Orr, F. M. - Theory of Gas Injection Processes Episode 4

4.2. SHOCKS 51 shock at τ shock at τ+∆τ Ci=Ci Ci=Ci ξ ξ+∆ξ Figure 4.6: Motion of a shock. original conservation equation. In other words, it is a statement that volume is conserved across the shock, just as Eq. 4.1.1 states that volume is conserved at locations where all the derivatives exist. Eq. 4.2.2 says that the velocity at which the shock propagates is set by the slope of a line that connects the two states on either side of the shock on a plot of F1 against C1 such as that shown in Fig. 4.1. Now we apply the jump condition to determine what happens at the leading edge of the dis-placement zone, where fast characteristics (the characteristics in Fig. 4.5 that have high values of dF1/dS1) intersect the characteristics for the initial composition. Point a in Fig. 4.7 is the initial composition, which is the composition on the downstream side of the leading shock, and points b, c, d, e, f, and g are possible composition points for the fluid on the upstream side of the shock. Any of the shock constructions shown in Fig. 4.7 satisfies Eq. 4.2.2. Hence some additional reasoning is required to select which shock is part of a unique solution to the flow problem. Two physical ideas play a role in that reasoning. The first is simply an observation that compositions that make up the downstream portion of the solution must have moved more rapidly than compositions that lie closer to the inlet. If not, slow-moving downstream compositions would be overtaken by faster compositions upstream. The idea is frequently stated [31] as a Velocity Constraint: Wave velocities in the two-phase region must decrease monoton-ically for zones in which compositions vary continuously as the solution composition path is traced from downstream compositions to upstream compositions . When the velocityconstraint is satisfied, the solutionwill be single-valued throughout. Composition variations that satisfy the velocity constraint are sometimes described as compatible waves, and the velocity constraint may also be called a compatibility condition. The second idea is that a shock can exist only if it is stable in the sense that it would form again if it were somehow smeared slightly from a sharp jump, as might happen if a small amount of physical dispersion were present, for example. That idea can be stated in terms of wave velocities [67, 83, 106] as an Entropy Condition: Wave velocities on the upstream side of the shock must be greater 52 CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT than (or equal to) the shock velocity and wave velocities on the downstream side must be less than (or equal to) the shock velocity. (In the examples considered here, the wave velocity can be equal to the shock velocity on only one side of the shock at a time.) For the application considered here, the condition really has nothing to do with the thermodynamic entropy function, but the name has been universally used in descriptions of solutions to hyperbolic conservation laws since the ideas behind entropy conditions were first derived for compressible fluid flow problems in which entropy must increase across the shock. Consider what would happen to a shock that was slightly smeared if the entropy condition were not satisfied. Slow-moving compositions upstream of the shock would be left behind by fast-moving compositions downstream of the shock, and as a result, the shock would pull itself apart. Hence, the entropy condition must be satisfied if a shock is to be stable. A shock that does satisfy the entropy condition is said to be self-sharpening. For a detailed discussion of the various mathematical forms in which entropy conditions can be expressed, see the review given by Rhee, Aris and Amundson [106, pp. 213–220 and pp. 341–348]. We now apply the velocity constraint and the entropy condition to obtain a unique solution for two-component displacement. Fig. 4.8 illustrates possible solutions for the leading shocks indicated in Fig. 4.7. Consider, for example, a shock that connects downstream composition a and upstream composition b. The top left panel of Fig. 4.8 shows the location of the shock at some fixed time and also shows how the solution would behave if the concentration of C1 increased smoothly upstream of the shock. The wave velocity, Λ, of the shock (Eq. 4.2.2) is given by the slope of the chord that connects points a and b on Fig. 4.7. That velocity is clearly less than one, and hence the a→b shock moves more slowly than the single-phase compositions downstream of the shock, which have unit velocity. The wave velocity of the composition just upstream of the shock is given by dF1/dC1 at point b. That velocity is lower still than the wave velocity of the shock. Thus, the a→b shock violates the entropy condition. As the C1 concentration upstream of the shock increases, however, the wave velocities increase to values greater than the shock velocity, a variation that produces compositions that violate the velocity constraint. Hence, a solution that includes a shock from a to b followed by a continuously varying composition violates both the velocity constraint and the entropy condition and can be ruled out, therefore. The a→c, a→d, and a→e shocks all satisfy the entropy condition, but all three violate the velocity constraint, as the profiles in Fig. 4.8 show. The a→g satisfies the velocity constraint, but it violates the entropy condition because the wave velocity of the upstream composition is lower than the shock velocity. Hence, the only remaining possible solution is that shown for the a→f shock. The point f is the point at which the chord drawn from point a is tangent to the overall fractional flow curve. The a→f shock does satisfy the entropy condition, but it does so in a special way. The wave velocity of the composition C1 of point f is equal to the shock velocity, because the shock velocity is given by the slope of the tangent a–f, and that chord slope is the same as dF1/dC1 at point f. A shock in which the shock velocity equals the wave velocity on one side of the shock is sometimes called a semishock [106, pp. 217–219] , an intermediate discontinuity [40], or a tangent shock [82]. Because the leading shock must be a semishock if it is to satisfy the velocity constraint and the entropy condition, the composition of the fluid on the upstream side of the shock can be found easily by solving 4.2. SHOCKS 53 1.0 a 0.8 fa a 0.6 0.4 0.2 b a a a d c a a 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Overall Volume Fraction of Component 1, C1 Figure 4.7: Possible shocks from the initial state at point a. II I dC1| = CII − CI . (4.2.3) The tangent construction described in Eq. 4.2.3 and shown in Fig. 4.7 is equivalent to the well-known Welge tangent construction [133] used to solve the problem of Buckley and Leverett [10] for water displacing oil. Just as a shock was required in order to make the solution single-valued at the leading edge of the transition zone, another shock is required at the trailing edge. The characteristics in Fig. 4.4 for the injection composition intersect the characteristics in Fig. 4.5 for slow moving compositions, C1, greater than the shock composition. Reasoning similar to that for the leading shock shows that the trailing shock also is a semishock, this time with the wave velocity on the downstream side of the shock equal to the shock velocity. In fact, similar arguments indicate that a shock must form any time the number of phases changes for the fractional flow relation used here. Fig. 4.9 shows the resulting tangent constructions for the leading (a→b) and trailing shock (c→d). Fig. 4.10 gives the completed solution profiles of S1 and C1. Each profile includes a zone of constant state with the initial composition ahead of the leading shock, a zone of continuous variation of overall composition and saturation between the leading shock and the trailing shock, and finally another zone of constant state with the injection composition behind the trailing shock. The solution in Fig. 4.10 is reported as a function of ξ/τ, which is the wave velocity of the corresponding 54 CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT 1 1 a-b a-e 0 0 0 1 0 1 ξ ξ 1 1 a-c a-f 0 0 0 1 0 1 ξ ξ 1 1 a-d a-g 0 0 0 1 0 1 ξ ξ Figure 4.8: Composition profiles for the leading shocks to various two-phase compositions. value of C1. In this homogeneous, quasilinear problem, the wave velocity of any composition is constant, and hence the position of any composition that originated at the inlet must be a function of ξ/τ only. In fact, Lax [67] showed that the solution to a quasilinear Riemann problem is always a function of ξ/τ only. The spatial position of a given composition C1 can be obtained simply by multiplying the corresponding value of ξ/τ by the value of τ at which the solution is desired. Another version of the solution is shown in Fig. 4.11, which includes a τ-ξ diagram and a plot of the C1 profile at τ = 0.60. Shown in the τ-ξ portion of Fig. 4.11 are the trajectories of the leading and trailing shocks and a few of the characteristics. The locations, ξ, of the shocks and the compositions associated with specific characteristics can be read directly from the t-x diagram for a particular value of τ, as Fig. 4.11 illustrates. From Fig. 4.11 it is easy to see that as the flow proceeds, the solution retains the shape shown in the profiles of Figs. 4.10 and 4.11, but the entire solution stretches as fast-moving compositions pull away from slow-moving ones. That behavior is typical of problems in which convective phenomena dominate the transport. Fig. 4.11 also illustrates the point that when the entropy condition is satisfied for a particular 4.2. SHOCKS 55 1.0 d a c b 0.8 0.6 0.4 0.2 a a 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Overall Volume Fraction of Component 1, C1 Figure 4.9: Leading and trailing shock constructions. shock, characteristics on either side of the trajectory of a shock either impinge on the shock trajec-tory or are at least parallel to the shock trajectory. In the case of the leading shock, for example, the characteristics of the initial composition, which lies downstream of the shock, intersect the shock trajectory, while characteristic just upstream of the shock overlaps the shock trajectory. The reverse is true at the trailing shock. Between the trajectories of the shocks is the fan of characteristics associated with the continuous variation of composition, which is known as a spreading wave, a rarefaction wave, or an expansion wave. Because the characteristics all emanate from a single point, the origin, they are also referred to as a centered wave. The change in slope of the characteristics in the spreading wave reflects the fact that the slope of the fractional flow curve drops rapidly over a fairly narrow range of composition (see Fig. 4.2). As a result, the wave velocity declines significantly during the relatively small compositionchange between the leading and trailing shocks. In the solution shown in Fig. 4.10 the overall compositions and saturations vary in the two-phase region, but the phase compositions do not. They are fixed by the specified phase equilibrium. It is the differing amounts of the two phases present and flowing that change the overall composition and fractional flow. ... - tailieumienphi.vn