The Compartments-in-Series Ordinary Differential Model
Khilar and Fogler (1987) divided a core into n-compartments as depicted in this article. The contents of these compartments are assumed well-mixed. Therefore, the composition of the flow stream leaving the compartments should be the same as the contents of the compartments. However, because particles having sizes comparable or larger than the pore throats are trapped within the porous media, the particle concentration of the stream leaving a compartment will be a fraction, Υ, of the concentration of the fluid in the compartment, Υ is called the particle transport efficiency factor. Pore surfaces are considered as the source of in-situ mobilized particles and the pore throats are assumed the locations of particle capture. A particle mass balance over a thin slice yields
mp and m*p denote the mass of particles captured at the pore throats and the indigeneous particles remaining on pore surfaces, respectively. In Eq. 10-65
Substituting Eqs. 10-66 and 67, and rearranging Eq. 10-65 becomes:
The mass balance of particles captured at the pore throats is given by:
The mass balance of indigeneous particles remaining on pore surfaces is
where mpo is the mass of particles initially available on pore surface. The rate of particle entrainment by the flowing phase is assumed both colloidally and hydrodynamically induced.
t is shear stress. as is the specific pore surface area, the colloidally induced release coefficient given by (Khilar and Fogler, 1983):
Single-Phase Formation Damage by Fines Migration and Clay Swelling
Cs is the salt concentration. Csc is the critical salt concentration for particle expulsion, hydrodynamically induced release coefficient given by (Gruesbeck and Collins, 1982):
fc is the critical shear stress required to mobilize particles on pore surface. The rate of capture of particles at pore throats is assumed proportional to the flowing phase particle concentration:
β, is the capture coefficient. Let ppfc be the critical particle concentration above which bridging at pore throats occur and particles cannot travel between pore bodies. If the particle concentration is below Рpfc , then no trapping at pore throats takes place. Therefore,
where B is a characteristic constant and K৹ is the initial permeability.
Simplified Partial Differential Model
Cernansky and Siroky (1985) considered injection of a low particle concentration suspension at a constant rate into porous media made of a bed of filaments. Neglecting the diffusion of particles and the contribution of the small amount of particles in the flowing suspension, they expressed the total mass balance of particles similar to Gruesbeck and Collins' (1982) simplified mass balance equation. Thus, for incompressible liquid and particles and constant injection rate, the total volumetric particle balance equation is given by:
Cernansky and Siroky (1985) expressed the net rate of particle deposition in porous media as the difference between the deposition by pore throat plugging and entrainment by hydrodynamic mobilization. Considering the critical shear stress necessary to mobilize the deposited particles in porous media, Civan et al. (1989) modified their rate equation as:
where D is the hydraulic tube diameter, and the pressure gradient is represented by Darcy's law:
where ɸ0 is the initial porosity. Thus, substituting Eqs. 10-88 through 10-90 into Eq. 10-87 yields:
Based on their experimental studies, Cernansky and Siroky (1985) proposed an empirical permeability-porosity relationship as:
where E and G are some empirical constants.
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