7.6 SUMMARY 211 Initial Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Injection Tie line Tie Line Length 0.8 0.6 Initial 0.4 Injection XO XO XO 1500 0.6 3000 2500 3500 0.4 0.0 1000 XO XO 1500 0.4 Injection XO XO XO Inj 0.4 XO XO XO XO XO XO XO XO XO 1500 Initial 0.6 0.2 XO XO 2000 XO 2500 3500 Initial Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Injection Tie line 0.8 XO 0.2 0.0 1000 1.0 Tie Line Length Tie Line Length Initial XO 3000 2500 (b) Oil 2, Gas 2, Wang Initial Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Injection Tie line 0.6 2000 Pressure (psia) (a) Oil 1, Gas 1, Wang 0.8 XO XO XO XO XO XO XO Pressure (psia) 1.0 Initial Inj 0.2 XO XO 2000 Initial Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Crossover Tie Line Injection Tie line 0.8 XO XO XO XO 0.2 0.0 1000 1.0 Tie Line Length 1.0 3000 Pressure (psia) (c) Oil 1, Gas 1, Jessen 3500 0.0 1000 XO 1500 2000 2500 3000 3500 Pressure (psia) (d) Oil 2, Gas 2, Jessen Figure 7.11: Tie line lengths for condensing/vaporizing gas drive systems described by Zick [140] (a)Crossover tie line has zero length at 2169 psia (147.6 atm) (b) Crossover tie line has zero length at 3013 psia (205.0 atm) (c) Crossover tie line has zero length at 2256 psia (153.5 atm) (d) Crossover tie line has zero length at 3070 psia (208.9 atm) (a) and (b) replotted from Wang [128], used with permission (c) and (d) recalculated with a slightly different fluid description from Jessen [42], used with permission 212 CHAPTER MULTICOMPONENT GAS/OIL DISPLACEMENTS • Whether or not tie line rarefactions appear in a displacement can be determined easily from the placement (on the vapor side or the liquid side of the two-phase region) and concavity of envelope curves • Once the key tie lines are determined for a given pair of injection gas and oil compositions, the full solution for a gas displacement can be determined relatively easily • Effects of volume change as components transfer between phases are similar in multicomponent systems to those observed for systems with two, three, and four components Use of scaled shock and rarefaction velocities allows straightforward construction of the complete solutions • A displacement is multicontact miscible if any of the key tie lines is a critical tie line (a tie line of zero length) • In displacements in which the injection gas is a mixture or the injection gas does not have the highest equilibrium K-value, it is likely that a crossover tie line will control miscibility Condensing/vaporizing gas drives, including most CO2 floods, are examples 7.7 Additional Reading The first solutions for systems with more than four components were obtained by Dindoruk [19] and Johns [54] Dindoruk reported solutions for a constant K-value system with five components and outlined the extension of his approach to systems with variable K-values Johns obtained solutions with variable K-values for a five-component system and for a six-component system, both for injection of a pure component A systematic procedure for calculating solutions for multicomponent systems with any number of components in the injection gas was reported by Jessen et al [48] Jessen [42] and Ermakov worked simultaneously to develop the systematic approach for finding multicomponent solutions when effects of volume change as components change phase are absent or present Numerous example solutions for systems with and without volume change are reported by Jessen [42], and additional solutions that show the effects of volume change are described by Ermakov [23] The analytical calculation of MMPs for multicomponent systems was considered first by Johns and Orr [55] Wang [128] and Wang and Orr [129] showed how to find the key tie lines for injection of a single-component gas and then used the key tie lines to find the MMP They then considered how to find the key tie lines and the MMP for a multicomponent injection gas [128] Jessen et al developed a much more efficient algorithm for finding the key tie lines [45, 42] and showed that predictions of minimum miscibility pressures based on the key tie line approach agreed very well with experimental data and with MMPs calculated by other methods Wang and Peck [131] reported results of additional tests of the accuracy of the key tie line approach They showed that predicted MMPs and MMEs for systems with widely varying injection gas compositions agreed very well with experimental observations Chapter Compositional Simulation by F M Orr, Jr and K Jessen The gas displacement problems considered in previous chapters can also be solved by numerical approaches, typically finite difference methods In fact, before the analytical solutions considered here were developed, the only way to solve such problems was to so numerically In this chapter, we compare analytical and numerical solutions and show that the numerical solutions converge to the analytical ones In addition, we consider the limitations of each approach to solving the flow equations Interpretation of numerical solutions requires careful attention to the effects of numerical dispersion on the solutions, the subject of the next section We also consider here numerical approaches to calculation of minimum miscibility pressure (MMP) or minimum injection gas enrichment for miscibility (MME) Several numerical schemes have been described, ranging from calculations based on mixing cells to extrapolation of numerical results from compositional simulations The analysis of composition paths for multicomponent displacements discussed in Chapter can be used to determine whether these numerical approaches yield accurate estimates of MMP or MME That analysis shows that calculations based on a single mixing cell yield incorrect estimates of MMP and MME when a crossover tie line controls development of miscibility Numerical approaches based on multiple mixing cells or compositional simulation can yield accurate estimates of MMP and MME if appropriate attention is paid to effects of numerical dispersion In Section 8.1, we consider how numerical dispersion arises in finite difference computations Section 8.2 shows how calculated composition paths are modified by the presence of numerical dispersion The impact of numerical dispersion on a particular displacement depends on the phase behavior of the gas/oil system Section 8.3 describes how the magnitude of the effect of numerical dispersion depends on phase behavior and suggests a way to determine which systems will be sensitive to the effects of dispersion Finally, numerical schemes for calculation of MMP and MME are discussed in Section 8.4 8.1 Numerical Dispersion Numerical solutions to material balance equations like Eqs 2.4.4 or 4.1.1 are usually obtained by finite difference methods Those solutions are always affected to some extent by truncation error In this section, we examine the sources of that truncation error Eq 4.1.1 can be solved easily with a fully explicit (forward time), backward space difference of the form 213 214 CHAPTER COMPOSITIONAL SIMULATION ∆τ n n (8.1.1) (F − Fi,k−1 ) ∆ξ i,k where k indicates the grid block and n the time level While there are several other differencing options available, Eq 8.1.1 illustrates well the issues that arise in finite difference solutions (see Mallison et al [73] for an evaluation of higher order computational schemes that improve accuracy for one dimensional solutions over the simple differencing scheme discussed here) It is possible to obtain more accurate finite difference solutions at a given grid resolution with the approaches Mallison describes, but the overall impacts of numerical dispersion are common to all the methods, so we use here the simplest of methods to illustrate the ideas In a displacement calculation, the compositions in each grid block are calculated at the new time step with Eq 8.1.1 A flash calculation is then performed for the composition in each grid block to determine the phase compositions and saturations, with which the fractional flows of the components can be calculated in preparation for the next time step Lantz [64] showed that even when the conservation equation being solved is dispersion-free, the truncation error in the finite difference version of the differential equation mimics a second order term in the original differential equation He showed that for finite difference form of Eq 8.1.1, the numerical eqivalent of the Peclet number for single-phase flow is [64, 127] n+1 n Ci,k = Ci,k − P e−1 = num ∆ξ ∆τ 1− ∆ξ , (8.1.2) and for two-phase flow is ∆ξ df1 ∆τ df1 1− (8.1.3) dS1 ∆ξ dS1 The exact form of the numerical Peclet number depends on the finite difference formulation for the time and space derivatives Peaceman [96, pp 74-81] reports expressions appropriate to the various implicit and explicit finite difference forms in common use For the remainder of this discussion, we will use the simple form of Eq 8.1.1, which permits calculation of compositions at the new time level without a matrix inversion and which illustrates in a straightforward way the impact of numerical dispersion on calculated composition paths Eqs 8.1.2 and 8.1.3 indicate that the inverse numerical Peclet number will be greater than zero as long as ∆τ < ∆ξ The P e−1 must be positive if the numerical calculation is to be stable num [96] If it is negative, the numerical solution will show nonphysical oscillations in compositions and saturations Hence, ∆τ must be less than ∆ξ for single-phase flow, and much less than ∆ξ df if dS1 > 1, as it will be for the S-shaped fractional flow curve appropriate to two-phase flow in a porous medium, particularly when the injection gas has a viscosity that is lower than that of the oil displaced As a result, effects of numerical dispersion can be reduced by reducing ∆ξ (and therefore ∆τ ), but they cannot be eliminated entirely for Eq 8.1.1 The limitation on time step size is a version of the Courant-Friedrichs-Levy (CFL) condition [15, 69], which states that the finite difference scheme of Eq 8.1.1 is unstable if P e−1 = num λp∆τ > 1, p = 1, nc − 1, (8.1.4) ∆ξ for each of the p eigenvalues Because the nontie-line eigenvalues generally have values close to one, it is the tie-line eigenvalue (df /dS) that determines the maximum stable value of the time step size 8.2 COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS 215 Table 8.1: Peclet Numbers for Finite Difference Simulations Grid Blocks 20 100 1000 10000 P e−1 0.0462 0.00926 0.000926 0.0000926 Pe 21.6 108 1080 10800 Numerical dispersion arises from the way fluids are treated in the finite difference representation Consider what happens in a first-contact miscible displacement, for example An increment of solvent is injected into the first grid block in the first time step At the end of that time step the fluids are mixed so that the block has uniform composition In the next time step, a fraction of the solvent injected in the first time step flows out of grid block to grid block 2, and that happens no matter how small the amount of solvent injected into grid block Thus some solvent propagates the entire length of grid block in the first time step, even though physical flow over the same distance would take longer than one time step In the second time step, some solvent leaves grid block by the same sequence of events Thus, there is a smearing of the composition profile that comes from the treatment of the blocks as mixing cells with finite size In the limit as the grid blocks become infinitely small, that smearing disappears 8.2 Comparison of Numerical and Analytical Solutions If the analytical solutions obtained in Chapters 4-7 are correct, then numerical solutions obtained with a finite difference calculation will converge to the analytical solutions as the number of grid blocks is increased To illustrate convergence and how numerical dispersion affects compositional simulation results, we consider first a simple ternary system with constant K-values (K1 = 2.5, K2 = 1.5, and K3 = 0.05), which might represent a CH4 /CO2 /C10 system at a relatively low pressure In this example, the initial oil mixture contains no CH4 , and the injection gas is pure CH4 The mobility ratio was fixed at M = 5, and the simulations were performed by solving Eq 8.1.1 with fixed ratio of time step size to grid block size, ∆τ /∆ξ = 0.1 Figure 8.1 compares the composition path obtained by compositional simulation for finite difference grids of 20, 100, 1000, and 10,000 grid blocks with the composition path of the analytical solution The corresponding saturation and composition profiles are also shown in Fig 8.1, and recovery curves for CO2 and C10 are shown in Fig 8.2 The analytical solution includes a leading semishock from the initial composition to a point on the initial tie line, a very short rarefaction along the initial tie line, a long nontie-line rarefaction that connects the initial tie line to the injection tie line, and a trailing shock to the injection composition Comparison of the finite difference (FD) and analytical (MOC) solutions shown in Figs 8.1 and 8.2 reveals several important points First, the FD solutions converge to the analytical solution, confirming that the two approaches are consistent However, the rate of convergence is not high, and very fine computational grids are required for this problem if the details of the solution are to be reflected accurately The FD solutions with 20 and 100 grid blocks show significant deviations of the calculated composition paths and composition profiles In this displacement, the CO2 that 216 CHAPTER COMPOSITIONAL SIMULATION CH4 a Injection Gas a Injection Gas Tie Line Dilution Line 20 Initial Tie Line 100 Oil C10 1000 a a a a CO2 a MOC 20 Blocks 100 Blocks 1000 Blocks 10,000 Blocks Sg aa a a a a CH4 aa a CO2 aa1000 a a 100 20 a C10 a 0.0 a a a 1.0 ξ/τ aa 2.0 Figure 8.1: Effects of numerical dispersion on a vaporizing gas drive for a ternary system with constant K-values, K1 = 2.5, K2 = 1.5, and K3 = 0.05 The initial composition is C1 = 0, C2 = 0.3760, C3 = 0.6240, and the injection gas is pure C1 For all simulations, ∆τ /∆ξ = 0.1, and M = 8.2 COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS 217 Fraction of CO2 and C10 Recovered 1.0 C10 CO2 0.8 0.6 0.4 Analytical Solution 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks 10,000 Grid Blocks 0.2 0.0 τ Figure 8.2: Component recovery for the displacements shown in Fig 8.1 is present initially in the oil is swept up into a bank (see the CO2 profile in Fig 8.1) That bank is poorly resolved with 20 and 100 grid blocks but is much better resolved with 1000 grid blocks With 10,000 grid blocks, however, the FD solution is nearly indistinguishable from the the MOC solution on the scale of the plots in Fig 8.2 Thus, the FD solution does, in fact, converge to the analytical solution, but it is clear that this problem is relatively sensitive to the effects of numerical dispersion (see Section 8.3) Second, the FD solutions can resolve the key tie lines, shocks and rarefactions that are important parts of the solution, but they so only if the FD grid includes enough grid blocks Regions of the solution where compositions are changing rapidly, the shocks and the nontie-line rarefaction, are the most difficult to capture in the FD solutions The leading and trailing shocks are resolved somewhat better at a given grid resolution because they are self-sharpening The nontie-line rarefaction is smeared much more The composition gradient is significant along the nontie-line path, and numerical dispersion acts to reduce that gradient When that rarefaction is smeared, the FD composition path follows closely a path that resembles the nontie-line paths obtained in the analytical solution As the grid is refined, the nontie-line portion of the FD solution approaches more closely the MOC nontie-line path Calculated component recovery (Fig 8.2) is also quite sensitive to grid resolution Because recovery at some late time in a displacement (often 1.1 or 1.2 pore volumes injected) is often used as an indicator of multicontact miscibility, it is important that effects of numerical dispersion on recovery be assessed when compositional simulations are used to estimate minimum miscibility pressure (see Section 8.4) Here it is enough to note that when numerical dispersion alters composition path significantly, it can also have a quite significant effect on calculated recovery, particularly in multicomponent systems at pressures or enrichments near the MMP or MME 218 CHAPTER COMPOSITIONAL SIMULATION The FD solutions reflect the interplay of dispersion (numerical in this case) and convection Fig 8.1 shows the dilution line that connects the initial oil composition to the injection gas composition The effect of dispersion is to move the solution composition path toward the dilution line If there were no flow at all, then mixtures of the initial and injection fluids would lie on the dilution line The effect of convection, and the accompanying chromatographic separations of components that take place as components partition between the flowing phases, is to push the composition path closer to the MOC solution, in which effects of dispersion are absent The Peclet number reflects the relative importance of the contributions of dispersion and convection (see Section 2.7) In this example, the Peclet number can be estimated with Eq 8.1.2 The value of df1 /dS1 is not constant throughout the solution but for the purposes of estimating the effects of dispersion, it is convenient to use the maximum value of df1 /dS1 in Eq 8.1.2, because the resulting value of the Peclet number will also determine whether the numerical computation is stable (it is if P e−1 is positive) That value is about 2.45 for M = and the relative permeability functions of Eqs 4.1.14-4.1.19 with Sor = Sgc = 0, the values used for all the constant K-value solutions discussed in this chapter Table 8.1 reports approximate Peclet numbers for the four FD simulations The solutions with P e > 1000 are close to the analytical solution The estimate given in Section 2.7 of the Peclet number appropriate to slim tube displacements (P e = 2500) suggests that about 2500 grid blocks would be required for the level of numerical dispersion in the FD calculation to approximate the physical value With that grid resolution, the FD solution would approximate closely the dispersionfree MOC solution, another indication that the use of the analytical solutions for 1D displacements like those performed in slim tubes is a reasonable approach Systems with K-values that depend on composition display similar behavior To examine the impact of numerical dispersion in simulations with variable K-values, Eq 8.1.1 was solved using the Peng-Robinson EOS with the pressure at which the phase behavior was evaluated held constant Fig 8.3 compares the analytical solution for a six-component displacement with no volume change(see Fig 7.5) with FD solutions calculated with 50, 500, and 5000 grid blocks In this system, the FD solutions converge reasonably rapidly to the analytical solutions as the grid is refined With 500 grid blocks, much more limited smearing of the shocks is observed, and with 5000 grid blocks, the FD solution is almost indistinguishable from the MOC solution The agreement is similar when the effects of volume change are included, as Fig 8.4 shows These examples demonstrate that FD compositional simulation can produce solutions that converge to the analytical solution if sufficiently fine grids are used The computational cost is much higher for the FD solutions, of course The FD solutions are needed, however, to deal with situations in which the pressure at which phase behavior is evaluated is not constant or when the injection composition is not constant, because the analytical solutions derived here are for Riemann problems only in which the initial and injection compositions are constant Finite differences are also used for two-dimensional and three-dimensional compositional simulations, for which analytical solutions are not available While computational cost for these one-dimensional calculations is not a problem, corresponding two- and three-dimensional simulations are often too slow to allow use of large numbers of grid blocks in each dimension It is rare, for example, to see use of as many as twenty grid blocks between wells in field-scale calculations Hence, it is likely that effects of numerical dispersion on calculated composition paths will be significant in multidimensional FD compositional simulations One approach to dealing with the difficulties that arise from numerical dispersion in FD calculations is to decouple the representation of the effects of reservoir heterogeneity, which control where low viscosity injected gas flows preferentially, from the kinds of chromatographic dtermination of 8.2 COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS 219 Sg MOC 50 Grid Blocks 500 Grid Blocks 5000 Grid Blocks C10 CO2 0.2 0.0 0.2 C16 CH4 0.0 0.2 C4 C20 0.2 0.0 0.0 ξ/τ ξ/τ Figure 8.3: Saturation and composition (mole fraction) profiles for displacement of a six-component oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71C) Effects of volume change as components change phase are not included in this example The solid line is the analytical solution, and the dotted lines are the FD solution See the discussion of Fig 7.6 for a description of the MOC solution for this fully self-sharpening example 220 CHAPTER COMPOSITIONAL SIMULATION MOC 50 Grid Blocks 500 Grid Blocks 5000 Grid Blocks vD Sg 0 C10 CO2 0.2 0.0 C16 CH4 0.2 0.0 0.2 0.2 C4 C20 0.0 0.0 ξ/τ ξ/τ Figure 8.4: Saturation and composition (mole fraction) profiles for the displacement of a sixcomponent oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71 C) This example includes the effects of volume change as components change phase The solid line is the analytical solution, and the dotted lines are the FD solutions 8.3 SENSITIVITY TO NUMERICAL DISPERSION 221 composition paths considered in this book through the use of streamline models [121, 8, 120, 16, 43] The effects of heterogeneity are represented by the streamlines If spatial variations in permeability dominate the flow, then it can be quite difficult to move streamlines away from zones of high permeability Indeed, one of the significant reasons that it can be difficult to achieve high overall reservoir sweep efficiency is that low viscosity gas finds and continues to flow preferentially in high permeability zones, and gravity segregation can partially mitigate or aggravate the effects of permeability variation, depending on injection rates and the spatial distribution of permeability When permeability variations dominate the flow, the locations of streamlines can be recomputed at relatively long intervals (in effect FD simulations recalculate streamlines every time step) The compositional effects can then be represented by one-dimensional solutions calculated for each streamline There is no attempt to represent the effects of component transport across streamlines (by diffusion, physical dispersion, and viscous crossflow), and the assumption that the displacement pressure is constant for the purposes of evaluation of phase behavior is clearly not strictly satisfied, however, so this approach also has limitations For problems in which the restrictions of Riemann problems (constant initial and injection compositions) are satisfied, the one-dimensional analytical solutions obtained in Chapters 4-7 can be used as the one-dimensional solution along streamlines Because those solutions are self-similar, they can be evaluated only once and applied repeatedly as the one-dimensional solutions are propagated along streamlines The streamline approach, particularly when analytical solutions can be applied along streamlines, can be orders of magnitude faster than FD compositional simulation if it is reasonable to update streamlines relatively infrequently, and it has the advantage that it is much less subject to the adverse effects of numerical dispersion [43] For an example of application of streamlines with analytical one-dimensional solutions to a field-scale displacement of condensate by CO2 , see Seto et al [109] If the initial or injection compositions are not constant, or if new wells are added, then a numerical one-dimensional solution is required[16] In such cases, attention to the sensitivity of calculated composition paths to numerical dispersion, the subject of the next section, will be needed 8.3 Sensitivity to Numerical Dispersion The example shown in Fig 8.1 is one that is relatively sensitive to the effects of numerical dispersion Fig 8.5 compares FD and MOC solutions for a system that is less sensitive to dispersion The only difference between the displacements in Figs 8.1 and 8.5 is the K-value of the intermediate component, which is K2 = 1.5 in Fig 8.1 and K2 = 0.5 in Fig 8.5 Recovery curves for the displacement illustrated in Fig 8.5 are reported in Fig 8.6 This displacement is vaporizing gas drive with a low-volatility intermediate component, and it is fully self-sharpening The only rarefaction occurs along the initial tie line, and a shock connects the initial and injection tie lines In this example, the FD composition paths for all the grid resolutions deviate much less from the MOC solution than did the the paths, profiles, and recovery shown in Figs 8.1 and 8.2 The composition paths all lie relatively close to the initial and injection tie lines and the intermediate (nontie-line) shock The intermediate shock is smeared more than the leading and trailing shocks, because it is only weakly self-sharpening Comparison of the results of Figs 8.5 and 8.6 with those of Figs 8.1 and 8.2 indicates that 222 CHAPTER COMPOSITIONAL SIMULATION CH4 a Injection Gas a a a Dilution Line Oil C10 a a Sg a a a a a a a CO2 1 MOC 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks a a a CH4 CO2 a a 1000 100 20 a a a a C10 a 0.0 a a a a 1.0 ξ/τ 2.0 Figure 8.5: Effects of numerical dispersion on a vaporizing gas drive for a ternary system with constant K-values, K1 = 2.5, K2 = 0.5, and K3 = 0.05 The initial composition is C1 = 0, C2 = 0.3760, C3 = 0.6240, and the injection gas is pure C1 For all simulations, ∆τ /∆ξ = 0.1, and M = 8.3 SENSITIVITY TO NUMERICAL DISPERSION 223 1.0 Fraction of C4 and C10 Recovered CO2 0.8 C10 0.6 0.4 MOC Solution 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks 0.2 0.0 τ Figure 8.6: Component recovery for the displacements shown in Fig 8.5 the sensitivity of calculated displacement behavior to the effects of numerical dispersion depends on the details of the phase behavior of the system because the two systems, which differ only in the K-value of the intermediate component, show very different rates of convergence to the MOC solution Indeed, the FD solution with 1000 grid blocks for K2 = 0.5 achieves approximately the level of agreement with the MOC solution that requires 10000 grid blocks when K2 = 1.5 The reasons for the differences lie in the orientation of the key tie lines with respect to the dilution line Systems in which the dilution line lies close to and nearly parallel to the key tie lines might be expected to show relatively low sensitivity to numerical dispersion, because any deviations in path toward the dilution line not depart dramatically from no-dispersion solution path On the other hand, systems in which the key tie lines cross the dilution line at large angles and the no-dispersion path includes portions that are well away from the dilution line, path deviations toward the dilution line caused by numerical dispersion can lead to much larger differences between the numerical and analytical paths Because component recovery depends strongly on composition path in displacements in which significant component transfers occur between phases, these displacements show much greater sensitivity to the effects of dispersion, whether physical or numerical Displacements that are multicontact miscible or nearly so often show significant sensitivity to numerical dispersion An indication of the sensitivity of a particular displacement to the effects of numerical dispersion is given by the magnitude of the deviation of the composition path from the dilution line Fig 8.7 illustrates one way in which that deviation can be measured, in terms of a dispersive distance, for a ternary system at its minimum miscibility pressure In this displacement, the no-dispersion composition path jumps from the initial oil composition to the critical point and then traces the vapor portion of the binodal curve to the injection gas tie line The perpendicular distance from the 224 CHAPTER COMPOSITIONAL SIMULATION aInjection Gas Dilution Line a a a Critical Point Dispersive Distance Oil 0 Figure 8.7: Schematic representation of the dispersive distance in cartesian coordinates for a ternary system Table 8.2: Oil and Gas Compositions for Displacements with Four Components (Mole Fractions) Oil A N2 CH4 C4 C10 xoil 0.0 0.5 0.1624 0.3376 ygas,1 1.0 0.0 0.0 0.0 ygas,2 0.1 0.9 0.0 0.0 dilution line to the critical point is a measure of the difference between the dispersion-dominated path (the dilution line) and the convection-dominated path When that distance is relatively large, the effect of disperion is relatively large, as the example in Figs 8.1 and 8.2 demonstrates When that distance is relatively small, the impact of dispersion on composition path is smaller, as in the displacements of Figs 8.5 and 8.6 To test whether the dispersive distance provides a reasonable quantitative measure of the effect of numerical disperion in a displacement, we consider first two four-component displacements The oil is a ternary mixture of CH4 , C4 , and C10 (see Table 8.2) Two injection gases, pure N2 , and a mixture with 90 mole percent CH4 and 10% N2 (Table 8.2) illustrate the differences in sensitivity that are reflected in the dispersive distance The composition path for displacement by pure N2 is shown in Fig 8.8 This phase diagram is is plotted for a displacement pressure of 305 atm at 344 K In this system, the initial oil tie line is the tie line that lies closest to the critical locus, and hence it controls development of miscibility The initial oil tie line is quite short, and hence this displacment is nearly miscible The MMP for this gas/oil pair is 309 atm Also shown in Fig 8.8 is a FD composition path computed with 100 grid blocks This is a system with a relatively large 8.3 SENSITIVITY TO NUMERICAL DISPERSION 225 CH4 MOC FD • Initial oil N2 • C4 C10 Figure 8.8: Displacement of Oil A by pure N2 at 4482 psia (305 atm) and 161 F (71 C), conditions that are quite close to the MMP, which is 4541 psia (309 atm) This displacement has a relatively large dispersive distance of 0.173 because the initial tie line, which controls miscibility, is relatively distant from the dilution line The FD path was calculated with 100 grid blocks dispersive distance (0.173), and the FD composition path shows significant effects of numerical dispersion Fig 8.9 compares oil recovery at 1.2 pore volumes injected, as a function of pressure, for the FD and analytical solutions The recovery estimates obtained in the low resolution FD solutions differ significantly from the analytical solution, and they converge relatively slowly to the analytical results In this system, careful refinement of the FD grid would be required to obtain an accurate estimate of the MMP Figs 8.10 and 8.11 show similar plots for displacement of the same oil by the second gas mixture containing 90 mole percent CH4 and 10% N2 Because the initial tie line is the same, the MMP is also unchanged, but the dispersive distance is much smaller The composition paths shown and the corresponding recovery curves show that this system is much less sensitive to the effects of numerical dispersion The FD recovery for Oil displaced by Gas is 30 percent below the MOC recovery at the MMP (Fig 8.8), but for Oil displaced by Gas 2, the FD recovery is about 15 percent below the MOC value Thus, qualitatively at least, the idea that the dispersive distance gives an indication of sensitivity to effects of numerical dispersion is supported by these results for simple systems Jessen et al [47] investigated the quantitative response of a number of multicomponent gas/oil displacements to variations in initial oil composition and injection gas composition that induced significant corresponding variations in dispersive distance Table 8.3 reports the compositions of the three oils used and three gas mixtures that were then diluted with varying fractions of N2 , CH4 , 226 CHAPTER COMPOSITIONAL SIMULATION Recovery at 1.2 PVI 1.0 0.9 0.8 0.7 100 grid blocks 500 grid blocks 1000 grid blocks 5000 grid blocks MOC 0.6 0.5 200 300 400 500 Pressure (atm) Figure 8.9: Difference in calculated recovery between finite difference solutions with with various grid resolutions and the analytical solution for displacement of Oil A by pure N2 at 4482 psia (305 atm) and 160 F (71 C) and CO2 The characterization of Oil B and the associated separator Gas B is reported by Jessen [42] It has a calculated bubblepoint pressure of 249 atm at 387 K Oil C was studied by Høier [36], who reports the EOS characterization used here At 368 K, the bubblepoint pressure of that oil is 251 atm Oil D is one of the examples reported by Zick [140] The characterization of the components in this oil is reported in Table 7.5 It has a calculated saturation pressure of 102 atm at 358 K To test the use of the dispersive distance to assess a priori the sensitivity of a FD simulation to dispersion, a series of simulations was carried out with a variety of injection gases for a range of pressures above and below the MMP For each displacement, the key tie lines were calculated, the shortest tie line was identified, and the dispersive distance was evaluated as the orthogonal distance between that tie line (of zero length) and the dilution line at the MMP Table 8.4 summarizes the oil and gas compositions used and reports the dispersive distances In Table 8.4, the mechanism is identified as V, for a vaporizing drive in which the initial oil tie line controls development of miscibility, or C/V, which indicates a condensing/vaporizing displacement in which one of the crossover tie lines is critical at the MMP Fig 8.12 shows the results of the simulations It shows the difference in calculated recoveries at 1.2 pore volumes injected between the FD result and the MOC solution with the pressure set at the MMP The recovery at the MMP for the MOC solution is 100 percent, of course The FD recoveries agree better with the MOC results (see Figs 8.10 and 8.11) at pressures well below the MMP, when composition paths are less strongly affected by phase behavior, and at pressures well above the MMP, when the twophase region is smaller and the negative impact of dispersion applies over a smaller fraction of the 8.3 SENSITIVITY TO NUMERICAL DISPERSION 227 CH4 • MOC FD • Initial oil N2 C4 C10 Figure 8.10: Displacement of Oil A by a gas mixture containing 10% N2 and 90% CH4 at 4482 psia (305 atm) and 161 F (71 C) This displacement has a smaller dispersive distance of 0.091 because the dilution line is now closer to the initial tie line Table 8.3: Oil and Gas Compositions of Multicomponent Displacements Comp N2 CO2 CH4 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C11 C16 C23 C33 Oil B xoil 0.00450 0.01640 0.45850 0.07150 0.06740 0.00840 0.03110 0.01030 0.01650 0.02520 0.12440 0.06320 0.05024 0.03240 0.01996 ygas 0.0049 0.0182 0.8139 0.0915 0.0467 0.0050 0.0124 0.0020 0.0026 0.0009 0.0019 Oil C Comp xoil N2 0.00785 CO2 0.00265 CH4 0.45622 C2 0.06092 C3 0.04429 i-C4 0.00865 n-C4 0.02260 i-C5 0.00957 n-C5 0.01406 C6 0.02097 C7+1 0.04902 C7+2 0.09274 C7+3 0.09880 C7+4 0.07362 C7+5 0.03804 Oil D xoil ygas CO2 CH4 C2 C3 0.04400 0.20041 0.04777 0.04012 0.2218 0.2349 0.2350 0.2745 n-C4 0.02992 0.0338 n-C5 C6 C7 C13 C19 C27 C38 0.02342 0.05873 0.22013 0.13398 0.09970 0.06215 0.03916 Comp 228 CHAPTER COMPOSITIONAL SIMULATION Recovery at 1.2 PVI 1.0 0.9 0.8 0.7 100 grid blocks 500 grid blocks 1000 grid blocks 5000 grid blocks MOC 0.6 0.5 200 300 400 500 Pressure (atm) Figure 8.11: Difference in calculated recovery between finite difference solutions with with various grid resolutions and the analytical solution for displacement of Oil A by a gas mixture containing 10% N2 and 90% CH4 at 4482 psia (305 atm) and 161 F (71 C) displacement length Hence, the differences in recoveries at the MMP are the maximum observed over the range of pressures investigated Fig 8.12 demonstrates a clear relationship between the magnitude of the dispersive distance and the difference in recovery that results from numerical dispersion, though there is considerable scatter about the two trend lines shown One of those lines, indicates, approximately, the difference in recovery for FD solutions with 100 grid blocks, and the second, the same difference for solutions obtained with 1000 grid blocks The differences are smaller for the fine grid simulations, but they can still be significant for systems with large values of the dispersive distance Fig 8.12 suggests that an assessment of the sensitivity of a particular gas/oil system to numerical dispersion can be made in two ways The sensitivity can be determined directly by numerical simulation alone simply by performing fine and coarse grid simulations at a series of pressures (preferably in the range of interest, including the MMP if miscible displacement is the goal) and monitoring the changes in predicted recovery as a function of grid resolution and pressure A much more computationally efficient approach is to use the analytical solutions of Chapter for the key tie lines to estimate the dispersive distance If the displacement pressure of interest is not the MMP, then the dispersive distance can be estimated by taking the orthogonal distance between the dilution line and midpoint of the section of the shortest tie line between the two equal eigenvalue points that bound the rarefaction along that tie line In practice, simply taking the midpoint of the shortest tie line will give a quite reasonable estimate, accurate enough for the purpose of determining whether the system in question is relatively sensitive, or not, to numerical dispersion Fig eee can then be used to determine whether a careful grid refinement study is warranted for 8.3 SENSITIVITY TO NUMERICAL DISPERSION 229 Table 8.4: Dispersive Distances for Gas Displacements No 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Oil A A A A A A B B B B B B B B C C C C C C C C D D D D D D D D D D D D Injection Gas N2 75% N2 /25 % CH4 50% N2 /50 % CH4 25% N2 /75 % CH4 10% N2 /90 % CH4 CH4 Gas B CH4 CO2 25% N2 /75 % CH4 30% N2 /70 % CH4 35% N2 /65 % CH4 40% N2 /60 % CH4 N2 CH4 25% N2 /75 % CH4 30% N2 /70 % CH4 40% N2 /60 % CH4 50% N2 /50 % CH4 65% N2 /35 % CH4 75% N2 /25 % CH4 N2 Gas D 90% Gas D + 10% CH4 80% Gas D + 20% CH4 60% Gas D + 40% CH4 40% Gas D + 60% CH4 20% Gas D + 80% CH4 CH4 90% Gas D + 10% CO2 80% Gas D + 20% CO2 70% Gas D + 30% CO2 50% Gas D + 50% CO2 25% Gas D + 75% CO2 T (K) 344 344 344 344 344 344 387 387 387 387 387 387 387 387 368 368 368 368 368 368 368 368 358 358 358 358 358 358 358 358 358 358 358 358 MMP (atm) 309 309 309 309 309 309 365 371 226 380 380 380 380 380 509 537 533 533 537 537 537 537 148 171 198 260 342 456 471 159 171 184 220 329 Mechanism V V V V V V C/V V C/V V V V V V C/V V V V V V V V C/V C/V C/V C/V C/V C/V V C/V C/V C/V C/V C/V Dispersive Distance 0.17357 0.17504 0.16269 0.16120 0.09095 0.00379 0.01470 0.04475 0.16515 0.12537 0.14961 0.16993 0.18518 0.19765 0.04307 0.17061 0.21000 0.27000 0.30161 0.31354 0.31254 0.30361 0.06930 0.06698 0.06686 0.05512 0.03463 0.02620 0.09057 0.06769 0.06618 0.06463 0.06099 0.05403 230 CHAPTER COMPOSITIONAL SIMULATION Difference in Recovery (%) 50 b c 40 (MOC-FD)/MOC, 100 grid blocks (MOC-FD)/MOC, 1000 grid blocks b 30 b 20 10 b bb bb bb b b b b b b 0.0 b c c b b b b b b bbb b c b b b b b b b c c c c 0.1 0.2 0.3 Dispersive Distance Figure 8.12: Difference in recovery between finite difference solutions with 100 and 1000 grid blocks and the analytical solution as a function of dispersive distance additional compositional simulations Haajizadeh et al [28] and Wang and Peck [131] described differences in sensitivity to numerical dispersion for systems with differing displacement mechanisms (condensing, vaporizing, condensing/vaporizing) and suggested that the observed sensitivity was associated with the displacement mechanism The results presented in Table 8.4 and Fig eee suggest that it is not the displacement mechanism that controls the sensitivity of the displacement simulation to dispersion but the distance between the composition path and the dilution line that is responsible For example, the vaporizing N2 displacements in Table 8.4 (in particular, displacements 1-4, 10-15, and 16-22) show relatively large dispersive distances and and significant sensitivity, while some vaporizing displacements by CH4 (displacements 6, 8, and 15) show relatively small dispersive distances One CH4 vaporizing displacement (29) shows more sensitivity, however Thus, the sensitivity of a particular displacement should be assessed directly for the specific gas/oil system 8.4 Calculation of MMP and MME MMP Calculation Using a Single Mixing Cell At least three computational approaches have been proposed for calculation of MMPs and MMEs: mixing cell calculations for a single cell, multiple mixing cell calculations and compositional simulations Next we examine the accuracy of single-cell methods The original versions were based on the analysis of ternary systems in which miscibility occurs if either the initial oil tie line or the injection gas tie line is a critical tie line The initial oil tie line can be identified by the following ... % CH4 N2 Gas D 90% Gas D + 10% CH4 80% Gas D + 20% CH4 60% Gas D + 40% CH4 40% Gas D + 60% CH4 20% Gas D + 80% CH4 CH4 90% Gas D + 10% CO2 80% Gas D + 20% CO2 70% Gas D + 30% CO2 50% Gas D + 50%... lines for injection of a single-component gas and then used the key tie lines to find the MMP They then considered how to find the key tie lines and the MMP for a multicomponent injection gas [128 ]... distance of 0.091 because the dilution line is now closer to the initial tie line Table 8.3: Oil and Gas Compositions of Multicomponent Displacements Comp N2 CO2 CH4 C2 C3 i-C4 n-C4 i-C5 n-C5 C6