Radial Filter Cake Model
A schematic of the formation of a filter cake over the sand face during over-balanced mud circulation in a wellbore is shown in this article. The radii of the mud slurry side cake surface, the sand face over which the cake is built up, and the external surface considered for the region of influence are denoted by rc, rw and re, respectively. The formation thickness is h. The particle mass balance equation is given by (Civan, 1994, 1998a)
The slurry shear-stress at the cake surface is given by the Rabinowitsch-Mooney equation (Metzner and Reed, 1955)
The radial volumetric flux of the carrier fluid is given by
Integration of Eq. 12-54 for conditions prevailing prior to and during filter cake formation leads to the following expressions, respectively (Civan, 1999a)
Thus, eliminating (Ρc - Ρe) between Eqs. 12-55 and 56, substituting Eq.12-46, and then solving for q, yields for Darcy flow ( βf =βC = O :
Substituting Eq. 12-57 and considering the initial condition given by Eq. 12-51, Eq. 12-49 can be solved using a numerical scheme, such as the Runge-Kutta-Fehlberg four (five) method (Fehlberg, 1969). The cumulative filtrate volume is given by Eq. 12-27. The pressure difference (Pc-Pe}, or the slurry injection pressure pc when the back pressure pe is prescribed, can be calculated by Eq. 12-56. When the inertial flow terms are negligible, equating Eqs. 12-55 and 12-56 and rearranging leads to (Civan, 1998a):
where q0 is the injection rate given by Eq. 12-55 for βf = 0 before the filter cake buildup and
Thus, substituting Eqs. 12-46 and 12-62 into Eq. 12-49 and rearranging yield the filtration flow rate equation as (Civan, 1998a):
The wall shear-stress is calculated by Eq. 12-47 for the varying cake radius, rc=rc(t}. The filter cake thickness is calculated by means of Eqs. 12-46 and 62. Equations 12-65 and 66 can be solved numerically using an appropriate method such as the Runge-Kutta method. However, for thin cakes, it is reasonable to assume that the wall-shear stress is approximately constant, because rc=rw. Then, Eq. 12-65 can be integrated as (Civan, 1998a):
For constant rate filtration, Eq. 12-49 subject to Eq. 12-51 can be integrated numerically for varying shear-stress is. When the filter cake is thin, the variation of the shear-stress is by the cake radius rc can be neglected and an analytical solution can be obtained as for dynamic filtration conditions (B ≠ 0) (Civan, 1999a):
The solution for static filtration conditions (B = 0) is obtained as (Civan, 1999a):
Eq. 12-68 and 12-69 apply irrespective of whether the flow is Darcy or non-Darcy.
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.