Orr, F. M. - Theory of Gas Injection Processes Episode 14 doc

Bibliography 251 [90] Orr, F.M., Jr and Taber J.J Use of Carbon Dioxide in Enhanced Oil Recovery Science, page 563, 11 May 1984 [91] Orr, F.M., Jr., Dindoruk, B., and Johns, R.T Theory of Multicomponent Gas/Oil Displacements Ind Eng Chem Res., 34:2661–2669, 1995 [92] Orr, F.M., Jr., Johns, R.T and Dindoruk, B Development of Miscibility in Four-Component CO2 Floods Soc Pet Eng Res Eng., 8:135–142, 1993 [93] Orr, F.M Jr., Silva, M.K., Lien, C.L and Pelletier, M.T Laboratory Experiments to Evaluate Field Prospects for Carbon Dioxide Flooding J Pet Tech., pages 888–898, April 1982 [94] Orr, F.M., Jr., Yu, A.D., and Lien, C.L Phase Behavior of CO2 and Crude Oil in LowTemperature Reservoirs Society of Petroleum Engineers Journal , pages 480–492, August 1981 [95] Pande, K.K Interaction of Phase Behaviour with Nonuniform Flow PhD thesis, Stanford University, Stanford, CA, December 1988 [96] Peaceman, D.W Fundamentals of Numerical Reservoir Simulation Elsevier Scientific Publishing, New York, 1977 [97] Pedersen, K.S., Fjellerup, J.F., Fredenslund, A., and Thomassen, P Studies of Gas Injection into Oil Reservoirs by a Cell to Cell Simulation Model, SPE 13832, 1985 [98] Pedersen, K.S., Fredenslund, A., and Thomassen, P Properties of Oils and Natural Gases Gulf Publishing Company, Houston, Texas, 1989 [99] Peng, D.Y and Robinson, D.B A New Two-Constant Equation of State Ind Eng Chem Fund., 15:59–64, 1976 [100] Perkins, T.K.and Johnston, O.C A Review of Diffusion and Dispersion in Porous Media Soc Pet Eng J., pages 70–84, March 1963 [101] Pope, G.A The Application of Fractional Flow Theory to Enhanced Oil Recovery Soc Pet Eng J., 20:191–205, June 1980 [102] Pope, G.A., Lake, L.W and Helfferich, F.G Cation Exchange in Chemical Flooding: Part - Basic Theory Without Dispersion Soc Pet Eng J., pages 418–434, December 1978 [103] Ratchford, HH and Rice, J.D Procedure for Use of Electrical Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium J Pet Tech., pages 19–20, October 1952 [104] Reamer, H.H and Sage, B.H Phase Equilibria in Hydrocarbon Systems Volumetric and Phase Behavior of the n-Decane–CO2 System J Chem Eng Data, 8(4):508–513, October 1963 [105] Reid, R.C., Prausnitz, J.M., and Sherwood, T.K The Properties of Gases and Liquids, 3rd ed McGraw Hill, New York, NY, 1977 252 Bibliography [106] Rhee, H., Aris, R and Amundson, N.R First-Order Partial Differential Equations: Volume I Prentice-Hall, Englewood Cliffs, NJ, 1986 [107] Rhee, H., Aris, R and Amundson, N.R First-Order Partial Differential Equations: Volume II Prentice-Hall, Englewood Cliffs, NJ, 1989 [108] Rhee, H.K., Aris, R., and Amundson, N.R On the Theory of Multicomponent Chromatography Phil Trans Roy Soc London, 267 A:419–455, 1970 [109] Seto, C.J., Jessen, K., and Orr, F.M., Jr Compositional Streamline Simulation of Field Scale Condensate Vaporization by Gas Injection, SPE 79690, SPE Reservoir Simulation Symposium, Houston, TX, February 3–5 2003 [110] Shaw, J and Bachu, S Screening, Evaluation , and Ranking of OIl Reservoirs Suitable for CO2 -Flood EOR and Carbon Dioxide Sequestration J Canadian Pet Tech., 41 (9):51–61, 2002 [111] Slattery, J.C Momentum, Energy, and Mass Transfer in Continua McGraw-Hill, New York, NY, 1972 [112] Stalkup, F.I Miscible Displacement Society of Petroleum Engineers, Dallas, 1983 [113] Stalkup, F.I Miscible Displacement Monograph 8, Soc Pet Eng of AIME, New York, 1983 [114] Stalkup, F.I Displacement Behavior of the Condensing/Vaporizing Gas Drive Process, SPE 16715, SPE Annual Technical Conference and Exhibition, Dallas, TX, September 1987 [115] Stalkup, F.I Effect of Gas Enrichment and Numerical Dispersion on Enriched-Gas-Drive Predictions Soc Pet Eng Res Eng., pages 647–655, November 1990 [116] Stein, M.H., Frey, D.D., Walker, R.D., and Pariani, G.J Slaughter Estate Unit CO2 Flood: Comparison between Pilot and Field Scale Performance J Pet Tech., pages 1026–1032, September 1992 [117] Taber, J.J., Martin, F.D., and Seright, R.D EOR Screening Critera Revisited–Part I: Introduction to Screening Criteria and Enhanced Recovery Field Projects Soc Pet Eng Res Eng., pages 189–198, August 1997 [118] Tanner, C.S., Baxley, P.T., Crump, J.G., and Miller, W.C Production Performance of the Wasson Denver Unit CO2 Flood, SPE 24156, SPE/DOE 8th Annual Symposium on Enhanced Oil Recovery, Tulsa, OK, April 22-24 1992 [119] Thiele, M.R Modeling Multiphase Flow in Heterogeneous Media Using Streamtubes PhD thesis, Stanford University, Stanford, CA, December 1994 [120] Thiele, M.R., Batycky, R.P., and Blunt, M.J A Streamline-Based 3D Field-Scale Compositional Simulator, SPE 38889, SPE Annual Technical Conference and Exhibition, San Antonio, TX, October 5-8 1997 [121] Thiele, M.R., Blunt, M.J., and Orr, F.M., Jr Modeling Flow in Heterogeneous Media Using Streamlines–II Compositional Displacements In Situ, 19(4):367–391, 1995 Bibliography 253 [122] van der Waals, J.D On the Continuity of the Gaseous and Liquid States In J.S Rowlinson, editor, J.D van der Waals: On the Continuity of the Gaseous and Liquid States, pages 83–140 North-Holland Physics Publishing, 1988 [123] Van Ness, H.C and Abbott, M.M Classical Thermodynamics of Nonelectrolyte Solutions With Applications to Phase Equilibrium McGraw-Hill, San Francisco, 1982 [124] Varotsis, N., Stewart, G., Todd, A.C and Clancy, M Phase Behavior of Systems Comprising North Sea Reservoir Fluids and Injection Gases J Pet Tech., pages 1221–1233, November 1986 [125] Wachmann, C The Mathematical Theory for the Displacement of Oil and Water by Alcohol Society of Petroleum Engineers Journal, 231:250–266, September 1964 [126] Walas, S.M Phase Equilibria in Chemical Engineering Butterworth Publishers, Stoneham, MA, 1985 [127] Walsh, B.W and Orr, F.M Jr Prediction of Miscible Flood Performance: The Effect Of Dispersion on Composition Paths in Ternary Systems IN SITU, 14(1):19–47, 1990 [128] Wang, Y Analytical Calculation of Minimum Miscibility Pressure PhD thesis, Stanford University, Stanford, CA, 1998 [129] Wang, Y and Orr, F.M., Jr Analytical Calculation of Minimum Miscibility Pressure Fluid Phase Equilibria, 139:101–124, 1997 [130] Wang, Y and Orr, F.M., Jr Calculation of Minimum Miscibility Pressure J Petroleum Science and Engineering, 27:151–164, 2000 [131] Wang, Y., and Peck, D.G Analytical Calculation of Minimum Miscibility Pressure: Comprehensive Testing and Its Application in a Quantitative Analysis of the Effect of Numerical Dispersion for Different Miscibility Development Mechanisms, SPE 59738, SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 2000 [132] Watkins, R.W A Technique for the Laboratory Measurement of Carbon Dioxide Unit Displacement Efficiency in Reservoir Rock, SPE 7474, SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct 1-3 1978 [133] Welge, H.J A Simplified Method for Computing Oil Recovery by Gas or Water Drive Trans., AIME, 195:91–98, 1952 [134] Welge, H.J., Johnson, E.F., Ewing, S.P.,Jr., and Brinkman, F.H The Linear Displacement of Oil from Porous Media by Enriched Gas J Pet Tech., pages 787–796, August 1961 [135] Whitson, C.H and Michelsen, M The Negative Flash Fluid Phase Equilibria, 53:51–71, 1989 [136] Wilson, G.M A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculation Paper 15C, presented at the 1969 AIChE 65th National Mtg., Cleveland, OH, 1969 254 Bibliography [137] Wingard, J.S Multicomponent, Multiphase Flow in Porous Media with Temperature Variation PhD thesis, Stanford University, Stanford, CA, November 1988 [138] Yellig, W.F., and Metcalfe, R.S Determination and Prediction of CO2 Minimum Miscibility Pressures J Pet Tech., pages 160–168, January 1980 [139] Zhu, J., Jessen, K., Kovscek, A.R., and Orr, F.M., Jr Analytical Theory of Coalbed Methane Recovery by Gas Injection Soc Pet Eng J., pages 371–379, December 2003 [140] Zick, A.A A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gas, SPE 15493, SPE Annual Technical Conference and Exhibition, New Orleans, LA, October 1986 Appendix A 255 APPENDIX A: Entropy Conditions in Ternary Systems In this appendix we consider the entropy condition for shocks in ternary systems The derivation of the entropy condition for the shock between tie lines follows that of Wang [128], which is based, in turn, on the approach used by Johansen and Winther [51] to study polymer displacements The derivation given here is for a specific system with constant K-values, but the patterns of behavior are the same for systems with variable K-values The use of the constant K-value example is an attempt to illustrate the abstract concept of an entropy condition in a concrete way Entropy conditions are statements about the stability of a shock, written in terms of the relative magnitudes of eigenvalues of compositions on either side of the shock and the shock velocity If a shock is stable, it must be self-sharpening In other words, if a stable shock were to be smeared slightly by some physical mechanism, it must sharpen again into a shock in the limit as that physical mechanism is removed Dispersion is one physical mechanism that can create a continuously varying composition in place of a jump in composition In a binary displacement, the requirement of a stable shock can be translated easily into a statement about the eigenvalues on either side of the shock For example, the discussion in Section 4.2 states that the eigenvalue on the upstream side of a shock must be greater than the shock velocity, and the eigenvalue on the downstream side must be less than the shock velocity For a ternary displacement, however, there are two eigenvalues at each point in the composition space, so the statement of shock stability in terms of those eigenvalues is necessarily more complex In this appendix we consider the statement of an entropy condition for each of the shocks that can appear in the solution for a ternary displacement, leading, trailing, and intermediate, and we show that if there is an intermediate shock, it is a semishock Leading Shock To illustrate the statement of the entropy condition for the various shocks, we consider a specific case: constant K-values, with K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The solution for this example is shown in Fig 5.16 The behavior of the leading shock, which connects a single-phase composition with a composition on the initial tie line, is exactly the same as that described for leading shock in a binary system (see Section 4.2) The leading shock is a shock that arises because of the behavior of λt, and it occurs along the extension of the initial tie line It is a semishock that is faster than the composition velocities on the downstream side of the shock, the right state for this shock (entropy conditions are frequently written in terms of left and right states, with the left state referring to upstream compositions and the right state to downstream compositions) Those velocities are are all one Fig A.1 shows the relationships between the shock velocity and the eigenvalues λt and λnt for the leading shock The leading shock has a velocity, ΛLR equal to λL, t which is indicated by the fact that the line drawn from the right state composition, R, to the left state composition L is tangent to the fractional flow curve The tie line eigenvalue, λt, is given by the slope of the fractional flow curve The nontie-line eigenvalue is given by Eq 5.1.24 (see Section 5.1) λnt = F1 − C1e F1 + p = C1 + p C1 − C1e (A.1) For constant K-values, the value of p, which is the negative of the volume fraction of component on the envelope curve, C1e , is given by Eq 5.1.49, 256 Appendix A p = −C1e = K1 − K2 K1 − K3 x2 x = 1, K2 − 1 − K3 γ (A.2) with γ given by Eq 5.1.50, − K3 K2 − (A.3) K1 − K2 K1 − K3 In the example considered here, an LVI vaporizing drive, K2 < 1, so γ is negative, as is p for any tie line The point labeled CL is the composition at the point at which the extension of the initial tie 1e line is tangent to the envelope curve (see Fig 5.12) Eq A.1 indicates that the slope of the line drawn from the left state composition, L, to CL is the nontie-line eigenvalue, λL Comparison of 1e nt the slopes for the leading shock and λnt indicates that the leading shock velocity, ΛLR is greater than λL on the upstream side of the shock Hence, the relationships among the shock velocity and nt eigenvalues are γ= < ΛLR = λL, t (A.4) λL nt (A.5) xR , a tangent drawn to the fractional 1 flow curve for the longer tie line does not intersect the fractional flow curve for the shorter tie line More care is required to show that a similar statement is true for more complex phase behavior and mobility ratio that is not constant.) Thus, there is no solution for a shock that lands at L2 , where λL = ΛLR , and satisfies the shock balance equations t Point L3 is an acceptable landing point, however At L3 , the slope of the fractional flow curve is lower than the slope of the shock line, and hence λL < ΛLR Variation along the injection gas t tie line to a trailing semishock point would be consistent with the velocity rule, and an immediate genuine shock to the injection composition is also allowed Hence, we conclude that at the landing point on the injection gas tie line, λL < ΛLR t Next we consider possible right states on the initial oil tie line Fig A.5 shows three possible jump points on that tie line Point R3 can be ruled out immediately At R3 , λR < ΛLR , as t comparison of the slope of the fraction flow curve at R3 and the slope of the shock line indicates In other words, the intermediate shock moves faster than the compositions on the rarefaction along the initial tie line, a situation that would violate the velocity rule Point R1 is also not an acceptable right state, although more effort is required to show that it Appendix A 261 Overall Fractional Flow of Component 1, F1 1.4 a 1.2 X L3 1.0 aa 0.8 R2 a a L1 R1 a 0.6 a R3 0.4 0.2 0.0 0.0 0.4 0.2 0.6 0.8 1.0 1.2 1.4 Overall Volume Fraction of Component 1, C1 Figure A.5: Shock constructions for jump points of the intermediate shock on the initial oil tie line is not permitted To work out the behavior of eigenvalues on either side of a shock for a right state at R1 , we consider a displacement in which a small amount of dispersion is present For a ternary displacement with a small dispersion coefficient , the conservation equations are ∂F1 ∂C1 + ∂τ ∂ξ ∂F2 ∂C2 + ∂τ ∂ξ = = ∂ C1 , ∂ξ ∂ C2 ∂ξ (A.11) (A.12) We will seek a solution of Eqs A.11 and A.12 for a shock traveling with wave velocity Λ subject L L to the boundary conditions that the compositions C1 = C1 and C2 = C2 on the far upstream side R R of the shock and C1 = C1 and C2 = C2 on the far downstream side satisfy the Rankine-Hugoniot conditions for a shock moving with velocity Λ In addition, we will require that the derivatives of C1 and C2 be zero far upstream and far downstream of the shock The entropy condition can be derived by requiring that the discontinuous solution (one with a shock) be the limit of a traveling wave solution as → A traveling wave solution to Eqs A.11 and A.12 has the form C1 = C1 (ζ) = C1 ( ξ − Λτ ), C2 = C2 (ζ) = C2 ( ξ − Λτ ) (A.13) Application of the chain rule gives the derivatives of C1 and C2 , Λ dCi ∂Ci =− ∂τ dζ i = 1, 2, (A.14) 262 Appendix A dCi ∂Ci =− ∂ξ dζ i = 1, 2, (A.15) and d2 Ci ∂ Ci =− ∂ξ dζ i = 1, (A.16) As a result Eqs A.11 and A.12 become ∂F1 dC2 dC1 ∂F1 dC1 d2 C1 + −Λ , = dζ ∂C1 dζ ∂C2 dζ dζ (A.17) d2 C2 ∂F2 dC2 dC2 ∂F2 dC1 + −Λ = dζ ∂C1 dζ ∂C2 dζ dζ (A.18) and Integration of Eqs A.17 and A.18 gives dC1 L L L = F1 − ΛC1 − F1 (C1 , C2 ) − ΛC1 dζ , (A.19) dC2 L L L = F2 − ΛC1 − F2 (C1 , C2 ) − ΛC2 dζ (A.20) and A shock that satisfies the Lax entropy condition is one that satisfies Eqs A.19 and A.20 with L L R R C1 (−∞) = C1 , C2 (−∞) = C2 , C1 (∞) = C1 , and C2 (∞) = C2 [51, 128] Wang showed that Eq A.20 can be recast into an equation for the x1 , so that the solution variables are C1 and x1 (see [128, Appendix C] for a detailed derivation) That equation is a dx1 = (xL − x1)(ΛLR − λ), dζ b (A.21) where a = (1 − K2 )(K1 − 1)x1 − (K2 − 1)(K3 − 1) + (K1 − 1)S L , (A.22) b = (1 − K2 )(K1 − 1)x1 − (K2 − 1)(K3 − 1) {1 + (K1 − 1)S} , F L − π(xL, x1 ) , λ= L − π(xL, x ) C1 1 (A.23) (A.24) (A.25) and π(xL, x1 ) = Eq A.19 can be rearranged to yield (K1 − K2 )(K1 − K3 ) x1 xL (K2 − 1)(K3 − 1) (A.26) Appendix A 263 0.5 0.4 CL1 CL3 a a x1 C1 dC1/dζ < dC1/dζ > a 0.4 0.2 dC1/dζ < a CR1 0.3 0.2 0.0 C+ CR3 0.6 0.8 1.0 C1 Figure A.6: Regions of positive and negative values of dC1 /dζ and trajectories with dC1 /dζ = dC1 L = (C1 − C1 )(Ω − ΛLR ), dζ (A.27) where Ω= L F1 − F1 L C1 − C1 (A.28) Ω is the slope of a line that connects any point along the fractional flow curve for the initial oil tie line to point L (see Fig A.5) In Eq A.28, F1 and C1 lie along the solution to Eqs A.19 and A.21 Those equations determine C1 (ζ) and x1 (ζ) Eq A.28 indicates that dC1 < 0, dζ dC1 > 0, dζ − < C1 < C1 , (A.29) − + C1 < C1 < C1 , (A.30) (A.31) and dC1 < 0, dζ + C1 < C1 < (A.32) − In these expressions, C1 refers to a trajectory in composition space (C1 ,x1 ) along which dC1 /dζ = 0, the boundary between the zone of positive values of dC1 /dζ at low values of C1 (ζ) Correspondingly, 264 Appendix A 1.6 a Overall Fractional Flow of Component 1, F1 CL 1e 1.4 a 1.2 X 1.0 L a 0.8 a CR 1e R a a 0.6 0.4 0.2 a 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Overall Volume Fraction of Component 1, C1 Figure A.7: Tangent constructions for a shock between tie lines The slope of the line from R to X gives the velocity, Λ, of the shock from R to L The slope of the line from R to CR gives the 1e value of λnt at R The slope of the line from L to CL is λnt at L Comparison of the various slopes 1e reveals the relative magnitudes of the shock velocity and eigenvalues + C1 refers to a second trajectory with dC1 /dζ = 0, this time at high values of C1 (ζ) Fig A.6 shows schematically the arrangement of zones of positive and negative dC1 /dζ On each of the initial and injection tie lines, the three regions of negative, positive, and negative dC1 /dζ exist for low, intermediate and high values of C1 (ζ) In order for a trajectory to connect left state L1 to R1 , dC1 /dζ would have to be negative, but there is no way for the trajectory to pass through the zone in which dC1 /dζ is positive Hence, there are no trajectories that connect L to R1 Therefore, a shock from left state L to a right state R1 for which λR > ΛLR is not permitted t The only remaining possibility is that there is a shock from L to R2 That shock is allowed It does not violate the velocity rule that prevented shocks to right state compositions for which λR < ΛLR because λR = ΛLR And it does not violate the entropy statement that prohibits a shock t t to a right state composition with λR > ΛLR Hence, the intermediate shock must be a semishock t with λR = ΛLR (see Section 7.2 for a continuity argument that confirms that the intermediate t shock is a semishock at which λn t = Λ on the shorter of the initial or injection tie lines) As a result, the statement of the entropy condition for the tie line eigenvalue is λL < λR = ΛLR t t (A.33) Finally, we consider the relative magnitudes of λR , λL , and ΛLR The fractional flow diagram nt nt L R for the intermediate shock is shown in Fig A.7 Direct evaluation of C1e , and C1e using Eq A.2 L R indicates that C1e > C1e as long as xL > xR Fig A.3 shows that xL is larger than xR for this 1 1 system (see Appendix C of Wang [128]) for a detailed proof that the statement must be true for slightly dispersed shock traveling to the right) Appendix A 265 L R As the locations of the tie-line intersection point in Figs A.3 and A.7 show, C1e > CiX > C1e The velocity of the intermediate shock is given by the slope of the line from R to X, and the R nontie-line eigenvalues, λL and λR are given by the slopes of the lines drawn from R and L to C1e nt nt L and C1e respectively Comparisons of those slopes indicates that λR < ΛLR < λL nt nt (A.34) Hence, the intermediate shock is self-sharpening with respect to the nontie-line eigenvalues upstream and downstream of the shock, as it should be if it replaces a nontie-line rarefaction that is prohibited by the velocity rule because λnt increases as the nontie-line path is traced upstream Summary The example of the LVI vaporizing gas drive considered in the appendix leads to the following statement of the entropy condition: λR < Λ < λL , nt nt and λL < Λ = λR t t (A.35) If instead we had considered a LVI condensing gas drive, the statement of the entropy condition would differ Here again, one set of characteristics is sharpening (the nontie-line eigenvalues) and one set is not, but the semishock occurs on the injection (left state) tie line instead of the right state (initial) tie line λR < Λ < λL , nt nt and Λ = λL < λR t t These are the expressions given as Eqs 5.2.25 and 5.2.26 (A.36) 266 Appendix B APPENDIX B: Details of Gas Displacement Solutions In this appendix, full details of all the solutions illustrated in Chapters 4-8 are reported Unless otherwise noted, the fractional flow functions used in the solutions have the form of Eqs 4.1.204.1.22 and Sor = Sgc = Chapter 4–Binary Displacements Table B.1: Displacement details for Fig 4.10 Binary gas displacement with no volume change, M = Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil Point d d c c-b c-b c-b c-b b a a C1 1.0000 1.0000 0.7794 0.7625 0.7250 0.6875 0.6500 0.6316 0.0500 0.0500 C2 0.0000 0.0000 0.2206 0.2375 0.2750 0.3125 0.3500 0.3684 0.9500 0.9500 S1 0.0000 0.0000 0.7725 0.7500 0.7000 0.6500 0.6000 0.5755 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2786 0.2786 0.3552 0.5643 0.8329 1.1614 1.3409 1.3546 1.0000 Appendix B 267 Table B.2: Displacement details for Fig 4.16 Binary gas displacement with volume change Fluid properties and phase compositions are reported in Table 4.1 Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil Point d d c c-b c-b c-b c-b c-b c-b c-b c-b c-b c-b b a a z1 1.0000 1.0000 0.7020 0.6833 0.6205 0.5692 0.5264 0.4902 0.4592 0.4323 0.4088 0.3881 0.3676 0.6316 0.0000 0.0000 z2 0.0000 0.0000 0.2980 0.3167 0.3795 0.4308 0.4736 0.5098 0.5408 0.5677 0.5912 0.6109 0.6324 0.3684 1.0000 1.0000 S1 0.0000 0.0000 0.8630 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000 0.4500 0.3941 0.5500 0.0000 0.0000 Flow Vel 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.5093 0.5093 ξ/τ 1.0000 0.0063 0.0063 0.0090 0.0217 0.0400 0.0662 0.1044 0.1605 0.2442 0.3710 0.5659 0.8329 0.9147 0.9147 0.5093 Table B.3: Displacement details for Fig 4.16 Binary gas displacement with no volume change Fluid properties and phase compositions are reported in Table 4.1 Compositions reported are in mole fractions Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil Point d d c c-b c-b c-b c-b c-b c-b c-b c-b c-b b a a zCO2 1.0000 1.0000 0.7894 0.7499 0.7011 0.6563 0.6150 0.5679 0.5415 0.5085 0.4779 0.4492 0.4243 0.0000 0.0000 zC10 0.0000 0.0000 0.2980 0.3795 0.4308 0.4736 0.5098 0.5408 0.5677 0.5912 0.6109 0.6109 0.3684 1.0000 1.0000 S1 0.0000 0.0000 0.8375 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3538 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.0118 0.0118 0.0217 0.0400 0.0662 0.1044 0.1605 0.2442 0.3710 0.5660 0.8693 1.2972 1.2972 1.0000 268 Appendix B Chapter 5–Ternary Displacements Table B.4: Displacement details for Fig 5.16 Composition path and profiles for a vaporizing gas drive with low volatility intermediate component K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The injection gas is pure CH4 , and the initial oil has composition C1 = 0.1, C2 = 0.5, and C3 = 0.4 Compositions in volume fractions Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil Point f f d d d d c c c-b c-b b b a a CH4 1.0000 1.0000 0.8806 0.8806 0.8806 0.8806 0.5781 0.5781 0.5710 0.5476 0.5294 0.5294 0.1500 0.1500 C4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2924 0.2924 0.2955 0.3057 0.3135 0.3135 0.2908 0.2908 C10 0.0000 0.0000 0.1194 0.1194 0.1194 0.1194 0.1295 0.1295 0.1335 0.1468 0.1570 0.1570 0.5592 0.5592 S1 1.0000 1.0000 0.8473 0.8473 0.8473 0.8473 0.5651 0.5651 0.5500 0.5000 0.4614 0.4614 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2878 0.2878 0.2878 0.7707 0.7707 0.7707 0.7707 0.8415 1.1111 1.3546 1.3546 1.3546 1.0000 Appendix B 269 Table B.5: Displacement details for Fig 5.17 Composition route, saturation, and composition profiles for a self-sharpening (HVI) condensing gas drive K1 = 2.5, K2 = 1.5, K3 = 0.05, and M = The injection gas has composition, C1 = 0.6, C2 = 0.4, and C3 = 0, and the initial oil has composition, C1 = 0.3, C2 = 0, and C3 = 0.7 Compositions reported are in volume fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Intermediate Shock Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c d-c d-c c b b a a CH4 0.6000 0.6000 0.4907 0.4769 0.4595 0.4420 0.4246 0.4071 0.4058 0.5916 0.5916 0.3000 0.3000 CO2 0.4000 0.4000 0.3590 0.3538 0.3472 0.3407 0.3341 0.3276 0.3271 0.0000 0.0000 0.0000 0.0000 C10 0.0000 0.0000 0.1503 0.1693 0.1933 0.2173 0.2413 0.2653 0.2671 0.4084 0.4084 0.7000 0.7000 S1 1.0000 1.0000 0.7395 0.7000 0.6500 0.6000 0.5500 0.5000 0.4963 0.3505 0.3505 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2454 0.2454 0.3255 0.4554 0.6247 0.8415 1.1111 1.1334 1.1334 1.4833 1.4833 1.0000 Table B.6: Displacement details for Fig 5.18 A condensing gas drive (LVI) with a nontie-line rarefaction K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The injection gas has composition, CCH4 = 0.8, CC4 = 0.2, and CC10 = 0., and the initial oil has composition, CCH4 = 0.3, CC4 = 0, and CC10 = 0.7 Compositions reported are in volume fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Equal Eig Point Nontie-line Rarefaction Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c c c-b c-b c-b c-b b b a a CH4 0.8000 0.8000 0.6543 0.6359 0.6127 0.5894 0.5883 0.5695 0.5579 0.5572 0.5737 0.6009 0.6009 0.3000 0.3000 C4 0.2000 0.2000 0.2661 0.2744 0.2850 0.2956 0.2960 0.2987 0.2816 0.2325 0.1271 0.0000 0.0000 0.0000 0.0000 C10 0.0000 0.0000 0.0796 0.0896 0.1023 0.1151 0.1156 0.1317 0.1605 0.2103 0.2992 0.3991 0.3991 0.7000 0.7000 S1 1.0000 1.0000 0.7395 0.7000 0.6500 0.6000 0.5977 0.5500 0.5000 0.4500 0.4000 0.3665 0.3665 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2454 0.2454 0.3255 0.4554 0.6247 0.6336 0.6429 0.6750 0.7267 0.7930 0.8418 1.5015 1.5015 1.0000 270 Appendix B Table B.7: Displacement details for Initial Oil Composition A in Fig 5.22 The viscosity ratio on the initial tie line is M = 3.115, and on the injection tie line, M = 4.586 The molar volumes used to convert mole fractions to volume fractions were CO2 , 150.978 cm3 /gmol, C4 , 101.886, C10 , 215.013 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.7030, xC4 = 0.0436, xC10 = 0.2534, yCO2 = 0.9566, yC4 = 0.0220, yC10 = 0.0215 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.6554, xC4 = 0., xC10 = 0.3446, yCO2 = 0.9817, yC4 = 0., yC10 = 0.0183 Compositions reported in the table are in volume fractions Segment Injection Gas Trailing Shock Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil A Point e e d d d c c-b c-b b a a CO2 1.0000 1.0000 0.9208 0.9208 0.9208 0.8608 0.8552 0.8501 0.8468 0 C4 0 0 0.0301 0.0306 0.0310 0.0313 0.1035 0.1035 C10 0 0.0792 0.0792 0.0792 0.1091 0.1142 0.1189 0.1219 0.8965 0.8965 S1 1.0000 1.0000 0.8131 0.8131 0.8131 0.6223 0.6000 0.5800 0.5669 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.1482 0.1482 0.1482 0.8048 0.8048 0.9106 1.0125 1.0825 1.0825 1.0000 Table B.8: Displacement details for Initial Oil Composition B in Fig 5.22 The viscosity ratio on the initial tie line is M = 1.957, and on the injection tie line, M = 4.586 The molar volumes used to convert mole fractions to volume fractions were CO2 , 150.978 cm3 /gmol, C4 , 101.886, C10 , 215.013 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.7636, xC4 = 0.0777, xC10 = 0.1586, yCO2 = 0.9237, yC4 = 0.0478, yC10 = 0.0285 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.6554, xC4 = 0., xC10 = 0.3446, yCO2 = 0.9817, yC4 = 0., yC10 = 0.0183 Compositions reported in the table are in volume fractions Segment Injection Gas Trailing Shock Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil B Point e e d d d c c-b c-b c-b b a a CO2 1.0000 1.0000 0.9557 0.9557 0.9557 0.8710 0.8709 0.8693 0.8677 0.8661 0 C4 0 0 0.0577 0.0577 0.0579 0.0583 0.0586 0.2204 0.2204 C10 0 0.0443 0.0443 0.0443 0.0714 0.0714 0.0727 0.0740 0.1753 0.7796 0.7796 S1 1.0000 1.0000 0.9202 0.9202 0.9202 0.6703 0.6700 0.6600 0.6500 0.6398 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.4243 0.4243 0.4243 0.8860 0.8860 0.8876 0.9372 0.9880 1.0408 1.0408 1.0000 ... Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Equal Eig Point Nontie-line Rarefaction Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c c c-b c-b c-b c-b b b... Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil Point d d c c-b c-b c-b c-b c-b c-b c-b c-b c-b c-b b a a z1 1.0000 1.0000 0.7020 0.6833 0.6205 0.5692 0.5264 0.4902 0.4592 0.4323... volume fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Intermediate Shock Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c d-c d-c c b b a a CH4 0.6000