# Compressive Cake Filtration Including Fines Invasion

The applicability of the majority of the previous models, such as those by Corapcioglu and Abboud (1990), Liu and Civan (1996), Tien et al. (1997) and Civan (1998b), is limited to low rate or low pressure difference filtration processes because these models facilitate Darcy's law to describe flow through porous media. However, filtration at high flow rates and high overbalance pressure differences may involve some inertial flow effects, especially during the initial period of the filter cake formation.

In the literature, the initial non-linear relationships of the filtrate volume versus the square root of time has been attributed to invasion and clogging of porous media by fine particles during filtrate flow into porous media prior to filter cake formation. The cumulative volume of the carrier fluid (filtrate) lost into porous media during this time is usually referred to as the spurt loss (Darley, 1975).

Based on an order of magnitude analysis of the relevant dimensionless groups of the general mass and momentum balances of the multiphase
systems involving the cake buildup, Willis et al. (1983) concluded that non-parabolic filtration behavior is not caused by non-Darcy flow. Instead, it is a result of the reduction of the permeability of porous media by clogging by fine particles.

Their claim is valid under the conditions of their experimental test conditions. The phenomenological models for filter cake buildup involving fine particle invasion have been presented by Liu and Civan (1996) and Civan (1998b) for low rate filtration. However, a close examination of most filtration data reveal some non-Darcian flow effect during the short, initial period of filtration depending on the magnitude of the filtration flow rate and/or the applied pressure difference.

The large flow rates encountered during this period usually promote a non-Darcy effect. Willis et al. (1983) investigated the non-parabolic filtration behavior, but concluded that the non-parabolic behavior is a result of the impairment of the permeability of porous media by invasion and clogging by fine particles rather than by the non-Darcy flow effect. This conclusion is justified for their experimental conditions, however, some reported experimental data appear to involve a non-Darcy flow effect during the initial period of filter cake buildup. Civan (1998b, 1999b) developed linear and radial filtration models and verified them by means of experimental data. These models are more generally applicable because of the following salient features:

1. A cake-thickness-averaged formulation leads to a convenient and computationally efficient representation of the filtration processes by means of a set of ordinary differential equations;

2. The nonhomogeneous-size particles of the slurry are classified into the groups of the large and fine particles, and the large particles form the cake matrix and the fine particles deposit inside the cake matrix;

3. The flow through porous cake and formation, which acts as a filter, is represented by Forchheimer's (1901) law to account for the inertial flow effects encountered during the early filtration period;

4. The dynamic and static filtration conditions encountered with and without the slurry flowing tangentially over the cake surface, respectively, are considered;

5. The variation of the filter cake porosity and permeability by compaction due to the drag of the fluid flowing through the cake matrix and deposition of fine particles within the cake matrix is considered;

6. An average fluid pressure is used to determine the fluid drag force applied to the cake matrix;

7. The formulations are presented for general purposes, but applied for commonly encountered cases involving incompressible particles and carrier fluids; and

8. The constant and variable rate filtration processes can be simulated.

The model presented in this section incorporates empirical constitutive relationships for the permeability and porosity variations of compressible
cakes retaining fine particles. The simulation of a series of filtration scenarios are presented to demonstrate the parametric sensitivity of the model. It is determined that permeability impairment by fine particles retainment and pore throat clogging in the filter cake is increasingly induced by cake compression.

It was also determined that constant pressure filtration limits the filtrate invasion more effectively than constant rate filtration and the non-Darcy flow effect is more significant during the initial period of the filter cake formation. The cake formation models developed in this section can be used for predicting the effects of the compressible filter cakes involving the drilling muds and fracturing fluids. The applications of the improved models are illustrated by typical case studies.

## 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 Well Bore 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.