A new model of the constricted unit cell type for granular porous media and collocation solution of the creeping Newtonian flow problem



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The constricted unit cell model for granular porous media developed by Payatakes, Tien and Turian (1973) is extended here to take in account the random orientation and interconnectivity of the flow channels. In the proposed model each unit cell corresponds to a pore (cavern) and has two coaxial constricted inlet and outlet ports ("throats"). The unit cells have random dimensions and orientations, the distributions of which can be determined from simple experimental measurements. The flow through a unit cell is assumed to be identical to that through a segment of the corresponding periodically constricted tube. A collocation solution of creeping newtonian flow through periodically constricted tubes is obtained. The profile of the wall of the type of tube considered is piecewise smooth, composed of symmetric parabolic segments and accords with the unit cell geometry of the porous media model. A transformation of the domain of interest into a rectangular one is obtained, which allows satisfaction of all boundary conditions. The collocation solution gives the stream function in terms of the new independent variables and can easily be converted to the original cylindrical coordinates. Axial velocity and radial velocity are obtained in analytical form, and the pressure drop is calculated both from integration of the energy dissipation function and of the Navier-Stokes equations. The results are compared with the finite-difference solution by Payatakes et al. (1973b) and are found in good agreement. Differences between the two solutions are attributed mainly to discretization error in the finite-difference solution. Based on the collocation solution, the new model is used to study the flow of fluids through granular randomly packed beds.. Experimental and theoretical predictions of permeability for two samples are compared and found in remarkable agreement. Intended applications of the model coupled with the present flow solution are in the modeling of processes taking place in the macroporous space of beds of monosized or nearly monosized grains (deep bed filtration, etc.). A modified version of this model will be applied to the modeling of secondary and tertiary oil recovery from oilbearing rocks.