Two-Phase Formation Damage by Fines Migration 257 10 20 30 40 Filtration Time (min) 50 60 • Experiment —r- Simulation Figure 11-7. Cumulative fluid loss vs. filtration time during mud filtration (Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers). 0 5 10 15 Core Length (cm) • Experiment Simulation Figure 11-8. Instantaneous to initial permeability ratio (or permeability alteration factor) vs. core length after one hour of filtration time during mud filtration (Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers). 258 Reservoir Formation Damage 50 100 Experiment 200 300 400 Filtration Time (min.) Simulation 500 600 — 2-Phase Row Figure 11-9. Cumulative fluid loss vs. filtration time during dynamic mud filtration (Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers). 100 200 300 400 Filtration Time (min.) 500 600 Experiment Simulation 2-Phase Flow Figure 11-10. Instantaneous to initial permeability ratio (or permeability alteration factor) vs. filtration time during dynamic mud filtration (Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers). Two-Phase Formation Damage by Fines Migration 259 Prediction of formation damage due to dynamic mud filtration in two- phase flow was also carried out to demonstrate the capacity and appli- cation of the model and to provide a comparison with single-phase flow results. If the core studied by Jiao and Sharma (1992) was saturated by oil prior to mud filtration, the invasion of the water-based mud would lead to two-phase flow of oil and water in the rock. The same model parameters determined above for mud filtration in single-phase flow were used. Additional data necessary for simulation, including connate water saturation, residual oil saturation and relative permeabilities, are adapted from the case of fines migration in two-phase flow that was also simulated above. The predicted mud filtration volume and permeability alteration in two-phase flow are also plotted in Figures 11-9 and 11-10, as denoted by the dashed lines. These results indicate that filtration volume and formation damage are significantly less when a water-based mud invades an oil-bearing formation. This is because the total mobility for simul- taneous two-phase flow of water and oil is usually less than that of single phase of water in formations, especially in Berea sandstones, which are generally strongly water-wet and have a very low permeability for water phase with the presence of oil in the formations. References Civan, F., "A Generalized Model for Formation Damage by Rock-Fluid Interactions and Particulate Processes," SPE Paper 21183, Proceedings of the SPE 1990 Latin American Petroleum Engineering Conference, October 14-19, 1990, Rio de Janeiro, Brazil, 11 p. Civan, F., Predictability of Formation Damage: An Assessment Study and Generalized Models, Final Report, U.S. DOE Contract No. DE-AC22- 90-BC14658, April 1994. Civan, F., "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE Paper No. 28709, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412. Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Symposium, Lafayette, Louisiana, February 14-15, 1996, pp. 311-326. Civan, F., "Convenient Formulations for Immiscible Displacement in Porous Media," SPE Paper 36701, Proceedings of the 71st SPE Annual Tech. Conf. and Exhibition, Denver, Colorado, October 6-9, 1996, pp. 223-236. Collins, E. R., Flow of Fluids Through Porous Materials, Penn Well Publishing Co., Tulsa, Oklahoma, 1961, 270 p. 260 Reservoir Formation Damage Craig, F. F., Jr., The Reservoir Engineering Aspects of Waterflooding, Third Printing, November 1980, Society of Petroleum Engineers of AIME, New York, 1971, 134 p. Dake, L. P., Fundamentals of Reservoir Engineering, Elsevier Scientific Publ. Co., New York, 1978, 443 p. Eleri, O. O., & Ursin, J-R., "Physical Aspects of Formation Damage in Linear Flooding Experiments," SPE 23784 paper, presented at the SPE Intl. Symposium on Formation Damage Control, Lafayette, Louisiana, February 26-27, 1992. Gruesbeck, C., & Collins, R. E., "Particle Transport Through Perforations," SPEJ, December 1982, pp. 857-865. Gruesbeck, C., & Collins, R. E., "Entrainment and Deposition of Fine Particles in Porous Media," SPEJ, December 1982, pp. 847-856. Jiao, D., & Sharma, M. M., "Formation Damage Due to Static and Dynamic Filtration of Water-Based Muds," SPE 23823 paper, presented at the SPE Intl. Symposium on Formation Damage Control, Lafayette, Louisiana, February 26-27, 1992. Khilar, K. C., & Fogler, H. S., "Water Sensitivity of Sandstones," SPEJ, February 1983, pp. 55-64. Ku, C-A., & Henry, Jr., J. D., "Mechanisms of Particle Transfer from a Continuous Oil to a Dispersed Water Phase, J. Colloid and Interface ScL, 1987, Vol. 116, No. 2, pp. 414-422. Liu, X., & Civan, F., "Characterization and Prediction of Formation Damage in Two-Phase Flow Systems, SPE 25429 paper, Proceedings of the SPE Production Operations Symposium, March 21-23, 1993, Oklahoma City, Oklahoma, March 21-23, 1993, pp. 231-248. Liu, X., & Civan, F, "Formation Damage and Skin Factors Due to Filter Cake Formation and Fines Migration in the Near-Wellbore Region," SPE 27364 paper, Proceedings of the 1994 SPE Formation Damage Control Symposium, February 9-10, 1994, Lafayette, Louisiana, pp. 259-274. Liu, X., & Civan, E, "Formation Damage by Fines Migration Including Effects of Filter Cake, Pore Compressibility and Non-Darcy Flow— A Modeling Approach to Scaling from Core to Field," SPE Paper #28980, SPE International Symposium on Oilfield Chemistry, February 14-17, 1995, San Antonio, TX. 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, & Evans, R. D., "Correlation of the Non-Darcy Flow Coefficient, J. of Canadian Petroleum Technology, Vol. 34, No. 10, 1995, pp. 50-54. Two-Phase Formation Damage by Fines Migration 261 Luan, Z., "Splitting Pseudospectral Algorithm for Parallel Simulation of Naturally Fractured Reservoirs," SPE Paper 30723, Proceedings of the Annual Tech. Conf. & Exhibition held in Dallas, TX, October 22-25. Muecke, T. W., "Formation Fines and Factors Controlling their Movement in Porous Media," JPT, pp. 147-150, Feb. 1979. Peng, S. J., & Peden, J. M., "Prediction of Filtration Under Dynamic Conditions," paper SPE 23824 presented at the SPE Intl. Symposium on Formation Damage Control held in Lafayette, LA, February 26- 27, 1992, pp. 503-510. Rahman, S. S., & Marx, C., "Laboratory Evaluation of Formation Damage Caused by Drilling Fluids and Cement Slurry," J. Can. Pet. Tech., November-December, 1991, pp. 40-46. Richardson, J. G., "Flow Through Porous Media," In: V. L. Streeter (Editor), Handbook of Fluid Dynamics, Section 16, McGraw-Hill, New York, 1961, pp. 68-69. Sarkar, A. K., "An Experimental Investigation of Fines Migration in Two- Phase Flow," MS Thesis, U. of Texas, Austin, 1988. Sarkar, A. K., & Sharma, M. M., "Fines Migration in Two-Phase Flow," JPT, May 1990, pp. 646-652. Sutton, G. D., & Roberts, L. D., "Paraffin Precipitation During Fracture Stimulation," JPT, September 1974, pp. 997-1004. Ucan, S., & Civan, R, "Simultaneous Estimation of Relative Permeability and Capillary Pressure for Non-Darcy Flow-Steady-State," SPE Paper 35271, Proceedings of the 1996 SPE Mid-Continent Gas Symposium, Amarillo, TX, April 29-30, 1996, pp. 155-163. Yokoyama, Y, & Lake, L. W., "The Effects of Capillary Pressure on Immiscible Displacements in Stratified Porous Media," SPE 10109 paper, presented at the 56th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, San Antonio, TX, October 5-7, 1981. Chapter 12 Cake Filtration: Mechanism, Parameters and Modeling* Summary Models for interpretation and prediction of incompressible and com- pressible filter cake thickness, and filtrate volume and rate data for linear and radial filtration cases, and at static and dynamic filtration conditions are presented. Effects of compressibility and small particle invasion and deposition inside the cake and formation, as well as the Darcy versus non- Darcy flow regimes, are considered. Methods and diagnostic charts for determining the model parameters from experimental filtration data are presented. Applications for radial and linear filtration cases are presented and the results are compared for constant rate and constant pressure drive filtration. Model assisted analyses of experimental data demonstrate the diagnostic and predictive capabilities of the models. The parametric studies indicate that the particle screening efficiency of the formation is an important factor on the filter cake properties and filtration rate, the differences between the linear and radial cake filtration performances are more pronounced, and the cake thickness and filtrate volume are smaller, for constant pressure filtration than constant rate filtration. The present thickness-averaged ordinary differential models are shown to reproduce the predictions of the previous partial differential model rapidly with significantly less computational effort. Because of the simplicity of the equations and reduction of computational effort, the thickness-averaged * Parts of this chapter have been reprinted with permission of the American Institute of Chemical Engineers and the Society of Petroleum Engineers from Civan (1998a,b, and 1999a, b). 262 Cake Filtration: Mechanism, Parameters and Modeling 263 linear and radial filter cake formation models offer significant advantages over the partial differential models for the analysis, design, and opti- mization of the cake filtration processes involving the well-bore and hydraulically created fracture surfaces. Simplified models considering incompressible particles and carrier fluids and analytical solutions for incompressible cakes without fines invasion are also presented. These models provide insight into the mechanism of cake filtration and offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial filtration processes. Introduction Cake filtration occurs inherently in many in-situ hydrocarbon reservoir exploitation processes. For example, hydraulic fracturing of petroleum bearing rock and overbalanced drilling of wells into petroleum reservoirs usually cause a cross-flow filtration, which leads to a filter cake build- up over the face of the porous rock and filtrate invasion into the reservoir (Civan, 1994, 1996). When the slurry contains particles of different sizes, the larger particles of the slurry form the skeleton of the filter cake and the smaller particles can migrate into and deposit within the porous cake formed by the large particles. Simultaneously, the cake may undergo a compaction process under the effect of the fluid drag as the suspension of smaller particles flow through the cake (Tien et al., 1997). Con- sequently, the porosity, permeability, and thickness of the cake vary, which in turn effect the performance of the filtration process. Static filtration occurs when a slurry is applied to a filter without cross-flow. Therefore, the particles are continuously deposited to form thicker filter cakes. Dynamic filtration involves some cross-flow. Therefore, the filter cake thickness varies until the particle deposition and erosion rates equal. Model assisted analyses, interpretation of experimental data and optimi- zation and simulation of the filtration processes are of continuing interest for the industry. The majority of the previous modeling efforts has been limited to linear filtration applications, in spite of the fact that many industrial filtration processes facilitate radial filtration applications. Linear filtration models can closely approximate radial filtration only when the thicknesses of the filter and filter cake are sufficiently small relative to the radius of the filter surface exposed to slurry. Otherwise, radial models should be used for radial filtration. Because of their simplicity, empirical correlations such as those reviewed by Clark and Barbat (1989) are frequently used for static and dynamic filtration. Xie and Charles (1997) have demonstrated that the use of a set of properly selected dimensionless groups leads to improved empirical correlations. Simple models, are preferred in many applications because 264 Reservoir Formation Damage of their convenience and the reduced computational effort. The applic- ability of the majority of the previously reported simple analytical models, such as by Collins (1961), Hermia (1982), and de Nevers (1992), are usually limited to linear and constant rate filtration. However, models for constant pressure filtration are also required for certain applications. Civan (1998a) developed and verified improved linear and radial filtration models applicable for incompressible cake filtration without fines invasion at static and dynamic conditions. Simplified models omit the internal details of the filtration processes and, therefore, may lead to incorrect results if applied for conditions beyond the range of the experimental data used to obtain the empirical correlations. In many applications, the phenomenological models describ- ing the mechanisms of the cake formation, based on the conservation laws and rate equations, are preferred for filter cake build-up involving small particle migration and deposition and cake compaction, because these models allow for extrapolation beyond the range of data used to test and calibrate the models. Chase and Willis (1992), Sherman and Sherwood (1993), and Smiles and Kirby (1993) presented partial differential models for compressible filter cakes without particle intrusion. Liu and Civan (1996) developed a partial differential model for incompressible filter cake build-up, and filtrate and fine particle invasion into petroleum bearing rock at dynamic condition. Tien et al. (1997) have developed a partial differential model for compressible filter cakes considering small particle retention inside the cake at static condition. The solutions of such partial differential models require complicated, time consuming, and com- putationally intensive numerical schemes. To alleviate this difficulty, Corapcioglu and Abboud (1990), Abboud (1993), and Civan (1994) have resorted to formulations facilitating cake thickness averaging. Con- sequently, the partial differential filtration models have been reduced to ordinary differential equations requiring much less computational effort. Such mathematically simplified models are particularly advantageous because ordinary differential equations can be solved rapidly, accurately, and conveniently by readily available and well established numerical methods. The thickness-averaged models developed by Corapcioglu and Abboud (1990) and Abboud (1993) consider a constant porosity and linear cake filtration at static condition. The constant porosity assumption was justified by their filtration experiments because they used very dilute suspensions of particles and low pressure filtration, near the atmospheric pressure. Their models would not be applicable for high pressure filtration of thick slurries considered by Tien et al. (1997). Further, they assumed the same values for the rates of deposition of the small and large particles over the progressing filter cake surface. This assumption is invalid for most applications. Cake Filtration: Mechanism, Parameters and Modeling 265 Civan (1998b) developed improved ordinary differential, linear and radial filtration models incorporating the effects of filter cake compaction, small particle invasion and retention at static and dynamic conditions. He applied filter cake thickness averaging by extending the methodology by Corapcioglu and Abboud (1990) and Civan (1994, 1996). The new models alleviate the aforementioned problems associated with the previous models. Civan (1998b) also derived the simplified forms of his models, considering that the particles and carrier fluid can be assumed incom- pressible for many practical applications. He presented the applications to radial and linear filtration processes and compared the results. The thickness-averaged ordinary differential filter cake model reproduced the predictions of the Tien et al. (1997) partial differential model rapidly with less computational effort. In most filtration models, the flow through porous media is represented by Darcy's law. Consequently, the applicability of these models is limited to filtration undergoing at low flow rate or low pressure difference conditions. Civan (1999a,b) also developed linear and radial filtration models incorporating a non-Darcy flow behavior and applicable at static and dynamic filtration conditions by extending Civan's (1998a,b) model considering Darcy behavior. The non-Darcy behavior is represented by Forchheimer's (1901) law. In this chapter, the filtration models are presented by including the non-Darcy effects. However, the models also apply for Darcy flow because the non-Darcy effects diminish at low flow rates. Civan (1998a) also developed and verified several methods for determining the parameters of these incompressible cake filtration models from experimental data by constructing diagnostic charts of linear types. However, some parameters should be either directly measured or deter- mined by a least-squares regression of experimental data with the filtration models as demonstrated by Civan (1998a,b). In this chapter, Civan's (1998a,b, 1999a,b) filtration models are presented. Incompressive Cake Filtration In this section, models for interpretation and prediction of incom- pressible 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 filtra- tion data are presented. Model assisted analyses of three sets of experi- mental data demonstrate the diagnostic and predictive capabilities of the model. These models provide insight into the mechanism of incompres- sible cake filtration and offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial filtration processes. 266 Reservoir Formation Damage Linear Filter Cake Model A schematic of the formation of a filter cake over a hydraulically created fracture is shown in Figure 12-1. Figure 12-2 shows the simplified, one-dimensional linear cake filtration problem considered in this section. The locations of the mud slurry side cake surface and the slurry and effluent side surfaces of the porous medium are denoted, respectively, by x c , x w , and x e . Consistent with laboratory tests using core plugs, the cross- sectional area is denoted by a and the core length by L = x e - x w . The mass balance of particles in the filter cake is given by (Civan, 1996, 1998a) (12-1) where p p is the particle density, t is time, e 5 is the volume fraction of particles of the cake that can be expressed as a function of the porosity <|> c of the cake as e,= l-<t> c (12-2) and R ps is the net mass rate of deposition of particles of the slurry to form the cake given by (Civan, 1998b, 1999a,b) RPS = k d u c c p -k e (e s p,) (T, -t cr )u(i s -T cr ) (12-3) 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 u c , normal to the filter surface, given by u c = q/a (12-4) where q is the carrier fluid filtration flow rate and a is the area of the cake surface. c p is the mass of particles contained per unit volume of the carrier fluid in the slurry. k d 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 shear- stress i s applied by the slurry to the cake surface exceeds a minimum critical shear stress i cr necessary for detachment of particles from the cake surface. The shear-stress is given by (Metzner and Reed, 1955) [...]... and Modeling 27 1 ( 12- 22) KcLf)aKf ( 12- 23) aK Alternatively, eliminating (pc-pe] then solving for 8 yields: between Eqs 12- 18 and 12- 19 and i- aKt 5= ( 12- 24) aK, Notice that Eq 12- 24 yields 8 = 0 when q = q0 Differentiating Eq 122 4 with respect to time and then substituting into Eq 12- 11 yields \LL c ~r ( 12- 25) a dt aK a2 ^ ' The initial condition for Eq 12- 25 is ( 12- 26) Substituting Eq 12- 20 and considering... 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 27 2 Reservoir Formation Damage The relationships between filtrate flow rate and cumulative filtrate volume are given by = \qdt ( 12- 27) = dQ/dt ( 12- 28) Note that Eqs 12- 24 and 25 simplify to Eqs 12- 29... 1999a) 2nhKf {rw r,., r ( 12- 55) and q\L 2nhK K 1 ( 12- 56) 1 (2nh) Thus, eliminating (/?c - pe) between Eqs 12- 55 and 56, substituting Eq 12- 46, and then solving for q, yields for Darcy flow ( f =P C = O : ( 12- 57a) and for non-Darcy flow: ( 12- 57b) in which 1 rw-8 1 ( 12- 58) r w 27 8 Reservoir Formation Damage ( 12- 59) 2nhK ( 12- 60) Substituting Eq 12- 57 and considering the initial condition given by Eq 12- 51,... given by InrJi ( 12- 48) 27 6 Reservoir Formation Damage Reservoir formation Filter cake Mud :•* slurry Figure 12- 4 Radial filter cake over a wellbore sandface (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers) Substituting Eqs 12- 2, 12- 9, 12- 46, and 12- 48 into Eq 12- 45 results in (1999a) dt rw-8 ' ( 12- 49) where ^yp^ ( 12 ~50) and B is given by Eq 12- 13 The initial... certain limit value, 8^, when the particle deposition and erosion rates equate Simultaneously, the filtration rate also reaches a limiting value, determined by Eq 12- 11 as: ( 12- 72) At this condition, Eq 12- 29 yields the limiting value of the filter cake thickness as: ( 12- 73) Consequently, substituting Eqs 12- 30, 12- 31, 12- 12 and 12- 13 for A, B, C, and D into Eqs 12- 72 and 12- 73 leads to the following relationships... 1998a): ( 12- 61) Equation 12- 61 can be written as: ( 12- 62) where = q0D/Kf ( 12- 63) where q0 is the injection rate given by Eq 12- 55 for $f = 0 before the filter cake buildup and ( 12- 64) Thus, substituting Eqs 12- 46 and 12- 62 into Eq 12- 49 and rearranging yield the filtration flow rate equation as (Civan, 1998a): = (-\/C)q2[Aqexp(C/q-D)-B]exp(C/q-D) ( 12- 65) subject to the initial condition given by: ( 12- 66)... 12- 49 is S = 0,f = 0 ( 12- 51) Forchheimer's (1901) equation for radial flow of the carrier fluid reads as K The radial volumetric flux of the carrier fluid is given by ( 12- 52) Cake Filtration: Mechanism, Parameters and Modeling «=-«2nrh 27 7 ( 12- 53) Thus, substituting Eq 12- 53 into Eq 12- 52 results in = M- 9 , ( 12- 54) Integration of Eq 12- 54 for conditions prevailing prior to and during filter cake formation. .. result, ke = 0 or 5 = 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): (l /2) 8 ( 12- 37) which results in Eqs 7-96 of Collins (1961) by invoking Eqs 12- 18 for $f =0, Eqs 12- 30, 12- 31, and 12- 12 and expressing the mass of suspended particles per unit volume of the carrier fluid... difference, integrating Eq 12- 17 for conditions existing prior to and during the process of formation of a filter cake leads to, respectively: loV-Lf • Pe = e aKf , + ( 12- 18) and ( 12- 19) | KcLfaKf p/ a Consequently, eliminating (pc -pe) between Eqs 12- 18 and 19, and then solving for q, yields for Darcy flow m^ =p c =0) ( 12- 20a) and for non-Darcy flow: 2a ( 12- 20b) in which ( 12- 21) Cake Filtration: Mechanism,... given by 5 = x,., - ( 12- 10) 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) ( 12- 11) where ( 12- 12) ( 12- 13) The initial condition for Eq 12- 11 is 8 = 0, t = 0 ( 12- 14) The rapid filtration flow of the carrier fluid through the cake and filter can be expressed by Forchheimer's (1901) equation a, jr-"-' ( 12 15) - The inertial . Modeling 27 1 K c L f )aK f ( 12- 22) aK ( 12- 23) Alternatively, eliminating (p c -p e ] between Eqs. 12- 18 and 12- 19 and then solving for 8 yields: aK t 5 = i- ( 12- 24) aK, Notice that Eq. 12- 24 . cumulative filtrate volume are given by = qdt ( 12- 27) = dQ/dt ( 12- 28) Note that Eqs. 12- 24 and 25 simplify to Eqs. 12- 29 and 12- 33 (Civan, 1998a), respectively, when the inertial . Eqs. 12- 2, 12- 9, 12- 46, and 12- 48 into Eq. 12- 45 results in (1999a) dt r w -8 ' where ( 12- 49) ^yp^ ( 12 ~ 50) and B is given by Eq. 12- 13. The initial condition for Eq. 12- 49