Will Vertical Mixing Improve Oil Recovery for Gravity Dominated Flows in Heterogeneous Reservoirs?
Authors: de Riz L. , Muggeridge A.H. 
Published: December 2007

Abstract
Oil recovery from gas injection is usually adversely affected by gravity. Shales are believed to improve recovery in this situation, principally because there is more opportunity for vertical mixing to disperse the gravity tongue. This paper uses analytical methods in conjunction with detailed simulation to valuate this hypothesis. We show that typical field scale simulations will give optimistic predictions for recovery. Vertical mixing does improve recovery, but not by the amount predicted by field scale simulations. In addition we show that theoretical methods tend to over-estimate the amount of vertical mixing as they neglect the effects of viscous cross-flow. Vertical mixing improves recovery by reducing the mobility ratio between the gas-oil mixture in the tongue and the bypassed oil. This suggests that the Todd and Longstaff model may be a better way to represent the effects of shales than an effective vertical permeability.

Introduction
Oil recovery from gas injection is usually adversely affected by gravity. The unfavorable mobility ratio and large density difference cause the gas to over-ride the oil in an unstable gravity tongue (figure 1). This causes early gas breakthrough and a reduced vertical sweep efficiency. This applies both to miscible and immiscible gas injection.
Recovery is improved in thick reservoirs or reservoirs with low vertical permeability as the tendency for gas to over-ride the oil is reduced. Vertical equilibrium cannot be established on the time-scale of the displacement.
Gas over-ride is also believed to be reduced when there are numerous discontinuous shales (see figures 2 and 3) becausethe shales act as baffles slowing down the segregation of gas and oilthe low vertical shale separation means the gravity tongue between shales is very thin and easily dispersed resulting in a more piston-like displacement between the shales.
In the limit assumption (2) results in a Stiles type of displacement in reservoirs with many continuous shales.
Even detailed field scale simulations cannot represent discontinuous shales with more than one or two grid block's vertical separation because of computer memory and CPU-time constraints. As a result the extent to which a gravity tongue is dispersed by diffusion, velocity dependent dispersion or capillary pressure has not been fully investigated. In effect detailed field scale simulations are assuming that (2) above is valid. Coarser grid simulations tend to use an effective vertical permeability to represent the shales influence on flow.
If the gravity tongue is not completely dispersed between shales then field scale simulations of oil recovery will be overly optimistic. This applies whether the shales are represented explicitly or by a reduced bulk vertical permeability. The extent of the inaccuracy will depend upon the level of vertical (transverse) mixing that occurs between shales. This paper uses theoretical analysis and detailed simulation to determine the level of transverse (vertical) mixing of gravity tongues flowing between shales. It examines the particular problem of gravity dominated miscible gas injection in which the principal mixing mechanisms are molecular diffusion and velocity dependent dispersion. First contact miscibility is assumed. The effect of capillary pressure in the immiscible limit for the same displacement is also investigated. Compositional effects have been ignored in the interests of simplicity.

Theoretical Analysis
Dietz's theory shows that the thickness of a gravity tongue as a fraction of reservoir thickness is independent of the reservoir thickness. After breakthrough, the fractional tongue thickness b at the production well is given by:
(1)
and the oil recovery is given approximately by
(2)
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