4.2 SHOCKS 51 Ci=CΠ i shock at τ shock at τ+∆τ Ci=CΙ i 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 displacement 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 monotonically for zones in which compositions vary continuously as the solution composition path is traced from downstream compositions to upstream compositions When the velocity constraint is satisfied, the solution will 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 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 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 Overall Fractional Flow of Component 1, F1 g f 0.8 a a e 0.6 0.4 a b 0.2 a a c d 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 F II − FiI dF1 II | = iII dC1 Ci − CiI (4.2.3) The tangent construction described in Eq 4.2.3 and shown in Fig 4.7 is equivalent to the wellknown 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 TWO-COMPONENT GAS/OIL DISPLACEMENT C1 a-b a-e 0 ξ ξ C1 a-c a-f 0 ξ ξ C1 a-d a-g 0 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 d a 1.0 a Overall Fractional Flow of Component 1, F1 c b a 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 trajectory 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 composition change 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 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 56 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT 1.0 d C1 c b 0.5 a 0.0 0.0 0.5 1.0 1.5 ξ/τ 1.0 d S1 c b 0.5 0.0 0.0 a 0.5 1.0 1.5 ξ/τ Figure 4.10: Solution composition and saturation profiles 4.3 Variations in Initial or Injection Composition In the binary gas/oil displacement problem, the leading shock forms because some two-phase mixtures of injected fluid with initial fluid move rapidly and overtake the initial composition The trailing shock forms because some Component can evaporate into the unsaturated injected vapor How fast the leading and trailing shocks move depends on the initial and injection compositions In this section we examine briefly the sensitivity of the solution to the binary displacement problem to changes in the initial and injection compositions Fig 4.12 shows a set of key points on the fractional flow curve Points a, b, c, and d are from the solution discussed in the previous section Points a and d are the initial and injection values, and points b and c are the tangent points for the leading and trailing semishocks Point e is the saturated vapor phase, and point i is the saturated liquid Point f corresponds to S1 = − Sor , and point h is that at which S1 = Sgc Point g is the intersection of the F1 = C1 line with the overall fractional flow curve Point j is the inflection point in the fractional flow curve It corresponds to the maximum in dF1 /dC1 shown in Fig 4.2 Fig 4.13 shows examples of the solutions that result when the initial composition is fixed at a C1 and the injection composition is varied The six panels in Fig 4.13 illustrate changes in the appearance of the composition profiles for injection compositions with decreasing volume fractions, inj C1 Fig 4.14 shows the corresponding characteristic (τ -ξ) diagrams The following observations can be made for injection compositions in the regions bounded by the key points in Fig 4.13: inj e d to e: For injection compositions in the range > C1 > C1 , the solution still 4.3 VARIATIONS IN INITIAL OR INJECTION COMPOSITION 57 71 68 Trailing Shock 1.0 = =0 0.8 C C a a a a τ 0.6 Leading Shock 0.4 0.2 0.0 1.0 C1 a a a a 0.5 0.0 0.0 0.5 1.0 ξ Figure 4.11: Evaluation of the solution at a specific time, τ , from the τ -ξ diagram includes leading and trailing semishocks connected by a spreading wave (see the top left panel of Fig 4.13), and the τ -ξ diagram shown in the corresponding panel in Fig inj 4.14 is qualitatively similar to Fig 4.11 As C1 is decreased, the trailing shock speed decreases, reaching zero when the injected fluid is vapor saturated with component inj e (C1 = C1 ) The leading portion of the solution is unchanged, however inj f e e to f: Compositions in the range C1 > C1 > C1 have zero wave velocity because e dF1 /dC1 = There is a trailing shock from the injection composition to C1 , but it has zero wave velocity, and hence the fan of characteristics in Fig 4.14 extends all the way to the ξ = axis The leading portion of the solution remains unchanged f to b: An injection composition between f and b has nonzero wave velocity, and as a result, the solution in the lower left panel of Fig 4.13 shows a zone of injection compositions at the upstream end that all propagate with the same wave velocity That portion of the solution has a set of parallel characteristics in Fig 4.14 that emanate from the ξ = axis The fan of characteristics that represents the spreading wave terminates at the characteristic that represents that propagation of the injection composition When inj b C1 = C1 , the entire solution upstream of the leading shock is that zone of constant 58 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT d a 1.0 a Overall Fractional Flow of Component 1, F1 c a a f e b a 0.8 0.6 a j ag 0.4 i a a 0.2 h a a 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Overall Volume Fraction of Component 1, C1 Figure 4.12: Composition ranges for variation of injection and initial compositions b composition with C1 = C1 The leading shock velocity is still unchanged, however inj g b b to g: A leading semishock is no longer possible for C1 > C1 > C1 , because continuous variation from the tangent shock point to the injection composition is prohibited by the velocity rule A leading shock to the injection composition followed by a set of constant compositions at the injection composition satisfies the entropy condition and velocity constraint The top right panel in Fig 4.14 indicates, for example, that the characteristics associated with the injection composition intersect the leading shock trajectory, as the characteristics associated with the initial composition, an indication that the entropy condition is satisfied The leading shock velocity is given by Eq 4.2.2 inj init with the known compositions C1 and C1 The leading shock velocity is now lower inj inj g b than that of the leading semishocks that form for C1 > C1 When C1 = C1 , the leading shock has unit velocity g to h: A leading shock directly from the injection composition to the initial composition is no longer possible because it would violate the entropy condition A shock from the initial composition to the injection composition would have a velocity less than one The characteristics of the initial composition (see the middle right panel of Fig 4.14)would not intersect the shock trajectory, and hence, the entropy condition would not be satisfied The only path available that does not violate the entropy condition is 4.3 VARIATIONS IN INITIAL OR INJECTION COMPOSITION 59 Cinj=0.560 d-e b-g C1 Cinj=0.975 0 ξ ξ 1 Cinj=0.920 Cinj=0.320 C1 e-f g-h 0 ξ ξ 1 Cinj=0.770 Cinj=0.215 C1 f-b h-i 0 ξ ξ Figure 4.13: Effect of changes in injection composition i a leading shock with unit velocity to the saturated liquid composition, C1 , followed by a slower trailing shock to the injection composition The characteristics of the injection composition intersect the trailing shock, and the characteristics of the initial composition intersect the leading shock The zone between the two shocks is what is known as a zone of constant state The composition i has two wave velocities: one is the leading shock velocity, and the other is the trailing shock velocity h to i: The situation is the same as for g-h, except that the trailing shock has zero velocity The trailing shock is a jump from the injection composition to the saturated i liquid composition, C1 Fig 4.13 indicates that the form of the solution changes significantly as the injection composition changes Solution behavior also changes if the initial composition is changed Fig 4.15 illustrates inj d what happens if the injection composition is fixed at C1 = C1 = 1, and the initial composition is varied in ranges bounded by the key points shown in Fig 4.12 Four composition intervals are 60 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT d-e b-g τ 0 e-f g-h τ 0 0 f-b h-i τ 0 0 ξ ξ Figure 4.14: τ -ξ diagrams for displacements illustrated in Fig 4.13 important: a to i: As the amount of component in the initial mixture increases, the velocity of the leading semishock also increases slightly (the slope of the tangent drawn from the initial composition to the fractional flow curve increases), and the composition on the upstream side of the leading shock decreases slightly below point b The remainder of the composition profile upstream of the leading shock is unaffected i to j: Increasing C1 from i toward j causes the leading shock speed to increase substantially and the shock composition to approach j init j to c: For C1 greater than the inflection composition, there is no leading shock The leading portion of the solution is simply a spreading wave init c c to e: When C1 > C1 , the trailing shock is no longer a semishock Instead that inj init evaporation shock is what is known as a genuine shock, a jump from C1 to C1 , with 4.4 VOLUME CHANGE 61 1 j-c Cinit=0.18 C1 a-i Cinit=0.62 0 1 ξ ξ 1 c-e Cinit=0.30 C1 i-j Cinit=0.80 0 1 ξ ξ Figure 4.15: Effect of changes in initial composition init velocity given by Eq 4.2.2 The trailing shock velocity increases as C1 is increased, init e reaching unit velocity when C1 = C1 Figs 4.13 and 4.15 indicate that solutions for binary gas/oil displacement show considerable variation as the injection and initial conditions are changed Many of the features of these binary solutions reappear in the multicomponent solutions that are considered in subsequent chapters, and hence a detailed understanding of the binary solutions is useful underpinning for the analysis of more complex multicomponent flows 4.4 Volume Change When components change volume as they transfer from one phase to another, volume is not conserved, and the appropriate balance equation on moles of component i is Eq 2.3.9 written for the two components, ∂G1 ∂H1 + ∂τ ∂ξ ∂G2 ∂H2 + ∂τ ∂ξ = 0, (4.4.1) = 0, (4.4.2) where Gi = xi1 ρ1D S1 + xi2 ρ2D (1 − S1 ), (4.4.3) 62 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT Hi = vD (xi1 ρ1D f1 + xi2 ρ2D (1 − f1 )) 4.4.1 (4.4.4) Flow Velocity The local flow velocity, vD , appears in both balance equations in the definition of Hi That velocity changes when components change volume as they transfer between phases or as the composition of a phase changes When compositions change along a tie line, however, the local flow velocity remains constant [22] When a composition variation remains on a single tie line within the two-phase region (as it must for this binary problem where there is only one tie line), the phase composition, xi1 and xi2 , and the dimensionless molar phase densities, ρ1D and ρ2D , remain constant at the values for the equilibrium phases As a result, substitution of the definitions for Gi and Hi (Eqs 4.4.3 and 4.4.4) into Eqs 4.4.1 and 4.4.2 followed by rearrangement gives ∂S1 ∂ + ∂τ ∂ξ vD f1 + x12 ρ2D x11 ρ1D − x12 ρ2D = 0, (4.4.5) ∂S1 ∂ + ∂τ ∂ξ vD f1 + x22 ρ2D x21 ρ1D − x22 ρ2D = (4.4.6) and Subtraction of Eq 4.4.6 from Eq 4.4.5 yields an expression for the spatial derivative of the velocity, x22 ρ2D x12 ρ2D − x11 ρ1D − x12 ρ2D x21 ρ1D − x22 ρ2D ∂vD = ∂ξ (4.4.7) It is convenient to rewrite Eq 4.4.7 in terms of the equilibrium K-values, x11 = K1 x12 and x21 = K2 x22 , which gives ρ2D ρ2D − K1 ρ1D − ρ2D K2 ρ1D − ρ2D ∂vD = ∂ξ (4.4.8) As long as K1 = K2 , which must be true if two phases are to form, Eq 4.4.8 shows that ∂vD = ∂ξ Hence, the local flow velocity is constant for composition variations in the two-phase region along a single tie line This behavior results from the fact that the phase densities remain constant for mixtures on a single tie line in the two-phase region 4.4.2 Characteristic Equations The characteristic equations can now be obtained just as they were in Section 4.1 Arguments similar to those given in Section 4.1 indicate that H1 is a function of G1 only, and hence, ∂G1 dH1 ∂G1 + = ∂τ dG1 ∂ξ (4.4.9) G1 is a function of ξ and τ , and therefore, ∂G1 dτ ∂G1 dξ dG1 = + dη ∂τ dη ∂ξ dη Comparison of Eqs 4.4.9 and 4.4.10 gives the characteristic equations, (4.4.10) 4.4 VOLUME CHANGE 63 dG1 dη dτ dη dξ dη = 0, (4.4.11) = 1, (4.4.12) = dH1 dG1 (4.4.13) Comparison of Eqs 4.4.11–4.4.13 with the corresponding equations for constant volume flow, Eqs 4.1.9–4.1.11, indicates that within the two-phase region, at least, the solutions for flow with and without volume change have similar structure The similarity can be seen more clearly if dH1 /dG1 is evaluated Differentiation of Eq 4.4.4 gives df1 dH1 = vD (x11 ρ1D − x12 ρ2D ) dG1 dG1 (4.4.14) Eq 4.4.3 can be rearranged to show that S1 is a function of G1 only, and therefore, df1 dS1 df1 df1 = = dG1 dS1 dG1 x11 ρ1D − x12 ρ2D dS1 (4.4.15) Substitution of Eq 4.4.15 into Eq 4.4.14 shows that df1 dH1 = vD dG1 dS1 (4.4.16) Hence for compositions within the two-phase region, the wave velocity is simply the wave velocity for constant volume flow scaled by the appropriate local flow velocity within the two-phase region The distinction between flow velocity and wave velocity is an important one The flow velocity is the total volumetric flow rate of all the phases per unit area The wave velocity is the speed at which a given composition propagates The two are very different When volume is not conserved, the flow velocity does not change when the composition varies along a single tie line, but it does change at shocks that enter or leave the two-phase region Hence, the next step is to determine how flow velocity varies across the shocks 4.4.3 Shocks Consider the trailing shock from the injection composition, Gd , to composition, Gc , in the two-phase 1 region A shock balance indicates that the shock wave velocity, Λcd , is given by Λcd = Hic − Hid , Gc − Gd i i i = 1, (4.4.17) Eqs 4.4.17 can be written for either component When volume is not conserved, the two equations are independent As a result, the shock balances can be solved for both the downstream c composition, Gc , and the flow velocity, vD To show how that is done, it is convenient to write i np Hi = vD αi = vD xij ρjD fj j=1 (4.4.18) 64 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT c To find vD , we write Eqs 4.4.17 for components and and eliminate Λcd , which gives d c c v d αd − vD αc vD αd − vD αc 1 = D Gd − Gc Gd − Gc 1 2 (4.4.19) d d vD is the injection flow velocity, and by definition (Eq 2.3.2), vD = αd and Gd are the injection 1 c data, so Eq 4.4.19 can be solved for vD once Gc and αc are determined, 2 c vD = αd (Gd − Gc ) − αd (Gd − Gc ) 1 2 αc (Gd − Gc ) − αc (Gd − Gc ) 1 2 (4.4.20) Application of the entropy condition and velocity constraint shows that if injection composition is single-phase vapor, the trailing shock is a semishock that satisfies c vD αd − v c αc df1 = i d D ci , dS1 Gi − Gi i = or (4.4.21) c When component is not present in the injection fluid, vD can be eliminated from Eq 4.4.21, and it can then be solved to find the composition at point c Otherwise, Eqs 4.4.20 and 4.4.21 can be c solved simultaneously for vD and point c Similar manipulations give expressions that can be solved for the composition at the leading a shock, Gb, and the flow velocity, vD , ahead of the shock The wave velocity of the leading shock is i Λab = Hia − Hib , Ga − Gb i i (4.4.22) where Ga and Hia are the initial values For an initial composition in the single-phase region, the i leading shock is a semishock determined by a αb (Ga − Gb ) − αb (Ga − Gb ) vD 2 1 , = a c vD α1 (Ga − Gb ) − αa (Ga − Gb ) 2 1 (4.4.23) and αa (v a /v b ) − αb df1 = i Da D b i , dS1 Gi − Gi i = or (4.4.24) b b c Therefore, when vD is known, as it is from the solution for the trailing shock because vD = vD , a b vD and Gi can be obtained by solving Eqs 4.4.23 and 4.4.24 Thus, in a binary displacement, only three flow velocities exist: the known injection velocity behind the trailing shock, a fixed flow velocity in the two-phase region, and a different flow velocity ahead of the leading shock 4.4.4 Example Solution To illustrate how volume change affects flow behavior, we consider displacement of a hydrocarbon, decane (C10 ), by a gas, carbon dioxide (CO2 ), at 500 psia (34 atm) and 160 F (71 C) Table 4.1 reports Peng-Robinson equilibrium phase compositions and the initial, injection and phase molar densities Table 4.2 gives compositions, wave velocities, and flow velocities for the solution with volume change, and Table 4.3 reports the corresponding values for the solution without volume change In both cases, the fractional flow curves were assumed to be Eqs 4.1.20-4.1.22, with 4.4 VOLUME CHANGE 65 Table 4.1: Equilibrium Phase Compositions and Fluid Properties at 500 psia (34 atm) and 160 F (71 C) Fluid xCO2 xC10 Initial Oil Equil Liq Equil Vap Injected Gas 0.2733 0.9976 1 0.7267 0.0024 ρ (gmol/l) 4.829 5.988 1.378 1.375 ρ (g/cm3) 0.6869 0.6910 0.0610 0.0605 µ (cp) 0.333 0.018 - Sor = Sgc = Overall mole fractions shown in Table 4.2 were calculated from the values of Gi by noting that zi = Gi np (4.4.25) ρjD Sj j=1 Overall compositions in Table 4.3 were calculated from volume fractions using the pure component densities in Table 4.1 according to ρci nc zi = S1 i=1 {ci1 S1 + ci2 (1 − S1 )} ρci ci1 + (1 − S1 ) nc , (4.4.26) ρci ci2 i=1 where the phase volume fractions are given by Eq 2.4.1 with the phase mole fraction data in Table 4.1 Fig 4.16 and the wave velocity (dξ/dτ ) data in Tables 4.2 and 4.3 show that the composition profiles have similar appearances in the displacements with and without volume change, but the flow proceeds more slowly when volume is not constant In particular, the velocity of the leading shock is much lower when effects of volume change are included In fact, it has a wave velocity less than one, which means that more than one pore volume must be injected for the leading shock to reach the outlet (at ξ = 1) when volume is variable The change in flow velocity occurs because CO2 occupies much less volume when it is dissolved in the liquid phase than it does in the vapor phase [19] When CO2 saturates the C10 present in the two-phase region, therefore, significant volume is lost, and the flow slows accordingly As Table 4.2 and Fig 4.16 show, the flow velocity, vD , ahead of the leading shock is only about half the injection velocity In both displacements there is a slow-moving trailing evaporation shock It moves slowly because the solubility of C10 in CO2 is small In other words, a large amount of CO2 must be injected to evaporate the remaining C10 The velocity change at the trailing shock is small, however The concentration of C10 in the vapor phase is so low that the volume change associated with the transfer of C10 to the vapor has minimal effect on the flow velocity The values of vD in Table 4.2 indicate again that vD remains constant for compositions in the two-phase region The wave velocities in Fig 4.16 and Tables 4.2 and 4.3 also indicate that the displacement of C10 by CO2 is relatively inefficient While the leading shock moves with appreciable velocity, 66 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT 1.0 d z1 c c No volume change Volume change 0.5 b b 0.0 0.0 a 1.0 0.5 a 1.5 ξ/τ S1 1.0 0.5 0.0 0.0 0.5 1.0 1.5 1.0 1.5 ξ/τ vD 1.0 0.5 0.0 0.0 0.5 ξ/τ Figure 4.16: Displacement of C10 by CO2 , with and without volume change as components transfer between phases 4.5 COMPONENT RECOVERY 67 Table 4.2: Displacement of C10 by CO2 with Volume Change Label a b c d zCO2 0.0000 0.3676 0.4088 0.5264 0.7020 1.0000 S1 0.0000 0.3941 0.5000 0.7000 0.8630 1.0000 dξ dτ 0.5097 0.9147 0.3710 0.0662 0.0063 1.0000 vD 0.5097 0.9999 0.9999 0.9999 0.9999 1.0000 τ < 1.0932 1.0932 2.6956 15.097 158.75 > 158.75 RC10 0.5574 0.6503 0.8087 1.0000 1.0000 Table 4.3: Displacement of C10 by CO2 without Volume Change Label a b c d zCO2 0.0000 0.7214 0.7842 0.8703 0.9294 1.0000 S1 0.0000 0.3539 0.5000 0.7000 0.8375 1.0000 dξ dτ 1.0000 1.2967 0.3710 0.0662 0.0118 1.0000 vD 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 τ < 0.7712 0.7712 2.6954 15.096 85.092 > 85.092 QC10 0.7712 0.8336 0.9137 1.0000 1.0000 somewhat higher CO2 concentrations (and saturations) move much more slowly As a result, C10 is recovered much more slowly after the arrival of the leading shock at the outlet 4.5 Component Recovery The amount of component i recovered at the outlet can be calculated from the composition and saturation profiles obtained as part of the solution to the Riemann problem Just as the spatial distribution of compositions is found by solving a differential material balance, the recovery of individual components is obtained from an integral balance over the flow length The amount of any component recovered from the porous medium is simply the amount present initially plus the amount of that component injected during the time the flow has taken place minus the amount of that component currently present in the porous medium When volume change is neglected, for a porous medium of dimensionless length ξ = 1, the resulting expression for Q1 , the volume of component recovered, is inj init Q1 = C1 + F1 τ − C1 dξ (4.5.1) Prior to the arrival of the leading shock, fluid leaves the porous medium with the fractional flow of the initial mixture Because the initial composition is constant, the recovery of each component is just τ Fiinit Breakthrough of injected fluid occurs at τBT , when the leading shock arrives at the outlet, where ξ = Accordingly, 68 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT Λab After breakthrough, the integral in Eq 4.5.1 can be evaluated as τBT = τ Λcd C1 dξ = (4.5.2) C1 dξ + inj = C1 Λcd τ + τ Λcd τ Λcd C1 dξ C1 dξ (4.5.3) (4.5.4) The integral in Eq 4.5.4 is evaluated through integration by parts, which gives τ Λcd C1 dξ = C1 ξ ]1 Λcd τ − out C1 c C1 ξdC1 , (4.5.5) out where C1 is the overall composition at ξ = at time τ Evaluation of the first term and substitution of Eq 4.1.12 for ξ in 4.5.5 followed by integration gives τ Λcd out c C1 dξ = C1 − C1 τ Λcd − out C1 c C1 τ dF1 dC1 , dC1 out c out c = C1 − C1 τ Λcd − τ (F1 − F1 ) (4.5.6) (4.5.7) out where F1 is the fractional flow at the outlet at time τ Substitution of Eqs 4.5.4 and 4.5.7 and the definition of Λcd (Eq 4.2.2) into Eq 4.5.1 gives the final expression for the recovery of component 1, init out out Q1 = C1 − C1 + τ F1 (4.5.8) Similar reasoning leads to the expression for the recovery of component 2, init out out Q2 = C2 − C2 + τ F2 (4.5.9) It is also easy to show that when volume is conserved, the difference between the total amount of fluid injected and the total volume of component produced must be the volume of component recovered, Q2 = τ − Q1 (4.5.10) Similar integral balances apply when effects of volume change are included The resulting expressions for recovery of component i are Ri = Ginit − Gout + τ Hiout i i (4.5.11) Fig 4.17 compares recovery of C10 for the example solutions displayed in Fig 4.16 Values reported in Tables 4.2 and 4.3 under the columns labeled τ are the arrival times of the corresponding compositions at the outlet at ξ = Also given are the values of recovery of C10 , RC10 or QC10 , reported as a fraction of the amount of C10 initially present 4.6 SUMMARY 69 1.0 Fraction of C10 Recovered 0.8 0.6 0.4 0.2 No volume change Volume change 0.0 τ Figure 4.17: Recovery of component 2, C10 in displacements with and without volume change Fig 4.17 and Tables 4.2 and 4.3 indicate again that breakthrough of injected CO2 occurs at about 0.77 pore volumes injected (PVI) without volume change, but when more than one pore volume has been injected, at 1.09 PVI, when account is taken of volume change The effect of volume change is largest when the displacement pressure is high enough that there is appreciable solubility of CO2 in the oil but low enough that there is significant difference between the partial molar volumes of CO2 in the vapor and liquid phases At still higher pressures, where the partial molar volumes can differ much less, the assumption of no volume change is often quite reasonable Recovery is lower when volume change is considered because the dissolved CO2 present in the liquid phase upstream of the leading shock occupies less volume Hence, more C10 remains in the undisplaced liquid in the transition zone when volume change is significant In both displacements, however, recovery of C10 is slow after breakthrough of injected CO2 In the terminology in widespread use in the oil industry, both displacements are immiscible There is a large region of two-phase flow, and large amounts of gas must be injected to recover small amounts of oil after breakthrough Even so, all of the C10 could eventually be recovered by evaporation However, the arrival times of the trailing shock (see Tables 4.2 and 4.3), 85 and 158 pore volumes injected, are so long that a recovery process based on evaporation of large amounts of undisplaced oil would be unattractively slow As the theory developed in the next two chapters shows, however, more efficient displacements can be designed for systems that contain more than two components 4.6 Summary In this chapter we develop the basic ideas of the method of characteristics: by calculating how fast a particular composition propagates through the one-dimensional porous medium, we can work out 70 CHAPTER TWO-COMPONENT GAS/OIL DISPLACEMENT the behavior of a displacement of an oil mixture by a gas mixture That basic idea will be applied several times more in subsequent chapters as systems with more components are described For displacements in binary systems, the following key ideas carry over into systems with more than two components: • The propagation (or wave) velocity for a composition inside the two-phase region is df /dS1 (when volume change as components change phase is neglected) • Any solution must satisfy a velocity constraint , which requires, for regions in which compositions are varying continuously, that compositions with high wave velocity lie downstream of compositions with lower wave velocity • A shock is required if the number of phases present changes (as the solution compositions are traced upstream or downstream) • A shock must satisfy an entropy condition , which requires that the shock be self-sharpening This means that compositions on the upstream side of the shock must travel at wave velocities greater than or equal to the shock speed, and compositions on the downstream side of the shock must move at wave velocities that are less than or equal to the shock speed • Displacement of a single-phase oil mixture by a single-phase gas mixture includes a leading shock from the oil composition to a mixture composition in the two-phase region and a shock from the gas composition to a different mixture composition in the two-phase region Both shocks are semishocks in which the wave speed of the shock matches the composition wave speed on the two-phase side of the shock The two shock compositions are connected by a continuous composition variation along the equilibrium tie line • Adding the effects of volume change as components change phase to the analysis does not change the patterns of displacement behavior, but the wave velocities of all the compositions change 4.7 Additional Reading Method of Characteristics Volume I of First Order Partial Differential Equations by Rhee, Aris, and Amundson [106] gives an excellent introduction to the method of characteristics in Chapter The method of characteristics is applied to chromatography problems closely related to the binary displacement problem in Chapter 5, and the Buckley-Leverett problem for waterflooding is also solved there The behavior of shocks and entropy conditions is discussed in some detail in Chapter and again in Chapter Binary Displacement without Mutual Solubility The original solution for a binary displacement was that of Buckley and Leverett [10] for displacement of oil by water Many authors have subsequently discussed the solution to that problem Welge [133] derived the tangent construction used to determine the leading shock velocity and composition For reviews of the theory of water/oil displacement that are closely linked to the approach taken here, see the discussions of Lake [62, Section 5-2], Rhee, Amundson and Aris [106, Section 5.6], and Bedrikovetsky [6, Chapter 1] Dake summarizes the conventional approach to the problem [17, pp 356-372]  ... (4. 4.10) 4. 4 VOLUME CHANGE 63 dG1 dη dτ dη dξ dη = 0, (4. 4.11) = 1, (4. 4.12) = dH1 dG1 (4. 4.13) Comparison of Eqs 4. 4.11? ?4. 4.13 with the corresponding equations for constant volume flow, Eqs 4. 1.9? ?4. 1.11,... CHAPTER TWO-COMPONENT GAS/ OIL DISPLACEMENT d-e b-g τ 0 e-f g-h τ 0 0 f-b h-i τ 0 0 ξ ξ Figure 4. 14: τ -? ? diagrams for displacements illustrated in Fig 4. 13 important: a to i: As the amount of component... substitution of the definitions for Gi and Hi (Eqs 4. 4.3 and 4. 4 .4) into Eqs 4. 4.1 and 4. 4.2 followed by rearrangement gives ∂S1 ∂ + ∂τ ∂ξ vD f1 + x12 ρ2D x11 ρ1D − x12 ρ2D = 0, (4. 4.5) ∂S1 ∂ +