5.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 111 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 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 observations 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 nontieline paths (see Fig 5.9) indicates that the point of tangency of the nontie-line path to a tie 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 = c d c d F1 − F1 C X − F1 C X − F1 ∂F1 = = = X − Cc c d X d ∂C1 C1 − C1 C1 C1 − C1 (5.5.1) 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 c d The fractional flow curves shown in Fig 5.19 are typical of systems in which y1 < y1 and c d M < M , 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 112 CHAPTER TERNARY GAS/OIL DISPLACEMENTS a Overall Fractional Flow of Component 1, F1 1.2 X 1.0 a d* a 0.8 af d ca b a 0.6 0.4 0.2 aa 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Overall Volume Fraction of Component 1, C1 Figure 5.19: Tangent constructions for a shock between tie lines 5.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 113 C1 f C fa a d a e a d a ac ab a ac aa C3 C2 f a Sg e aa d d a a c fa a d b a a a ξ a a C3 Sg b C2 ad c a b a a a ξ HVI Vaporizing Drive LVI Vaporizing Drive C1 C1 b a a a a C3 C2 e a a c a Sg d C3 a b ξ C2 e a a b a a a HVI Condensing Drive d Sg b a a e ac d a a e a a ad c c a b a a b ξ a a LVI Condensing Drive Figure 5.20: Structure of solutions for condensing and vaporizing gas drives 114 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Table 5.4: Nontie-Line Shocks and Rarefactions Envelope Curve and Tie Line Intersections Vapor Side Vapor Side Liquid Side Liquid Side Intermediate K-Value K2 < K2 < K2 > K2 > Process Name LVI Condensing LVI Vaporizing HVI Condensing HVI Vaporizing Shortest Tie Line Injection Gas Initial Oil Injection Gas Initial Oil Composition Variation Rarefaction Shock Shock 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 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 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 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 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 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 5.5 STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 115 C C fa fa d a d a ac ab2 ac ab a a2 a1 a C3 fa a d 1.0 ad ca Sg 0.0 0.0 C3 0.5 a a1 1.5 fa a d b1 a 1.0 C2 2.0 0.0 0.0 2.5 ad ca Sg 1.0 C2 ab2 aa2 0.5 1.0 ξ C C fa d4 a d a a4 a ac a3 a C3 C2 Sg d 0.0 0.0 C3 1.0 ad fa a d4 ca 0.5 C2 a3 a 1.0 1.5 2.0 ξ (c) Initial mixture a3 2.5 Sg a 2.5 (b) Initial mixture a2 fa fa 2.0 ξ (a) Initial mixture a1 1.0 1.5 0.0 0.0 0.5 a a a4 1.0 1.5 2.0 2.5 ξ (d) Initial mixture a4 Figure 5.21: Effect of variations in initial composition All initial compositions lie on the same initial tie line or its extension (a) a1 is a single phase liquid (b) a2 has a gas saturation of percent (c) a3 has a gas saturation of 30 percent (d) a4 has an initial gas saturation of 80 percent 116 CHAPTER TERNARY GAS/OIL DISPLACEMENTS CO2 a a a a a a aa C10 A B a a 1.0 a C C4 a a a a a a B a C aa A a a a Sg a a 0.0 0.0 0.2 0.4 0.6 0.8 a aa a 1.0 1.2 ξ Figure 5.22: Changes in solution composition route as the initial mixture is enriched in the intermediate 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 A A B B B A: CCO2 = 0, CC4 = 0.195950, CC10 = 0.804050, Point B: CCO2 = 0, CC4 = 0.373735, CC10 = C C C 0.626265, and Point C: CCO2 = 0, CC4 = 0.475887, CC10 = 0.524113 5.6 MULTICONTACT MISCIBILITY 5.5.1 117 Effects of Variations in Initial Composition 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 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 a1 in the single-phase region to a2 with a gas saturation of 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 5.6 Multicontact Miscibility 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) 118 5.6.1 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Vaporizing Gas Drives Figure 5.22 shows what happens as the initial mixture is enriched in Component 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 Fia − Fib , Cia − Cib Λint = i Fic − Fid , Cic − Cid Λtrail = i Fid − Fie Cid − Cie i = 1, nc (5.6.1) Now consider what happens in the limit as the initial mixture is enriched enough in Component 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 Fib = Cib Note also that Fia = Cia because the initial mixture is single-phase Therefore, it must be that Λlead = i Fia − Fib C a − Cib = ia = 1, Cia − Cib Ci − Cib i = 1, nc (5.6.2) Because the critical tie line has zero length, Point c must also be the plait point, and hence Fic = Cic 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 λnt = 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, Fid = Cid Also, the injection mixture is single-phase, so Fie = Cie Thus, it must also be that Λint = i Fic − Fid C c − Cid = ic = 1, Cic − Cid Ci − Cid Λtrail = i Fid − Fie C d − Cie = id = 1, Cid − Cie Ci − Cie i = 1, nc , (5.6.3) and i = 1, nc (5.6.4) 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 5.6 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 5.6.2 Condensing Gas Drives 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, Fid = Cid 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 Fic = Cic to the intersection of the equivelocity curve and the initial tie line, where Fib = Cib The intermediate shock velocity is then Λtrail = i Fib − Fic C b − Cic = ib = 1, Cib − Cic Ci − Cic i = 1, nc (5.6.5) Finally, the leading shock connects a point on the equivelocity curve where Fib = Cib , and at the initial composition, Fia = Cia , 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 5.6.3 Multicontact Miscibility in Ternary Systems 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 120 CHAPTER TERNARY GAS/OIL DISPLACEMENTS In 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 5.7 Volume Change 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 ∂H1 + ∂τ ∂ξ ∂G2 ∂H2 + ∂τ ∂ξ ∂G3 ∂H3 + ∂τ ∂ξ = 0, (5.7.1) = 0, (5.7.2) = 0, (5.7.3) where Gi = xi1 ρ1D S1 + xi2 ρ2D (1 − S1 ), (5.7.4) Hi = vD (xi1 ρ1D f1 + xi2 ρ2D (1 − f1 )) (5.7.5) Manipulations similar to those of Sections 4.4 and 5.1 yield the eigenvalue problem, {[H(u)] − λ[G(u)]} u = 0, (5.7.6) where ⎡ ∂H1 ∂z1 ⎢ H(u) = ⎣ ∂H12 ∂z ∂H3 ∂z1 ⎡ ∂G ∂z ⎢ ∂G1 [G(u)] = ⎣ ∂z1 ∂G3 ∂z1 ∂H1 ∂z2 ∂H2 ∂z2 ∂H3 ∂z2 ∂G1 ∂z2 ∂G2 ∂z2 ∂G3 ∂z2 ∂H1 ∂z3 ∂H2 ∂z3 ∂H3 ∂z3 ∂G1 ∂z3 ∂G2 ∂z3 ∂G3 ∂z3 ∂H1 ∂vD ∂H2 ∂vD ∂H3 ∂vD ⎤ ⎥ ⎦, (5.7.7) ⎤ ⎥ ⎦, (5.7.8) and uT = [dz1 , dz2, vD ] (5.7.9) In these expressions, zi 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, vD , must also be determined 5.7 VOLUME CHANGE 121 The analysis of paths and shocks for ternary systems with volume change follows the same reasoning as the analysis for flow without volume change Tie lines are still paths, and there are also nontie-line paths that are quite similar in appearance to those for systems without volume change [19, 20] The composition route for a solution is assembled using shocks and rarefactions that are selected using the velocity constraint and the entropy condition as tools Additional complexity arises, of course, from the fact that flow velocity is not constant It changes across phase-change shocks, and it varies along a nontie-line path, though flow velocity remains fixed for composition variations that remain in the single-phase region along a single tie line (see Section 4.4) In this section, we consider what happens in fully self-sharpening ternary displacements In these flows, all flow velocity changes occur at shocks, because the only rarefactions present occur along tie lines The full problem, with nontie-line rarefactions along which flow velocity changes, is considered in Chapter for systems with four components As in problems without volume change, two kinds of shocks are important, those across which the number of phases changes, and those that occur between tie lines in the two-phase region We begin by showing that the key results of the analysis for flow without volume change involving tie line extensions and intersections also hold when there is volume change on mixing In Section 5.2 we showed that a shock that connects a single-phase composition with a two-phase composition must occur along the extension of a tie line We now consider whether that statement is also true when components change volume as they transfer between phases The appropriate shock balance is Eq 4.4.17 Λ= HiI − HiII , GI − GII i i i = 1, nc , (5.7.10) where I and II refer to the single-phase and two-phase sides of the shock On the single-phase side of the shock, I I HiI = vD xI ρI = vD GI i D i i = 1, nc (5.7.11) Substitution of Eq 5.7.11 into 5.7.10 gives xI = i HiII − ΛGII i , I ρI (vD − Λ) D i = 1, nc (5.7.12) Substitution of the definitions for HiII and GII shows that i II xI = AxII + Byi i i i = 1, nc, (5.7.13) where A= II II II ρII [vD (1 − fV ) − Λ(1 − SV )] LD I ρI (vD − Λ) D (5.7.14) II II I ρII [vD fV ) − Λ(1 − SV I)] LD I ρI (vD − Λ) D (5.7.15) and B= 122 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Thus, each single-phase mole fraction is a linear combination of the equilibrium phase mole fractions, and the statement that phase-change shocks occur along tie line extensions holds true when effects of volume change are present Hence, the two key tie lines identified previously, the injection gas and initial oil tie lines, are the important tie lines for a displacement whether or not there is volume change That result means that the discussion of Section 4.4 provides considerable guidance for the behavior of multicomponent systems with volume change Flow velocities can change substantially across phase-change shocks, but they not change for variations along a single tie line Hence, flow velocities not change for the portions of the displacements that occur along the initial oil and injection gas tie lines They may change for the leading shock, the trailing shock, and for a rarefaction along the nontie-line path or the corresponding shock between the initial and injection tie lines if a rarefaction is not possible It is commonly observed, however, that the only significant change in flow velocity occurs at the leading shock, for exactly the reasons described in Section 4.4 The important result that extensions of tie lines connected by a shock must intersect also holds when volume is not conserved [19] The argument is similar to that given in Section 5.2.3 for flows without volume change Eq 5.7.10, written for a shock that connects two tie lines containing points c and d, can be rearranged (with the addition of GX to both sides) to give i Hic + Λcd GX − Gc = Hid + Λcd GX − Gd , i i i i i = 1, nc (5.7.16) As before, we set each side of Eq 5.7.16 equal to HiX , and write the resulting expressions in the standard form of a shock balance, Λcd = HiX − Hic H X − Hid = iX , GX − Gc Gi − Gd i i i i = 1, nc (5.7.17) If the composition point associated with GX and HiX is restricted to lie in the single-phase i X region, then HiX = vD GX The left side of Eq 5.7.16 can be expanded using the definitions of Hic i and Gc to obtain and expression for GX , i i c GX = Dρc xc + Eρc D yi , i LD i V i = 1, nc, (5.7.18) with the coefficients D and E defined as D= c c c vD (1 − fV ) − Λcd (1 − SV ) , X − Λcd vD (5.7.19) c c c vD fV − Λcd SV X vD − Λcd (5.7.20) and E= In these expressions, the superscript c indicates that the quantities with the superscript are evaluated at point c or on the tie line that contains that point Eq 5.7.17 is the equation of a straight line in a space of molar concentration (measured in moles per unit volume) It states that the molar concentration GX is a linear combination of the molar concentrations of the equilibrium phases i Thus, GX lies on the extension a line that connects the equilibrium concentrations on one side of i the shock Identical manipulations of the right side of Eq.5.7.16 give a similar expression, 5.7 VOLUME CHANGE 123 d GX = Jρd xd + Kρd D yi , i LD i V i = 1, nc , (5.7.21) with J= d d d vD − fV − Λcd − SV X vD − Λcd , (5.7.22) and K= d d d vD fV − Λcd SV X vD − Λcd (5.7.23) Eq 5.7.21 states that GX also lies on the extension of a second line in molar concentration space i associated with the equilibrium compositions on the tie line on the other side of the shock Hence we have shown that the idea of intersecting tie lines applies to a tie line represented in terms of molar concentrations when components change volume as they change phase Eqs 5.7.16 and 5.7.21 also imply that the extensions of the tie lines themselves must intersect To see why this must be true, consider a line in concentration space, ρzi = (1 − β)ρL xi + βρV yi , i = 1, nc , (5.7.24) where β is a parameter that determines position along the line If the concentration point in question is in the single-phase region, then β will be greater than one (all vapor) or less than zero (all liquid) Summation of Eq 5.7.24 over the nc components gives an expression for ρ, ρ = (1 − β)ρL + βρV (5.7.25) Thus, ρ is a molar density that is an average of the phase molar densities Substitution of Eq 5.7.25 into Eq 5.7.24 gives an expression that can be solved for β, β= ρL (zi − xi ) ρL (zi − xi ) + ρV (zi − yi ) (5.7.26) Eq 5.7.26 actually indicates that β is a volume fraction, specifically the vapor saturation, S1 = SV Consider a tie line molar balance, zi = yi V + xi (1 − V ) (5.7.27) Eq 5.7.27 can be rearranged to solve for the mole fraction vapor, V , V = zi − xi yi − xi (5.7.28) The volume fraction of vapor is given by SV = and substitution of 5.7.28 into 5.7.29 yields V ρV (1−V ) V ρV + ρL , (5.7.29) 124 CHAPTER TERNARY GAS/OIL DISPLACEMENTS SV = ρL (zi − xi ) = β ρL (zi − xi ) + ρV (zi − yi ) (5.7.30) Finally, substitution of Eqs 5.7.25 and 5.7.26 into 5.7.24 gives zi = (1 − SV ) ρL SV ρ V xi + yi (1 − SV ) ρL + SV ρV (1 − SV ) ρL + SV ρV (5.7.31) Thus, we have shown that an equation of the form of 5.7.24, which places a molar concentration on a line connecting equilibrium concentrations, implies that the overall mole fraction, zi , lies on the extension of a tie line that connects the equilibrium mole fractions Eqs 5.7.17 and 5.7.21 have exactly the form of Eq 5.7.24, so they require also that tie lines intersect in mole fraction space This argument demonstrates that the extensions of tie lines connected by a shock must intersect, whether or not components change volume as the transfer between phases[19] The intersection shock balances given in Eq 5.7.17 are complicated expressions, but they can be reduced to remarkably simple forms [19] that are much more convenient for use in constructing solutions Consider a shock that connects two tie lines, designated A and B The appropriate shock balances are Λcd = HiA − GX H B − GX i i = i , GA − GX GB − GX i i i i i = 1, nc (5.7.32) We now define a shock velocity scaled by the flow velocity at tie line A and make use of the notation of Eq 4.4.18, X αA − v A GX Λ Λ = A = i A v Xi , v Gi − Gi ∗ i = 1, nc (5.7.33) Eq 5.7.33 can be rearranged to solve for the ratio of flow velocities, Λ ∗ GA − GX αA vX i i = i − , X X A v Gi Gi i = 1, nc (5.7.34) Next, we eliminate the velocity ratio, v X /v A , using versions of Eq 5.7.34 written for components and The result is Λ∗ = Λ αA GX − αA GX = vA GA GX − GA GX 1 (5.7.35) Substition of the definitions of αi (Eq 4.4.18) and Gi (Eq 4.4.3) [23] yields the final expression, Λ∗ = XA Λ f A − S1 = , A XA vA S1 − S1 (5.7.36) and similar manipulations with the shock balance for tie line B (Eq 5.7.32) gives Λ∗ = XB Λ v B f B − S1 = A , B XB vA v S1 − S1 (5.7.37) 5.7 VOLUME CHANGE 125 XA In these expressions, S1 refers to the vapor saturation at the intersection of the two tie lines XB measured on tie line A, while S1 is the intersection saturation measured on tie line B Because tie XA XB lines not intersect inside the two-phase region, the values of S1 and S1 must be negative or greater than one Eqs 5.7.36 and 5.7.37 allow straightforward calculation of shock velocities given the composition of the intersection point Solution construction follows the sequence described in Section 5.5, with the addition of calculation of flow velocities In a vaporizing gas drive, like that illustrated in Fig 5.23 for example, the solution is obtained as follows Manipulations similar to those used to derive Eq 5.7.36 yield an expression that can be used to determine the leading shock, Λ∗ = a Λ f b − S1 λb df b = = bt = , b b a dS1 vD S1 − S1 vD (5.7.38) a b where S1 is the saturation measured along the tie line that contains point b S1 must be negative or greater than, because point a lies in the single-phase region The intermediate shock is determined by solving Eq 5.7.36 set equal to λc The landing point of the intermediate shock on the injection t tie line can be found by writing shock balances for components and and eliminating the ratio of flow velocities across the shock The result is αd λc df c αc αd − αd αc 3 = ct = vD dS1 Gc − Gd − αd Gc − Gd 1 3 (5.7.39) d c c c The ratio of flow velocities, vD /vD , can then be found from Eq 5.7.37 with vD = vD The scaled velocity of the trailing shock is given by an expression like Eq 5.7.38 It is equal to the scaled tie d line eigenvalue, λe /vD , if the trailing shock is a semishock If it is a genuine shock, the speed is t set by the saturation and compositions at point d Once all the composition points are known, the d a flow velocities, vD and vD can be determined from αe = i − Λ∗ef f vd Gi Ge i Gi −1 , i = 1, nc, (5.7.40) a αb vD = i − Λ∗ab b Ga vD i Gb i −1 , Ga i i = 1, nc (5.7.41) f and An example of the effect of volume change in a ternary displacement is shown in Fig 5.23 Fig 5.23 compares composition routes for displacements saturation profiles and flow velocities for displacements with and without volume change The solution shown was obtained using the Peng-Robinson equation of state to calculate phase behavior at 1000 psia (68 atm) and 160 F (71 C) The ternary diagrams in Fig 5.23 show that in both cases, the leading shock occurs along the tie-line extension through the initial oil composition, and the trailing shock does so along the tie line that extends through the injection composition In this LVI vaporizing gas drive, both composition routes include an intermediate shock from the initial tie line to the injection tie line In this example, rarefactions occur on both the initial oil and injection gas tie lines The two key tie lines are the same, but the composition points that make up the solutions with and without volume change differ 126 CHAPTER TERNARY GAS/OIL DISPLACEMENTS CO2 a CO2 a f a a e d d aa C4 a a a a a a C4 (b) No Volume Change 0.2 0.4 a a a 0.6 ξ 0.8 a a a 1.0 (c) Saturation Profiles 1.2 vD 1.2 a a Sg 0.0 0.0 aa C10 (a) Volume Change 1.0 f a a ac ab ac a b C10 e 0.0 0.0 Volume change No volume change 0.2 0.4 0.6 ξ 0.8 1.0 1.2 (d) Flow Velocity Figure 5.23: Composition routes and saturation and flow velocity profiles for displacement of a C4 /C10 mixture by CO2 at 1000 psia (68 atm) and 160 F (71 C) 5.8 COMPONENT RECOVERY 127 The saturations shown in Fig 5.23c demonstrate that the profiles are similar in structure, whether or not there is volume change The last panel of Fig 5.23 shows the local flow velocity It is constant when volume is conserved, but it changes across each shock when volume is not conserved In this example, however, the only significant change in flow velocity occurs across the leading shock The dissolution of CO2 in undisplaced oil in the transition zone leads to a substantial reduction in volume because dissolved CO2 occupies much less volume at these conditions than does CO2 vapor The result is a leading shock velocity that is actually less than one: the leading shock arrives at the outlet after one pore volume of pure CO2 has been injected Experimental observations confirm that when the displacement pressure is low enough that the CO2 density is relatively low but high enough that there is appreciable solubility of CO2 in the oil, breakthrough after one pore volume injection is observed [89] Similar behavior is observed when the injected gas is CH4 , though the change in flow velocity is smaller because the solubility of CH4 in the oil is lower and the volume change is also smaller In N2 displacements the effects are smaller still because the solubility of nitrogen in the oils is also lower For solutions that confirm those statements see Dindoruk [19] 5.8 Component Recovery Calculation of component recovery for ternary systems is based on exactly the same material balance that was used in Section 4.5 for binary systems The volume of component i recovered is the volume of component i present initially plus the volume of component i injection through time, τ , minus the volume of component i present at time τ Hence, the recovery is given by Qi = Ciinit + Fiinit τ − Ci dξ (5.8.1) Integration by parts gives Qi = Ciinit − Ciout + τ Fiout , (5.8.2) the same expression obtained previously (Eq 4.5.8) The corresponding expression when components change volume as they transfer between phases is Eq 4.5.11, Ri = Ginit − Gout + τ Hiout i i (5.8.3) Given the solutions obtained in the previous sections, it is quite easy to determine recovery at the times at which the various key points in the composition profiles arrive at the outlet (ξ = 1) Consider, for example, the vaporizing gas drive illustrated in Figs 5.23 Because the shock velocities and composition points are all determined as the solution is found, the recovery calculation is straightforward Table 5.5 reports component recoveries at the times of arrival at the outlet of the key segments of the composition profiles in Fig 5.23 Recovery curves for the examples in Fig 5.23 are shown in Fig 5.24 for the two components present in the initial oil mixture The recovery of each component is scaled by Cia , the initial volume fraction or Ga, the initial concentration of each component In that form, the dimensionless recovery i of each component is proportional to τ until the leading shock reaches the outlet It is equal to τ when there is no volume change, but when effects of volume change are included, the recovery is scaled by the flow velocity at the outlet In the displacement of Fig 5.23, breakthrough of 128 CHAPTER TERNARY GAS/OIL DISPLACEMENTS Table 5.5: Component Recovery at Selected Times τ Λab λc t λd nt λd t Λef Arrival at ξ = No Volume Change Volume Change Fib Cia − Cib + Λab Fc Cia − Cic + λi c t Fid a − Cd + Ci i λd nt Fd Cia − Cid + λi d t Fif Cia − Cif + Λef i Ga − Gb + Λab i i c Hi Ga − Gc + λ c i i Leading Shock Intermediate Shock Zone of Constant State Trailing Rarefaction Trailing Shock Hb t Ga i − Gd i + Ga − Gd + i i Ga − Gf + i i d Hi d λnt d Hi λd t f Hi Λef 1.0 Fraction of C4 and C10 Recovered C4 C10 0.8 C4 C10 0.6 0.4 0.2 No Volume Change Volume Change 0.0 τ Figure 5.24: Component recovery for the displacements shown in Fig 5.23 5.9 SUMMARY 129 injected CO2 occurs just after one pore volume injected because CO2 loses volume as it dissolves in undisplaced oil After breakthrough, the recovery curves for the two components differ, because the chromatographic separations that occur during the displacement cause the two components to propagate at different speeds toward the outlet In this example, in which pure CO2 displaces a mixture of C4 and C10 , all of the C4 present initially is recovered when the intermediate shock arrives at the outlet Recovery of C10 is complete when the trailing shock arrives, though in this example, the slow evaporation of C10 into the flowing CO2 requires 17.4 and 26.4 pore volumes for the flows without and with volume change While recovery is slow in this immiscible case, much faster recovery and correspondingly higher displacement efficiency is possible when the pressure is high enough that multicontact miscibility occurs 5.9 Summary A number of key ideas have been developed in this chapter that will be the basis for understanding displacements with more than three components The analysis of ternary as well as multicomponent displacements is built on the following concepts: • Eigenvalues represent the velocity at which a given overall composition propagates • Eigenvectors represent directions in composition space along which compositions vary in a way that satisfies the conservation equations • Composition paths are integral curves of the eigenvectors Tie lines are paths In addition, there is a set of nontie-line paths Every composition point in the two-phase region lies on a single tie-line path and a single nontie-line path • The composition route that is the solution to any displacement problem must lie on paths that connect the initial oil composition to the injection gas composition • A composition route can enter or leave the two-phase region only via a shock A shock from a single-phase composition into the two-phase region must land on the tie line that extends through the single-phase composition • The extensions of tie lines connected by a shock must intersect • The shocks and rarefactions that are assembled to build the solution for a given set of injection and initial compositions must satisfy a velocity rule (fast compositions lie downstream of slow ones in regions of continuous composition and saturation variation), and shocks must satisfy an entropy condition (wave velocities of compositions are faster than the shock and those downstream are slower: shocks that are perturbed slightly must sharpen again into a shock) • The solution composition route may switch from a tie-line path to a nontie-line path (or vice versa) as long as the velocity rule is satisfied If the wave velocities associated with the tie-line and nontie-line paths are not the same, that composition point appears as a zone of constant state in the associated solution profile • The important features of a ternary gas displacement are determined by the two key tie lines: the tie line that extends through the initial oil composition and the tie line that extends through the injection gas composition 130 CHAPTER TERNARY GAS/OIL DISPLACEMENTS • Multicontact miscibility occurs if either of the key tie lines is a critical tie line (a tie line tangent to the binodal curve at the plait point) If the injection gas tie line is critical, the displacement is a condensing gas drive If the initial oil tie line is the critical tie line, the displacement is a vaporizing gas drive • Effects of volume change as components transfer between phases not change the basic patterns of displacement behavior observed for flows without volume change The same key tie lines are important, though specific compositions on those tie lines change The rate of recovery can be strongly influenced by volume change when it changes local flow velocity appreciably All of these ideas apply still in displacements with more than three components While the geometry of tie lines is more complex and will require some study, the approach to the problem of multicomponent displacements draws heavily on the description of ternary flows Thus, a detailed understanding of ternary displacements is required background for understanding the multicomponent flows considered in subsequent chapters 5.10 Additional Reading The first analytical solution for a gas/oil displacement was published in a remarkable 1961 paper by Welge et al [134] That paper considered flow of ternary mixtures (condensing gas drives), and it included the effects of volume change as components transfer between phases While the mathematical development is a challenge to follow, it sets the stage for much work to follow Wachmann [125] solved the somewhat simpler problem of ternary flow for oil/water/alcohol systems in which effects of volume change were ignored Work on ternary theory then paused until the 1970s when a group of investigators at Shell Development Company set out to describe the behavior of surfactant/oil/water systems The analysis was aimed at understanding how phase behavior and resulting low interfacial tensions could be used to recover oil left behind by waterflooding That theory was reported in a series of papers by Larson and Hirasaki [66], Larson [65], Helfferich [31], and Hirasaki [35] Helfferich’s 1981 paper [31] states the theory in a general form that makes clear the fact that the underlying mathematical description of the interaction of phase behavior and two-phase flow is common to a variety of displacement processes, despite differences in the details of the phase equilibrium The first proof that wave velocities remain constant along nontie-line paths when tie lines meet at a point was obtained by Cer´ and Zanotti [11] e Application to gas/oil systems came again with the work of Dumor´ et al [22], who considered e three-component gas injection processes with effects of volume change as components transfer between phases Detailed reviews of related work on surfactant and polymer flooding systems are given by Pope [101] and Johansen [50] Bedrikovetsky [6, pp 295-313] gives a version of the theory that emphasizes graphical constructions involving the fractional flow curves for the two key tie lines Detailed descriptions with numerous examples of ternary displacements without volume change are given by Pande [95], Johns [54] and Dindoruk [19] The proof that tie lines connected by a shock must intersect (for systems with no volume change) is due to Johns [54, 56], though a proof for a special case was reported by Orr et al [92] Johns also described the classification of the four types of ternary displacements (LVI and HVI, self-sharpening and spreading nontie-line paths) and showed how the intermediate K-value determines self-sharpening behavior Dindoruk [19] used ... (5 .7. 27) Eq 5 .7. 27 can be rearranged to solve for the mole fraction vapor, V , V = zi − xi yi − xi (5 .7. 28) The volume fraction of vapor is given by SV = and substitution of 5 .7. 28 into 5 .7. 29... composition route that avoids the two-phase region 5.6.2 Condensing Gas Drives The analysis of condensing gas drives is similar In condensing gas drives, the injection gas is enriched with the intermediate... shock balance is Eq 4.4. 17 Λ= HiI − HiII , GI − GII i i i = 1, nc , (5 .7. 10) where I and II refer to the single-phase and two-phase sides of the shock On the single-phase side of the shock, I I HiI