Reservoir Formation Damage Episode 2 Part 3 doc

Correlation of Jiao and Sharma 1994 predicted filter cake thickness data Civan, F., 1998; reprinted by permission of the AlChE, ©1998aAlChE.. Correlation of Jiao and Sharma 1994 predicte

Equation 12-33 can be rearranged in a linear form as:

d(\\ — = i = n H 1 dq A B

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 toobtain the following expressions:

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, Kc

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 beestimated by Eq 12-6, but the ideal theory may not yield a correct value asexplained previously by Ravi et al (1992) and in this chapter Therefore,

Ravi et al (1992) suggested that icr 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(rc/rw)

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,

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/<70] 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 cakethickness versus the filtration time data Once these coefficients aredetermined, 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 kd and ke The discussion of the linear filtration about the

determination of icr 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 qx Then, substituting Eq.

12-46 into Eqs 12-49 and 62 yields the following relationships, respectively:

+ D) (12-82)The filter cake permeability is determined by Eq 12-64 as:

et al (1997) They have resorted to a model assisted estimation of theparameters because there is no direct method of measurement for some

of these parameters

The numerical solutions of the present models require the information

on the characteristics of the slurries, particulates, carrier fluids, filters andfilter 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 thefiltrate volumes or rates and do not offer a complete set of suitable datathat is needed for full scale experimental verification of the presentmodels 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 radialfiltration, because these data provide more information than the otherreported studies The data is presented in Table 12-1 in consistent Darcyunits, which are more convenient for flow through porous media

Linear Filtration Applications

Jiao and Sharma (1994) carried out linear filtration experiments usingconcentrated bentonite suspensions They only measured the filtratevolume and predicted the filter cake thickness using a simple algebraicmodel These data are given in their Figures 3 and 10, respectively InFigures 12-5 to 12-7, their data are plotted according to the linearplotting schemes presented in the previous section for determination ofparameters As can be seen from these figures, the coefficients of Eqs.12-76, 12-29, and 12-11 obtained by the least-squares regression methodand the corresponding coefficients of regression are given, respectively, by:

A/C = 8.297min/cm6, B/C = 0.1136 cm-3, R 2 = 0.8713 (12-84)

C = 0.0034cm4 /min, D = 0.0076cm, R2 = 0.949 (12-85)

A = 0.0229 cnT2, 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, inferringthe possibility of larger measurement errors involved in the filtrate volumedata Another source of errors may be due to the three-point finitedifference numerical differentiation of the filtrate volume data to obtainthe filtrate flow rate data used to construct Figure 12-5 The data necessaryfor Figure 12-5 were obtained by a series of numerical procedures, first

Figure 12-5 Correlation of Jiao and Sharma (1994) experimental data (Civan,

F., 1998; reprinted by permission of the AlChE, ©1998a AlChE All rightsreserved)

60

Figure 12-6 Correlation of Jiao and Sharma (1994) predicted filter cake

thickness data (Civan, F., 1998; reprinted by permission of the AlChE, ©1998aAlChE All rights reserved)

Figure 12-7 Correlation of Jiao and Sharma (1994) predicted filter cake

thickness data (Civan, R, 1998a; reprinted by permission of the AlChE, ©1998AlChE 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 q0 = 0.096 mL/min by a

three-point forward differentiation of the measured, initial filtrate volumedata This data is expected to involve a larger error because of thepossibility of relatively larger errors involved in the early filtrate volumedata The noisy data had to be smoothed prior to numerical differentiation,which may have introduced further errors Because of the propagation ofthe 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 qx =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 qx estimated by an extrapolation of the

derivatives of the filtrate volume data beyond the range of the

experi-mental data is qM =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 cakethickness as 5^ =0.24cm The predicted cake thickness data presented

in Figure 10 of Jiao and Sharma (1994) indicates a value of mately 0.17cm Therefore, their prediction of the limiting filter cakethickness appears to be an underestimate compared to the 0.24cm valueobtained by Civan (1998a)

approxi-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, and12-78 form a set of alternative equations to determine the deposition

and erosion rate constants, kd and ke 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 Tcr parameters had to be estimated andused with Eqs 12-70 and 12-75 to match the filtration data over theperiod of the filtration process Then, the <j)c and icr values obtained this way were used in Eqs 12-70 and 12-75 to calculate the kd and ke 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 tcr =1.25 xlO3 dyne/cm2 Whereas, the prevailingshear stress calculated by Eq 12-5 is only T = 16 dyne/cm2 Under theseconditions, theoretically the cake erosion should not occur becauseT«:icr 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 Recallthat we used this value in Eq 12-72 to calculate the limiting flow rate

of qx =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 flowrate 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 anumerical result of the least-squares regression of Eq 12-11 to data,

because the coefficient of regression R2 = 0.9873 is very close to one Hence, it can be inferred that 1 > icr 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

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 themodel represents the measured filtrate volumes over the complete range

of 600 min of filtration time as closely as the quality of their experimentaldata permits However, they did not measure the cake thickness, butpredicted 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 asuspension of lucite in water As shown in Table 12-1, they reported only

a few parameter values They only provide some measured filtrate flowrate and cake thickness data in their Table 2 However, the filtration timedata is missing Therefore, a full scale simulation of their filtration process

as a function of time could not be carried out by Civan (1998a) Onlythe linear plotting of the measured data according to Eq 12-29 could beaccomplished 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

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).

Predicted in the present study

Predicted by Jiao and Sharma(1994)

100 200 300 400

Filtration Time, t, min.

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).

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).

Radial Filtration Applications

Fisk et al (1991) conducted radial filtration experiments using aseawater-based partially hydrolized polyacrylamide mud Their Figure 4provides the measured dynamic and static filtrate volumes versus filtrationtime data Judging by their Figure 4, their static filtration data containsonly 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 theprevious 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 missingparameter 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

Figure 12-11 Comparison of the predicted and measured filtrate volumes for

radial filtration of a sea-water based partially hydrolized polyacrylamide drillingmud (Civan, F., 1998; reprinted by permission of the AlChE, ©1998a AlChE.All rights reserved)

filtrate volumes with reasonable accuracy in view of the uncertaintiesinvolved in the estimated values of the missing data Fisk et al (1991)did not report any results on the filter cake thickness and therefore acomparison 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 preting experimental data, estimating the model parameters, and simulat-ing the linear and radial, incompressive cake filtration processes at staticand dynamic filtration conditions The simplified forms of these modelsconform with the well-recognized simplified models reported in the literature.These models are capable of capturing the responses of typical laboratoryfiltration tests while providing insight into the governing mechanisms

inter-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 differencefiltration processes because these models facilitate Darcy's law to describeflow through porous media However, filtration at high flow rates andhigh overbalance pressure differences may involve some inertial floweffects, especially during the initial period of the filter cake formation

In the literature, the initial non-linear relationships of the filtrate volumeversus the square root of time has been attributed to invasion and clogging

of porous media by fine particles during filtrate flow into porous mediaprior to filter cake formation The cumulative volume of the carrier fluid(filtrate) lost into porous media during this time is usually referred to asthe spurt loss (Darley, 1975)

Based on an order of magnitude analysis of the relevant dimensionlessgroups of the general mass and momentum balances of the multiphasesystems involving the cake buildup, Willis et al (1983) concluded thatnon-parabolic filtration behavior is not caused by non-Darcy flow Instead,

it is a result of the reduction of the permeability of porous media byclogging by fine particles Their claim is valid under the conditions oftheir experimental test conditions The phenomenological models for filtercake buildup involving fine particle invasion have been presented by Liuand Civan (1996) and Civan (1998b) for low rate filtration However, aclose examination of most filtration data reveal some non-Darcian floweffect during the short, initial period of filtration depending on themagnitude of the filtration flow rate and/or the applied pressure difference

The large flow rates encountered during this period usually promote anon-Darcy effect Willis et al (1983) investigated the non-parabolicfiltration behavior, but concluded that the non-parabolic behavior is aresult of the impairment of the permeability of porous media by invasionand 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 floweffect during the initial period of filter cake buildup.

Civan (1998b, 1999b) developed linear and radial filtration models andverified them by means of experimental data These models are moregenerally applicable because of the following salient features:

1 A cake-thickness-averaged formulation leads to a convenient andcomputationally 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 intothe groups of the large and fine particles, and the large particles formthe 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 theinertial flow effects encountered during the early filtration period;

4 The dynamic and static filtration conditions encountered with andwithout the slurry flowing tangentially over the cake surface, respec-tively, are considered;

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

com-6 An average fluid pressure is used to determine the fluid drag forceapplied to the cake matrix;

7 The formulations are presented for general purposes, but applied forcommonly encountered cases involving incompressible particles andcarrier fluids; and

8 The constant and variable rate filtration processes can be simulated.The model presented in this section incorporates empirical constitutiverelationships for the permeability and porosity variations of compressiblecakes retaining fine particles The simulation of a series of filtrationscenarios are presented to demonstrate the parametric sensitivity of themodel It is determined that permeability impairment by fine particlesretainment and pore throat clogging in the filter cake is increasinglyinduced by cake compression It was also determined that constantpressure filtration limits the filtrate invasion more effectively than con-stant rate filtration and the non-Darcy flow effect is more significant

during the initial period of the filter cake formation The cake formationmodels developed in this section can be used for predicting the effects

of the compressible filter cakes involving the drilling muds and ing fluids

fractur-The applications of the improved models are illustrated by typicalcase studies

Radial Filtration Formulation

Consider that a slurry is applied to the inner surface of a drum filterand the filtrate leaves from its outer surface (see Figure 12-4) The modeldeveloped here is also equally applicable for the reverse operation The

filter cake is located between the filter inner surface radius rw (cm) OVer

which the cake is formed, and the slurry side cake surface radius rc (cm) and its thickness is denoted by h = rw -r c The external surface radius of the filter from which the filtrate leaves is re (cm) and the filter width is

indicated by w(cm), such that the area of the inner filter surface over

which the cake is formed is 2nrw w The slurry flows over the cake surface at a tangential or cross-flow velocity of v f (cm/s) and the filtrate

flows into the filter at a filtration velocity of M/(cm3/cm3>s) normal tothe filter face due to the overbalance of the pressure between the slurryand the effluent sides of the filter The flowing suspension of particles

and the filter cake (solid) are denoted by the subscripts / and s,

respec-tively The carrier phase (liquid) and the particles are denoted, respectively,

by / and p Following Tien et al (1997), the slurry is considered to

contain particles larger than the filter medium pore size that form the filtercake and the particles smaller than the pore sizes of the filter cake andthe filter medium, which can migrate into the cake and the filter to depositthere All particles (small plus large) are denoted by p, and the large

and small particles are designated by pi and p2, respectively.

Civan (1998b, 1999b) developed the filtration models by consideringthe cake-thickness averaged volumetric balance equations for

1 The total (fine plus large) particles of the filter cake;

2 The fine particles of the filter cake;

3 The carrier fluid of the suspension of fine particles flowing throughthe filter cake; and

4 The fine particles carried by the suspension of fine particles flowingthrough the filter cake

The radial mass balances of all particles forming the cake, the small ticles retained within the cake, the carrier fluid, and the small particlessuspended in the carrier fluid are given, respectively, by (Civan, 1998b):