Incompressive Cake Filtration
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:
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