Jr., A., Carpenter, T. D., & Dewers, T. A., "Dissolution and Time-Dependent Compaction of Albite Sand: Experiments at 100°C and 160°C in pH-buffered Organic Acids and Distilled Water," Tectonophysics, Vol. 295, 1998, pp. 93-115.
Holstad, A., "Mathematical Modeling of Diagenetic Processes in Sedimentary Basins," Mathematical Modeling of Flow Through Porous Media," Bourgeat, A. P., Carasso, C., Luckhaus, S., and Mikelic, A., (Eds.), World Scientific Publ. Co. Pte. Ltd., 1995, pp. 418-428.
Hunkeler, F. and Bohni, H., "Determination of Pit Growth Rates on Aluminum Using a Metal Foil Technique," Corrosion, Vol. 37(11), 1981, 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, 1990, pp. 81/1-21.
Leetaru, H. E., "Reservoir Characteristics and Oil Production in the Cypress and Aux Vases Formations at Storms Consolidated Field in White County, Illinois," Illinois Petroleum Series 150, 1996, Department of Natural Resources, Illinois State Geological Survey, 47 p.
Leontaritis, K. J., Amaefule, J. O., & Charles, R. E., "A Systematic Approach for the Prevention and Treatment of Formation Damage Caused by Asphaltene Deposition," SPE 23810 paper, Proceedings of the SPE International Symposium on Formation Damage Control, Lafayette, Louisiana, February 26-27, 1992, pp. 383-395.
Liu, X., & Ortoleva, P., "A General-Purpose, Geochemical Reservoir Simulator," SPE 36700 paper, Proceedings of the 1996 SPE Annual Technical Conference and Exhibition, Denver, Colorado, October 6-9, 1996, pp. 211-222.
Liu, X., Ormond, A., Bartko, K., Li, Y., & Ortoleva, P., "A Geochemical Reaction-Transport Simulator for Matrix Acidizing Analysis and Design," J. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 181-196.
Majors, J., "Crystallization and the Bottom Line," Chemical Processing, Vol. 62, No. 2, 1999, pp. 55-59.
Merino, E., & Dewers, T., "Implications of Replacement for Reaction-Transport Modeling," Journal of Hydrology, Vol. 209, 1998, pp. 137-146.
Oddo, J. E., & Tomson, M. B., "Why Scale Forms and How to Predict It," SPE Production Facilities, February 1994, pp. 47-54.
Ortoleva, P., Chadam, J., Merino, E., & Sen, A., "Geochemical Self-Organization II: The Reactive-Infiltration Instability," Amer. J. ScL, Vol. 287, 1987, pp. 1008-1040.
Putnis, A., & McConnell, J. D. C., Principles of Mineral Behaviour, Blackwell Scientific Publ., Boston, 1980.
Raines, M. A., & Dewers, T. A., "Mixed Transport/Reaction Control of Gypsum Dissolution Kinetics in Aqueous Solutions and Initiation of Gypsum Karst," Chemical Geology, Vol. 140, 1997, pp. 29-48.
Raines, M. A., & Dewers, T. A., "Mixed Kinetics Control of Fluid-Rock Interactions in Reservoir Production Scenarios," J. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 139-155.
Reddy, M. M., "Effect of Magnesium Ion on Calcium Carbonate Nucleation and Crystal Growth in Dilute Aqueous Solutions at 25° Celsius, in Studies in Diageneses, Denver, F. A. Mumpton (Ed.), Colorado, U.S. Geological Survey," Bulletin 1578, 1986, pp. 169-182.
Reddy, M. M., "Carbonate Precipitation in Pyramid Lake, Nevada, Probable Control by Magnesium Ion," in Mineral Scale Formation and Inhibition (Z. Amjad, ed.), Plenum Press, New York, 1995, pp. 21-32.
Rege, S. D., & Fogler, H. S., "Competition Among Flow, Dissolution and Precipitation in Porous Media," AIChE J., Vol. 35, No. 7, 1989, pp. 1177-1185.
Richardson, S. M., & McSween, H. Y, Geochemistry: Pathways and Processes, Prentice Hall, Inc., New York, New York, 1989.
Ring, J. N., Wattenbarger, R. A., Keating, J. F., & Peddibhotla, S., "Simulation of Paraffin Deposition in Reservoirs," SPE Production & Facilities, February 1994, pp. 36-42.
Roberts, B. E., "The Effect of Sulfur Deposition on Gaswell Inflow Performance," SPE Reservoir Engineering, May 1997, pp. 118-123.
Schneider, G. W., "A Geochemical Model of the Solution-Mineral Equilibria Within a Sandstone Reservoir," M.S. Thesis, The University of Oklahoma, 1997, 157 p.
Stumm, W., & Morgan, J. J., Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, John Wiley and Sons, New York, New York, 1996.
Todd, B. J., Willhite, G. P., & Green, D. W., "A Mathematical Model of In-situ Gelation of Polyacrylamide by a Redox Process," SPE Reservoir Engineering, February 1993, pp. 51-58.
Walton, A. G., "Nucleation in Liquids and Solutions," Nucleation, Zettlemoyer, A. C. (Ed.), M. Dekker, Inc., New York, New York, 1969, pp. 225-307.
Zhu, T., & Tiab, D., "Improved Sweep Efficiency by Selective Plugging of Highly Watered Out Zones by Alcohol Induced Precipitation," JCPT, Vol. 32, No. 9, November 1993, pp. 37-43.
