Construct a rarefaction along the shortest tie line that connects the equal-eigenvalue point or the intermediate shock point to the tangent shock point for the single-phase composition o
Trang 15.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 111
7 Construct a rarefaction along the shortest tie line that connects the equal-eigenvalue point
or the intermediate shock point to the tangent shock point for the single-phase composition (oil or gas) associated with the shortest tie line
8 Determine whether a tie-line rarefaction can occur on the longest tie line A rarefaction can occur if variation along the tie line from the landing point of the intermediate shock or the intersection of the nontie-line path with the tie line to the tangent shock point satisfies the velocity rule If so, the shock from the single-phase composition to the longest tie line is a semishock If not, a genuine shock is constructed from the landing or intersection point to the single-phase composition
The fact that solution construction must begin on the shortest tie line arises from two ob-servations about the geometry of composition paths and shocks The first observation applies to displacements in which a rarefaction connects the two key tie lines In that case, there are two points at which the appropriate nontie-line path intersects the two key tie lines Both points must be switch points at which the velocity rule is satisfied The argument given in Section 5.3 indicates that one of the two points must be the equal-eigenvalue point The geometry of nontie-line paths (see Fig 5.9) indicates that the point of tangency of the nontie-nontie-line path to a tie nontie-line (which is the point at which eigenvalues are equal) occurs on the shorter of the two tie lines If the equal-eigenvalue point on the longer tie line were selected instead, the paths traced would not reach the shorter tie line Solution construction can proceed by the steps outlined above once the equal-eigenvalue point is found on the shorter of the two key tie lines
The second observation applies to displacements in which the two key tie lines are connected
by a shock In that case, the nontie-line rarefaction is replaced by a semishock with a wave speed that matches the tie-line eigenvalue on the same tie line that includes the equal-eigenvalue point for the rarefaction path, again, the shorter of the two key tie lines
Figure 5.19 illustrates the construction of a semishock between tie lines (the example shown is
the c→d shock in Fig 5.16) The shock balance for the intermediate shock (written for component
1) is
Λcd= F
c
1 − F d
1
C c
1− C d
1
= C
X
1 − F c
1
C X
1 − C c
1
= C
X
1 − F d
1
C X
1 − C d
1
= ∂F1
Fig 5.19 shows the appropriate tangent construction: a chord drawn from point X, the intersection
point of the two tie lines, to a tangent point on the fractional flow curve for the shorter tie line
locates point c, the point that satisfies Eq 5.5.1 The intersection of the same chord with the fractional flow curve for the longer tie line gives point d.
The fractional flow curves shown in Fig 5.19 are typical of systems in which y c
1 < y1d and
M c < M d, both reasonable physical assumptions In such systems, it is possible to construct a tangent to the fractional flow curve for the shorter tie line that also intersects the fractional flow curve for the longer tie line If, on the other hand, the tangent had been drawn to the fractional
flow curve for the longer tie line, to point d∗ in Fig 5.19, it would not intersect the curve for the
shorter tie line In that case, there would be no solution to Eq 5.5.1 Hence, the tangent must
be constructed to the shorter tie line, and therefore it is appropriate to start solution construction with the shorter tie line
Trang 20.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Overall Volume Fraction of Component 1, C1 a
a a
a a
a
a
a
b c
d
f
X
d *
Figure 5.19: Tangent constructions for a shock between tie lines
Trang 35.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 113
C1
C2
C3
a aa
aa a
a
b c
d f
0
1
Sg
a a a a
a a
a
b c
d d
e f
HVI Vaporizing Drive
C1
C2
a
a a
a
bc
d f
0
1
Sg
ξ
a
a a a
a
a a
a a a
a
b c
d d f
LVI Vaporizing Drive
C1
C2
C3
a
a
aa a a
b
cd e
0
1
Sg
a a a a
a
a
b b
c d
e
HVI Condensing Drive
C1
C2
C3
a
a
a a
a
e
0
1
Sg
a a
a a a
a
b b
c d e
LVI Condensing Drive
Figure 5.20: Structure of solutions for condensing and vaporizing gas drives
Trang 4Table 5.4: Nontie-Line Shocks and Rarefactions Envelope Curve and Intermediate Process Shortest Composition Tie Line Intersections K-Value Name Tie Line Variation
Vapor Side K2< 1 LVI Condensing Injection Gas Rarefaction Vapor Side K2< 1 LVI Vaporizing Initial Oil Shock
Liquid Side K2> 1 HVI Condensing Injection Gas Shock
Liquid Side K2> 1 HVI Vaporizing Initial Oil Rarefaction
Tie-line length also indicates whether a displacement is condensing or vaporizing When the initial oil tie line is the shorter of the two key tie lines, the displacement is a vaporizing gas drive When the injection gas tie line is the shorter tie line, the displacement is a condensing gas drive The steps outlined above determine the following segments of the solution for displacements
in which a single-phase vapor displaces a single-phase oil, listed in order from the downstream to upstream locations Fig 5.20 illustrates the important composition variations and profile segments
1 Leading Shock, a→b A leading shock is always present if the initial composition is a
single-phase liquid In a vaporizing gas drive (initial oil tie line is shorter than the injection gas tie line), it will always be a semishock In a condensing gas drive (injection gas tie line is shorter than the initial oil tie line), it may be a semishock or a genuine shock
2 Tie-Line Rarefaction, b-c In a vaporizing gas drive, this rarefaction along the initial oil
tie line is always present It connects the landing point of the leading semishock with the point at which the nontie-line composition variation begins, either the equal-eigenvalue point
or the semishock point of the intermediate shock In a condensing gas drive, this segment is missing if the leading shock is a genuine shock, as it often is
3 Composition Variation between Tie Lines, c-d or c→d If the composition variation
is a rarefaction, the wave velocity on the nontie-line path will match the tie line eigenvalue on the shorter tie line (at the equal-eigenvalue point), and there will be a zone of constant state associated with the point at which the nontie-line path intersects the longer tie line If the composition variation is a shock, the shock velocity will match the tie-line eigenvalue on the shorter tie line, and there will be a zone of constant state associated with the shock landing point on the longer tie line
4 Tie-Line Rarefaction, d-e In a condensing gas drive, this segment, which connects the nontie-line path switch point on the injection gas tie line (point d to the trailing shock point (point e), is always present That shock is always a semishock In a vaporizing drive, this
segment is present only if the trailing shock is a semishock Otherwise, this segment is missing
5 Trailing Shock, e→f A trailing shock is always present as long as the injection gas is a
single-phase mixture In a condensing drive, it is always a semishock In a vaporizing drive
it may be a semishock or a genuine shock
Trang 55.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 115
C1
C2
a
a a
a 1
bc
d f
0.0
1.0
Sg
0.0 0.5 1.0 1.5 2.0 2.5
ξ a
a a
a a a
a 1
b 1
c
d d f
(a) Initial mixture a1
C1
C2
C3
a a a
a a
a2
bc2
d f
0.0
1.0
Sg
0.0 0.5 1.0 1.5 2.0 2.5
ξ
a
a a a
a a a
a a a
a2
b 2
c
d d f
(b) Initial mixture a2
C1
C2
C3
a
a a
a a3 c
d f
0.0
1.0
Sg
0.0 0.5 1.0 1.5 2.0 2.5
ξ
a
a a a
a
a 3
c
d d f
(c) Initial mixture a 3
C1
C2
C3
a
aa
a4
d4f
0.0
1.0
Sg
0.0 0.5 1.0 1.5 2.0 2.5
ξ
a a
a4
d4 f
(d) Initial mixture a 4 Figure 5.21: Effect of variations in initial composition All initial compositions lie on the same
initial tie line or its extension (a) a 1 is a single phase liquid (b) a 2 has a gas saturation of 5
percent (c) a 3 has a gas saturation of 30 percent (d) a 4 has an initial gas saturation of 80 percent
Trang 6C4
a a a
a
a a
a
a aa
0.0
1.0
Sg
ξ
a
a a
a a
a
a
a a
a a
a
a
a a
a a
C
Figure 5.22: Changes in solution composition route as the initial mixture is enriched in the
inter-mediate component for displacements conducted at 1600 psia (109 atm) and 160 F (71 C) Phase
behavior was calculated with the Peng-Robinson equation of state Initial compositions are Point
A: C CO A 2 = 0, C C A4 = 0.195950, C C A10 = 0.804050, Point B: C CO B 2 = 0, C C B4 = 0.373735, C C B10 =
0.626265, and Point C: C CO C 2 = 0, C C C4 = 0.475887, C C C10 = 0.524113
Trang 75.6 MULTICONTACT MISCIBILITY 117
The patterns shown in Fig 5.20 may change if the initial composition changes To illustrate the variations in patterns of shocks and rarefactions as the initial composition varies, we consider the displacement of Fig 5.16, and vary the initial composition along the extension of the initial tie line and the tie line itself Fig 5.19 shows the fractional flow curves for the initial and injection tie lines Fig 5.21 shows composition paths and gas saturation profiles for the four patterns of displacement behavior
If the initial composition lies between points a and b in Fig 5.19, the variations in patterns all
involve the leading shock and rarefaction along the initial tie line The discussion of Figs 4.12 and
4.15 describes the changes that are observed for initial composition variations between a and b For example, for initial compositions between a and the inflection point on the fractional flow curve for the initial tie line (point j in Fig 4.12), the patterns of shocks do not change, though the wave velocity of the leading shock increases as the initial composition moves from a to j Comparison
of the positions of the leading shocks in Figs 5.21a and 5.21b shows that the leading shock speeds
up significantly as the composition is moved from a 1 in the single-phase region to a 2 with a gas saturation of 5 percent in the two-phase region The intermediate and trailing shock compositions are unchanged, however
For initial compositions between the inflection point on the initial tie line (Point j on Fig 4.12) and the tangent shock point for the intermediate shock (Point c in Figs 5.16 and 5.19), the leading
shock is missing, and the leading portion of the profile begins with a rarefaction along the initial tie line, but the trailing portions of the displacement are unchanged Fig 5.21c illustrates that situation for an initial gas saturation of 30 percent
For initial compositions between the semishock point for the intermediate shock (Point c on
Figs 5.16 and 5.19) and the vapor composition on the initial tie line, a tangent intermediate shock is
no longer possible Variation along the initial tie line from the initial composition to Point c would
violate the velocity rule Instead, there is an immediate shock to the injection tie line, but that shock is a genuine shock Fig 5.21d shows that the entire displacement consists of an evaporation shock from the initial composition to the injection composition when the initial gas saturation is
80 percent This behavior is typical of condensate systems, in which the saturation of liquid is quite low Thus, for initial compositions in the two-phase region, one or more of the segments that appear in the displacment of a liquid phase by a vapor phase may be missing
In this section we consider how displacement efficiency in a gas injection process changes depending
on the relative locations of the injection gas and initial oil tie lines Displacement efficiency depends quite strongly on the relative locations of the key tie lines with respect to the plait point, as we
now show To illustrate the important ideas behind developed miscibility or multicontact miscibility,
consider first a vaporizing gas drive that is fully self-sharpening (in other words, the two key tie lines are connected by a shock)
Trang 85.6.1 Vaporizing Gas Drives
Figure 5.22 shows what happens as the initial mixture is enriched in Component 2 in a vaporizing
gas drive Fig 5.22 shows the compositions of the three initial mixtures a, b, and c, the composition
routes, and the resulting saturation profiles for the three displacements Those solutions show that
as the initial mixture is enriched in the intermediate component, the leading shock slows, and the intermediate and trailing shocks speed up, and displacement efficiency improves, because the amount of oil phase present in the transition zone decreases
To see why this behavior occurs, consider the shock balances for the leading, intermediate, and trailing shocks are
Λlead i = F
a
i − F b i
C a
i − C b i
, Λint i = F
c
i − F d i
C c
i − C d i
, Λtrail i = F
d
i − F e i
C d
i − C e i
i = 1, n c (5.6.1) Now consider what happens in the limit as the initial mixture is enriched enough in Component
2 that the initial tie line becomes the critical tie line that is tangent to the binodal curve at the
plait point The plait point terminates the equivelocity curve, where f1 = S1 As a result, Point
b, the landing point of the leading shock, lies on the equivelocity curve, and F i b = C i b Note also
that F i a = C i a because the initial mixture is single-phase Therefore, it must be that
Λlead i = F
a
i − F b i
C a
i − C b i
= C
a
i − C b i
C a
i − C b i
Because the critical tie line has zero length, Point c must also be the plait point, and hence
F i c = C i c And as the initial mixture is enriched in the intermediate component enough to approach
the critical tie line, Point d on the injection gas tie line approaches the vapor locus of the binodal
curve, reaching it when the initial tie line is the critical tie line If a nontie-line path connects the initial and injection tie lines, it becomes the binodal curve in the limit, and all compositions
on that path have λ nt = 1 If a shock connects the two tie lines, it is replaced in the limit by
an indifferent wave with unit velocity, because a path switch at the equal-eigenvalue point (the critical point) followed by a rarefaction along the binodal curve now does not violate the velocity constraint However, shock velocities go smoothly to the limit because exactly the same velocity
that is obtained for a shock from the critical point to Point d on the binodal curve on the injection
tie line At that point, S1= f1 and therefore, F i d = C i d Also, the injection mixture is single-phase,
so F i e = C i e Thus, it must also be that
Λint i = F
c
i − F d i
C c
i − C d i
= C
c
i − C d i
C c
i − C d i
and
Λtrail i = F
d
i − F e i
C i d − C e
i
= C
d
i − C e i
C i d − C e
i
Thus, we have proved that in the limit as the inital tie line becomes the critical tie line, all three shocks coalesce into a single shock and move with unit velocity, and the resulting displacement is piston-like That displacement moves all of the oil ahead of the shock, and hence all of the initial oil is recovered at one pore volume injected when the shock reaches the outlet Similar arguments apply when the intermediate shock is replaced by a rarefaction In that case, the rarefaction along
Trang 95.6 MULTICONTACT MISCIBILITY 119
the nontie-line path follows the vapor portion of the binodal curve, which has unit velocity, when the initial tie line is a critical tie line Here again, the entire transition zone between injected fluid and initial oil moves with unit velocity, and no oil is left undisplaced It is this perfectly efficient
displacement that is meant by the term multicontact miscibility While the intial and injection
mixtures are not miscible in the sense that they could be mixed in any proportions and form only one phase, the combination of two-phase flow and phase equilibrium causes chromatographic separations that lead to a composition route that avoids the two-phase region
The analysis of condensing gas drives is similar In condensing gas drives, the injection gas is enriched with the intermediate component, and it is the injection gas tie line that becomes the critical tie line with sufficient enrichment The appropriate shock balances are again those given
in Eqs 5.6.1 When the injection tie line is the critical tie line, Point d is the plait point, where
f1 = S1, and as a result, F i d = C i d Eq 5.6.4 indicates, therefore that the trailing shock has unit velocity
If a rarefaction connects the injection and initial tie lines, it becomes, in the limit as the injection tie line reaches the critical tie line, the equivelocity curve All compositions on that curve have unit velocity, so that indifferent wave is equivalent to a unit velocity shock If, on the other hand, the nontie-line path is self-sharpening, the shock from the injection tie line to the initial tie line
connects the plait point, where F i c = C i c to the intersection of the equivelocity curve and the initial
tie line, where F i b = C i b The intermediate shock velocity is then
Λtrail i = F
b
i − F c i
C b
i − C c i
= C
b
i − C c i
C b
i − C c i
Finally, the leading shock connects a point on the equivelocity curve where F i b = C i b, and at
the initial composition, F i a = C i a, and therefore, Eq 5.6.2 shows that this shock also has unit velocity Thus, when the injection gas tie line is a critical tie line in a condensing gas drive, all the compositions present in the solution route move with unit velocity, and the displacement is piston-like
The analysis of this section indicates that multicontact miscible displacement occurs in ternary systems when either the initial tie line or the injection tie line is a critical tie line, a tie line of zero length that is tangent to the binodal curve at the critical point Condensing gas drives are
multicontact miscible when the injection gas tie line is the critical tie line, and vaporizing gas drives are multicontact miscible when the initial oil tie line is critical
It is relatively easy to adjust composition of the injection gas in field applications, so it is
common to determine a minimum enrichment for miscibility (MME) for condensing gas drives It
is important to note, however, that the MME depends on the displacement pressure, because the size of the two-phase region for a given ternary system as well as the locations of the plait point and the critical tie line depend on pressure Thus, as reservoir pressure changes during the life of a displacement process, the MME will also change While reservoir temperature is usually taken to
be fixed, any change due, for example, to injection of cold water, would also change the MME
Trang 10In oil field settings it is generally impossible to adjust the composition of the initial oil, and hence
in vaporizing gas drives, it is the pressure that is adjusted to find the minimum miscibility pressure (MMP) for a given injection gas composition In ternary systems, the MMP in a vaporizing gas drive does not depend on the injection gas composition In systems with more components, however,
it is possible for the MMP to depend on injection gas composition Thus, in field displacements it
is often reasonable to consider both adjustment of reservoir pressure and injection gas composition
as ways to achieve the efficient displacement that results from multicontact misibility
When components change volume as they transfer from one phase to another, the appropriate
balance equation on moles of component i is Eq 2.3.9 written for the three components,
∂G1
∂τ +
∂H1
∂G2
∂τ +
∂H2
∂G3
∂τ +
∂H3
where
G i = x i1ρ 1D S1+ x i2ρ 2D(1− S1), (5.7.4)
H i = v D (x i1ρ 1D f1+ x i2ρ 2D(1− f1)). (5.7.5) Manipulations similar to those of Sections 4.4 and 5.1 yield the eigenvalue problem,
where
H (u) =
⎡
⎢
∂H1
∂z1
∂H1
∂z2
∂H1
∂z3
∂H1
∂v D
∂H2
∂z1
∂H2
∂z2
∂H2
∂z3
∂H2
∂v D
∂H3
∂z1
∂H3
∂z2
∂H3
∂z3
∂H3
∂v D
⎤
[G(u)] =
⎡
⎢
∂G1
∂z1
∂G1
∂z2
∂G1
∂z3 0
∂G2
∂z1
∂G2
∂z2
∂G2
∂z3 0
∂G3
∂z1
∂G3
∂z2
∂G3
∂z3 0
⎤
and
u T = [dz1, dz2, v D ]. (5.7.9)
In these expressions, z i is the overall mole fraction of component i Only two of the mole fractions are
independent, because they sum to one, but all three of the conservation equations are independent,
and they are needed because the flow velocity, v D, must also be determined