Reservoir Formation Damage Episode 2 Part 3 doc

282 Reservoir Formation Damage Equation 12-33 can be rearranged in a linear form as: d(\\ 1 dq A B — = i = n H dt(q dt q + (12-76) Thus, the intercept (B/C) and slope (-A/C) of the straight-line plot of Eq. 12-76 can be used with Eqs. 12-30, 12-31, 12-12, and 12-13 to obtain the following expressions: (B/C), k d (A/C)(t-T c > (12-77) , _ d~ (12-78) Comparing Eqs. 12-75 and 12-77 yields an alternative expression for determination of the limit filtrate rate as: q x =(B/C)/(A/C) (12-79) Eq. 12-79 can be used to check the value of q x obtained by Eq. 12-72. Equation 12-74 can be used to determine the filter cake permeability, K c . Equations 12-70 and 12-75 or 12-77 and 12-78 can be used to calculate the particle deposition and erosion rates k d and k e , if the cake porosity <J) C and the critical shear stress i cr are known. ty c can be measured. i cr can be estimated by Eq. 12-6, but the ideal theory may not yield a correct value as explained previously by Ravi et al. (1992) and in this chapter. Therefore, Ravi et al. (1992) suggested that i cr should be measured directly. Radial Filtration Given the filter cake thickness 8, the progressing surface cake radius r c can be calculated by Eq. 12-46. Then a straight line plot of ln(r c /r w ) vs. (l/q) data according to Eq. 12-62 yields the values of C and D as the slope and intercept of this line, respectively. A straightline plot of [d8/df] versus \q/(r w -8)] data according to Eq. 12-49 yields the values of A and B as the slope and intercept of this line, respectively. At static filtration conditions, v = 0 and T = 0 according to Eq. 12-47. Therefore, Cake Filtration: Mechanism, Parameters and Modeling 283 5 = 0 according to Eq. 12-13. Consequently, substituting B = 0 and Eq. 12-63, Eq. 12-65 can be expressed in the following linear form: (12-80) = [\n(A/C)-2CK f /q 0 ] Thus, a straightline plot of ln[-g 3 dq/dt\ versus (l/q] yields the values of (2C) and [ln(A/C)-2C£y/<7 0 ] as the slope and intercept of this line, respectively. This allows for determination of the A and C coefficients only. The determination of a full set of A, B, C, and D from Eqs. 12-49 and 12-65 requires both the filtrate flow rate (or volume) and the cake thickness versus the filtration time data. Once these coefficients are determined, then their values can be used in Eqs. 12-50, 12-13, 12-63, and 12-64 to determine the values of the deposition and erosion rate constants k d and k e . The discussion of the linear filtration about the determination of i cr by Eq. 12-6 is valid also in the radial filtration case. At dynamic equilibrium, the filter cake thickness and the filtrate flow rate attain certain limiting values 8^ and q x . Then, substituting Eq. 12-46 into Eqs. 12-49 and 62 yields the following relationships, respectively: Aq_=B(l-SJr w ) (12-81) + D) (12-82) The filter cake permeability is determined by Eq. 12-64 as: K c = DK f /\n(r e /r w ) (12-83) The equations and the linear plotting schemes developed in this section allow for determination of the parameters of the filtration models, mentioned at the beginning of this section, from experimental filtrate flow rate (or volume) and/or filter cake thickness data. The remaining parameters should be either directly measured or estimated. In the following appli- cations, the best estimates of the missing data have been determined by adjusting their values to fit the experimental data. This is an exercise similar to several other studies, including Liu and Civan (1996) and Tien et al. (1997). They have resorted to a model assisted estimation of the parameters because there is no direct method of measurement for some of these parameters. 284 Reservoir Formation Damage Applications The numerical solutions of the present models require the information on the characteristics of the slurries, particulates, carrier fluids, filters and filter cakes, the actual conditions of the tests conducted, and the measure- ments of all the system parameters and variables. The reported studies of the slurry filtration have measured only a few parameters and the filtrate volumes or rates and do not offer a complete set of suitable data that is needed for full scale experimental verification of the present models. Civan (1998a) used the Willis et al. (1983) and Jiao and Sharma (1994) data for linear filtration, and the Fisk et al. (1991) data for radial filtration, because these data provide more information than the other reported studies. The data is presented in Table 12-1 in consistent Darcy units, which are more convenient for flow through porous media. Linear Filtration Applications Jiao and Sharma (1994) carried out linear filtration experiments using concentrated bentonite suspensions. They only measured the filtrate volume and predicted the filter cake thickness using a simple algebraic model. These data are given in their Figures 3 and 10, respectively. In Figures 12-5 to 12-7, their data are plotted according to the linear plotting schemes presented in the previous section for determination of parameters. As can be seen from these figures, the coefficients of Eqs. 12-76, 12-29, and 12-11 obtained by the least-squares regression method and the corresponding coefficients of regression are given, respectively, by: A/C = 8.297min/cm 6 , B/C = 0.1136 cm- 3 , R 2 = 0.8713 (12-84) C = 0.0034cm 4 /min, D = 0.0076cm, R 2 = 0.949 (12-85) A = 0.0229 cnT 2 , B = 0.0003 cm/min, R 2 = 0.9873 (12-86) The coefficients of regressions very close to 1.0 indicate that the pre- sent equations closely represent the data. The coefficient of regression R 2 =0.8713 indicated by Figure 12-5 and Eq. 12-84 is lower than those indicated by Figures 12-6 and 12-7 and Eqs. 12-85 and 12-86, inferring the possibility of larger measurement errors involved in the filtrate volume data. Another source of errors may be due to the three-point finite difference numerical differentiation of the filtrate volume data to obtain the filtrate flow rate data used to construct Figure 12-5. The data necessary for Figure 12-5 were obtained by a series of numerical procedures, first Cake Filtration: Mechanism, Parameters and Modeling 285 t 1 0.16 0.12 = 8.297x-0.1136 R 2 = 0.8713 0.08 0.04 0.015 0.02 0.025 0.03 Filtrate Flow Rate, q, ml/min 0.035 0.04 Figure 12-5. Correlation of Jiao and Sharma (1994) experimental data (Civan, F., 1998; reprinted by permission of the AlChE, ©1998a AlChE. All rights reserved). 5 CD O 0.15 y = 0.0034X - 0.0076 R 2 = 0.949 0.05 20 30 40 50 Reciprocal Filtrate Flow Rate, 1/q, min/ml 60 Figure 12-6. Correlation of Jiao and Sharma (1994) predicted filter cake thickness data (Civan, F., 1998; reprinted by permission of the AlChE, ©1998a AlChE. All rights reserved). 286 Reservoir Formation Damage 0.002 jg 0.0015 c co o 0.001 O *7 C T3 O 0.0005 y = 0.0229X - 0.0003 R 2 = 0.9873 * Predicted — Linear (Predicted) 0.02 0.04 0.06 Filtrate Flow Rate, q, ml/min 0.08 0.1 Figure 12-7. Correlation of Jiao and Sharma (1994) predicted filter cake thickness data (Civan, R, 1998a; reprinted by permission of the AlChE, ©1998 AlChE. All rights reserved). to calculate q = dQ/dt from the filtrate volume Q data, and then (\/q) and [d/dt(l/q)]. The initial filtrate volume rate is obtained as q 0 = 0.096 mL/min by a three-point forward differentiation of the measured, initial filtrate volume data. This data is expected to involve a larger error because of the possibility of relatively larger errors involved in the early filtrate volume data. The noisy data had to be smoothed prior to numerical differentiation, which may have introduced further errors. Because of the propagation of the significantly larger measurement errors involved in the early filtrate volume data, the first two of the [d/dt (l/q)] values degenerated and deviated significantly from the expected straightline trend. Therefore, these two data points formed the outliers for linear regression and had to be discarded. Substituting the values given in Eq. 12-84 into Eq. 12-79 yields the limiting filtrate flow rate as q x =0.014mL/min. On the other hand, substituting the values given in Eq. 12-86 into Eq. 12-72 yields q x = 0.013mL/min. These two values obtained from the filtrate flow rate and cake thickness data, respectively, are very close to each other. The limiting filtrate volume rate q x estimated by an extrapolation of the Cake Filtration: Mechanism, Parameters and Modeling 287 derivatives of the filtrate volume data beyond the range of the experi- mental data is q M =0.017mL/min and close to the values obtained by the regression method. This is an indication of the validity of the filtration model. Using q x =0.014mL/min in Eq. 12-29 yields the limiting filter cake thickness as 5^ =0.24cm. The predicted cake thickness data presented in Figure 10 of Jiao and Sharma (1994) indicates a value of approxi- mately 0.17cm. Therefore, their prediction of the limiting filter cake thickness appears to be an underestimate compared to the 0.24cm value obtained by Civan (1998a). The above obtained values can now be used to determine the values of the model parameters as following. The filter cake permeability can be calculated by Eq. 12-74. Equations 12-70, 12-71, 12-75, 12-77, and 12-78 form a set of alternative equations to determine the deposition and erosion rate constants, k d and k e . Here, Eqs. 12-70 and 12-75 were selected for this purpose. However, Jiao and Sharma (1994) do not offer any data on the cake porosity ty c and the critical shear stress i cr necessary for detachment of the particles from the progressing cake surface. Therefore, the § c and T cr parameters had to be estimated and used with Eqs. 12-70 and 12-75 to match the filtration data over the period of the filtration process. Then, the <j) c and i cr values obtained this way were used in Eqs. 12-70 and 12-75 to calculate the k d and k e values. Using the slurry tangential velocity of v = 8.61cm/s, the typical parti- cle diameter of d = 2.5 x 10" 4 cm, and the particle separation distance of / = 2. x 10~ 7 cm in Eq. 12-5, the critical shear stress for particle detachment is estimated to be t cr =1.25 xlO 3 dyne/cm 2 . Whereas, the prevailing shear stress calculated by Eq. 12-5 is only T = 16 dyne/cm 2 . Under these conditions, theoretically the cake erosion should not occur because T«:i cr . Therefore, the value of the coefficient B should be zero. In contrast, as indicated by Eq. 12-86, the present analysis of the data has led to a small but nonzero value of B = 3.x 10" 4 cm/min. Recall that we used this value in Eq. 12-72 to calculate the limiting flow rate of q x =0.013mL/min. This value was shown to be very close to the q x =0.014mL/min value calculated by Eq. 12-79 and the approximate value of q x =0.017mL/min obtained by extrapolating the filtrate flow rate data beyond the range of the experimental data. Thus, it is reasonable to assume that B = 3. x 10" 4 cm/min is a meaningful value and not just a numerical result of the least-squares regression of Eq. 12-11 to data, because the coefficient of regression R 2 = 0.9873 is very close to one. Hence, it can be inferred that 1 > i cr and the cake erosion occurred in the actual experimental conditions of Jiao and Sharma (1994). In view of this discussion, it becomes apparent that the theoretical value obtained by Eq. 12-6 is not realistic. 288 Reservoir Formation Damage The Jiao and Sharma (1994) data and the missing parameter values, which have been approximated by fitting the experimental data, are given in Table 12-1. The results presented in Figure 12-8 indicate that the model represents the measured filtrate volumes over the complete range of 600 min of filtration time as closely as the quality of their experimental data permits. However, they did not measure the cake thickness, but predicted it using a simple algebraic model. As shown in Figure 12-9, the cake thicknesses predicted by Jiao and Sharma (1994) and Civan (1998a) are close to each other. Willis et al. (1983) conducted linear filtration experiments using a suspension of lucite in water. As shown in Table 12-1, they reported only a few parameter values. They only provide some measured filtrate flow rate and cake thickness data in their Table 2. However, the filtration time data is missing. Therefore, a full scale simulation of their filtration process as a function of time could not be carried out by Civan (1998a). Only the linear plotting of the measured data according to Eq. 12-29 could be accomplished. As indicated by Figure 12-10, the best linear fit of Eq. 12-29 with the least-squares method has been obtained with a coefficient of regression of R 2 = 0.9921, very close to 1.0. This reconfirms the validity of the filtration model. Predicted in the present study »- Measured by Jiao and Sharma(1994) 100 200 300 400 Filtration Time, t, min 500 600 Figure 12-8. Comparison of the predicted and measured filtrate volumes for linear filtration of fresh water bentonite suspension (Civan, R, 1998; reprinted by permission of the AlChE, ©1998a AlChE. All rights reserved). Cake Filtration: Mechanism, Parameters and Modeling 289 Predicted in the present study Predicted by Jiao and Sharma(1994) 100 200 300 400 Filtration Time, t, min. 500 600 Figure 12-9. Comparison of the predicted cake thicknesses for linear filtration of fresh water bentonite suspension (Civan, R, 1998a; reprinted by permission of the AlChE, ©1998 AlChE. All rights reserved). 0.2 0.15 Q) £ O !c 0> CD O 0.05 - - y = 0.0655x - 0.0006 R 2 = 0.9921 1 1.5 2 Reciprocal Filtrate Flow Rate, 1/q, 10 4 s/m 3 2.5 Figure 12-10. Correlation of Willis et al. (1983) measured filter cake thickness data (Civan, F., 1998; reprinted by permission of the AlChE, ©1998a AlChE. All rights reserved). 290 Reservoir Formation Damage Radial Filtration Applications Fisk et al. (1991) conducted radial filtration experiments using a seawater-based partially hydrolized polyacrylamide mud. Their Figure 4 provides the measured dynamic and static filtrate volumes versus filtration time data. Judging by their Figure 4, their static filtration data contains only three distinct measured values. This data is insufficient to extract meaningful information on the values of the A and C coefficients by regression of Eq. 12-80, because the calculation of In [-q~ 3 dq/dt\ requires a two step, sequential numerical differentiation—first to obtain the filtrate flow rate q = dQ/dt by differentiating the filtrate volume Q, and then differentiating q to obtain dq/dt. On the other hand, their dynamic filtration data is limited to the filtrate volume. As explained in the previous section on the determination of parameters, the determination of all coefficients of A, B, C, and D by means of Eqs. 12-49 and 12-65 requires both the filtration volume and filter cake thickness measurements. Therefore, the Fisk et al. (1991) radial filtration data has more missing parameter values, which had to be approximated as given in Table 12-1. Figure 12-11 shows, the model predicts the measured dynamic and static 10 20 30 40 Filtration Time, t, min 50 60 Figure 12-11. Comparison of the predicted and measured filtrate volumes for radial filtration of a sea-water based partially hydrolized polyacrylamide drilling mud (Civan, F., 1998; reprinted by permission of the AlChE, ©1998a AlChE. All rights reserved). Cake Filtration: Mechanism, Parameters and Modeling 291 filtrate volumes with reasonable accuracy in view of the uncertainties involved in the estimated values of the missing data. Fisk et al. (1991) did not report any results on the filter cake thickness and therefore a comparison of the cake thicknesses could not be made by Civan (1998a) in the radial filtration case. Conclusion The models presented in this section offer practical means of inter- preting experimental data, estimating the model parameters, and simulat- ing the linear and radial, incompressive cake filtration processes at static and dynamic filtration conditions. The simplified forms of these models conform with the well-recognized simplified models reported in the literature. These models are capable of capturing the responses of typical laboratory filtration tests while providing insight into the governing mechanisms. 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. [...]... ( 12- 124 ) The volumetric flux and flow rate are related by 2n rh ( 12- 125 ) 30 0 Reservoir Formation Damage where h is the formation thickness Thus, invoking Eq 12- 125 into Eq 12- 124 yields M- (q 2nhk(r ( 12- 126 ) The pressure differences over the filter cake and porous media can be expressed by integrating Eq 12- 126 , respectively, as (Civan, 1999b): 1 2 (2nh) 2nhkf * In I 1 r r,., (2nh )2 ( 12- 127 ) ( 12- 128 )... respectively Substituting Eqs 12- 92 to 12- 96 into Eqs 12- 87 to 12- 90 leads to the following volumetric balance equations, respectively (Civan, 1999b): ( 12- 97) dt dt ep2s] = 2rcN°p2s ( 12- 98) ( 12- 99) w = 2rc \ ' > slurry -2r w (w,) filrate («,) \ ' I _, 29 6 Reservoir Formation Damage ( 12- 100) Eqs 12- 97 through 100 can be solved numerically subject to the initial conditions given by: ( 12- 101) Linear Filtration... Corapcioglu and Abboud (1990) correspond to the present Eqs 12- 1 03, 12- 109, and 12- 106, respectively, with some differences Equation 12- 1 03 simplifies to their Eq 18, 29 8 Reservoir Formation Damage assuming pp is constant and substituting Eq 12- 1 12 The present Eqs 12- 106 and 12- 109 simplify to their Eqs 24 and 28 , substituting Eq 12- 1 13 for ep2l«el Also, Corapcioglu and Abboud (1990) did not distinguish... fine particles flowing through the cake matrix and porous formation are assumed the same Then, adding Eqs 12- 127 and 12- 128 , and rearranging and solving, yields (Civan, 1999b) for Darcy flow \fof =p c =0j: ( 12- 129 a) and for non-Darcy flow: = —-£— = 2nrji Mslurr ( 12- 129 b) 2a in which ^- P £ r,, J l r, r ( 12- 130 ) Cake Filtration: Mechanism, Parameters and Modeling 27 1 h V \'w Y = -(p c -p e ) 30 1 ( 12- 131 )... respectively, as €,=1-0 ( 12- 1 12) €/=E/+ep2;=0 ( 12- 1 13) where (j) is the average cake porosity (cm 3 /cm 3 ) The following expressions for the small particle volume flux and mass per carrier fluid volume can be written according to Eqs 12- 94 through 96, respectively, as: " *2/ = «/ e P2 //e,= «, . 8 .29 7min/cm 6 , B/C = 0.1 136 cm- 3 , R 2 = 0.87 13 ( 12- 84) C = 0.0 034 cm 4 /min, D = 0.0076cm, R 2 = 0.949 ( 12- 85) A = 0. 022 9 cnT 2 , B = 0.00 03 cm/min, R 2 = 0.98 73 ( 12- 86) The coefficients . of Eq. 12- 76 can be used with Eqs. 12- 30 , 12- 31 , 12- 12, and 12- 13 to obtain the following expressions: (B/C), k d (A/C)(t-T c > ( 12- 77) , _ d~ ( 12- 78) Comparing Eqs. 12- 75 and 12- 77 . = 2nrji 2a in which Mslurr ( 12- 129 b) ^- P £. r,, J__l r, r. ( 12- 130 ) Cake Filtration: Mechanism, Parameters and Modeling 30 1 27 1 h V 'w ( 12- 131 ) Y = -(p c -p e ) ( 12- 1 32 ) Although