Particle Deposition Rates
The rate of deposition of the particles of the slurry over the slurry side cake surface is assumed proportional to the particle mass flux approaching the filter cake. The rate of erosion of the particles from the slurry side cake surface is assumed proportional to the tangential, excess shear stress above the critical stress necessary for particle mobilization. Therefore, for dynamic filtration involving cross flow, the net mass rate of all particles (large plus small) deposition per unit area of the slurry side cake surface is given by the difference between the deposition and erosion rates as (Civan, 1996, 1998b):
where ut is the carrier fluid filtration flux normal to the cake surface (cm3/ cm 2 • s) and Cpl is the slurry particle concentration expressed as the particle mass per unit volume of the carrier fluid in the slurry. For small particles retention over the slurry side of the cake, an expression similar to Eq. 12-134 can be written as:
in which (Cp2i)slur denotes the mass of small particles per volume of the carrier fluid in the slurry. The net mass rate of deposition of small particles within the filter cake is given by:
which is similar to Eq. 33 of Corapcioglu and Abboud (1990), but the deposition and the mobilization terms are more consistently expressed. The rate expressions given by Eqs. 12-134 through 136 for the deposition of the total (fine plus large) and fine particles of the slurry over the progressing cake surface and the retention of the fine particles of the flowing suspension within the cake matrix can be expressed in terms of the volumetric rates, respectively, as (Civan, 1999b):
In Eqs. 12-10 through 12, the slurry shear-stress, ts, acting over the progressing cake surface is estimated using the Rabinowitsch-Mooney equation (Metzner and Reed, 1955). This equation can be expressed for linear and radial flow cases, respectively, as follows:
where ḱr and n' are the consistency (dyne/cm2/sn') and flow (dimensionless) indices, which are equal to the fluid viscosity, jo, and unit for Newtonian fluids, respectively, and v is the tangential velocity of the slurry over the filter cake surface. For static filtration, v = 0 and therefore T = 0, and the second term on the right side of Eq. 12-134 drops out, leading to an expression similar to Corapcioglu and Abboud (1990) and Tien et al. (1997).
tcr is the minimum slurry shear-stress necessary for detachment of particles from the progressing cake surface. Following Ravi et al. (1992), the critical shear stress necessary for detachment of the deposited particles from the progressing cake surface can be estimated according to Potanin
and Uriev (1991) by Eq. 12-6. However, the actual critical stress can be substantially different than predicted by Eq. 12-6, because the ideal theory
neglects the effects of the other factors, including aging (Ravi et al., 1992), surface roughness, and particle stickiness (Civan, 1996) on the particle detachment.
Therefore, Ravi et al. (1992) recommend that the critical shear stress be determined experimentally. U(ts-tcr) is the Heaviside unit step function. It is equal to zero when ts < tcr and one for Ts≥tcr. k°d and k°e are the rate coefficients for the total (fine plus large) particles deposition and detachment at the progressing cake surface. k°2 and k°2 are the rate coefficients for the fine_ particles deposition and detachment at the rogressing cake surface. ҟd and ҟe are the cakethickness-average rate coefficients for the deposition and mobilization of the fine particles within the filter cake matrix.
References
Abboud, N. M., "Formation of Filter Cakes with Particle Penetration at the Filter Septum," Paniculate Science and Technology, Vol. 11, 1993, pp. 115-131.
Adin, A., "Prediction of Granular Water Filter Performance for Optimum Design," Filtration and Separation, Vol. 15, No. 1, 1978, pp. 55-60.
Arshad, S. A., "A Study of Surfactant Precipitation in Porous Media with Applications in Surfactant-Assisted Enhanced Oil Recovery Processes," Ph.D. Dissertation, University of Oklahoma, 1991, 285 p.
Chase, G. G., & Willis, M. S., "Compressive Cake Filtration," Chem. Engng. ScL, Vol. 47, No. 6, 1992, pp. 1373-1381.
Chen, W., "Solid-Liquid Separation via Filtration," Chemical Engineering, Vol. 104, February 1997, pp. 66-72.
Civan, F., "A Multi-Phase Mud Filtrate Invasion and WellBore Filter Cake Formation Model," SPE 28709 paper, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, Veracruz, Mexico, October 10-13, 1994, pp. 399-412.
Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Control Symposium held in Lafayette, Louisiana, February 14-15, 1996, pp. 311-326.
Civan, F, "Incompressive Cake Filtration: Mechanism, Parameters, and Modeling," AIChE J., Vol. 44, No. 11, November 1998a, pp. 2379-2387.
Civan, F., "Practical Model for Compressive Cake Filtration Including Fine Particle Invasion," AIChE J., Vol. 44, No. 11, November 1998b, pp. 2388-2398.
Civan, F., "Predictive Model for Filter Cake Buildup and Filtrate Invasion with Non-Darcy Effects," SPE 52149 paper, Proceedings of the 1999 SPE Mid-Continent Operations Symposium held in Oklahoma City, Oklahoma, March 28-31, 1999a.
Civan, F., "Phenomenological Filtration Model for Highly Compressible Filter Cakes Involving Non-Darcy Flow," SPE 52147 paper, Proceedings of the 1999 SPE Mid-Continent Operations Symposium held in Oklahoma City, Oklahoma, March 28-31, 1999b.
Clark, P. E., & Barbat, O., "The Analysis of Fluid-Loss Data," SPE 18971 paper, Proc., SPE Joint Rocky Mountain Regional/Low Permeability Reservoirs Symposium and Exhibition, Denver, Colorado, March 6-8, 1989, pp. 437-444.
Collins, E. R., Flow of Fluids Through Porous Materials, Penn Well Publishing Co., Tulsa, Oklahoma, 1961, 270 p.
Corapcioglu, M. Y., & Abboud, N. M., "Cake Filtration with Particle Penetration at the Cake Surface," SPE Reservoir Engineering, Vol. 5, No. 3, August 1990, pp. 317-326.
Dake, L. P., Fundamentals of Reservoir Engineering, Elsevier Scientific Publishing Co., New York, 1978, 443 p.
Darcy, H., "Les Fontaines Publiques de la Ville de Dijon," Dalmount, Paris (1856).
Darley, H. C. H., "Prevention of Productivity Impairment by Mud Solids," Petroleum Engineer, September 1975, pp. 102-110.
de Nevers, N., "Product in the Way Processes," Chemical Engineering Education, Summer 1992, pp. 146-151.
Donaldson, E. C., & Chernoglazov, V, "Drilling Mud Fluid Invasion Model," J. Pet. Sci. Eng., Vol. 1, No. 1, 1987, pp. 3-13.
Fehlberg, E., "Low-Order Classical Runge-Kutta Formulas with Stepsize Control and their Application to Some Heat Transfer Problems," NASA TR R-315, Huntsville, Alabama, July 1969.
Fisk, J. V., Shaffer, S. S., & Helmy, S., "The Use of Filtration Theory in Developing a Mechanism for Filter-Cake Deposition by Drilling Fluids in Laminar Flow," SPE Drilling Engineering, Vol. 6, No. 3, September 1991, pp. 196-202.
Forchheimer, P., "Wasserbewegung durch Boden," Zeitz. ver. Deutsch Ing. Vol. 45, 1901, pp. 1782-1788.
Hermia, J., "Constant Pressure Blocking Filtration Laws—Application to Power-Law Non-Newtonian Fluids," Trans. IChemE, Vol. 60, 1982, pp. 183-187.
Jiao, D., & Sharma, M. M., "Mechanism of Cake Buildup in Crossflow Filtration of Colloidal Suspensions," J. Colloid and Interface Sci., Vol. 162, 1994, pp. 454-462.
Jones, S. C., & Roszelle, W. O., "Graphical Techniques for Determining Relative Permeability from Displacement Experiments," Journal of Petroleum Technology, Trans AIM E, Vol. 265, 1978, pp. 807-817.
Liu, X., & Civan, F, "Formation Damage and Filter Cake Buildup in Laboratory Core Tests: Modeling and Model-Assisted Analysis," SPE Formation Evaluation J., Vol. 11, No. 1, March 1996, pp. 26-30.
Liu, X., Civan, F, and Evans, R. D., "Correlation of the Non-Darcy Flow Coefficient," Journal of Canadian Petroleum Technology, Vol. 34, No. 10, December 1995, pp. 50-54.
Metzner, A. B., & Reed, J. C., "Flow of Non-Newtonian Fluids—Correlation of the Laminar, Transition, and Turbulent Flow Regions," AIChE J., Vol. 1, No. 4, 1955, pp. 434-440.
Peng, S. J., & Peden, J. M., "Prediction of Filtration Under Dynamic Conditions," SPE 23824 paper, presented at the SPE Intl. Symposium on Formation Damage Control held in Lafayette, Louisiana, February 26- 27, 1992, pp. 503-510.
Potanin, A. A., & Uriev, N. B., "Micro-rheological Models of Aggregated Suspensions in Shear Flow," /. Coll. Int. ScL, Vol. 142, No. 2, 1991, pp. 385-395.
Ravi, K. M, Beirute, R. M., & Covington, R. L., "Erodability of Partially Dehydrated Gelled Drilling Fluid and Filter Cake," SPE 24571 paper, Proceedings of the 67th Annual Technical Conference and Exhibition of the SPE held in Washington, DC, October 4-7, 1992, pp. 219-234.
Sherman, N. E., & Sherwood, J. D., "Cross-Flow Filtration: Cakes With Variable Resistance and Capture Efficiency," Chemical Engineering Science, Vol. 48, No. 16, 1993, pp. 2913-2918.
Smiles, D. E., & Kirby, J. M., "Compressive Cake Filtration—A Comment," Chem. Engng. ScL, Vol. 48, No. 19, 1993, pp. 3431-3434.
Tien, C., Bai, R., & Ramarao, B. V., "Analysis of Cake Growth in Cake Filtration: Effect of Fine Particle Retention, AIChE J., Vol. 43, No. 1, January 1997 pp. 33-44.
Tiller, F. M., & Crump, J. R., "Recent Advances in Compressible Cake Filtration Theory," in Mathematical Models and Design Methods in Solid-Liquid Separation, A. Rushton, ed., Martinus Nijhoff, Dordrecht, 1985.
Willis, M. S., Collins, R. M., & Bridges, W. G., "Complete Analysis of Non-Parabolic Filtration Behavior," Chem. Eng. Res. Des., Vol. 61, March 1983, pp. 96-109.
Xie, X., & Charles, D. D., "New Concepts in Dynamic Fluid-Loss Modeling of Fracturing Fluids," J. Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 29-40.
