Fractional Flow Theory Applicable to Non-Newtonian Behavior in EOR Processes

Abstract

The method of characteristics, or fractional-flow theory, is extremely useful in understanding complex Enhanced Oil Recovery (EOR) 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 EOR 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 distance from injection well. The fractional flow curve is then a function of position and we analyze the characteristic equations for two applications-polymer and foam floods. For polymer flooding, we present a semi-analytical solution for the changing fractional flow curve where characteristics and shocks collide. The semi-analytical solution is shown to give good agreement with the finite-difference simulation thus helping us understand the development and resolution of shocks. We discuss two separate cases of foam injection with or without preflush. We observe that the fractional flow solutions are more accurate than finite-difference simulations on a comparable grid and hence the method can be used to calibrate simulators. For 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, to greater accuracy than with conventional simulation. 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 EOR. It can also be an input to streamline simulations.

Injectivity Errors in Simulation of Foam EOR

“…Here we address both issues; for simplicity we exclude the dry-out effect that is the focus of Section 3.1. Rossen et al (2011) describe a method for modeling 1D dynamic two-phase displacements with non-Newtonian phase viscosities using the MOC. Unlike the model in Section 3.1, illustrated in Fig.…”

“…Although the method employs a numerical solution of equations derived from the MOC, these equations can be solved to an arbitrary level of precision (e.g., to the cm scale or below), much more accurately than is feasible with conventional simulation. To simplify the model and focus on nonNewtonian effects, Rossen et al (2011) exclude the effect of water saturation on foam stability, specifically the abrupt collapse of foam at a limiting water saturation S w n described in Section 3.1 above. Thus their study includes the simultaneous effects of changing water saturation and non-Newtonian viscosity, but it excludes by far the largest effect of changing water saturation in a SAG process, the dry-out effect.…”

“…In the foam model of Rossen et al (2011), gas mobility is reduced by foam by a factor that is independent of water saturation but depends on total superficial velocity like a power-law fluid with power-law exponent ½. In addition, mobility depends on water saturation because the relative permeabilities of water and gas depend on water saturation.…”

“…Details of the foam model are in Appendix B. Rossen et al (2011) note that an important implication of this model is the effect on injectivity, but they do not calculate injectivity. In this paper we use the results of Rossen et al for water saturation as a function of time and radial position to determine injectivity for a shear-thinning foam in a SAG process.…”

“…Foam is also shear-thinning in many cases. This paper considers the two problems in turn: non-Newtonian effects and foam dry-out.In studying non-Newtonian effects we use the method-of-characteristics approach of Rossen et al (2011), which resolves both changing saturations and non-Newtonian rheology with great precision near the wellbore, and compare to conventionally computed injectivity using the Peaceman equation in a grid block. By itself, the strongly non-Newtonian rheology of the "low-quality" foam regime makes a significant difference to injectivity of foam.…”