The Plugging-Nonplugging Parallel Pathways Model
The porous media realization is based on the plugging and nonplugging pathways concept according to Gruesbeck and Collins (1982). 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 deposits is assumed to occur by jamming and blocking of pore throats when several particles approach narrow flow constrictions. Deposits that are sticky and deformable usually seal the flow constrictions (Civan, 1990, 1994, 1996). Therefore, conductivity of a flow path may diminish without filling the pore space completely. Fluid seeks alternative flow paths until all the flow paths are eliminated.
Then the permeability diminishes even though the porosity may be nonzero. Another important issue is the criteria for jamming of pore throats. As demonstrated by Gruesbeck and Collins (1982) experimentally for perforations, the probability of jamming of flow constrictions depends strongly on the particle concentration of the flowing suspension and the flow constriction-to-particle diameter ratio. The pore plugging mechanisms are analyzed considering an infinitesimally small width slice of the porous core. The total cross-sectional area, A, of the porous slide can be separated into two parts:
1 the area Ap, containing pluggable paths in which plug-type deposition and pore filling occurs, and
2 the area, Anp, containing nonplugging paths in which nonplugging surface deposition occurs.
Thus, the total area of porous media facing the flow is given by:
The fraction of the plugging pathways is a characteristic property of porous media and the particles of the critical size, comparable or larger than the pore throat size (Gruesbeck and Collins, 1982; Schechter, 1992).The pore size distribution of the porous medium and the size distribution of the particles determine its value. However, its value varies because the nonplugging pathways undergo a transition to become plugging during formation damage. The volumetric flow rate, q, can also be expressed as a sum of the flow rates, qp and qnp, through the pluggable and nonpluggable paths as:
The volumetric flows and volumetric fluxes are related by the following expressions:
Kp and Knp represent the permeabilities of the pluggable and nonpluggable fractions of the core. Assuming that the plugging and nonplugging paths are interconnective and hydraulically communicating, the pressure gradients are taken equal:
Then, it can be shown that, the average permeability of the porous medium is given by (Civan, 1992; Schechter, 1992):
and the superficial flows in the plugging and nonplugging pathways are given respectively, by:
Let Φpo and Φnpo denote the initial pore volume fractions of the plugging and nonplugging pathways of the porous media (Civan, 1995) and Kp and Knp represent the fractions of the bulk volume occupied by the deposits in the respective pathways. Thus, the instantaneous porosities in the plugging and nonplugging pathways are given by:
The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships by Civan (1994) by generalizing the expressions given by Gruesbeck and Collins (1982):
where n1 and n2 are the permeability reduction indices, α is a coefficient and Kpo and Knpo are the permeabilities at the reference porosities Φpo and Φnp of the plugging and nonplugging pathways, respectively. Eq. 5-58 represents the snow-ball effect of plugging on permeability, while Eq. 5-59 expresses the power-law effect of surface deposition on permeability. Eqs. 5-58 and 5-59 have been also verified by Gdanski and Shuchart (1998) and Bhat and Kovscek (1999), respectively, using experimental data. Bhat and Kovscek (1999) have shown that the power-law exponent in Eq. 5-59 can be correlated as a function of the coordination number and the pore body to throat aspect ratio, applying the statistical network theory for silica deposition in silicaous diatomite formation. Note that, for n2 <0 and Φnp/Φnpg «1, Eq. 5-59 simplifies to the expression given by Gruesbeck and Collins (1982):
Thus, substitution of Eqs. 5-58 and 59 into Eq. 5-49 results in the following expression for the permeability of the porous media (Civan, 1994, 1996):
When E = 0, it yields a nonlinear model as:
In view of the Gruesbeck and Collins (1982) plugging and nonplugging pathways approach, Civan (2000) concluded that the E coefficient can be analogous to the fraction of the non-plugging pathways and Eqs. 5-63 and 64 can be attributed to the nonplugging and plugging pathways in porous media, respectively.
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