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5.5. STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 111 7. Construct a rarefaction along the shortest tie line that connects the equal-eigenvalue point or the intermediate shock point to the tangent shock point for the single-phase composition (oil or gas) associated with the shortest tie line. 8. Determine whether a tie-line rarefaction can occur on the longest tie line. A rarefaction can occur if variation along the tie line from the landing point of the intermediate shock or the intersection of the nontie-line path with the tie line to the tangent shock point satisfies the velocity rule. If so, the shock from the single-phase composition to the longest tie line is a semishock. If not, a genuine shock is constructed from the landing or intersection point to the single-phase composition. The fact that solution construction must begin on the shortest tie line arises from two ob-servations about the geometry of composition paths and shocks. The first observation applies to displacements in which a rarefaction connects the two key tie lines. In that case, there are two points at which the appropriate nontie-line path intersects the two key tie lines . Both points must be switch points at which the velocity rule is satisfied. The argument given in Section 5.3 indicates that one of the two points must be the equal-eigenvalue point. The geometry of nontie-line paths (see Fig. 5.9) indicates that the point of tangency of the nontie-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 cd F1 −F1 C1 −F1 C1 − F1 ∂F1 C1 −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. The fractional flow curves shown in Fig. 5.19 are typical of systems in which yc < yd and Mc < Md, 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 5. TERNARY GAS/OIL DISPLACEMENTS a 1.2 X 1.0 d* a a a f d 0.8 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 C1 de a d a c b a b C3 aa C2 C3 a a C2 1 a e a a a d a c 1 a a d d b c a b 0 a 0 a ξ ξ HVI Vaporizing Drive C1 LVI Vaporizing Drive C1 e b a e b a d a a a c a a C3 C2 C3 C2 1 e a d c a b 0 ξ 1 e a b d a 0 c a b b a ξ HVI Condensing Drive LVI Condensing Drive Figure 5.20: Structure of solutions for condensing and vaporizing gas drives. 114 CHAPTER 5. 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 < 1 K2 < 1 K2 > 1 K2 > 1 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. 1. Leading Shock, a→b. A leading shock is always present if the initial composition is a single-phase liquid. In a vaporizing gas drive (initial oil tie line is shorter than the injection gas tie line), it will always be a semishock. In a condensing gas drive (injection gas tie line is shorter than the initial oil tie line), it may be a semishock or a genuine shock. 2. Tie-Line Rarefaction, b-c. In a vaporizing gas drive, this rarefaction along the initial oil tie line is always present. It connects the landing point of the leading semishock with the point at which the nontie-line composition variation begins, either the equal-eigenvalue point or the semishock point of the intermediate shock. In a condensing gas drive, this segment is missing if the leading shock is a genuine shock, as it often is. 3. Composition Variation between Tie Lines, c-d or c→d. If the composition variation is a rarefaction, the wave velocity on the nontie-line path will match the tie line eigenvalue on the shorter tie line (at the equal-eigenvalue point), and there will be a zone of constant state associated with the point at which the nontie-line path intersects the longer tie line. If the composition variation is a shock, the shock velocity will match the tie-line eigenvalue on the shorter tie line, and there will be a zone of constant state associated with the shock landing point on the longer tie line. 4. Tie-Line Rarefaction, d-e. In a condensing gas drive, this segment, which connects the nontie-line path switch point on the injection gas tie line (point d to the trailing shock point (point e), is always present. That shock is always a semishock. In a vaporizing drive, this segment is present only if the trailing shock is a semishock. Otherwise, this segment is missing. 5. Trailing Shock, e→f. A trailing shock is always present as long as the injection gas is a single-phase mixture. In a condensing drive, it is always a semishock. In a vaporizing drive it may be a semishock or a genuine shock. 5.5. STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS 115 C1 C1 d a d a c c b a 2 a2 C3 a1 a C2 C3 C2 1.0 a 1.0 a a d a d d d c a b1 c a b2 0.0 a1 0.0 a2 0.0 0.5 1.0 1.5 2.0 2.5 ξ 0.0 0.5 1.0 1.5 2.0 2.5 ξ (a) Initial mixture a1 f a d a a a 3 (b) Initial mixture a2 f a d4 a a4 a C3 C2 C3 C2 1.0 a 1.0 a d d c a a3 0.0 0.0 f a d4 a a a4 0.0 0.5 1.0 1.5 2.0 2.5 ξ 0.0 0.5 1.0 1.5 2.0 2.5 ξ (c) Initial mixture a3 Figure 5.21: Effect of variations in initial composition. (d) Initial mixture a4 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 5 percent. (c) a3 has a gas saturation of 30 percent. (d) a4 has an initial gas saturation of 80 percent. ... - tailieumienphi.vn
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