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An analytical solution for describing the non-Darcy displacement of a Newtonian fluid by a non-Newtonian fluid in porous media has been developed. The two-phase non-Darcy flow is described using the Barree-Conway model under multiphase conditions. A power-law non-Newtonian fluid, whose viscosity is a function of the flow potential gradient and the phase saturation, is considered. The analytical solution is similar to the Buckley-Leverett theoretical solution, which can be regarded as an extension of the Buckley-Leverett theory to the non-Darcy flow of non-Newtonian fluids. The analytical results revel how non-Darcy displacement by a non-Newtonian fluid is controlled not only by relative permeabilities but also by non-Darcy flow coefficients as well as non-Newtonian rheological constitutive parameters and injection rates. The comparison among Darcy, Forchheimer and Barree-Conway models is also discussed. For application, the analytical solution is then applied to verify a numerical simulator for modeling multi-phase non-Darcy flow of non-Newtonian fluids.

Non-Darcy displacement by a non-Newtonian fluid in porous media according to the Barree-Conway model