In this section, models for interpretation and prediction of incompressible filter cake thickness, and filtrate volume and rate data for linear and radial filtration cases, at static and dynamic conditions, are presented. Methods for determining the model parameters from experimental filtration data are presented. Model assisted analyses of three sets of experimental data demonstrate the diagnostic and predictive capabilities of the model. These models provide insight into the mechanism of incompressible cake filtration and offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial filtration processes.

Linear Filter Cake Model

The locations of the mud slurry side cake surface and the slurry and effluent side surfaces of the porous medium are denoted, respectively, by хc, хw, and хe. Consistent with laboratory tests using core plugs, the crosssectional area is denoted by a and the core length by L = хe - хw. The mass balance of particles in the filter cake is given by (Civan, 1996, 1998a)

where Ρp is the particle density, t is time, εs is the volume fraction of particles of the cake that can be expressed as a function of the porosity ɸc of the cake as

and Rps is the net mass rate of deposition of particles of the slurry to form the cake given by (Civan, 1998b, 1999a,b)

The first term on the right of Eq. 12-3 expresses the rate of particle deposition as being proportional to the mass of particles carried toward the filter by the filtration volumetric flux uc, normal to the filter surface, given by

where q is the carrier fluid filtration flow rate and a is the area of the cake surface. cp is the mass of particles contained per unit volume of the carrier fluid in the slurry. κd is the deposition rate coefficient. The second term on the right of Eq. 12-3 expresses the rate of erosion of the cake particles from the cake surface on the slurry side. Erosion takes place only when the shearstress is applied by the slurry to the cake surface exceeds a minimum critical shear stress icr necessary for detachment of particles from the cake surface. The shear-stress is given by (Metzner and Reed, 1955)

where k' and ṅ denote, respectively, the consistency constant and flow index. The critical shear-stress is dependent on various factors, including surface roughness and particle stickiness (Civan, 1998a,b) and aging (Ravi et al., 1992) and it should be measured directly. The deposition and erosion rate constants depend on the properties of the particles and carrier fluid, and the conditions of the slurry, such as particle concentration, flow rate, and pressure. Ravi et al. (1992) have determined that the following equation proposed by Potanin and Uriev (1991) predicts the critical shear stress with the same order of magnitude accuracy of their experimental measurements:

where H = 3.0x10-13erg is the Hamaker coefficient, d(cm) is the average particle diameter, and l(cm) is the separation distance between the particle surfaces in the filter cake. However, the values calculated from Eq. 12-6 is only a first order accurate estimate because Eq. 12-6 has been derived from an ideal theory. The ideal theory does not take into account the effect of the other factors, such as aging (Ravi et al., 1992), surface roughness, and particle stickiness (Civan, 1996), on the particle detachment. Therefore, the actual value of the critical shear stress may be substantially different than that predicted by Eq. 12-6 using the particle size and separation distance data. Hence, Ravi et al. (1992) recommend experimental determination of the critical shear stress.

(Ԑs Ρp) is the mass of particles contained per unit bulk volume of the slurry side cake surface. The erosion rate is related also to the particle content of the cake (Ԑs Ρp) and erosion cannot occur if there is no cake, that is if (Ԑs Ρp)с = 0 . Here, the cake properties are assumed constant.

Then, Eq. 12-3 can be simplified to Civan's (1999a) equation as

in which H(Ԑs) = 0 when Ԑs=0 (no cake) and H(Ԑs) = l when Ԑs>0. The function H(Ԑs) can be expressed in terms of the cake thickness, ẟ, as H(ẟ) = 0 when ẟ = 0 and H(ẟ) = l when ẟ>0, because Ԑs=0 when ẟ = 0. The filter cake thickness ẟ is given by

Note the slurry side filter surface position xw is fixed. Substituting Eqs. 12-2, 4, 9 and 10, Eq. 12-1 can be written as (Civan,1998a)

The rapid filtration flow of the carrier fluid through the cake and filter can be expressed by Forchheimer's (1901) equation

The inertial flow coefficient is given by the Liu et al. (1995)

where β is the inertial flow coefficient in cm-1, K is the permeability in Darcy, and T is the tortuosity (dimensionless). Substituting Eq. 12-4 into Eq. 12-15 yields

As explained by Civan (1998a,b), the instantaneous carrier fluid filtration flow rate q is the same everywhere in the cake and filter irrespective of whether the process is undergoing a constant pressure or a constant rate filtration. In the following, the formulations for variable and constant rate filtration processes are derived. For variable rate filtration occurring under an applied pressure difference, integrating Eq. 12-17 for conditions existing prior to and during the process of formation of a filter cake leads to, respectively:

Consequently, eliminating (Ρc -Ρe) between Eqs. 12-18 and 19, and then solving for q, yields for Darcy flow (βf =βc =0)

Alternatively, eliminating (Ρc-Ρe] between Eqs. 12-18 and 12-19 and then solving for ẟ yields:

Notice that Eq. 12-24 yields ẟ = 0 when q = q0. Differentiating Eq. 12-24 with respect to time and then substituting into Eq. 12-11 yields

Substituting Eq. 12-20 and considering the initial condition given by Eq. 12-14, Eq. 12-11 can be solved using a numerical scheme, such as the Runge-Kutta-Fehlberg four (five) method (Fehlberg, 1969). Eqs. 12-25 and 12-26 can also be solved numerically using the same method. The relationships between filtrate flow rate and cumulative filtrate volume are given by

Note that Eqs. 12-24 and 25 simplify to Eqs. 12-29 and 12-33 (Civan, 1998a), respectively, when the inertial effects are neglected, that is, for βc=βf=0.

Then, the analytical solutions for the filtrate flow rate and cumulative volume as well as the filter cake thickness can be derived as demonstrated by Civan (1998a, 1999a). The analytical solution of Eqs. 12-33 and 34 is (Civan, 1998a)

Eliminating q between Eqs. 12-29 and 12-35 yields another expression as:

in which, usually, ẟ =0at t = 0 (i.e., no initial filter cake). Eq. 12-36 is different from Eq. 7-96 of Collins (1961) because Collins did not consider the filter cake erosion. Therefore, Collins' equation applies for static filtration. To obtain Collins' result, κe = 0 or B = 0 must be substituted in Eq. 12-11. Thus, eliminating q between Eqs. 12-29 and 12-11, and then integrating, yields the following equation for the filter cake thickness (Civan, 1998a):

which results in Eqs. 7-96 of Collins (1961) by invoking Eqs. 12-18 for βf =0, Eqs. 12-30, 12-31, and 12-12 and expressing the mass of suspended particles per unit volume of the carrier fluid in terms of the volume fraction, ɞp, of the particles in the slurry according to:

Civan (1998a) derived the expressions for the filtrate flow rate and the cumulative filtrate volume by integrating Eq. 12-33 for Β = 0 and applying Eq. 12-27, respectively, as:

Eq. 12-39 expresses that the filtrate rate declines by time due to static filter cake build-up. Donaldson and Chernoglazov (1987) used an empirical decay function:

in which β is an empirically determined coefficient. For constant rate filtration, Eq. 12-11 subject to Eq. 12-14 can be integrated numerically for varying shear-stress Ts. When the shear-stress is constant or does not vary significantly, an analytical solution can be obtained as (Civan, 1999a):

in which Β = 0 because T = 0 for static conditions and B ≠ 0 because t ≠ 0 for dynamic conditions. The cumulative filtrate volume is given by, for both the static and dynamic filtration

Then, the pressure difference (Ρc-Ρe) or the slurry injection pressure Ρc, when the back pressure at the effluent side of the porous filter media Ρe is prescribed, can be calculated by Eq. 12-19. The following conventional filtration equation (Hermia, 1982; de Nevers, 1992) can be derived by invoking Eq. 12-43 into Eq. 12-40:


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.