Determination of Model Parameters and Diagnostic Charts
Only five of these parameters may not be directly or conveniently measurable with the conventional techniques. These are the permeability Kc and porosity ɸc of the filter cake, and the deposition and erosion rate constants kd and ke, and the critical shear-stress icr for the particles.
However, given the experimental measurements of the filtrate volume Q(cm3), or rate q (cm3/s), and the filter cake thickness 6 as functions of the filtration time t, some of these parameters can be determined by means of the diagnostic charts constructed as described in the following. These are presented separately for the linear and radial filtration processes obeying Darcy's law according to Civan (1998a).
Linear Filtration
A plot of Eq. 12-11 for d&fdt versus q yields a straight line. Substituting the slope (A) and intercept (-B) of this line into Eqs. 12-12 and 13 yields, respectively, the following expressions for the particle deposition and erosion rate constants:
In dynamic filtration, the filter cake thickness attains a certain limit value, ẟ∞, when the particle deposition and erosion rates equate. Simultaneously, the filtration rate also reaches a limiting value, determined by Eq. 12-11 as:
At this condition, Eq. 12-29 yields the limiting value of the filter cake thickness as:
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 for the cake permeability and the ratio of the erosion and deposition rate constants, respectively, as:
Equation 12-33 can be rearranged in a linear form as:
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:
Comparing Eqs. 12-75 and 12-77 yields an alternative expression for determination of the limit filtrate rate as:
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 kd and ke, if the cake porosity ɸc and the critical shear stress tcr are known. ɸc can be measured. icr 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 icr should be measured directly.
Radial Filtration
Given the filter cake thickness ẟ, the progressing surface cake radius rc 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 [dẟ/dt] versus [q/(rw -ẟ)] 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:
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 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 ẟ∞ and q∞. Then, substituting Eq. 12-46 into Eqs. 12-49 and 62 yields the following relationships, respectively:
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 applications, 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.
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