Stability of Piston-Like Displacements of Water by Steam and Nitrogen in Porous Media

Downhole steam generation leads to consideration of reservoir fluid displacement by a mixture of steam and nitrogen. The linear stability analysis of the steam condensation front has been generalized to include a noncondensing gas. Roughly speaking, the addition of nitrogen increases the likelihood of having fingers, but, compared with the no-nitrogen case, the fingers will grow more slowly. Introduction The theory of the stability of flows through porous media has been a subject of interest for more than 25 years, dating back to the pioneering work of Dietz, Chuoke et al., and Saffman and Taylor. They considered injecting one fluid (e.g., water) to force a second fluid (e.g., oil) out of a porous medium. The primary result was that instabilities (fingering) occurred when the driving fluid was more mobile than the driven fluid. Hagoort included multiple fluid phases. Miller generalized the original work to include steam driving water (liquid). He showed that the thermodynamic phase transition (steam to water) introduces two stabilizing effects. The first effect introduces a water/steam velocity ratio as a multiplier of the mobility ratio. This factor is less than one because of the volume change upon condensation. The second effect is the cooling of incipient steam fingers by the surrounding water, which retards their growth. Baker anticipated these effects in a qualitative way to explain his experiments, which showed a more stable displacement by steam than was expected on the basis of mobility ratios alone. Armento and Miller also have considered the stability of the in-situ combustion front in porous media. Their work deals with a region where steam is generated. This paper reformulates Miller's results for a condensation front in a more useful form including general numerical results and extends the theory to include injection of a noncondensing gas (e.g., nitrogen) together with the steam. Depending on the particular situation, the presence of nitrogen can be either stabilizing or destabilizing. The motivation for the generalization comes from enhanced oil recovery projects where the exhaust gases from the steam generator are injected into the reservoir along with the steam. This paper considers perturbations on a flat condensation front that is perpendicular to its velocity. The gravitational force along this velocity is included, but the component of the gravitational force perpendicular to the velocity is not. Thus we include the effect of gravity on fingering, but we do not discuss the gravity override problem. In Stability Analysis we present two steps:determination of the motion of a flat condensation front (details are in the Appendix) andevaluation of the characteristic time for growth or decay of a perturbation of that front. In Results wegive the results for a specific reservoir;discuss the sensitivity of these results to the important reservoir parameters (flow velocities and absolute permeabilities),show that, if surface tension and gravitation are unimportant, the stability condition is independent of the absolute permeability and absolute flow rates, anddiscuss the longest wavelength for a stable perturbation. In the final section we discuss the main conclusions. Stability Analysis We consider a homogeneous porous medium with fluids in two regions as illustrated in Fig. 1. A steam/nitrogen mixture is injected at the left, and water (liquid) and nitrogen are produced at the fight. The linear stability analysis proceeds in two main stages and follows the general methods as discussed by Chandrasekhar and the specific application of Miller. First, we assume that the condensation front is flat, moves with constant velocity, v, and has properties that vary with z alone. SPEJ P. 625^