The Plugging-Nonplugging Parallel Pathways Partial Differential Model
Gruesbeck and Collins (1982a) developed a partial differential model based on the concept of parallel flow of a suspension of particles through plugging and nonplugging pathways, as depicted in this article. Relatively smooth and large diameter flowpaths mainly involve surface deposition and are considered nonplugging. Flowpaths that are highly tortuous and having significant variations in diameter are considered plugging. In the plugging pathways, retainment of particles occurs by jamming and blocking of pore throats when several particles approach narrow flow onstrictions.
Sticky and deformable deposits usually seal the flow constrictions (Civan, 1994, 1996). Therefore, conductivity of a flow path may diminish without filling the pore space completely. Thus, the fluid seeks alternative flow paths until all the flow paths are eliminated. Then the permeability diminish even though the porosity may be nonzero. Another important issue is the criteria for jamming of pore throats. As demonstrated by Gruesbeck and Collins (1982b) experimentally for perforations, the probability of jamming of flow constrictions strongly depends on the particle concentration of the flowing suspension and the flow constrictionto-particle diameter ratio. Gruesbeck and Collins (1982a) assumed that the liquid and particles have constant physical properties. The porous media is incompressible, homogeneous and isotropic.
There is hydraulic communication through the interconnectivity of the plugging and nonplugging pathways and therefore the pressure gradients and the particle concentrations of the suspension flowing through the plugging and nonplugging pathways are the same. The volume flux through the core is constant and only the external particle invasion is considered. The flow through porous media was assumed to obey the Darcy Law. In this section, the Gruesbeck and Collins (1982) model is presented with the modifications and improvements made by Civan (1995). (J)PO and §npg denote the initial pore volume fractions of the plugging and nonplugging pathways of the porous media (Civan, 1994, 1995).
These values can be determined experimentally for a given porous media and the particle size distribution. εp and εnp represent the fractions of the bulk volume occupied by the deposits. Thus, the instantaneous porosities are:
The fractions of the bulk volume containing the plugging and nonplugging pathways can be approximated, respectively, by:
However, Gruesbeck and Collins (1982) assume a constant value for ƒp (and therefore for ƒnp =l-ƒp), which is a characteristic of the porous medium and the particles. Total instantaneous and initial porosities are given, respectively, by:
The total deposit volume fraction and the instantaneous available porosity are given, respectively, by:
The rate of deposition in the plugging pathways is given by assuming the pore filling mechanism:
when the pore throat-to-particle diameter ratio is below its critical value. Civan (1990, 1994) recommended the following empirical correlation:
which is determined empirically as a function of the particle Reynolds number:
The rate of deposition in the nonplugging tubes is given as the difference between the rates of surface deposition and sweeping of particles (Civan, 1994):
where ðnp is the volume fraction of particles in the suspension of particles flowing through the nonplugging pathways. kd and ke are the surface deposition and mobilization rate constants. ke = 0 when tw<tcr. ɲe is the fraction of the uncovered deposits that can be mobilized from the pore surface, estimated by:
tcr is the minimum shear stress necessary to mobilize the surface deposits. inp is the wall shear-stress in the nonplugging tubes, given by the Rabinowitsch-Mooney equation (Metzner and Reed, 1955):
and the mean pore diameter is given by:
where C is an empirical shape factor. It can be shown that Eqs. 10-101 and 105 simplify to the deposition rate equations given by Gruesbeck and Collins (1982):
This requires that the effects of the permeability and porosity changes be negligible, the fraction of the uncovered deposits be unity, the suspension of particles be Newtonian, and the particle volume fractions of the suspensions flowing through the plugging and nonplugging pathways be the same, that is,
The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships (Civan, 1994):
Then, the average permeability of the porous medium is given by:
Note that Eq. 10-116 was derived independently by Civan (1992) and Schechter (1992) and is different than the expression given by Gruesbeck and Collins (1982). The superficial flows in the plugging and nonplugging pathways are given, respectively, by:
Considering that the physical properties of the particles and the carrier liquid are constant, the volumetric balance of particles in porous media is given by:
Substituting Eq. 10-100 into Eq. 10-119, and then rearranging, an alternative convenient form of Eq. 10-119 can be obtained as:
Following Gruesbeck and Collins (1982), Eq. 10-120 can be simplified for cases where e and a are small compared to ɸ0 and unity, respectively, and for constant injection rate, as:
The initial particle contents of the flowing solution and porous media are assumed zero:
where L is the length of porous medium. The particle content of the suspension of particles injected into the porous media is prescribed as:
Alternatively, the pressures of the inlet and outlet ends of the porous media instead of the flow rate can be specified. Then, the volumetric flux can be estimated by the Darcy law:
Then, the pressure obtained by solving Eqs. 10-126 through 10-128 is substituted into Eq. 10-124 to determine the volume flux. The preceding formulation of Eq. 10-119 or 120 applies to the overall system following Gruesbeck and Collins' (1982) assumption that the particle concentrations in the plugging and nonplugging pathways are the same according to Eq. 10-113. When different concentrations are considered, Eq. 10-120 should be applied separately for the plugging and nonplugging paths, respectively, as suggested by Civan (1995):
k is a particle exchange rate coefficient. A solution of Eqs. 10-129 through 132 along with the particle deposition rate equations, Eqs. 10-101 and 105, yields the particle volume fractions in the plugging and nonplugging flow paths.
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