Summary
Three-phase flow often occurs in reservoirs, particularly for secondary or
tertiary oil recovery methods such as miscible gas or chemical flooding. In
these cases, there is often significant mutual solubility of components in the
phases. Unfortunately, there is insufficient understanding of how three
partially miscible phases can affect flow. Furthermore, there are currently no
benchmark analytical solutions available to validate numerical simulations for
this complex flow regime.
In this research, compositional solution routes are developed by the method
of characteristics (MOC) for 1D, dispersion-free flow wherein up to three
partially miscible flowing phases may be present. The method is applied to a
water/alcohol/oil system that exhibits a large three-phase region in laboratory
experiments. Unique solutions are found based on continuity arguments,
shock-jump conditions, entropy constraints, and velocity constraints. The
analytical solutions are compared to fine-grid finite-difference simulations to
verify that they converge to the same dispersion-free limit.
The results show that composition routes within the three-phase region often
exhibit one phase below its residual saturation so that only two phases are
flowing. As miscibility is approached, cumulative oil recovery initially
declines because of the development of constant states in the solution, which
cause the leading shock to speed up. We show that multicontact miscibility is
developed at the critical point of one two-phase region and along the boundary
of the three-phase region where all shocks and waves flow at a dimensionless
velocity of one. Last, we show that injectivity changes by a factor of two for
the specific relative permeabilities and injection compositions used.
Introduction
Flow in a reservoir often consists of three or more flowing phases. For
example, injection of CO 2 can form three hydrocarbon phases under realistic
reservoir temperature and pressure conditions. 1 Three hydrocarbon phases are
believed to occur in many west Texas CO 2 floods, and the numerous phases can
have an adverse effect on wellbore injectivity. 2 Although three-phase flow is
believed to occur often, many fundamental questions remain about the effect of
three-phase flow on oil recovery, particularly its effect on the miscible gas
process.
Helfferich 3 first presented a general analytical theory using the MOC for
multicomponent multiphase displacements. Construction of solutions to specific
gas displacements has been ongoing ever since. For example, the analytical
theory for two-phase partially miscible gas displacements has been studied
extensively for floods with an arbitrary number of components. 4--6
Many researchers have also studied immiscible three-phase three-component
MOC displacements. 7--14 In three-phase immiscible systems the mass-balance
equations governing flow may be elliptic, hyperbolic, or nonstrictly hyperbolic
depending on the relative permeability model used. 7 Gonzales and Araujo 11 and
Sahni et al. 12 have studied the effect of relative permeabilities on
composition routes. Recently Juanes and Patzek 13,1.proposed new relative
permeabilities for which the mass-balance equation is hyperbolic at every
interior point of the three-phase region. The elliptic and strictly hyperbolic
cases are beyond the scope of this paper; however, displacements with elliptic
regions in the mass-balance equation have been studied extensively. 7
There has been considerable mathematical research dedicated to understanding
nonunique solutions in three-phase immiscible flow. 15--18 Nonuniqueness
results from the introduction of transitional shocks when Lax and Liu entropy
conditions fail to provide a composition route. 7 In the example cases studied
here, no transitional shocks were encountered.
Several researchers have examined the numerical simulation of three-phase
partially miscible flow. Orr, 19 for example, outlined a numerical simulator
that allows for partially miscible flow of up to four components and four
phases. Pongpitak 20 used this simulator to model three-phase waterflooding for
cores saturated with the alcohol n-butanol (NBA) and C 16. The simulations
significantly underestimated the oil recovery compared with core experiments.
Fanchi 21 showed the relative unimportance of small three-phase regions using a
similar simulator on pseudoternary representations of CO 2 injection with crude
and synthetic oils. Gardner et al. 22 obtained an excellent match to slim-tube
experiments with numerical simulation using pseudoternary representations of CO
2 injected into Wasson crude. Moreover, Gardner et al. 22 showed that the
numerical composition route is dependent upon the level of numerical
dispersion.
There is very little in the literature on the development of MOC solutions
to three-phase partially miscible flow. Giordano and Salter 23 compared
numerical simulations, corefloods, and one MOC solution for a partially
miscible three-phase water/NPA/DMP system. The MOC and simulated composition
routes agreed well, although the match to corefloods was only qualitative. The
development of this one MOC solution was not thoroughly explained in their
paper.
In this paper, the analytical theory using MOC is extended to
three-component, three-phase partially miscible flow for a variety of injection
compositions. Unique solutions are found for a three-phase partially miscible
flow problem with a large three-phase region. The MOC solutions are compared
with fine-grid finite-difference solutions. Changes in injectivity and
cumulative oil recovery with alcohol enrichment are also studied.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
12 January 2004
- Revised manuscript received:
17 February 2005
- Manuscript approved:
24 February 2005
- Version of record:
15 June 2005
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