The idea is frequently stated [31] as a Velocity Constraint: Wave velocities in the two-phase region must decrease monoton-ically for zones in which compositions vary continuously as th
Trang 14.2 SHOCKS 51
Ci=CiΙ
Ci=CiΠ
shock at τ shock at τ+∆τ
Figure 4.6: Motion of a shock
original conservation equation In other words, it is a statement that volume is conserved across the shock, just as Eq 4.1.1 states that volume is conserved at locations where all the derivatives exist Eq 4.2.2 says that the velocity at which the shock propagates is set by the slope of a line
that connects the two states on either side of the shock on a plot of F1 against C1 such as that
shown in Fig 4.1
Now we apply the jump condition to determine what happens at the leading edge of the dis-placement zone, where fast characteristics (the characteristics in Fig 4.5 that have high values of
dF1/dS1) intersect the characteristics for the initial composition Point a in Fig 4.7 is the initial composition, which is the composition on the downstream side of the leading shock, and points b,
c, d, e, f, and g are possible composition points for the fluid on the upstream side of the shock Any
of the shock constructions shown in Fig 4.7 satisfies Eq 4.2.2 Hence some additional reasoning is required to select which shock is part of a unique solution to the flow problem
Two physical ideas play a role in that reasoning The first is simply an observation that compositions that make up the downstream portion of the solution must have moved more rapidly than compositions that lie closer to the inlet If not, slow-moving downstream compositions would
be overtaken by faster compositions upstream The idea is frequently stated [31] as a
Velocity Constraint: Wave velocities in the two-phase region must decrease
monoton-ically for zones in which compositions vary continuously as the solution composition path is traced from downstream compositions to upstream compositions
When the velocity constraint is satisfied, the solution will be single-valued throughout Composition
variations that satisfy the velocity constraint are sometimes described as compatible waves, and the velocity constraint may also be called a compatibility condition.
The second idea is that a shock can exist only if it is stable in the sense that it would form again if it were somehow smeared slightly from a sharp jump, as might happen if a small amount of physical dispersion were present, for example That idea can be stated in terms of wave velocities [67, 83, 106] as an
Entropy Condition: Wave velocities on the upstream side of the shock must be greater
Trang 252 CHAPTER 4 TWO-COMPONENT GAS/OIL DISPLACEMENT
than (or equal to) the shock velocity and wave velocities on the downstream side must
be less than (or equal to) the shock velocity (In the examples considered here, the wave velocity can be equal to the shock velocity on only one side of the shock at a time.)
For the application considered here, the condition really has nothing to do with the thermodynamic entropy function, but the name has been universally used in descriptions of solutions to hyperbolic conservation laws since the ideas behind entropy conditions were first derived for compressible fluid flow problems in which entropy must increase across the shock Consider what would happen
to a shock that was slightly smeared if the entropy condition were not satisfied Slow-moving compositions upstream of the shock would be left behind by fast-moving compositions downstream
of the shock, and as a result, the shock would pull itself apart Hence, the entropy condition must
be satisfied if a shock is to be stable A shock that does satisfy the entropy condition is said to
be self-sharpening For a detailed discussion of the various mathematical forms in which entropy
conditions can be expressed, see the review given by Rhee, Aris and Amundson [106, pp 213–220 and pp 341–348]
We now apply the velocity constraint and the entropy condition to obtain a unique solution for two-component displacement Fig 4.8 illustrates possible solutions for the leading shocks indicated
in Fig 4.7 Consider, for example, a shock that connects downstream composition a and upstream composition b The top left panel of Fig 4.8 shows the location of the shock at some fixed time and
also shows how the solution would behave if the concentration of C1 increased smoothly upstream
of the shock The wave velocity, Λ, of the shock (Eq 4.2.2) is given by the slope of the chord that
connects points a and b on Fig 4.7 That velocity is clearly less than one, and hence the a→b
shock moves more slowly than the single-phase compositions downstream of the shock, which have
unit velocity The wave velocity of the composition just upstream of the shock is given by dF1/dC1
at point b That velocity is lower still than the wave velocity of the shock Thus, the a→b shock
violates the entropy condition As the C1 concentration upstream of the shock increases, however, the wave velocities increase to values greater than the shock velocity, a variation that produces
compositions that violate the velocity constraint Hence, a solution that includes a shock from a
to b followed by a continuously varying composition violates both the velocity constraint and the
entropy condition and can be ruled out, therefore
The a→c, a→d, and a→e shocks all satisfy the entropy condition, but all three violate the velocity constraint, as the profiles in Fig 4.8 show The a→g satisfies the velocity constraint, but
it violates the entropy condition because the wave velocity of the upstream composition is lower
than the shock velocity Hence, the only remaining possible solution is that shown for the a→f
shock
The point f is the point at which the chord drawn from point a is tangent to the overall fractional flow curve The a→f shock does satisfy the entropy condition, but it does so in a special way The
wave velocity of the composition C1 of point f is equal to the shock velocity, because the shock
velocity is given by the slope of the tangent a–f, and that chord slope is the same as dF1/dC1 at
point f A shock in which the shock velocity equals the wave velocity on one side of the shock is
sometimes called a semishock [106, pp 217–219] , an intermediate discontinuity [40], or a tangent shock [82] Because the leading shock must be a semishock if it is to satisfy the velocity constraint
and the entropy condition, the composition of the fluid on the upstream side of the shock can be found easily by solving
Trang 34.2 SHOCKS 53
0.0 0.2 0.4 0.6 0.8 1.0
Overall Volume Fraction of Component 1, C1
a
a a
a
a
a
a
a
a
a
a
b c d
e
f
g
Figure 4.7: Possible shocks from the initial state at point a
dF1
dC1| II = F
II
i
C II
i − C I i
The tangent construction described in Eq 4.2.3 and shown in Fig 4.7 is equivalent to the well-known Welge tangent construction [133] used to solve the problem of Buckley and Leverett [10] for water displacing oil
Just as a shock was required in order to make the solution single-valued at the leading edge of the transition zone, another shock is required at the trailing edge The characteristics in Fig 4.4 for the injection composition intersect the characteristics in Fig 4.5 for slow moving compositions,
C1, greater than the shock composition Reasoning similar to that for the leading shock shows that
the trailing shock also is a semishock, this time with the wave velocity on the downstream side of the shock equal to the shock velocity In fact, similar arguments indicate that a shock must form any time the number of phases changes for the fractional flow relation used here
Fig 4.9 shows the resulting tangent constructions for the leading (a→b) and trailing shock
(c→d) Fig 4.10 gives the completed solution profiles of S1 and C1 Each profile includes a zone of
constant state with the initial composition ahead of the leading shock, a zone of continuous variation
of overall composition and saturation between the leading shock and the trailing shock, and finally another zone of constant state with the injection composition behind the trailing shock The
solution in Fig 4.10 is reported as a function of ξ/τ , which is the wave velocity of the corresponding
Trang 454 CHAPTER 4 TWO-COMPONENT GAS/OIL DISPLACEMENT
0
1
C1
ξ a-b
0
1
C1
ξ a-c
0
1
C1
ξ a-d
0 1
ξ a-e
0 1
ξ a-f
0 1
ξ a-g
Figure 4.8: Composition profiles for the leading shocks to various two-phase compositions
value of C1 In this homogeneous, quasilinear problem, the wave velocity of any composition is constant, and hence the position of any composition that originated at the inlet must be a function
of ξ/τ only In fact, Lax [67] showed that the solution to a quasilinear Riemann problem is always
a function of ξ/τ only The spatial position of a given composition C1 can be obtained simply by
multiplying the corresponding value of ξ/τ by the value of τ at which the solution is desired Another version of the solution is shown in Fig 4.11, which includes a τ -ξ diagram and a plot
of the C1 profile at τ = 0.60 Shown in the τ -ξ portion of Fig 4.11 are the trajectories of the leading and trailing shocks and a few of the characteristics The locations, ξ, of the shocks and
the compositions associated with specific characteristics can be read directly from the t-x diagram
for a particular value of τ , as Fig 4.11 illustrates From Fig 4.11 it is easy to see that as the flow
proceeds, the solution retains the shape shown in the profiles of Figs 4.10 and 4.11, but the entire solution stretches as fast-moving compositions pull away from slow-moving ones That behavior is typical of problems in which convective phenomena dominate the transport
Fig 4.11 also illustrates the point that when the entropy condition is satisfied for a particular
Trang 54.2 SHOCKS 55
0.0 0.2 0.4 0.6 0.8 1.0
Overall Volume Fraction of Component 1, C1
a
a
a
a
a
b
c
d
Figure 4.9: Leading and trailing shock constructions
shock, characteristics on either side of the trajectory of a shock either impinge on the shock
trajec-tory or are at least parallel to the shock trajectrajec-tory In the case of the leading shock, for example, the characteristics of the initial composition, which lies downstream of the shock, intersect the shock trajectory, while characteristic just upstream of the shock overlaps the shock trajectory The reverse is true at the trailing shock
Between the trajectories of the shocks is the fan of characteristics associated with the continuous
variation of composition, which is known as a spreading wave, a rarefaction wave, or an expansion wave Because the characteristics all emanate from a single point, the origin, they are also referred
to as a centered wave The change in slope of the characteristics in the spreading wave reflects
the fact that the slope of the fractional flow curve drops rapidly over a fairly narrow range of composition (see Fig 4.2) As a result, the wave velocity declines significantly during the relatively small composition change between the leading and trailing shocks In the solution shown in Fig 4.10 the overall compositions and saturations vary in the two-phase region, but the phase compositions
do not They are fixed by the specified phase equilibrium It is the differing amounts of the two phases present and flowing that change the overall composition and fractional flow
Trang 656 CHAPTER 4 TWO-COMPONENT GAS/OIL DISPLACEMENT
0.0 0.5
1.0
C1
ξ/τ
a
b c
d
0.0 0.5
1.0
S1
ξ/τ
a
b c
d
Figure 4.10: Solution composition and saturation profiles
In the binary gas/oil displacement problem, the leading shock forms because some two-phase mix-tures of injected fluid with initial fluid move rapidly and overtake the initial composition The trailing shock forms because some Component 2 can evaporate into the unsaturated injected vapor How fast the leading and trailing shocks move depends on the initial and injection compositions In this section we examine briefly the sensitivity of the solution to the binary displacement problem
to changes in the initial and injection compositions
Fig 4.12 shows a set of key points on the fractional flow curve Points a, b, c, and d are from the solution discussed in the previous section Points a and d are the initial and injection values, and points b and c are the tangent points for the leading and trailing semishocks Point e is the
saturated vapor phase, and point i is the saturated liquid Point f corresponds to S1= 1−S or, and
point h is that at which S1 = S gc Point g is the intersection of the F1 = C1 line with the overall
fractional flow curve Point j is the inflection point in the fractional flow curve It corresponds to
the maximum in dF1/dC1 shown in Fig 4.2.
Fig 4.13 shows examples of the solutions that result when the initial composition is fixed at
C1a and the injection composition is varied The six panels in Fig 4.13 illustrate changes in the
appearance of the composition profiles for injection compositions with decreasing volume fractions,
C1inj Fig 4.14 shows the corresponding characteristic (τ -ξ) diagrams The following observations
can be made for injection compositions in the regions bounded by the key points in Fig 4.13:
Trang 74.3 VARIATIONS IN INITIAL OR INJECTION COMPOSITION 57
0.0 0.5
1.0
C1
ξ
0.0 0.2 0.4 0.6 0.8 1.0
Trailing Shock
Leading Shock
C1 = 0.68
C1 = 0.71
0
a
a
Figure 4.11: Evaluation of the solution at a specific time, τ , from the τ -ξ diagram.
includes leading and trailing semishocks connected by a spreading wave (see the top
left panel of Fig 4.13), and the τ -ξ diagram shown in the corresponding panel in Fig 4.14 is qualitatively similar to Fig 4.11 As C1inj is decreased, the trailing shock speed
decreases, reaching zero when the injected fluid is vapor saturated with component 2
(C1inj = C1e) The leading portion of the solution is unchanged, however.
dF1/dC1 = 0 There is a trailing shock from the injection composition to C1e, but it has
zero wave velocity, and hence the fan of characteristics in Fig 4.14 extends all the way
to the ξ = 0 axis The leading portion of the solution remains unchanged.
f to b: An injection composition between f and b has nonzero wave velocity, and as a
result, the solution in the lower left panel of Fig 4.13 shows a zone of injection composi-tions at the upstream end that all propagate with the same wave velocity That portion
of the solution has a set of parallel characteristics in Fig 4.14 that emanate from the
ξ = 0 axis The fan of characteristics that represents the spreading wave terminates at
the characteristic that represents that propagation of the injection composition When
C1inj = C1b, the entire solution upstream of the leading shock is that zone of constant
Trang 858 CHAPTER 4 TWO-COMPONENT GAS/OIL DISPLACEMENT
0.0 0.2 0.4 0.6 0.8 1.0
Overall Volume Fraction of Component 1, C1
a a a a
a
a a
a a
a
a
i h
g j
f e d
b
c
Figure 4.12: Composition ranges for variation of injection and initial compositions
composition with C1 = C1b The leading shock velocity is still unchanged, however.
contin-uous variation from the tangent shock point to the injection composition is prohibited
by the velocity rule A leading shock to the injection composition followed by a set of constant compositions at the injection composition satisfies the entropy condition and velocity constraint The top right panel in Fig 4.14 indicates, for example, that the characteristics associated with the injection composition intersect the leading shock tra-jectory, as do the characteristics associated with the initial composition, an indication that the entropy condition is satisfied The leading shock velocity is given by Eq 4.2.2
with the known compositions C1init and C1inj The leading shock velocity is now lower
than that of the leading semishocks that form for C1inj > C1b When C1inj = C1g, the
leading shock has unit velocity
g to h: A leading shock directly from the injection composition to the initial
com-position is no longer possible because it would violate the entropy condition A shock from the initial composition to the injection composition would have a velocity less than one The characteristics of the initial composition (see the middle right panel of Fig 4.14)would not intersect the shock trajectory, and hence, the entropy condition would not be satisfied The only path available that does not violate the entropy condition is
Trang 94.3 VARIATIONS IN INITIAL OR INJECTION COMPOSITION 59
0
1
C1
ξ
d-e
C1inj =0.975
0
1
C1
ξ
e-f
C1inj =0.920
0
1
C1
ξ
f-b
C1inj =0.770
0 1
ξ
b-g
C1inj =0.560
0 1
ξ
g-h
C1inj =0.320
0 1
ξ
h-i
C1inj =0.215
Figure 4.13: Effect of changes in injection composition
a leading shock with unit velocity to the saturated liquid composition, C i
1, followed by
a slower trailing shock to the injection composition The characteristics of the injection composition intersect the trailing shock, and the characteristics of the initial composi-tion intersect the leading shock The zone between the two shocks is what is known as
a zone of constant state The composition i has two wave velocities: one is the leading
shock velocity, and the other is the trailing shock velocity
h to i: The situation is the same as for g-h, except that the trailing shock has zero
velocity The trailing shock is a jump from the injection composition to the saturated
liquid composition, C1i.
Fig 4.13 indicates that the form of the solution changes significantly as the injection composition changes Solution behavior also changes if the initial composition is changed Fig 4.15 illustrates
what happens if the injection composition is fixed at C1inj = C1d = 1, and the initial composition
is varied in ranges bounded by the key points shown in Fig 4.12 Four composition intervals are
Trang 1060 CHAPTER 4 TWO-COMPONENT GAS/OIL DISPLACEMENT
0
1
d-e
0
1
e-f
0
1
ξ f-b
0 1
b-g
0 1
g-h
0 1
ξ h-i
Figure 4.14: τ -ξ diagrams for displacements illustrated in Fig 4.13.
important:
a to i: As the amount of component 1 in the initial mixture increases, the velocity of
the leading semishock also increases slightly (the slope of the tangent drawn from the initial composition to the fractional flow curve increases), and the composition on the
upstream side of the leading shock decreases slightly below point b The remainder of
the composition profile upstream of the leading shock is unaffected
sub-stantially and the shock composition to approach j.
leading portion of the solution is simply a spreading wave
evaporation shock is what is known as a genuine shock, a jump from C init
1 to C1inj, with