Non‐Newtonian flow in porous media

Abstract

This paper describes a preliminary study of the linear flow of a non-Newtonian fluid, a water solution of Dextran (a polysaccharide), in porous media. A modification of Darcy's law, which uses capillary rheology data, is developed to describe non-Newtonian flow in underground reservoirs. The generalization, in effect, replaces the porous media with a capillary of equivalent radius proportional to the square root of the ratio of permeability to porosity. The constant of proportionality a . should be independent of permeability and porosity for a given type of rock. This has been partially confirmed experimentally. In principle, a capillary rheogram and a single core test permit evaluation of ao. Then non-Newtonian flow can be predicted in this type of rock regardless of porosity, permeability, or flow rate.The flow of a Newtonian fluid (a fluid with a constant viscosity) through porous media is in good agreement with Darcy's law as expressed by Equation (1) with gravitational forces neglected.(1)where V, is the fluid superficial velocity vector, V p is the pressure gradient, f i is the fluid viscosity, and k, a property of the structure of the porous medium, is the permeability with consistent units of squared length. In the petroleum industry, it is customary to measure V, in cm./sec., f i in centipoise, and V p in atm./cm. with the resulting units of k called the darcy. The conversion is 1 darcy = 9.87 X 10-e sq. cm. This paper describes an extension of Darcy's equation (1) to the case in which the fluid does not follow Newton's law of a constant viscosity (ratio of shear stress to deformation rate); rather, the fluid belongs to a class called Stokesian fluids in the literature (1 ) . Such fluids have the property that the nonhydrostatic components of the fluid stress are unique nonlinear functions of the deformation rate components with the result that the measured "viscosity" varies with the shear stress prevailing in the viscometer. The bulk of the paper develops a one-dimensional theory and compares it with preliminary experimental results for an aqueous solution of a sugar polymer available commercially under the trade name Dextran. The model is developed by direct analogy with the results obtained for flow of the fluid through a uniform capillary and utilizes the rheogram obtained from the latter. A twodimensional analysis, not given here, showed that the dispersive or nonaxial component of the velocity enters the flow equation only as a product with an unknown normal coefficient of viscosity. It should also be pointed out that our choice of the model was guided by two restrictions:( 1 ) The model must give good predictions over a wide shear stress range, and (2) the model must be in a form suitable for routine calculations. The first restriction eliminates some of the stress-deformation rate relations proposed in the literature for Stokesian fluids such as the power law model; the second eliminates others such as the Eyring model.Finally, the flow of non-Newtonian fluids in porous media is important to th...

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