The method of characteristics, or fractional-flow theory, is extremely useful in understanding complex IOR processes and in calibrating simulators. One limitation has been its restriction to Newtonian rheology except in rectilinear flow. Its inability to deal with non-Newtonian rheology in polymer and foam IOR has been a serious limitation. We extend fractional-flow methods for two-phase flow to non-Newtonian fluids in one-dimensional cylindrical flow, where rheology changes with radial position r. The fractional-flow curve is then a function of r. At each position r the "injection" condition is the result of the displacement upstream; one can plot the movement of characteristics and shocks downstream as r increases. We show examples for IOR with non-Newtonian polymer and foam. For continuous injection of foam, one can map the 1D displacement in time and space exactly for the non-Newtonian foam bank, and approximately, but with great accuracy, for the gas bank ahead of the foam. The fractional-flow solutions are more accurate that finite-difference simulations on a comparable grid and can be used to calibrate simulators. Fractional-flow methods also allow one to calculate changing mobility near the injection well to a greater accuracy than simulation. For polymer and SAG (alternating-slug) foam injection, characteristics and shocks collide, making the fractional-flow solution complex. Nonetheless, one can solve exactly for changing mobility near the well, again to greater accuracy than with conventional simulation. For polymer solutions that are Newtonian at high and low shear rates but non-Newtonian in between, the non-Newtonian nature of the displacement depends on injection rate. The fractional-flow method extended to non-Newtonian flow can be useful both for its insights for scale-up of laboratory experiments and to calibrate computer simulators involving non-Newtonian IOR. It can also be an input to streamline simulations.


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