5.2 SHOCKS 91 C1 Switch at a a a λa nt λa t 0 ξ/τ C1 Switch at b a λb=λb t nt 0 ξ/τ C1 Switch at c a a λc t λc nt 0 ξ/τ Figure 5.11: Composition profiles for path switches Points a, b, and c are those shown in Fig 5.9 Path switches at points a and b satisfy the velocity constraint, but a switch at point c violates it and gives a solution profile that is multivalued 92 CHAPTER TERNARY GAS/OIL DISPLACEMENTS drop in wave velocity from unit velocity in the single-phase region to zero velocity on the tie line just inside the two-phase region In fact, any continuous variation would place compositions with zero wave velocity downstream of faster moving two-phase compositions upstream, which would violate the velocity constraint Therefore, a shock must form, just as it did in the the binary displacements of Chapter If a shock forms, then it must satisfy the jump condition derived in Section 4.2 (Eq 4.2.2), which actually applies for any number of components It states that Λ= FiII − FiI , CiII − CiI i = 1, nc (5.2.1) If the fluid on one side of the shock is single-phase, then FiI = CiI , and hence, Eq 5.2.1 can be rearranged to give CiI = FiII − ΛCiII , 1−Λ i = 1, nc (5.2.2) Substitution of the definitions of CiII and FiII in terms of the phase compositions (Eqs 5.1.3 and 5.1.4) yields (Helfferich [31]) CiI = yi f1 − ΛS1 1−Λ + xi (1 − f1 ) − Λ(1 − S1 ) , 1−Λ i = 1, nc (5.2.3) Eq 5.2.3 indicates that the overall volume fraction of component i in the single-phase mixture is a linear combination of the volume fractions of component i in the equilibrium liquid and vapor Hence, the overall composition of the single-phase mixture must lie on a straight line determined by the equilibrium phase compositions That line is a tie line, of course Thus, we have shown that a shock that connects a single-phase composition with a two-phase composition must occur along the extension of a tie line The fact that a shock can enter or leave the two-phase region only along a tie line extension identifies two key tie lines They are the injection tie line and the indextie line!initial oil initial tie line, which have extensions that pass through the injection and initial compositions The same two tie lines also turn out to be important in problems with more than three components It frequently happens (see Appendix A) that application of the entropy condition and the velocity constraint requires that a phase-change shock be a semishock in which the shock velocity matches the tie line eigenvalue at the landing point in the two-phase region In such cases, the shock velocity is given by λII = t 5.2.2 II df1 F II − FiI = Λi = iII II dS1 Ci − CiI i = 1, nc (5.2.4) Shocks and Rarefactions between Tie Lines A shock that connects two points within the two-phase region arises when continuous variation along a nontie-line path is not possible because eigenvalues increase as the path is traced from upstream compositions to downstream ones (a shock along a tie line connecting two points within 5.2 SHOCKS 93 the two-phase region is possible, but not unless the initial or injection composition is in the twophase region) Continuous variation along such a path would violate the velocity constraint, and hence a shock must form Such shocks are sometimes called self-sharpening waves To determine when self-sharpening waves occur, we consider how eigenvalues vary along a nontie-line path Let η be a parameter that varies monotonically along the nontie-line path Differentiation of the expression for λnt in Eq 5.1.24 with respect to η gives dλnt = dη C1 + p = dp dF1 + dη dη C1 + p − F1 + p (C1 + p)2 dp dC1 + , dη dη dp dp dF1 dC1 + − λnt + dη dη dη dη (5.2.5) The derivative dF1 /dη can be related to λt and ent by the following manipulations: ∂F1 dC1 ∂F1 dy1 dF1 = + , dη ∂C1 dη ∂y1 dη = λt + ∂F1 dy1 ∂y1 dC1 dC1 dη (5.2.6) The value of dy1 /dC1 is given by ent (Eq 5.1.25), λnt − λt dy1 = ∂F dC1 ∂y (5.2.7) Substitution of Eqs 5.2.6 and 5.2.7 into Eq 5.2.5 gives the desired result, e C1 − F1 dp F1 − C1 dC1 dλnt = = , e dη (C1 + p)2 dη (C1 − C1 )2 dη (5.2.8) e where C1 denotes the overall volume fraction of component on the envelope curve Nontie-line paths not cross the equivelocity curve (where C1 = F1 ), so the sign of dλnt/dη on a particular e path is determined by the sign of dC1 /dη Eq 5.2.8 makes it easy to determine when a shock must connect two tie lines Whether λnt increases or decreases as a nontie-line path is traced can be determined easily if the envelope curve e can be drawn (or even just sketched) by finding whether C1 increases or decreases as the path is traced The following example describes the patterns of shock and rarefaction behavior that are possible in ternary systems Constant K-Values To illustrate how these ideas apply to a simple system, we assume that K-values are independent of composition Also, we choose η = y1 For constant K-values, the slope and intercept of each tie line are obtained by inserting the definitions of the K-values into the definitions of α and φ (Eq 5.1.6), which gives α= and K2 − x2 , K1 − x1 (5.2.9) 94 CHAPTER TERNARY GAS/OIL DISPLACEMENTS φ= K1 − K2 x2 K1 − (5.2.10) e e Expressions for C1 and dC1 /dη can now be obtained from Eq 5.1.12 by differentiating Eqs 5.2.9 and 5.2.10 and using the equation for the liquid portion of the binodal curve (Eq 3.5.3), K1 x1 + K2 x2 + K3 (1 − x1 − x2 ) = (5.2.11) e The resulting expression for C1 is e C1 = y1 K1 K1 − K2 − K2 K1 − K3 , − K3 (5.2.12) e and the expression for dC1 /dη is e dp 2y1 dC1 =− = dη dη K1 K1 − K2 − K2 K1 − K3 − K3 (5.2.13) Eq 5.2.12 is an explicit expression for the envelope curve , a segment of parabola when Kvalues are constant Fig 5.12 shows envelope curves for the two possible situations, one in which the intermediate component (C2 ) partitions preferentially into the vapor phase (K2 > 1) and one in which the intermediate component prefers the liquid phase (K2 < 1) If K-values are found from an equation of state, an expression for the envelope curve is not easy to find, but the curve can be sketched easily if a few tie lines are known The argument that follows applies whether K-values are constant or not Consider the two tie lines shown in Fig 5.12a, in which K2 < Suppose that tie line A is the initial oil tie line, and tie line B is the injection gas tie line For gas displacing oil, some displacement composition route must connect the two tie lines If a nontie-line path is traced from e tie line A to tie line B, C1 decreases as the composition changes from the downstream (oil) tie line e to the upstream (injection gas) tie line Therefore, dC1 /dη < 0, and Eq 5.2.8 indicates that λnt decreases as the nontie-line path is traced upstream Such a variation satisfies the velocity rule, and hence a nontie-line rarefaction is permitted If, on the other hand, tie line B is the initial oil tie line, and tie line A is the injection gas tie e line, a shock is required As the nontie-line path is traced upstream from tie line B, C1 increases e as does η, dC1 /dη > 0, and according to Eq 5.2.8, λnt increases as the path is traced upstream That composition variation would violate the velocity rule, which requires that wave velocities of compositions upstream be lower than those downstream, and therefore composition variations along the nontie-line path are self-sharpening Hence a shock is required Similar reasoning for tie lines C and D in Fig 5.12b reveals that when K2 > 1, nontie-line paths are self-sharpening when tie line D is the injection gas tie line, and a rarefaction occurs when C is the injection tie line Table 5.1 summarizes those patterns While the examples in Fig 5.12 are for constant K-values, the patterns described are similar when K-values depend on composition If the nontie-line path is self-sharpening, then a shock must connect the two tie lines Table 5.1 indicates that whether a shock connects the initial oil and injection tie lines can be determined easily from the magnitude of K2 or equivalently, from the location of the envelope curve for systems with constant K-values The vast majority of gas/oil systems described by one of the equations of state in common use have envelope curves like those 5.2 SHOCKS 95 C1 C D C3 C2 b High volatility intermediate component C1 A B C3 C2 a Low volatility intermediate component Figure 5.12: Envelope curves for ternary systems with constant K-values: (a) low volatility intermediate (LVI) component (K2 < 1), (b) high volatility intermediate (HVI) indexHVI component (K2 > 1) 96 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Table 5.1: Nontie-Line Shocks and Rarefactions for Constant K-Value Systems Envelope Curve and Tie Line Intersections Vapor Side Vapor Side Liquid Side Liquid Side Intermediate K-Value K2 < K2 < K2 > K2 > Path Direction A→B B→A C→D D→C Composition Variation Rarefaction Shock Shock Rarefaction C1 a (y1 ,y2 ) (x1 ,x2 ) a C3 I1 C2 a Figure 5.13: Tie lines that extend through a point shown in Fig 5.12 in which the envelope curve moves away from the tie line as the tie-line length increases, and for those systems, Table 5.1 is appropriate However, it is possible to imagine systems in which the envelope curve moves toward the tie line as the length of the tie lines increases In such systems, the patterns of shocks and rarefactions are reversed The more general classification of the two types of envelope curves is included in the discussion of multicomponent systems in Chapter Hand’s Rule Fig 5.13 shows another simple approximation of tie line behavior, known as Hand’s Rule , that is frequently used in models of surfactant flooding [62] and is occasionally used for gas/oil systems [84, 127] In Fig 5.13, the tie lines all meet at a single point, I1 In a representation of phase behavior like that shown in Fig 5.13, the binodal curve is specified independently, and equilibrium phase compositions are found as the intersection of a tie line with the binodal curve The slope of a tie line in Fig 5.13 through a vapor composition point (y1 , y2 ) is 5.2 SHOCKS 97 α(η) = y2 , y1 − I1 (5.2.14) and the intercept is φ(η) = − y2 I1 = −I1 α(η) y1 − I1 (5.2.15) According to Eq 5.1.12 then, p= dφ dη dα dη = dα dη −I1 dα dη = −I1 (5.2.16) As a result, dp/dη = 0, and therefore, dλnt /dη = as well (Eq 5.2.8) Hence, when tie lines e meet at a point, λnt remains constant as a nontie-line path is traversed (Cer´ and Zanotti [11]) A composition variation along such a path is neither spreading nor self-sharpening Instead it is known as an indifferent wave, but it has the same appearance as a shock on composition profiles because all the compositions in the wave move at the same velocity 5.2.3 Tie-Line Intersections and Two-Phase Shocks We now prove an important result that will be useful in ternary displacements and will be even more useful in systems with more than three components If a shock connects two compositions, A and B, in the two-phase region then its wave velocity is given by ΛAB = FiA − FiB , CiA − CiB i = 1, nc (5.2.17) FiA − ΛAB CiA = FiB − ΛAB CiB (5.2.18) Rearrangement of Eq 5.2.17 gives Each side of Eq 5.2.18 is some overall composition expressed as a volume fraction of each component, which we call JiX , so that JiX = FiA − ΛAB CiA , (5.2.19) JiX = FiB − ΛAB CiB , (5.2.20) and We now ask what conditions JiX must satisfy if Eqs 5.2.19 and 5.2.20 are to be consistent with the original shock balance, Eq 5.2.17 Eqs 5.2.19 and 5.2.20 can be written in terms of the phase compositions on the two tie lines by substituting the definitions of FiA , FiB , CiA and CiB (Eqs 5.1.3 and 5.1.4) The result is A A A A A JiX = yi f1 − ΛAB S1 + xA (1 − f1 ) − ΛAB (1 − S1 ) , i and i = 1, nc, (5.2.21) 98 CHAPTER TERNARY GAS/OIL DISPLACEMENTS B B B B B JiX = yi f1 − ΛAB S1 + xB (1 − f1 ) − ΛAB (1 − S1 ) , i i = 1, nc (5.2.22) Eq 5.2.21 indicates that the mixture composition, JiX lies on the straight line that connects the A liquid phase composition, xA , with the vapor phase composition, yi , because the right side of Eq i 5.2.21 is a linear combination of the phase compositions That straight line is the tie line, of course Similarly, Eq 5.2.22 shows that the same mixture composition must also lie on the extension of B the tie line that connects xB to yi If the same overall composition is to lie on the extension of i both tie lines, the tie line extensions must intersect We have shown, therefore, that extensions of tie lines connected by a shock must intersect Tie lines can intersect only at a composition point that lies in the single-phase region Suppose that overall composition is CiX and consider the following shock balances constructed as if the shocks were from the single-phase point, CiX , to the composition points A and B on the tie lines being considered, ΛAB = CiX − FiA , CiX − CiA ΛAB = CiX − FiB , CiX − CiB i = 1, nc (5.2.23) Each shock balance can be rearranged to give FiA − ΛAB CiA = CiX (1 − ΛAB ), FiB − ΛAB CiB = CiX (1 − ΛAB ), i = 1, nc (5.2.24) Setting the left sides of the two expressions in Eq 5.2.24 equal and rearranging gives the original shock balance, Eq 5.2.17 Thus, the shock velocity can be found easily if the intersection point of the tie lines is known from Eq 5.2.23 The derivations of Eqs 5.2.21 and 5.2.22 were not restricted to three components Instead, they apply to systems with any number of components In ternary systems, all tie lines intersect (unless they are parallel, an unlikely event in gas/oil systems), so the intersection of two tie lines can be found easily by solving two simultaneous equations of the form of Eq 5.1.5 written for tie lines A and B In systems with four or more components, the tie-line intersection requirement will be used to identify key tie lines in addition to finding shock velocities 5.2.4 Entropy Conditions When more than two components are present, the entropy condition used to determine unique shock compositions are more complex The entropy conditions for the leading and trailing shocks, which occur along tie lines are similar to those for the binary displacements of Chapter See Appendix A for a statement of those entropy conditions Lax [67] stated the appropriate requirements for shocks like those that connect two tie lines in strictly hyperbolic multicomponent systems (a problem is strictly hyperbolic if the eigenvalues are everywhere distinct) The displacements considered here are not strictly hyperbolic, however, because there is a pair of points on every tie line where the eigenvalues are equal The basic idea of the entropy condition for two-component conservation equations like those solved in this chapter is that one set of characteristics is self-sharpening and the other is not [107, p 55] For nonstrictly hyperbolic problems, Isaacson’s statement of the entropy condition [39] is appropriate (see also Keyfitz and Kranzer [58] for more discussion of 5.3 EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES 99 entropy conditions): for ternary systems, the entropy condition is satisfied for left and right states indicated by L and R if, for eigenvalues λt and λnt : λR < Λ < λL , nt nt and λL < Λ = λR t t (5.2.25) λR < Λ < λL , nt nt and Λ = λL < λR t t (5.2.26) or The entropy condition for one example of a ternary displacement is derived in Appendix A and is applied to show that shocks between tie lines are semishocks at which the shock velocity is equal to the tie-line eigenvalue on one side of the shock 5.3 Example Solutions: Vaporizing Gas Drives In this section, we describe solutions for a gas displacement situation known as a vaporizing gas drive The name arises from the fact that vaporization of the intermediate component from the oil into the fast flowing vapor phase can lead to very efficient displacement if enough of the intermediate component is present in the oil (see Section 5.6) In typical ternary phase diagrams for gas/oil systems, component is chosen to be the component with the highest K-values, and it is plotted at the top vertex of the ternary diagram Component is generally chosen to be the component with the smallest K-value and is plotted at the lower left vertex of the ternary diagram On such phase diagrams, displacements in which the injection gas tie line lies to the left of the initial oil tie line are known as vaporizing gas drives in the standard terminology A second type of displacement known as a condensing gas drive /indexcondensing gas drive is described in Section 5.4 Condensing gas drives derive their name from the transfer of intermediate component from the injected gas to the liquid phase being contacted by the injected gas Condensing gas drives occur when the injection tie line lies to the right of the initial tie line on a ternary diagram with components ordered as described for a vaporizing gas drive Whether the displacement is a condensing or vaporizing gas drive, the unique composition route that connects the initial and injection compositions is selected by applying the velocity constraint, the entropy condition, and a requirement that the solution be continuous with respect to variations in the initial and injection data High Volatility Intermediate Component In the examples in this section, we ignore effects of volume change as components transfer between phases, and we assume that equilibrium K-values are independent of composition For the first example, we set K1 = KCH4 = 2.5, K2 = KCO2 = 1.5, and K3 = KC10 = 0.05 We will use the term high volatility intermediate (HVI) component to highlight the fact that K2 > Phase viscosities are also assumed to be independent of composition with µoil /µgas = Phase relative permeability functions are assumed to have the form of Eqs 4.1.13–4.1.19 with Sgc = Sor = The injected fluid is assumed to be pure CH4 , and several initial oil compositions listed in Table 5.2 are considered Fig 5.14 shows the solution composition route and the corresponding saturation and composition profiles for displacement of mixture a by pure CH4 Compositions, saturations and wave velocities of key points in the solution are reported in Table 5.3 100 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Table 5.2: Initial and Injection Compositions Fluid Injection Gas Initial Oil a Initial Oil A1 Initial Oil A2 Volume Fractions CCH4 CCO2 CC10 1.000 0.000 0.000 0.150 0.291 0.559 0.000 0.211 0.789 0.050 0.550 0.400 Table 5.3: Displacement of Several Initial Oil Mixtures by pure CH4 Point f e d c b a b1 A1 f2 e2 d2 c2 b2 A2 CCH4 1.0000 0.8178 0.7865 0.3466 0.3312 0.1500 0.3416 1.0000 0.8302 0.8302 0.0927 0.0879 0.0500 CCO2 0.0000 0.0000 0.0000 0.3947 0.3866 0.2908 0.3920 0.2115 0 0.6987 0.6821 0.5500 CC10 0.0000 0.1822 0.2135 0.2587 0.2822 0.5592 0.2665 0.7885 0.1698 0.1698 0.2086 0.2300 0.4000 S1 1.0000 0.7395 0.6856 0.4886 0.4372 0.4715 1.0000 0.7607 0.7607 0.4778 0.4191 ξ/τ 1.0000 0.2454 0.3593-1.1020 1.1800 1.5180 1.0000 1.2879 1.0000 1.0000 0.2468 0.2468-1.0797 1.2476 1.6474 1.0000 5.3 EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES 101 CH4 f a ea d a Dilution Line aac b a a C10 CO2 f a Sg ea a d d a CH4 ab aa f a e a c d a a a d c a b ca b CO2 f e d a a C10 0.0 f ea d a a da daca 1.0 ξ/τ a a a a a a a a ab 2.0 Figure 5.14: Composition route and saturation profile for a vaporizing gas drive with a high volatility intermediate component The injection gas is pure CH4 , and the composition of the oil is C1 = 0.15, C2 = 0.29 and C3 = 0.46 K-values are assumed to be K1 = 2.5, K2 = 1.5, and K3 = 0.05, and the viscosity ratio is constant at µoil /µgas = 102 CHAPTER TERNARY GAS/OIL DISPLACEMENTS The solution route enters the two-phase region with a shock from the initial composition a across the equivelocity curve (where F1 = C1 ) to point b As Fig 5.14 shows, point b lies on the tie line that extends through point a That shock is a semishock, so the shock velocity, Λab, equals b the tie-line eigenvalue, df1 /dS1 Upstream of the leading shock is a continuous variation (also called a rarefaction) along the initial tie line Thus, the leading portion of the solution is qualitatively the same as would be observed for a binary displacement like those described in Chapter At point c, the solution route switches to the nontie-line path Point c is the equal-eigenvalue point , where the nontie-line path is tangent to the initial tie line Point c is the only point where a path switch is allowed As Section 5.1.4 showed, compositions that lie above point c are not allowable path switch points because they violate the velocity rule Points below c are acceptable switch points on the initial oil tie line, but the corresponding switch point at the injection gas tie line would violate the velocity rule Hence the switch must occur at point c, the equal-eigenvalue point, and the upper branch of the nontie-line path must be traced if there is to be an allowable path switch at the intersection of the nontie-line path and the injection gas tie line According to the arguments of Section 5.2 summarized in Table 5.1 ( See also Eqs 5.2.8 and 5.2.13), eigenvalues decline as the nontie-line path is traced toward the injection tie line, and hence a rarefaction connects the injection gas and initial oil tie lines The saturation profile of Fig 5.14 shows that while the saturation changes appreciably as the nontie-line path is traced the corresponding wave spreads only slightly because λnt changes only slightly along the nontie-line path (see also Table 5.3) Point d is the point at which the nontie-line path intersects the injection tie line At that point, the solution route switches to the injection tie line, which creates a zone of constant state (see Section 5.3) The composition at point d has two wave velocities, a fast one given by the nontie-line eigenvalue, λd , and a slow one given by the tie-line eigenvalue, λd Therefore point d, nt t which is a single point on the ternary diagram, appears as a region in the profile Upstream of point d the solution is identical to the trailing portion of a binary displacement As the middle panel of Fig 5.14 shows, CO2 is not present at point d on the injection tie line, so the portion of the path on the injection tie line must be the same as that of a binary displacement Between points d and e, there is a continuous variation along the injection tie line, and there is a semishock from point e to point f, the injection composition The trailing shock moves relatively rapidly in this example because the K-value of C10 is large enough that the capacity of pure CH4 to vaporize pure C10 is significant The concentration profiles of Fig 5.14 show that the CH4 concentration increases steadily as the solution path is traced from the initial composition upstream to the injection composition in a profile that mirrors that of the gas saturation, S1 , in Fig 5.14 The profile pattern for C10 is similar to an inverted version of the profile for CH4 The C10 concentration declines monotonically as the solution path is traced, and the C10 concentration drops to zero behind the trailing evaporation shock The CO2 profile, however, differs qualitatively from those of CH4 and C10 There is a bank of fluid enriched with CO2 between points b and d The CO2 that enriches that bank was separated chromatographically from the two-phase mixtures that lie upstream of the CO2 bank (between d and e) Another view of that enrichment is given by the composition route shown in the ternary diagram of Fig 5.14 The overall compositions in the leading portion of the transition zone (a→b–c) contain larger fractions of CO2 , the intermediate component, than mixtures that lie on the dilution line that connects the initial mixture with the injection mixture In contrast, mixtures in the trailing 5.3 EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES 103 portion (d–e→f) contain less CO2 than dilution mixtures The CO2 present in the leading portion of the transition zone must have been vaporized from the initial oil, because there is no CO2 present in the injection fluid That enrichment of the flowing vapor phase is what is meant by the term vaporizing gas drive Now consider how the solution would change if the initial composition were changed along the extension of the same tie line that extends through point a If all the CH4 were removed, the initial composition would be that of point A1 in Table 5.3 and Fig 5.15 The resulting composition path and profiles are also shown in Fig 5.15 Comparison of Figs 5.14 and 5.15 and the wave velocities reported in Table 5.3 indicates that the velocity of the leading semishock, Λab changes Removing the CH4 slows the leading shock slightly, as Table 5.3 shows Upstream of point b1 , however, the solution is identical to that for initial composition a Thus, the overall pattern of the displacement is controlled by the initial and injection tie lines, while details such as leading and trailing shock velocity are determined by the location of the initial or injection composition on the tie-line extension If the amount of CO2 in the initial mixture is increased, as it is for initial mixture A2 for example, an additional changes to the composition profiles occur The leading shock now moves significantly faster, and the nontie-path is displaced toward the CH4 /CO2 side of the ternary diagram When the mixture A2 is displaced, the nontie-line path lands on the injection tie line at point d2 , which lies just above the tangent point for a trailing shock, e1 A rarefaction from d2 to e1 would violate the velocity constraint, because λt increases for decreasing CH4 along the injection tie line A shock from d2 to e1 , followed by a semishock from e1 to f is also not permitted because the shock from d2 to e1 would be slower than the trailing semishock, another violation of the velocity constraint Therefore, the final segment of the solution must be a genuine shock directly from d2 to the injection composition f Λ df = f d F1 − F1 f d C1 − C1 = d − F1 d − C1 (5.3.1) Such a shock forms whenever the intersection of the nontie-line path with the injection tie line lies above the semishock point on that tie line, and it moves with a higher wave velocity than the trailing semishock because there is less C10 to be evaporated Low Volatility Intermediate Component If the K-value of the intermediate component is less than one (K2 < 1), (LVI, low volatility intermediate) a rarefaction along the nontie-line path between the initial and injection tie lines is not permitted: λnt increases as the nontie-line path is traced upstream (see Section 5.2.2, Table 5.1) Consider a displacement with K1 = 2.5, K2 = 0.5, K3 = 0.05 Those K-values might represent a system of CH4 , C4 , and C10 , for example Figure 5.16 shows the solution for a displacement of an oil mixture with composition C1 = 0.1, C2 = 0.5, and C3 = 0.4 by pure CH4 with a fixed viscosity ratio of five The leading segment of the displacement is just as in the previous example: a tangent shock along the tie line that extends through the initial composition (point a), followed by a rarefaction along the initial tie line That segment ends at another tangent shock, one that connects the initial tie line to the injection tie line In Section 5.2.3, we proved that two tie lines connected by a shock 104 CHAPTER TERNARY GAS/OIL DISPLACEMENTS CH4 f a a e1 a d2 d1 a a a c1 b1 C10 Sg A1 a f a d a 2a e1 d b2 c2 d a2 a d1 a c2 aa c1 b1 aA1 a a a CO2 b2 a aA2 a a CH4 aa A2 a a a a a a a CO2 a a a a a a a a a a C10 a 0.0 a a a a a a aa 1.0 ξ/τ a a 2.0 Figure 5.15: Composition path and profiles for displacement of oils A1 and A2 by pure CH4 5.3 EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES 105 a CH4 f a d a ac a b a a C10 C4 fa a Sg d a d ca ab aa a a a CH4 a a a C4 a a a a a C10 0.0 a a a a a 1.0 ξ/τ 2.0 Figure 5.16: Composition path and profiles for a vaporizing gas drive with low volatility intermediate component K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The injection gas is pure CH4 , and the initial oil has composition C1 = 0.1, C2 = 0.5, and C3 = 0.4 106 CHAPTER TERNARY GAS/OIL DISPLACEMENTS must intersect, and that the velocity of the shock can be calculated easily using the intersection composition (Eq 5.2.23): Λcd = CiX − Fic df c , = λc = t dS CiX − Cic i = 1, nc (5.3.2) Eq 5.3.2 depends only on a single variable, the saturation on the initial tie line, and solving it is no more difficult than solving the tangent construction for the leading shock The landing point on the injection tie line is determined from CiX − Fid = λc , t X − Cd Ci i i = 1, nc (5.3.3) Again, this equation depends only on the saturation on the injection tie line, so it can be solved easily as well Figure 5.16 shows that there is no rarefaction present on the injection tie line: the landing point of the intermediate shock lies above the tangent point for a trailing shock As a result, the trailing shock is a genuine shock from the landing point on the injection tie line (point d) to the injection composition (point f) Thus, only one rarefaction is present in this displacement, and the remainder of the displacement behavior is determined by the velocities of the three shocks No integration is required to solve this problem Instead, three relatively simple nonlinear equations in a single variable are solved to determine the shock compositions and velocities 5.4 Example Solutions: Condensing Gas Drives Condensing gas drives are so named because some of the intermediate component present in the injection gas stream condenses into the liquid phase present in the transition zone Just as in the vaporizing gas drives, there are two versions of the solution In HVI systems, there is a shock between the injection and initial tie lines in condensing gas drives In LVI systems, on the other hand, a rarefaction connects those tie lines High Volatility Intermediate Component To illustrate the features of a condensing drive with a self-sharpening nontie-line path, we consider an initial mixture that contains only CH4 and C10 displaced by a mixture of CH4 and CO2 Here again we use constant equilibrium K-values with the same fractional flow function used in Section 5.3 Fig 5.17 shows the composition route, saturation profile, and composition profiles Composition e is the injection composition, in this case CCH4 = 0.6, CCO2 = 0.4 In a condensing gas drive the solution path is traced from the injection composition to the initial composition From the injection composition, the solution composition route enters the two-phase region with a shock to a composition on the tie line that extends through the injection composition The trailing shock d is a semishock with velocity Λde = df1 /dS1 Downstream of the trailing shock is a rarefaction along the injection tie line Evaluation of dλnt /dη with Eqs 5.2.8 and 5.2.13 indicates that λnt increases as nontie-line paths are traced from the injection tie line to the initial tie line A rarefaction along one of those paths would violate the velocity constraint, and hence, a shock is required Application of the entropy 5.4 EXAMPLE SOLUTIONS: CONDENSING GAS DRIVES 107 CH4 ae b a a ac d a a C10 e a a c a d Sg a CO2 a b b a aa a a a a CH4 a a a a a CO2 a a C10 a 0.0 a a a a 1.0 ξ/τ a 2.0 Figure 5.17: Composition route, saturation, and composition profiles for a self-sharpening (HVI) condensing gas drive K1 = 2.5, K2 = 1.5, K3 = 0.05, and M = The injection gas has composition, C1 = 0.6, C2 = 0.4, and C3 = 0, and the initial oil has composition, C1 = 0.3, C2 = 0, and C3 = 0.7 108 CHAPTER TERNARY GAS/OIL DISPLACEMENTS condition (see Section 5.2.4) shows that this shock is also a semishock, for which the wave velocity is Λbc = Fib − Fic df c = dS1 Cib − Cic (5.4.1) Eq 5.4.1 holds for any component, and because the concentration of CO2 is zero at point b, it is convenient to write Eq 5.4.1 for CO2 , Λbc = c FCO2 df c = c CCO2 dS1 (5.4.2) Eq 5.4.2 can be solved easily for the overall composition of point c, which must lie on the injection tie line Eq 5.4.1 written for CH4 , b b c c FCH4 − Λbc CCH4 = FCH4 − Λbc CCH4 , (5.4.3) can then be solved for the overall composition at point b In this example, the landing point b is below the overall composition at which a leading semishock lands on the initial tie line, and hence there is no rarefaction along the initial tie line Instead, there is a genuine shock from point b to the initial composition, point a The leading shock velocity is given by the shock material balance, Λab = b a FCH4 − FCH4 b a CCH4 − CCH4 (5.4.4) A rarefaction and semishock on the initial tie line can occur, but only if the fraction of CO2 in the injection mixture is low, in this example below about 5% In this condensing gas drive, then, the solution consists of a single rarefaction wave along the injection tie line and three shocks Because the rarefaction occurs on a single tie line, it is identical to the rarefaction that occurs in the binary displacements of Chapter The trailing and intermediate shocks are both semishocks, each of which is determined by solving a single nonlinear equation for the composition on one side of the shock Thus, the solution for this ternary displacement can be found by a procedure that is similar to and not much more difficult than that used to solve binary displacement problems Fig 5.17 shows that the trailing portion of the composition route is richer in CO2 , the intermediate component, than mixtures that form on the dilution line that connects the initial and injection compositions, while the downstream portion of the route is leaner in CO2 Another representation of the same idea is given in the CH4 profile shown in Fig 5.17, which indicates that there is a CH4 bank between the locations of the a→b and b→c shocks Thus, mixtures flowing at the leading edge of the transition zone are depleted in CO2 because it has condensed into the liquid phase present at the upstream end of the transition zone The CO2 present there must have condensed from the injection gas because no CO2 is present in the initial fluid It is that transfer of intermediate component from the injection gas that is the source of the name, condensing gas drive Here again, the components present in the injection and initial fluids have separated chromatographically due to the interaction of phase equilibrium with multiphase flow Low Volatility Intermediate Component 5.4 EXAMPLE SOLUTIONS: CONDENSING GAS DRIVES 109 CH4 ae ad ac b a a a C10 C4 e a a Sg d CH4 c a a a b b a a a a a a a a C4 1 a a a a a C10 a a 0.0 a a a a 1.0 ξ/τ 2.0 Figure 5.18: A condensing gas drive (LVI) with a nontie-line rarefaction K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The injection gas has composition, C1 = 0.8, C2 = 0.2, and C3 = 0., and the initial oil has composition, C1 = 0.3, C2 = 0, and C3 = 0.7 110 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Fig 5.18 shows the last of the four types of displacement that occur in ternary gas drives indexLVI!condensing When K2 < 1, tie line intersections occur on the vapor side of the twophase region, and a rarefaction occurs along the nontie-line path that connects the injection gas tie line to the initial oil tie line Otherwise, the pattern of composition and saturation variations is qualitatively similar to the self-sharpening condensing gas drive in Fig 5.17 There is a trailing semishock (e→d) preceded by a rarefaction along the injection gas tie line (d–c), a switch to the nontie-line path at point c, a rarefaction along which λnt increases (as it must as the solution route is traced toward the initial composition), and a leading genuine shock (b→a) 5.5 Structure of Ternary Gas/Oil Displacements The analysis of Sections 5.1 and 5.2 and the examples of vaporizing and condensing gas drives in Sections 5.3 and 5.4 reveal patterns of displacement behavior that can be catalogued easily in terms of the lengths of the two key tie lines and the location of tie line intersections with respect to the two-phase region (see Table 5.4) for systems that exhibit envelope curves with shapes like those shown in Fig 5.12 Similar patterns reappear in displacements with more than three components, and hence it is worth reviewing them Most gas/oil systems show envelope curves in which the envelope curve approaches the tie line as the tie-line length decreases, but the possibility that systems exist in which the envelope curve approaches the tie line as tie-line length increases cannot be ruled out That situation is considered in Chapter For typical ternary systems in which a single-phase vapor displaces a single-phase oil, the patterns of Table 5.4 are all that arise To see how the patterns arise, it is useful to summarize how a solution can be constructed for any ternary gas drive The solution to a ternary gas drive problem can be constructed with the following steps: Identify the tie lines that extend through the injection gas and initial oil compositions If, as is commonly the case, the injection gas and initial oil compositions lie in the single-phase region, the key tie lines can be found by a negative flash [135, 129] A negative flash is just a flash calculation for a mixture that is single-phase If it converges, it will give the compositions of the phases on the tie line that extends through the single-phase composition Construct semishocks from the injection gas and initial oil compositions to determine the tangent shock points on the injection gas and initial oil tie lines Determine whether a rarefaction or a shock connects the two key tie lines Table 5.4 summarizes the possible situations Or, Eq 5.2.13 can be used It gives the correct variation of λnt even if K-values are not constant Identify the shortest key tie line Solution construction begins with that tie line (see the argument below for the reasoning behind this statement) If a rarefaction connects the two tie lines, integrate the nontie-line path from the equaleigenvalue point on the shortest tie line to determine where it intersects the other tie line If an intermediate shock connects the two tie lines, construct a semishock for which the shock velocity equals λt on the shortest tie line  ... 0.3947 0.3 866 0.2908 0.3920 0.2115 0 0 .69 87 0 .68 21 0.5500 CC10 0.0000 0.1822 0.2135 0.2587 0.2822 0.5592 0. 266 5 0.7885 0. 169 8 0. 169 8 0.20 86 0.2300 0.4000 S1 1.0000 0.7395 0 .68 56 0.48 86 0.4372 0.4715... 1.0000 0. 760 7 0. 760 7 0.4778 0.4191 ξ/τ 1.0000 0.2454 0.359 3-1 .1020 1.1800 1.5180 1.0000 1.2879 1.0000 1.0000 0.2 468 0.2 46 8-1 .0797 1.24 76 1 .64 74 1.0000 5.3 EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES... from the injection gas because no CO2 is present in the initial fluid It is that transfer of intermediate component from the injection gas that is the source of the name, condensing gas drive