This work presents a fractional flow theory analysis of gravity-dominated, immiscible displacement in porous media. This effort represents the first to extend Buckley-Leverett theory to account for the effects of countercurrent flow while fluid injection and production is ongoing. The driving force for production is ongoing. The driving force for countercurrent flow is gravity, phase density differences, and a non-horizontal flow dimension. Our analysis is limited by the usual fractional flow theory assumptions, most importantly one-dimensional flow. Countercurrent flows are possible if the ratio of gravity to viscous forces is sufficiently large. Our analysis characterizes the gravity to viscous force ratio by a dimensionless Gravity Number. Our effort yields a more general and unified theory of immiscible displacement. We present new graphical solution methods to predict displacement performance for arbitrary initial and injected conditions.
The behavior of immiscible displacement in porous media depends on viscous and gravity forces. In general, one-dimensional flows may be uni-directional or countercurrent. Flow may also take place in either open or closed systems. Open-systems imply fluid injection into and/or fluid withdrawal from the system. Closed systems, in contrast, preclude injection or production. Viscous-dominated flows take place exclusively in open systems and are characterized by uni-directional flow. The theory of viscous-dominated, immiscible displacement was first introduced by Buckley and Leverett and Welge. Gravity-dominated flows, in contrast, may occur in either open or closed systems and may involve either uni-directional or countercurrent flow. The driving force for countercurrent flow is gravity, phase density differences, and a dipping reservoir. phase density differences, and a dipping reservoir. Gravity-dominated flows have been discussed by several investigators. Martin, Templeton et al. and Coats et al.' have discussed countercurrent flows; however, they limited their studies to systems precluding any fluid injection or production, i.e precluding any fluid injection or production, i.e closed systems. Richardson and Blackwell, Hagoort, and Joslin have discussed the performance of gravity-dominated floods; however, their applications were restricted to uni-directional flows in open systems. A unified theory accounting for the effects of countercurrent flow in both open and closed systems. i.e.. m systems with or without fluid injection, has not been presented to date. The purpose of this work is to present the general theory of immiscible displacement in porous media covering viscous- and gravity-dominated flows. This effort is the first to address such a wide range of displacements.
The development presented herein is based on the following assumptions or idealizations:1. A one-dimensional, isothermal, homogeneous permeable medium.2. At most, there are three components: oil, water, and gas.3. At most, there are two flowing phases: either water and oil or oil and gas.4. The water component will not partition into the oil and gas phases.5. The oil component will not partition into the water or gas phases.6. The gas component will not partition into the water or oil phases.7. Darcy's law applies.5. No adsorption of components by the solidphase.9. No dissipation, i.e., no capillary pressure. diffusion, dispersion, or viscous fingering.10. The system is incompressible and the effects of pressure on the phase behavior and fractional flow are negligible.11. The initial fluid distributions are uniform.12. The relative permeability relationships are constant and not subject to hysteresis.13. If the system is open, the injected condition is constant.