Orr, F. M. - Theory of Gas Injection Processes Episode 6

5.2. SHOCKS 91 1 Switch at a a a λnt λa 0 0 1 2 3 ξ/τ 1 Switch at b a λt =λnt 0 0 1 2 3 ξ/τ 1 Switch at c a a λc λnt 0 0 1 2 3 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 5. 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 4. 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 FII −FI CII −CI i = 1,nc. (5.2.1) If the fluid on one side of the shock is single-phase, then FI = CI, and hence, Eq. 5.2.1 can be rearranged to give FII −ΛCII i 1 − Λ i = 1,nc. (5.2.2) Substitution of the definitions of CII and FII in terms of the phase compositions (Eqs. 5.1.3 and 5.1.4) yields (Helfferich [31]) f1 −ΛS1 (1 − f1) − Λ(1 −S1) i 1 −Λ 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 II dfII FII − FI dSII CII − CI i = 1,nc. (5.2.4) 5.2.2 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 two-phase 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. Dif- ferentiation of the expression for λnt in Eq. 5.1.24 with respect to η gives dλnt 1 dF1 dp F1 + p dC1 dp dη C1 + p dη dη (C1 + p)2 dη dη 1 dF1 dp dC1 dp C1 + p dη dη nt dη dη The derivative dF1/dη can be related to λt and ~ent by the following manipulations: dF1 ∂F1 dC1 ∂F1 dy1 dη ∂C1 dη ∂y1 dη ∂F1 dy1 dC1 t ∂y1 dC1 dη The value of dy1/dC1 is given by ~ent (Eq. 5.1.25), dy1 λnt − λt dC1 ∂y1 Substitution of Eqs. 5.2.6 and 5.2.7 into Eq. 5.2.5 gives the desired result, dλnt C1 −F1 dp F1 −C1 dCe dη (C1 + p)2 dη (C1 −Ce)2 dη (5.2.5) (5.2.6) (5.2.7) (5.2.8) where C1 denotes the overall volume fraction of component 1 on the envelope curve. Nontie-line paths do not cross the equivelocity curve (where C1 = F1), so the sign of dλnt/dη on a particular path is determined by the sign of dCe/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 can be drawn (or even just sketched) by finding whether Ce 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 illustratehow 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 K2 −1 x2 K1 −1 x1 and (5.2.9) 94 CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS φ = K1 −K2x2. (5.2.10) 1 Expressions for Ce and dCe/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), K1x1 + K2x2 + K3(1 −x1 − x2) = 1. The resulting expression for Ce is y2 K1 −K2K1 −K3 1 K2 1 −K2 1 −K3 and the expression for dCe/dη is dCe dp 2y1 K1 −K2K1 −K3 dη dη K1 1 − K2 1 − K3 (5.2.11) (5.2.12) (5.2.13) Eq. 5.2.12 is an explicit expression for the envelope curve , a segment of parabola when K-values 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 < 1. 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 tie line A to tie line B, Ce decreases as the composition changes from the downstream (oil) tie line to the upstream (injection gas) tie line. Therefore, dCe/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 line, a shock is required. As the nontie-line path is traced upstream from tie line B, Ce increases as does η, dCe/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 inter-mediate (LVI) component (K2 < 1), (b) high volatility intermediate (HVI) indexHVI component (K2 > 1). ... - tailieumienphi.vn