Orr, F. M. - Theory of Gas Injection Processes Episode 9

6.3. SYSTEMS WITH VARIABLE K-VALUES 151 C3 C3 • • • • CH4 •• •• • • • • • • • CH •• •• 6 • • • C6 C16 (a) Vertical Ruled Surface C16 (b) Horizontal Ruled Surfaces Figure 6.11: Ruled surfaces of tie lines ([54, p. 207], used with permission), [56]. that intersect the initial tie line with the “horizontal” shock surface of tie lines that intersect the injection tie line. The sequence of steps for constructing a solution for a system with constant K-values can also be used for systems with variable K-values, except for the procedure used to determine the crossover tie line. Three situations are possible. If the crossover tie line is connected to the initial oil and injection gas tie lines by shocks, then a set of tie line intersection equations can be solved for the crossover tie line. If a shock connects either the injection gas or initial oil tie line to the crossover tie, then the crossover tie line is at the intersection of a rarefaction surface and a shock surface. Tie lines on the shock surface must intersect the injection gas or initial oil tie line. The rarefaction surface is generated by stepwise integration of some nontie-line path from the initial oil or injection gas tie line. If rarefactions connect the three tie lines, then the crossover tie line is the intersection of two rarefaction surfaces that are determined by integration. Table 6.1 can be used to determine whether a rarefaction or shock occurs for each segment. Fig. 6.12 shows the composition route for a displacement of an oil that is a mixture of CH4, C6, and C16 by a gas mixture that contains CH4 enriched with propane (C3). The geometry of the tie lines (see Table 6.1) indicates that there is a rarefaction between the initial tie line and the crossover tie line and a shock from the crossover tie line to the injection tie line. Fig. 6.13 shows the corresponding saturation and composition profiles. The crossover tie line is identified by integrating an arbitrary nontie-line path upwards from the initial tie line to find the tie line that intersects the injection tie line. Once the crossover tie line is determined, the actual nontie-line path is traced from the appropriate equal-eigenvalue point on the crossover tie line to the initial tie line. The semishock from the crossover tie line to the injection gas tie line is calculated by solving the shock balance for component C6 for the semishock point on the crossover tie line. That shock balance is the simplest to use because C6 is missing on the upstream side of the shock. The landing point on the injection tie line is found from a shock balance for one of the other components. 152 CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS C3 • • • CH4 f • •• • b •a C6 C16 Figure 6.12: Displacement with variable K-values. The initial oil contains 20 mole percent CH4, and 40 mole percent each of C6 and C16. The injection gas contains 90 mole percent CH4 and 10 mole percent C3. The Peng-Robinson EOS was used to calculate phase equilibrium at 2000 psia (136 atm and 200 F (93 C). See Johns [54, pp. 195–197] for EOS parameters and details of the solution. In this displacement again, there is a fast-moving CH4 bank at the leading edge of the transition zone (just upstream of the a → b shock). No C3 is present in that bank, but it appears in the flowing mixtures as the b - c rarefaction is traced. All of the C6 evaporates at the shock from the crossover tie line to the injection tie line (d → e). The trailing shock, across which all the C16 evaporates, moves very slowly because the solubility of C16 in pure CH4 is low. For displacements in which all the tie lines are connected by shocks (fully self-sharpening), solu-tion construction is simpler because no integration is required. Given an injection gas composition and an initial oil composition, the tie lines that extend through those compositions can be identified by a negative flash [135, 129]. The crossover tie line can then be found by solving for the tie line in the interior that intersects both the injection tie line and the initial tie line. That tie line must satisfy (see Eq. 6.1.27) an equation that specifies the intersection of the crossover tie line with the injection tie line, xinj n1 + (Kinj − 1)Linjo = xcr {1 + (Kcr −1)Lcr}, i = 1,nc, (6.3.1) and must also satisfy a similar expression that determines the intersection between the initial and crossover tie lines, xinit n1 + (Kinit − 1)Linito = xcr {1 + (Kcr − 1)Lcr}, i = 1,nc. (6.3.2) In Eqs. 6.3.1 and 6.3.2, the superscripts inj, cr, and init refer to the injection, crossover, and initial tie lines, and L is the overall mole fraction on a tie line. These expressions are nonlinear because the unknown K-values on the crossover tie line depend on xcr. They can be solved by guessing 6.3. SYSTEMS WITH VARIABLE K-VALUES 153 1 0 1 1 0 0 1 1 0 0 0.0 1.0 2.0 0.0 1.0 2.0 ξ/τ ξ/τ Figure 6.13: Saturation and composition (mole fraction) profiles for the displacement of Fig. 6.12 [54, p. 198]. a set of K-values and then iterating. Once the crossover tie line is determined, the shocks to the injection and initial tie lines can be found as in the previous examples. Fig. 6.14 shows the composition route for a displacement that is similar to the displacement of Figs. 6.12 and 6.13 except that propane is replaced by ethane (C2). Replacing C3 with C2 moves the tie line intersections for the segment connecting the initial and crossover tie lines from the vapor side of the two-phase region to the liquid. As a result, the nontie-line paths become self-sharpening (Table 6.1), and the nontie-line rarefaction in Fig. 6.12 is replaced by a shock in Fig. 6.14. Otherwise, the displacement profiles are similar, as comparison of Figs. 6.13 and 6.15 shows. The crossover tie line is connected to the initial and injection tie lines by semishocks, and the only rarefaction present in the profiles is the rarefaction along the crossover tie line. Because the crossover tie line is the shortest tie line, solution construction starts there. The construction of the semishocks to the initial and injection tie lines is no more difficult than shock construction for ternary systems, or even for a Buckley-Leverett problem. The key step is to find the crossover tie line and the corresponding tie-line intersection points. Once they are known, the shock constructions require only solution of a nonlinear equation in the saturation on the crossover tie line to determine the semishock points on that tie line (c and d in Fig. 6.14) and the shock velocity. A similar nonlinear expression in the saturation on the initial or injection tie line determines the landing points on those tie lines. 154 CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS C2 • • • CH4 f • d c • • C6 C16 Figure 6.14: Displacement with variable K-values. The initial oil contains 20 mole percent CH4, and 40 mole percent each of C6 and C10 . The injection gas contains 90 mole percent CH4 and 10 mole percent C2. The Peng-Robinson EOS was used to calculate phase equilibrium at 2000 psia (136 atm and 200 F (93 C). See Johns [54, pp. 195, 213, 215] for EOS parameters and details of the solution. 1.0g e,f e d c b 0.0 1.0 b a 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 1.0 2.0 0.0 1.0 2.0 ξ/τ ξ/τ Figure 6.15: Saturation and composition (mole fraction) profiles for the displacement of Fig. 6.14 [54, p. 214]. 6.4. CONDENSING/VAPORIZING GAS DRIVES 155 6.4 Condensing/Vaporizing Gas Drives In an important 1986 paper, Zick [140] used numerical compositional simulation results to demon-strate that gas displacement processes in a multicomponent system can display simultaneously fea-tures that resemble the behavior of vaporizing and condensing displacements described for ternary systems. Zick referred to such displacements as condensing/vaporizing gas drives. The analysis of four-component solutions reveals why Zick’s terminology is an accurate portrayal of the flow behavior. Consider, for example, the displacement illustrated in Figs. 6.12 and 6.13. Consider also two related ternary displacements illustrated in Fig. 6.16. If pure CH4 displaces oil O1, a mixture of CH4, C6, and C16, the displacement is the ternary vaporizing gas drive shown in Fig. 6.16a. If, on the other hand, the original oil composition is a mixture of C16 and CH4 (O2), and the injection gas is CH4 enriched with 10 percent C3, the displacement is the condensing gas drive shown in Fig. 6.16b. The four-component displacement is a combinationof segments that are similarto the vaporizing and condensing ternary displacements. Fig. 6.17 illustratesthat idea. The composition route for the ternary vaporizing gas drive for pure CH4 displacing oil O1 is shown in the base of the quaternary diagram. If 10 percent C3 is added to the injection gas, the segment of the ternary route that includes the rarefaction along the initial tie line, the intermediate shock from the initial tie line to the injection tie line, and the trailing shock move upward into the interior of the quaternary diagram. The same segments are still present: a rarefaction along the crossover tie line, a shock from the crossover tie line to the injection tie line, and a trailing shock to the injection gas composition. This vaporizing segment is connected to the initial oil tie line by a condensing segment that is essentially equivalent to the leading portions of the ternary condensing gas drive shown in Fig. 6.16b. The ternary displacement includes a shock from the initial oil composition along the initial tie line, a nontie-line rarefaction from the initial tie line to the injection tie line, and a tie-line rarefaction along the injection gas tie line. The quaternary system includes a leading shock along the initial tie line, a nontie-line rarefaction from the initial tie line to the crossover tie line, and a tie-line rarefaction along the crossover tie line. Thus, this four-component displacement consists of a leading condensing segment connected to a trailing vaporizing segment by the crossover tie line. The leading segment is similar to a ternary condensing drive, but with the crossover tie line as the injection gas tie line, and the trailing segment is closely related to a ternary vaporizing gas drive with the crossover tie line as the initial tie line. The crossover tie line links the two segments together in the quaternary system. The example shown in Figs. 6.12 and 6.17 are not the only possible combination of condensing and vaporizing segments. Fig. 6.18 illustrates why. Two surfaces of tie lines are shown in Fig. 6.18, both associated with a specific crossover tie line (the two surfaces shown are the same surfaces illustrated in Figs. 6.1 and 6.2). The crossover tie line is the tie line in the interior of the quaternary diagram that extends to point b on the C1 = 0 face and point e on the C4 = 0 face. That tie line is at the intersection of the two surfaces. The points a, b, and c all lie on extensions of tie lines in the vertical surface to the C1 = 0 face, and the points d, e, and f are on extensions of tie lines in the horizontal surface to the C4 = 0 face. Imagine that points a, b, and c are initial oil compositions and points d, e, and f are injection gas compositions. Consider what composition route patterns arise if the crossover tie line is fixed but there are various combinations of injection gas and initial oil compositions. Displacement of oil a by gas ... - tailieumienphi.vn