The Pennsylvania State University. The Graduate School. Department of Energy and Mineral Engineering MATHEMATICS OF MULTIPHASE MULTIPHYSICS

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1 The Pennsylvania State University The Graduate School Department of Energy and Mineral Engineering MATHEMATICS OF MULTIPHASE MULTIPHYSICS TRANSPORT IN POROUS MEDIA A Dissertation in Energy and Mineral Resources Engineering by Saeid Khorsandi 2016 Saeid Khorsandi Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2016

2 The dissertation of Saeid Khorsandi was reviewed and approved* by the following: Russell T. Johns Professor of Petroleum and Natural Gas Engineering Dissertation Advisor Chair of Committee Turgay Ertekin Professor of Petroleum and Natural Gas Engineering Luis F. Ayala H. Professor of Petroleum and Natural Gas Engineering Graduate Program Officer Wen Shen Professor of Mathematics Alberto Bressan Professor of Mathematics *Signatures are on file in the Graduate School ii

3 Abstract Modeling complex interaction of flow and phase behavior is the key for modeling local displacement efficiency of many EOR processes. The interaction is more complex for EOR techniques that rely on mass transfer between phases such as those that occur during miscible gas floods. Accurate estimation of local displacement efficiency is important for successful design of enhanced oil recovery (EOR) processes. Displacement efficiency can be estimated by experimental and computational methods. The dispersion-free displacement efficiency is 100% at a pressure above the minimum miscibility pressure (MMP) for gas flooding processes. Slim-tube experiments are one of the most reliable experimental approaches for MMP calculation. Computational methods including simulation, mixing cell and method of characteristics (MOC) solutions rely on accurate EOS fluid characterization. MOC is the fastest and the only solution method which is not affected by dispersion. However, current MOC methods have significant limitations in converging to the correct solution. In addition, the assumptions made in MOC may not be correct for some fluids, which can cause errors as large as 5000 psia in calculated MMPs. Current MOC solutions are simplified by assuming that only shocks connect key tie lines. Likewise the velocity condition cannot be applied directly to the shock-jump approximate approach. These simplifications reduce the computation time but result in decreased reliability of shock-jump approximation methods as well. We examined the assumptions of MOC for the case where the two-phase region splits at a critical point. This is referred to hence as bifurcating phase behavior. In this case, the assumption that the non-tie-line eigenvalues change monotonically between two key tie lines is incorrect. The correct solution is constructed for ternary displacements with bifurcating phase iii

4 behavior by honoring all constraints required for a unique solution velocity, mass balance, entropy and solution continuity. The solution is further validated using simulation and the mixing cell method. The simulation results are highly affected by dispersion for some cases such that the results of simulation and analytical solutions match only after using a very large number of grid blocks. The construction of the entire composition route using conventional MOC solutions is very challenging as the number of components increases. The other option is to separate the phase behavior from flow and then solve the tie-lines independent of fractional flow. We examined and developed this approach in detail here. We developed a global Riemann solver for ternary displacements and later extended the splitting technique to multicomponent displacements. Our approach does not suffer from the singularities present in Pires et al. (2006) and Dutra et al. (2009). The solution in tie-line space is constructed for a variety of fluid models including pseudoternary displacements with bifurcating phase behavior, and real fluid displacements (Zick 1986, Metcalfe and Yarborough 1979). Finally the MMP is calculated for several multicomponent (> 4) fluids using the analytical solution based solely on solving the continuous tie-line problem, where tie-line rarefactions and shocks can exist in tie-line space. Thus, we eliminate the need for the shock jump approximation assumption in determining the MMP. The splitting technique is used to construct analytical solutions for low salinity polymer flooding considering wettability alternation caused by cation exchange reactions. The solutions are validated using numerical simulation and experimental data. The solutions demonstrate that multiple salinity shocks form in low salinity injection and the fast moving salinity shock does not change the surface composition and wettability. In contrast, oil is recovered as a wettability front slowly moves in the reservoir and reduces the residual oil saturation. The wettability front creates an oil bank which will be gradually produced. iv

5 Table of Contents List of Figures... viii List of Tables... xv Nomenclature... xvi Acknowledgments... xix Chapter 1 Introduction Description of problem Research objectives Structure of the dissertation... 5 Chapter 2 Background and literature review Enhanced oil recovery Phase behavior and fluid properties Petrophysics Volumetric sweep efficiency Local displacement efficiency Displacement mechanisms of gas floods Experimental methods for estimating MMP Computational methods for estimating MMP Three-hydrocarbon-phase displacements Displacement mechanism of low salinity polymer flooding Hyperbolic system of equations Method of characteristics Finite difference estimation of solution Fractional flow theory Limitations of current MOC solutions for gas flooding Summary Chapter 3 Gas flooding mathematical model Conservation law Tie lines MOC solution for gas flooding Ternary compositional routes for complex phase behavior Bifurcating phase behavior Composition-route construction Features of displacements with bifurcating phase behavior Summary Chapter 4 Three-component global Riemann solver using splitting of equations v

6 4.1 Basic analysis, precise assumptions, and the main results Basic wave behavior The C-waves The β-waves Global solutions of Riemann problems Connecting C-waves with β-shock Connecting β-rarefaction wave to C-waves Global existence and uniqueness of solutions for Riemann problems Numerical Simulations with Front Tracking Initial and injection tie-line selection Example ternary displacements Summary Chapter 5 Tie-line routes for multicomponent displacements Mathematical model Tie-line space Thermodynamic definition Composition space parametrization Example ruled surface routes Constant K-value displacement Example four-component displacements Summary Chapter 6 Robust and accurate MMP calculation using an equation-of-state Riemann solver in tie-line space Pseudoternary ruled surfaces Estimation of ruled surfaces Riemann solver in tie-line space Validation of estimate Riemann solver MMP calculation for Nc displacements Four-component displacements Five-component displacement Twelve-component displacement by Zick (1986) Eleven-component displacement by Metcalfe and Yarborough (1979) Bifurcating phase behavior Summary Chapter 7 Application of splitting technique to low salinity polymer flooding Mathematical model Immiscible oil/water flow Cation Exchange Reaction Network Reactive Transport Model Wettability alteration Polymer Flooding Model Numerical solution Analytical solution vi

7 7.2.1 Decoupled system of equations Reactive transport solution Polymer transport solution Maping fractional flow Front tracking algorithm Matching wettability front retardation independent of reactions Matching reactions independent of fractional flow Results Two phase CEC without wettability alteration Low salinity waterflooding Low salinity polymer experiment Low salinity slug injection with varying slug size Summary Chapter 8 Conclusions Summary and conclusions Future research Application of splitting to compositional simulation WAG injection and hysteresis Fluid characterization Displacement mechanism for combined EOR techniques References Appendix A Tie-line derivatives Appendix B Switch condition Appendix C Shock composition paths Appendix D Reservoir simulation in tie-line space Appendix E Object oriented design of phase equilibrium calculation vii

8 List of Figures Figure 2-1: Scanning electron microscope (SEM) image of Berea sandstone core (Schembre and Kovscek 2005) Figure 2-2: (Top) Initial data for Buckley-Leverett problem. (Bottom) Characteristics for Buckley- Leverett solution. The characteristic line for the shock is shown in red line Figure 3-1: Comparison of composition route of UTCOMP and our simulator with MOC assumptions Figure 3-2: Comparison of tie lines of UTCOMP and our simulator with MOC assumptions Figure 3-3: Geometric construction of the nontie-line and tie-line eigenvalues. The tie-line eigenvalue is equal to the slope of the curve at any point in the two-phase region, while the nontie-line eigenvalue is equal to the slope of the line from h to a two-phase composition. These two eigenvalues are equal at the two umbilic points, where the line from h is tangent to the overall fractional flow curve Figure 3-4: Phase behavior of pseudoternary system showing the split of the two-phase region with pressure at 133 o F Figure 3-5: Three phase behavior at 40 o F and 1000 psia Figure 3-6: Critical locus of C 1N 2 and CO Figure 3-7: K-values at psia, 133ᵒF along the line where C concentration is zero Figure 3-8: a) Region of tie-line extensions that intersect within the single-phase liquid region at 16,000 psia, and b) values of tie-line parameters in Eq. (3.13) for all tie lines in positive composition space. The dashed line in figure b) represents a composition in figure a) as shown (see Eq. (3.13)). The intersection of the lines with the solid curve in figure b) shows the tie lines that pass through that composition. Point 2 lies on one of the envelope curves where successive tie-lines intersect Figure 3-9: Non-tie line paths and watershed points Figure 3-10: Triangle of shocks and continuity of solution viii

9 Figure 3-11: Tie-line eigenvalue from the oil to gas tie lines along the red non-tie line path shown in Figure Figure 3-12: Analytical and numerical composition profile using 20,000 grid blocks for displacement of oil 1 in Figure 3-10 by pure CO 2 at 16,000 psia Figure 3-13: Effect of numerical dispersion on the composition route for oil 2 at 16,000 psia Figure 3-14: K-values for the two-phase regions at 21,000 psia (see Figure 3-4) along the C 1N 2-CO 2 axis of the ternary diagram Figure 3-15: Discontinuous dispersion-free composition routes for point A Figure 3-16: Analytical composition profiles showing a discontinuity in the dispersion-free displacement of oil A. The discontinuity is verified by simulation for various initial compositions near point A with 400,000 grid blocks. The unstable solution shown in the figures for composition A does not satisfy the entropy condition Figure 3-17: Discontinuity in recovery calculated by simulation Figure 3-18: Example of three intersecting tie lines through an oil composition Figure 3-19: Analytical composition routes for oils 1-4 at 16,000 psia Figure 3-20: Analytical composition profiles for the heavy pseudocomponent for four oil compositions displaced by pure CO 2 at 16,000 psia Figure 3-21: Analytical composition profiles for the light pseudocomponent for four oil compositions displaced by pure CO 2 at 16,000 psia Figure 3-22: Analytical composition profiles for the CO 2 component for four oil compositions displaced by pure CO 2 at 16,000 psia Figure 3-23: Recoveries for oil 1 and 4 from numerical simulation with 20,000 grid blocks. The displacement is both condensing and vaporizing for oil 1, but only vaporizing for oil 4. Oil 4 has no MMP at because the displacement remains liquid-liquid at very high pressure Figure 4-1: Illustration of three-component phase diagram with constant K-values K1, K2, K3 = (0. 05, 1. 5, 2. 5). Left plot uses the C1, C2) coordinate, which the right plot uses the (C, β) ix

10 coordinate. The two red curves are the boundary of the two-phase region, and green lines are tie lines Figure 4-2: Integral curves for the β-family in the phase plane (C, β), corresponding to the case in Figure 4-1. Here, the red curves are the boundary of the two-phase region and are called binodal curves Figure 4-3: Functions F(C, β) and F(C; β, a). a=0.2 on the left plot Figure 4-4: Solutions to Riemann problems for C-waves. Left: If CL < CR, the lower convex envelope L M1 M2 R gives a shock L M1, a rarefaction fan M1 M2 and a shock M2 R. Right: If CL > CR, the upper concave envelope L M R gives a rarefaction fan L M and then a shock M R Figure 4-5: Illustration for β shock. Here the red and blue curves are graphs for FL and FR, and is the point ( σβ 1, σβ 1). The green line has slope σβ. Then, CL and CR must be selected from the corresponding graphs of FL and FR that intersect with the green line Figure 4-6: The set IL and JL are the x and y coordinates for the thick curves in (L1)-(L4). The set IR and JR are the x and y coordinates for the thick curves in (R1)-(R3) Figure 4-7: Riemann solver for the special case, where a tie-line is tangent to the two phase region, plots of the functions C F(., βl) and C F(., βr), where blue curve is for the left state, and red curve is for the right state Figure 4-8: Two possible relations between the curves C1, C2, C3 and C Figure 4-9: Three situations for different locations of CL and the corresponding sets of IL (with thick three line on L) and IL (with thick red line on R) Figure 4-10: Case 2, when C3 < CL < C2, the β-wave path consists of two β-rarefaction waves with a Ccontact in between Figure 4-11: Estimation of large β-rarefaction with smaller waves (Left) and convergence of results to the correct solution (Right) Figure 4-12: Comparison of the composition path calculated by finite difference simulation and front tracking x

11 Figure 4-13: Comparison of the composition profiles calculated by finite difference simulation using 10,000 grid blocks and front tracking with ε = Figure 4-14: Fronts for variation of initial condition where two slugs are injected Figure 4-15: Composition profiles at t = 0. 0 (bottom),0.1,0.2,0.3,0.4 and 0.5 (top) Figure 4-16: Analytical solution for ternary displacement in composition space. Point a is on the envelope curve and C1a = 1/λβ Figure 4-17: Tie-line coefficients for ternary phase behavior of Figure 1 in tie-line space Figure 4-18: Analytical solution for three-component displacements showing shocks and rarefactions in Lagrangian coordinates Figure 4-19: Injection and initial compositions considered for bifurcating phase behavior (Ahmadi et al. 2011). Three tie lines extend through composition I1. Points a and b lie on the envelope curve (see Khorsandi et al. 2014) Figure 4-20: Tie-line coefficients for three-component bifurcating phase behavior in tie-line space, where negative sign in Eqs. (3) is used Figure 4-21: Eigenvalue for three-component displacements with bifurcating phase behavior. Dashed line is a shock Figure 4-22: Analytical solution showing shocks and rarefactions in Lagrangian coordinates Figure 5-1: Phase diagram of water. There are four possible two-phase states at atmospheric pressure as shown by red squares (from Chaplin 2003) Figure 5-2: The new coordinates are demonstrated with red arrows Figure 5-3: Quaternary displacement with three possible crossover solutions based solely on shock-jump MOC (Yuan and Johns, 2005). The two-phase region for each ternary face is outlined by the blue and purple dashed lines Figure 5-4: Phase diagram in tie-line space and ruled surfaces for constant K-values Figure 5-5: Eigenvalues along the line β2 = Figure 5-6: Parametrization of the tie-line path for automatic construction of tie-line path for quaternary phase behavior of Table xi

12 Figure 5-7: Phase diagram in tie-line space and ruled surfaces for four-component displacements in Table 5.2 at 2900 psia and 160 o F Figure 5-8: Quaternary phase diagram with bifurcating phase behavior generated based on Ahmadi et al. (2011) at 8000 psia and 133 o F Figure 5-9: Four-component displacement of Figure 5-8 in tie-line space at 8000 psia and 133 o F Figure 6-1: Parametrization of ruled surfaces. The solution can be constructed by solving Γ1L = Γ2R. 160 Figure 6-2: Projection of tie-line routes to C1 C10 plane for four component displacement of Table Figure 6-3: Projection of tie-line routes to xc1 xc26 35 plane for the four component displacement of I3 by J1 in Table 4.5 with bifurcating phase behavior at 8000 psia and 133 o F Figure 6-4: Projection of tie-line routes to xc1 xc20 plane for six component displacement in Table 6.2 at 1000 psia and 160 o F Figure 6-5: The tie line route parameters for the displacement of I 1 by J 1 in Table Figure 6-6: Key tie-line lengths calculated by analytical solution and shortest tie-line length calculated by simulation for the displacement in Table Figure 6-7: Four-component displacements with bifurcating phase behavior. The shock-jump MOC over predict MMP by almost 4000 psi Figure 6-8: The key tie line length for five-component displacement. The shortest tie-line length is calculated by simulation Figure 6-9: The tie-line route at different pressures for five-component displacements Figure 6-10: Tie-line length variation with pressure calculated using approximate Riemann solver for displacement in Table 6.4 at 185 o F. MMP is estimated to be at 3097 psia Figure 6-11: Tie-line length variation form injection to initial tie-line at different pressures and 185 o F with compositions in Table Figure 6-12: Shortest tie-line length variation with pressure for displacement from Johns and Orr (1996) xii

13 Figure 6-13: MMP calculaiton for complex phase behavior of Mogensen et al. (2008). The shock-only MOC, improved shock-only MOC and mixing cell results are copied form Ahamdi et al. (2011) Figure 7-1: Mapping of fractional flow curve to the composition solution. Left figure uses the standard approach as is solved for the fractional flow problem for polymer flooding. Right figure demonstrates the wave velocities for the three different Riemann problems Figure 7-2: Mapping of fractional flow curve to composition solution. Left figure uses the same approach as fractional flow for polymer flooding. Right figure demonstrate the wave velocities for the three different Riemann problems Figure 7-3: Piecewise linear approximation of fractional flow is commonly used in front tracking algorithms Figure 7-4: The piecewise estimate of the fractional flow curve converts the rarefactions to small shocks. The error of approximation decreases as the number of the linear pieces of fractional flow is increased Figure 7-5: The interaction of shocks in a water flooding displacement with variable initial condition. The initial condition should be approximated with a piecewise constant function Figure 7-6: The single phase CE reactions are converted to two-phase transport. The slope of dashed lines are equal to cation front velocities Figure 7-7: Comparison of single- and two-phase transport of Mg + +. Wettability alteration is not included in this model. The simulation results are shown with dotted lines Figure 7-8: Comparison of single- and two-phase adsorbed concentration of Na at xd = 1 for the floods of Figure 7-7. The surface composition is not affected by the anion shock. The simulation results are shown with dotted lines Figure 7-9: Comparison of analytical solution results (solid line) and simulation results (dotted line) for high salinity and low salinity injection considering the effect of wettability alteration. The analytical solution with no CEC over predicts the effect of low salinity injection xiii

14 Figure 7-10: Solutions for low salinity water flooding. Left figure shows the analytical solution with original CEC and the right figure shows the analytical solution without CEC. The wettability front velocity is over estimated in the right figure Figure 7-11: Walsh diagram for low salinity polymer injection. Fractional flows are shown for oil wet (OW), oil wet with polymer (OWP) and water wet with polymer (WWP). The anion and polymer shocks have the same velocity. The wettability front is very slow Figure 7-12: Analytical solution and simulation results matched experimental data (Shaker Shiran and Skauge 2013). CEC and oil wet Sor were not provided for the experimental data and they are the only two fitting parameters used to match the low salinity flood Figure 7-13: Walsh diagram for low salinity flood followed by polymer injeciton. The fractional flows are shown for oil wet (OW), water wet (WW), and water wet polymer (WWP). The solution is not self similar and the results are calculated by the front tracking algorithm Figure 7-14: Low salinity pre-flush. The yellow area shows the high salinity water and blue area represents the polymer flooded region Figure 7-15: Saturation fronts for 1D low salinity slug injection. The low salinity slug size is 0.2 PV for left figure and 0.6 PV for the right figure. The Na + significantly reduces at the front shown by the red line so that wettability alteration occurs across this line. The shaded region represents the water with very low salinity Figure 7-16: Water saturation profiles for different low salinity slug sizes after 15 PVI calculated by MOC with cation exchange reaction Figure 7-17: Comparison of S or decrease from the analytical solutions to simulation and experimental results (Seccombe et al. 2008) for different low salinity slug sizes after 15 PVI. The simulation model used 100 grid blocks xiv

15 List of Tables Table 3.1 Component properties of the fluid system of Orr et al. (1993) Table 3.2 Component properties of the fluid system of Ahmadi et al. (2011) Table 4.1 Fluid characterization for the ternary system Table 4.2 Binary interaction coefficients for the ternary system Table 4.3 Initial condition for example problem Table 4.4 Injection condition for example problem Table 4.5 Component properties and compositions for bifurcating phase behavior developed based on Ahmadi et al. (2011) Table 5.1 The compositions and K-values for example case (Yuan et al. 2005) Table 5.2 Component properties and compositions for displacement by Dutra et al. (2009) Table 6.1 The compositions for four-component example with component properties in Table Table 6.2 Input properties for six-component MMP calculations from Johns (1992) Table 6.3 Input properties for five-component MMP calculations Table 6.4 The compositions for 12-component displacement from Zick (1986) Table 7.1 Water composition for the single- and two-phase displacements Table 7.2 Reaction parameters. CEC2 and CEC3 are calculated by matching experiments Table 7.3 The Corey relative permeability parameters for the experiments xv

16 Nomenclature ( ) Thermodynamic activities of a species [ ] The concentration of a solid species A Tie-line coefficient matrix B Tie-line coefficient matrix C Overall mole fraction C s The concentration of a species C s The adsorbed concentration of a species D Composition domain D Dispersion coefficient F Overall fractional flow F p Molar rate of the primary species p F q Molar rate of the secondary species q H Shock velocity in tie-line space H Vertical distance between injection and production wells I Composition set along a tie line J Jacobian matrix J Composition set along a tie line K K-value K Permeability K eq,r Equilibrium constant of reaction r K ij Derivatives of K-value respect to phase mole fraction L Horizontal distance between injection and production wells L j Liquid phase j for three phase fluids M p Molar density of the primary species p M q Molar density of the secondary species q N p The number of the primary species N sec The number of secondary reactions Q p Total molar rate of primary species p R Coefficient matrix in tie-line space S Phase saturation S Normalized phase saturation a Parameters used in viscosity model for polymer c Mole fraction in a phase e Eigenvector f Fractional flow of a phase f Fugacity of component g Gravitational constant h Envelope parameter xvi

17 k rα k rα k rα,ww k rα,ow Relative permeability of phase α Endpoint relative permeability of phase α Water wet endpoint relative permeability of phase α Oil wet endpoint relative permeability of phase α nc n α n p Pe t x x z i ρ ci ρ j α β ε λ Λ Α Β Γ γ μ ξ φ ψ φ i ρ ci ρ j Number of components Exponent in Corey s model for phase α Number of phases Peclet number time length Mole fraction in phase Mixture mole fraction Fugacity coefficient of component i Constant density of component i Molar or volumetric density of phase j Tie-line coefficient Tie-line coefficient Dispersion coefficient Eigenvalue Shock velocity Tie-line coefficient matrix Tie-line coefficient matrix Tie line Tie line space parameter Viscosity Shock layer moving coordinate Lagrangian coordinate Lagrangian coordinate Porosity Fugacity coefficient of component i Constant density of component i Molar or volumetric density of phase j Superscripts A Upstream shock B Downstream shock D Downstream I Independent L Left R Right U Upstream xvii

18 e ini inj l v Subscripts D g i i i ij inj j nt o P r t w Envelope curve Initial Injeciton Liquid Vapor Dimensionless Gas Component Initial Tie-line parameter index Component i in phase j Injection Phase Non-tie line Oil Production Residual Tie line Water Abbreviations CEP Critical end point EOR Enhanced oil recovery EOS Equation of state EVC Equal velocity curve FCM First contact miscible LSW Low salinity water flooding MCM Multi-contact miscible MIE Multi ion exchange MMP Minimum miscibility pressure MOC Method of characteristics ODE Ordinary differential equation PDE Partial differential equation PVI Pore volume injected RR Rachford-Rice equations TL Tie line WS Watershed point xviii

19 Acknowledgments I would like to express my deepest gratitude to my supervisor, Professor Russell T. Johns, who contributed immensely to my education and research throughout my studies at Penn State. I would also like to thank Dr. Wen Shen who helped me to better understand mathematics of conservations laws. I am grateful to my PhD committee members Dr. Alberto Bressan, Dr. Turgay Ertekin and Dr. Luis F. Ayala H. for their helpful comments and suggestions during my PhD studies. I greatly appreciate Dr. Changhe Qiao who helped with numerical simulation of low salinity floods. I also would like to thank members of the Enhanced Oil Recovery JIP, especially Dr. Dindoruk for his great comments on my research. Financial support for this research was provided by the Enhanced Oil Recovery Joint Industry Project at the EMS Energy Institute, Pennsylvania State University. I would like to express my gratitude to all my friends who helped me throughout my PhD studies. There are too many names to remember; however I wish to single out Liwei Li, Kaveh Ahmadi, Mohsen Rezaveisi, Payam Kavousi, Bahareh Nojabaei, Nithiwat Siripatrachai, Aboulghasem Kazemi Nia, Saeedeh Mohebinia, and Soumyadeep Ghosh for their support. I am also thankful for the encouragement from all of my family members. xix

20 Chapter 1 Introduction This chapter provides a brief review of motivations, and objectives of this research followed by the structure of this dissertation. 1.1 Description of problem Hyperbolic systems of equations can be used to examine enhanced oil recovery techniques with dominant convection of one or more phases. Purely convective flow models for EOR are usually developed assuming negligible dispersive mixing caused by compressibility, dispersion, diffusion, and capillary dissipation. These assumptions are reasonable for 1D displacements and slim-tube experiments. Hyperbolic systems of equations are used in modeling EOR techniques such as polymer flooding, gas flooding, and surfactant flooding. The other physical processes that are frequently modeled with hyperbolic equations are compressible gas flow, shallow water flow, traffic flow and chromatographic separation. Analytical solutions of the EOR problems can explain the result of complex interactions of phase behavior and transport. Therefore the solution can be used to understand displacement mechanisms, and calculate the optimum design parameters for floods as a function of reservoir properties. For examples, Walsh and Lake (1989) calculated the optimum simultaneous water assisted gas derive (SWAG) ratio based on fractional flow theory. In addition, Johns et al. (1993) explained the combined condensing/vaporizing gas drive by developing analytical solutions for gas floods. Furthermore, the other applications of analytical solutions are to benchmark numerical simulations (for example Mallison et al. 2005), improve the finite difference estimate of the flux 1

21 for convective displacements,calculate convective displacements along streamlines in a streamline simulator (Juanes and Lie 2008, Thiele 2001), improve the phase equilibrium calculations (Voskov and Tchelepi 2009), and MMP calculation (Johns and Orr 1996). The hyperbolic systems of equations are difficult to solve both numerically and analytically because of discontinuous jumps of solutions known as shocks. Shocks cannot be modeled with the strong form of PDEs, and the weak form or integral form of the equations should be used. The integral form of the PDEs is difficult to work with, instead the Rankine- Hougoniot condition is commonly used to construct the shocks. Multiple solutions, however, may satisfy the Rankine-Hougoniot condition. Therefore additional conditions are required to select the correct physical solution. In the petroleum engineering literature, the procedure to construct the solution is usually referred as the fractional flow theory (Buckley and Leverett 1942, Pope 1980, Helfferich 1981). The systems of equations for compositional displacements are usually not strictly hyperbolic and the composition space contains multiple umbilic points where the eigenvalues are not distinct. As a result, mathematical theories developed for strictly hyperbolic equations should be used catiouslyfor compositional flooding. Analytical solutions for 1-D displacements using MOC are used frequently in two-phase displacements to calculate the MMP (Buckley and Leverett 1942, Helfferich and Klein 1970, Helfferich 1981, Dumore et al. 1984, Orr 2007). The MOC method can estimate MMPs very quickly and accurately provided that the fluid characterization with a cubic EOS is reliable. The displacement mechanism and miscibility development for two-phase displacements are well known (Orr et al. 1993, Johns et al. 1993) and have been confirmed by experimental results and numerous applications. The approach for MMP estimation of multicomponent displacements is currently based on the paper by Johns and Orr (1996), which developed a graphical method to analytically calculate key tie lines for an 11-component displacement. The MOC approach was 2

22 simplified by Wang and Orr (1997) and Jessen et al. (1998) who used the assumption that shocks exist from one key tie line to the next. The complex phase behaviors encountered in real floods make the solution for compositional displacements even more complex. MOC solutions are currently reliable for simpler two-phase systems and the assumption of the solutions are not correct for some phase behaviors (Ahmadi et al. 2011). Analytical solutions are developed for three-phase partially miscible flow in ternary (LaForce and Johns 2005) and quaternary systems (LaForce et al. 2008, LaForce et al. 2010). There are no analytical solutions however for three phase displacements with four or more components and there are no analytical solutions for bifurcating phase behavior, which is closely related to three phase behavior. The analytical solution for injection of a mixture of gas, can be complicated by the existence of multiple tie lines that satisfy the geometric construction (Yuan and Johns 2005, Ahmadi et al. 2011b). The solution can be estimated using numerical methods; however, 1D simulations can be difficult to make due to problems with relative permeability, phase labeling and the simulation results are usually affected by dispersion (Stalkup 1987, Johns et al. 2002). In addition, the simulations do not calculate the key tie lines and structure of the solution. A more reliable alternative is the mixing cell method (Ahmadi and Johns 2011) which has been shown to match well the MMPs from slim-tube experiments for a variety of complex phase behavior. Furthermore, mixing cell models have been extended to three-phase displacements (Li et al. 2015). This does not negate the value of MOC theory, but it serves to underscore the more practical nature of using mixing-cell models as a more robust method of MMP estimation. We used conventional MOC to construct composition routes for ternary displacements with bifurcating phase behavior. The incorrect assumptions were revised and we combined entropy and velocity conditions into a switch condition which helps to eliminate many of the incorrect solutions. Further, we show that for such complex ternary displacements the MMP does 3

23 not exist for some oil and gas compositions. In those cases, L 1-L 2 behavior exists for all higher pressures even though the displacement efficiency is greater than 95% so that effectively a pressure is reached where high displacement efficiency occurs. However, the solution could not be extended to displacements with more than three components with bifurcating phase behavior. The limitations of current analytical solutions for gas flooding and MMP estimation techniques are revisited in this research. We simplify the analytical solutions by mathematically splitting the phase behavior equations from the flow equations. Johns (1992) and Dindoruk (1992) provided evidence that MMPs are likely independent of fractional flow. Johns (1992) demonstrated that tie lines connected by a shock must intersect at a composition outside of the two-phase region, and in many cases outside of positive composition space. Entov (2000) suggested the possibility of deriving an auxiliary problem which is only a function of tie line parameters and independent of fractional flow curve. Pires et al. (2006) later develop a Lagrangian coordinate that splits the phase behavior from flow in a set of auxiliary equations. However the coordinate transformation has singular points and the composition path cannot be calculated using the transformed coordinate. We completed the splitting procedure by developing a global Riemann solver for ternary displacements. The spiliting technique is extended to displacements up to 12 components. The developed ruled surface routes are used successfully to calculate MMP. The current fractional flow theory is not sufficient to handle the problems related to novel EOR techniques, such as low salinity and slug injection. In this research we examined the analytical solution for slug injection problems for gas flooding and low salinity polymer flooding. The splitting is used to construct the analytical solution for low salinity polymer flooding. The low salinity pre-flush is commonly used to protect polymer from the degrading effect of reservoir high salinity formation water. Furthermore, low salinity water can change wettability of the rock and increase oil recovery. The conventional MOC solutions cannot solve the low salinity 4

24 polymer flooding. The splitting technique is used to construct solutions for low-salinity polymer flooding. 1.2 Research objectives The analytical solutions for compositional displacements are oversimplified and limited boundary conditions are considered. The objectives of the present research are to: 1. Construct composition routes for displacements with complex phase behavior when twophase regions bifurcate into two separate two-phase regions by satisfying velocity and entropy conditions. 2. Improve the coordinate transformation technique to split phase behavior from flow by developing a two-step Riemann solver for ternary gas floods. Extend the Riemann solver in tie-line space for multicomponent systems. 3. Develop a more robust and accurate MMP method using the Riemann solver in tie-line space. Test the new algorithm for fluids where MMP could previously not be calculated. 4. Determine composition paths once tie-line solutions are formed by applying fractional flow into the tie lines. 5. Develop a front tracking algorithm for gas floods using general Riemann solvers. 6. Construct solutions for low salinity polymer flooding considering the wettability alteration mechanism model based on the cation exchange reactions. 1.3 Structure of the dissertation The application of hyperbolic equations in petroleum industry are discussed in Chapter 2. Mathematical models of gas floods are described in Chapter 3 followed by a description of 5

25 limitations of current MOC solutions for these systems. we provide the analytical solution in compositions space for bifurcating phase behavior. Chapter 4 describes the new splitting technique and the two-step Riemann solver for gas flooding. Chapter 5 demonstrates our Riemann solver in tie-line space and Chapter 6 demonstrates the application of tie-line space Riemann solver for MMP calculation. The splitting technique is used in Chapter 7 to construct solutions for low salinity polymer floods. Chapter 8 presents the main conclusions of this research and suggestions for future research.. 6

26 Chapter 2 Background and literature review This chapter reviews the applications of hyperbolic system of equations to examine compositional displacements. The first section provides a review of key affecting parameters in EOR. The second and third sections discusses the displacement mechanisms for gas flooding and low salinity polymer flooding. The fourth section reviews MOC and demonstrates challenges of developing analytical solutions for EOR hyperbolic equations. 2.1 Enhanced oil recovery Fossil fuels will remain an important energy source for the foreseeable future (U.S. EIA, 2013). Considering the limited petroleum resources and plenty of mature fields, EOR is an option to increase recovery factor of reservoirs by up to 40% and 20% compared to primary and secondary recoveries, respectively. The enhanced oil recovery techniques include injection of fluids which are not usually present in reservoir (Lake et al. 2014). These methods can be classified into thermal, chemical and miscible flooding. The EOR techniques can be combined to increase recovery and efficiency. Gas flooding recently became the most widely used and prolific enhanced oil recovery (EOR) technique (Oil and Gas Journal 2014), and is increasingly being considered for CO 2 storage (Li et al. 2015a). Injection of CO 2 and other gases to recover trapped oil is expected to increase, including its use for oil shale (Sheng 2015) or heavy oil (Okuno and Xu 2014) reservoirs. Furthermore, CO 2 injected in an EOR-storage process may help to maintain pore pressure, thereby reducing the risk of induced seismicity that is a concern in large scale geologic 7

27 storage of CO 2 in saline formations (Hitzman 2013, Zoback and Gorelick 2012, National Academy of Sciences 2012). The incremental oil recovery from CO 2 flooding is estimated to increase ultimate oil recovery by about 7-23% of the original oil-in-place (OOIP) (Jarrell et al., 2002), which makes CO 2 EOR and storage a potentially profitable process. Effective use of CO 2 for EOR and storage is possible through future energy policy and resource development planning based on the regional- and national-scale evaluation of the potential for the process (Orr, 2009; Godec et al., 2013). An accurate estimate of EOR efficiency is essential for screening and optimization purposes. As the number of EOR processes increases the chance of encountering more complex reservoirs and phase behavior increases, therefore prediction of EOR efficiency will be more complex. The efficiency of EOR techniques can be defined as a function of volumetric sweep efficiency and local displacement efficiency (Lake et al. 2014). RF = E v E D. (2.1) Volumetric sweep efficiency, E v, is defined as the volume of oil contacted over the amount of oil in place while local displacement efficiency, E D, is defined as the ratio of produced oil over the amount of contacted oil. Sweep and local displacement efficiencies are complex functions of fluid and petrophysical properties of the reservoir, geometry of the reservoir, well pattern and injection rates. We first provide a short review of fluid and petrophysical property models. Then the volumetric and local displacement efficiencies are discussed. In this research, we focus more on the effect of fluid phase behavior on displacements efficiencies. The key features of petrophysical models are discussed. 8

28 2.1.1 Phase behavior and fluid properties Accurate models of fluid properties as a function of pressure, temperature and composition are essential to examine and simulate EOR processes. The fluid models can be classified to the main groups of black oil and compositional fluid models. The phase behavior computation using a black oil model is usually much faster than for a compositional model, which makes black oil modeling more popular. Recently, Nojabaei et al. (2014) applied the black oil model to miscible gas floods. However, the black oil models cannot capture the complex mass ransfer between phases in highly compositional processes. The phase behavior is usually complicated in enhanced oil recovery processes. Miscible gas flooding encounters one of the most complex phase behaviors, because gas flooding relies on the significant mass transfer between different phases and the fluid composition significantly changes in gas floods. In addition, the high recovery factors can be achieved if the compositional path is close to the critical points and the phase behavior is more complex close to a critical point. Finally, wide ranges of gas composition are used as the solvent composition can be used to improve economics and recovery of the floods. For example dry gas and enriched mixtures of intermediates are used in Prudhoe Bay field in Alaska (McGuire, 2001) and CO 2 injection has been used widely in west Texas (Mizenko 1992, Orr and Taber 1984, Tanner 1992, Stein 1992). CO 2 is widely used as a solvent in gas injection because miscibility can be achieved at lower pressures, and CO 2 has a liquid like density. The phase behavior of a mixture of CO 2 and hydrocarbon components is more complicated compared to hydrocarbon mixtures. The mixture of CO 2 and reservoir oil can form a second hydrocarbon liquid at reservoir conditions. Although CO 2 floods usually have complex phase behavior, CO 2 flooding has very positive features. For example, underground CO 2 storage can be used to mitigate the CO 2 effect on global warming. Also the second hydrocarbon liquid in three-phase CO 2 floods acts as an extraction agent to 9

29 recover residual oil (Okuno, 2011). In addition the phase behavior can be affected by chemical reactions with rock (Venkatraman et al. 2015) or capillary pressure in tight rocks with a pore radius on the order of 10 nm (Nojabaei et al. 2013). The fluid cannot be considered as continuum fluid in pores smaller than 10 nm (Li et al. 2014). Phase behavior calculations based on EOS are more complex, however they provide better predictions. PR EOS (Peng and Robinson 1976) is the most commonly used EOS in the petroleum industry. PCSAFT (Gross and Sadowski 2001), however has been used for modeling asphaltene deposition (Mohebninia et al. 2014). Compositional fluid models are as accurate and predictive as the fluid characterization. Jaubert et al. (2002) demonstrated that accuracy of computed MMP increases by including swelling and multi-contact experimental results in fluid characterization. Egwuenu et al. (2008) demonstrated that tuning of MMP and MME increases fluid characterization quality with smaller number of components. The measured and calculated MMP should match closely for different computational techniques, otherwise, there might be an error in fluid characterization or the assumptions of the solutions are not valid (Ahmadi et al. 2011). Flash calculation Reservoir fluids may split into multiple phases and the equilibrium phase compositions should be computed using flash calculation algorithms. Phase equilibrium calculations using EOS are a time consuming part of compositional reservoir simulations (Chang, 1990), and the most time consuming part in a slim-tube simulator. Thermodynamic equilibrium is calculated based on the first and the second of laws of thermodynamic. The phases are at equilibrium when the fluid has the maximum entropy. For a fluid at constant pressure and temperature, the maximum 10

30 entropy is equivalent to the minimum Gibbs energy. The Gibbs energy of the fluid can be calculated by PR EOS (Robinson and Peng, 1978). An important step in multiphase flash calculations is to determine the number of equilibrium phases. The number of phases are determined iteratively using a series of stability tests and flash calculations (Michelsen 1982, Li and Firoozabadi 2012). Good initial estimates are necessary for convergence especially for three-phase calculations as improper K-value estimates may not have a solution at all. Typically initial estimates of K-values are available from the stability analysis of a two-phase mixture or better yet from the previous time step in compositional simulation where convergence of a three-phase flash was already obtained (Mohebbinia, 2013). Prior K-value estimates in simulation are not necessarily available however when the number of equilibrium phases change (Okuno et al. 2010). Li et al. (2015b) used flash results from previous contacts in their three-phase mixing cell to generate initial K-value estimates for flash calculations. The procedure for determining how many phases form is also clouded by the possibility of finding a false two-phase solution that is used for subsequent stability analysis. The assumption of a maximum number of phases can significantly reduce simulation time, however the assumption may result in discontinuous phase compositions. After determining the number of phases, the equilibrium compositions can be calculated by direct minimization of Gibbs energy (Nichita et al. 2002) or by searching for the phases with equal chemical potential (Rachford and Rice 1952). Whitson and Michelsen (1989) improved the Rachford-Rice algorithm by allowing negative values for phase mole fractions. The negative flash calculation is essential for analytical solution of displacement problems. The Rachford-Rice equation has several poles and is useful for a limited range of compositions. Several authors derived new objective functions which are compared by Li et al. (2012). Li and Johns (2007) and later Li et al. (2012) developed a constant K-value flash to calculate tie lines without calculating saturations. Their method can be used to calculate all tie lines that pass through a composition. 11

31 Juanes (2008) also developed a similar method to find all tie lines that extend through a singlephase composition when K-values are constant, but that method is not practical as the number of components increase. Three-phase equilibrium calculations are computationally time-consuming and difficult to make, especially near critical-end points. The robustness of flash calculations depends on the formulation and the solution algorithm. One important part of the solution algorithm is Rachford- Rice (RR) iteration. Michelsen (1994) first proposed an algorithm to solve the multiphase RR equations as a minimization of a convex objective function. Okuno et al. (2010) developed the multiphase RR algorithm as a minimization of a non-monotonic convex function with N c linear constraints. Their RR method is guaranteed to converge to the correct phase splits (if one exists) because no poles are within the feasible region defined by the linear constraints. Okuno et al. s method is applied as part of the developed three-phase mixing cell code in this paper. In addition Li et al. (2015b) developed a new phase labeling technique based on phase compositions. The labeling approach was used in a mixing cell algorithm, but has yet to be extended to reservoir simulators. In this research, we used Li-Johns (Li et al. 2012) algorithm for negative two phase flash calculations. Furthermore, we used tie-line tables for ternary displacements instead of iterative flash calculations. The parametrization of composition space and developing tie-line tables can help to improve reservoir simulation speed (Voskov and Tchelepi 2008). The extension of tie line tables to multicomponent displacements is more complicated and not very efficient for explicit simulators (Rezaveisi et al. 2014). 12

32 2.1.2 Petrophysics Geologic formations are the product of millions of years of physical and chemical processes. Thus reservoirs have complex structures at pore (Figure 2-1) and field scale. Multiphase flow in porous media is affected by different forces such as capillary and gravity. Therefore relative permeability models for multiphase transport are not very well developed. The reservoir rock properties vary significantly across the reservoir. The degree of permeability variance (Dykstra-Parsons coefficient) and the correlation lengths are important parameters to characterize reservoir heterogeneity (Lake et al. 2014). The porosity variation in a reservoir can be calculated as a function of permeability variation using the Carmen-Kozeny equation (Fitts 2002). The reservoir heterogeneity affects sweep efficiency, and the relative permeability models affect front velocities, however, the structure of analytical solutions usually remains the same. Therefore in this research we use simple relative permeability models (Brooks and Corey 1966) while ignoring capillary dissipation, trapping and hysteresis. The assumptions should be considered carefully for application of our solutions to other problems Volumetric sweep efficiency The sweep efficiency of an EOR process is a complex function of reservoir and injected fluid, petrophysical properties, heterogeneity, geometry of reservoir, well pattern, and injection rates. Achieving high sweep efficiencies can be a challenge for gas floods. The sweep efficiency from gas flooding is lower than water flooding for many reservoirs. The injected gas usually finds a shortcut or channel to production wells instead of uniformly sweeping the reservoir owing to high permeability paths, gravity tonguing or viscous instabilities. This is especially true when 13

33 vertical wells are used because permeability tends to be layered orthogonal to the direction of flow. Well patterns are usually aligned optimally with reservoir heterogeneity to maximize sweep efficiency, but early breakthrough cannot be avoided, just delayed. The impact of high permeability channels on flow is accentuated because gas viscosity is usually much lower than the oil viscosity at reservoir conditions. Therefore gas flows easier toward the production wells, increasing the mobility of fluids in the high permeability layers as flow occurs. Channeling becomes more severe for reservoirs with large permeability variations and greater longitudinal correlation lengths (Araktingi et al. 1993). As formations become more homogeneous and less impacted by gravity, viscous fingers can form also causing early breakthrough (Chang et al. 1994, Fayers and Newley 1998, Christie et al. 1990). Gravity forces can also decrease sweep efficiency by pushing the injected fluid to the top or bottom of the reservoir depending on fluid density contrasts (Rossen and Duijn 2004). Gravity can overcome heterogeneity completely if the flooding process is gravity stable (Perry 1982), greatly simplifying the process. Sweep efficiency for gas injection in vertical wells can be improved somewhat by using a water-alternating-gas injection scheme, which decreases the effective mobility ratio, or by injection of more dense fluids (water) at the top of the reservoir or less dense fluids (gas) at the bottom (Salimi et al. 2012, Sobers et al. 2013). The well pattern and injectivity can affect the sweep efficiency and economics of EOR as well. EOR is economically viable when solvent can be injected at relatively high flow rates producing significant incremental oil. Several factors influence injectivity of fluids into a well including reservoir permeability, relative permeability, injection well location with respect to flow barriers, injection well type, and reactions between the rock and injection fluids (Qiao et al. 2015b, Cinar et al. 2009, Xiao et al. 2011, Spiteri et al. 2005). Horizontal wells have greater contact area with the reservoir compared to vertical wells, generally giving them greater injectivity when the injection rate is distributed evenly along the well (Ganjdanesh et al. 2014, 14

34 Al-Khelaiwi and Davies 2007). Flow into or out of horizontal wells, however, are more affected by unstable viscous fingering owing to less permeability variation longitudinally and flow tends to be larger near the heel of the well where pressure drops are greatest. Estimation of sweep efficiency for field cases usually requires detailed simulation of flow in the reservoir. However, fast estimation of sweep efficiency without detailed simulation are necessary for screening of reservoirs for EOR. Several methodologies have been developed to identify and screen suitable reservoirs for EOR (Bachu et al. 2004, Taber et al. 1997a,b, Advanced Resources International, Inc., 2005). Zhang et al. (2010) estimated CO 2 storage capacity based on the assumption that the volume occupied by oil will become available for CO 2. Wood et al. (2008) developed dimensionless groups for continuous gravity-stable CO 2 flooding of homogeneous reservoirs with high vertical permeabilities and large dip angles. They estimated oil recovery and CO 2 storage for this process using injection and production with vertical wells. Heterogeneities were not included because the gas-oil interface during injection was assumed to be perfectly horizontal as it moves downward (completely gravity stable flow). Zhou et al. (1997) studied the scaling groups for multiphase flow in simple heterogeneous reservoirs. Fluid and reservoir properties that are important for sweep efficiency calculation can be decreased by developing scaling groups. The scaling groups can be used to develop screening techniques. Li et al. (2015a) developed the scaling groups for gravity assisted CO 2 EOR and storage using horizontal wells. The scaling groups are as follows. L H K z K x V DP λ xd Effective aspect ratio Dykstra-Parsons coefficient Correlation length coefficient in x-direction 15

35 λ zd μ o μ g ρ o ρ g ρ w ρ o H(ρ o ρ w )g P p P inj Correlation length coefficient in z-direction Mobility ratio (CO 2-oil) Buoyancy ratio Buoyancy number 1 S hr S wi 1 S hr S wr Normalized initial oil saturation o μ o k rw o μ w k ro S hr S wr Mobility ratio (water-oil) Residual hydrocarbon saturation Residual water saturation The first four scaling groups describe reservoir structure and rock properties. The effective aspect ratio takes into consideration both the length to height ratio and the vertical to horizontal permeability ratio; it is a measure of the rate of fluid communication in the horizontal direction to that in the vertical direction. L and H are the horizontal and vertical distance between injection and production wells. The flow characteristics, like viscous fingering, and channeling, depend primarily on the permeability heterogeneity of the reservoir Local displacement efficiency Core flood and slim tube experiments ideally have sweep efficiency equal to one, therefore based on Eq. (2.1), the recovery factor of these experiments are equal to local displacement efficiencies. In addition, the 1D floods can be used to study displacement mechanisms of EOR. Hyperbolic equations can be used to estimate local displacement efficiency and to estimate the recovery factor of EOR in a real reservoir by applying corrections for sweep efficiency. In addition, the dimension of flow can be reduced by assuming vertical equilibrium (Yortsos 1995) 16

36 or no vertical communication. The effect of viscous instability can be estimated using a correction to fractional flow (Koval 1963). Two phase immiscible displacements can be modeled with the Buckley-Leverett equation. The recoveries can be estimated easily using fractional flow theory for constant injection and initial conditions. The displacements mechanism are more complex for miscible displacements. The following sections discuss the displacements mechanisms in gas flooding for two- and three-phase displacements. Two-phase displacements are discussed with more detail in the next section. 2.2 Displacement mechanisms of gas floods Oil is displaced by injected gas at the pore level by mechanical displacement, development of miscibility, oil swelling, and viscosity reduction (Chung et al. 1988). The mass transfer of components between the gas and oil phases as miscibility is developed is the key for high efficiency of gas floods (Johns et al. 1993, Orr 2007). Miscibility can be achieved by increasing injection pressure above the minimum miscibility pressure (MMP) or enriching the injection gas with intermediate-weight hydrocarbons. Frist contact miscible (FCM) fluids dissolve in each other in any ratio to form a single phase on first contact, however multi-contact miscibility (MCM) develops in the reservoir as equilibrium fluids contact several times. There are three known mechanisms of MCM development. In a condensing gas drive the intermediate components condense from the gas to the liquid. The lightened oil becomes miscible with the injected gas after several contacts. On the other hand in a vaporizing gas drive, the injection gas becomes richer in intermediate components as the injection gas contacts reservoir oil and the rich gas becomes miscible with initial oil. Zick (1986) showed through experimental and simulation results that both vaporizing and condensing mechanisms happen simultaneously for some real gas drives. In other words, in a condensing/vaporizing gas drive, miscibility occurs by transferring 17

37 light intermediates from the oil to the gas and transferring intermediates from the gas to the oil. Johns et al. (1993) developed the analytical theory for condensing/vaporizing gas drives and proved its existence. They showed that MMP occurs when one of the key tie-line lengths becomes zero. Later, Johns et al. (2002a) quantified the condensing/vaporizing mechanism. However, the pressure required for miscibility should not be in excess of safe reservoir operational conditions (i.e., above fracture pressure), and there may be no pressure at which CO 2 miscibility is achieved with some heavy oils. Achieving miscibility is not as critical for gravity stable or gravity assisted floods, however, since there is ample contact time for gas to vaporize and swell the oil, and for film drainage to occur (Perry 1982). Dissipation forces in the reservoir can decrease recovery of a miscible flood by mixing fluids into the two-phase zone (Solano et al. 2001, Johns et al. 2002b, Jessen et al. 2004). Reservoir mixing is caused by molecular diffusion and is enhanced by any mechanism that increases the contact area between the reservoir oil and injected gas (Johns and Garmeh 2010). MMP is defined at zero mixing; therefore, experimental and computational techniques try to eliminate dispersion. These experimental and computational techniques are discussed in the following sections Experimental methods for estimating MMP Slim-tube experiments are widely accepted as the best experimental procedure to determine the MMP for miscible gas floods (Jarrell et al. 2002). Dispersion and mixing always decrease the recovery factor; therefore, MMP is usually determined as the bend in the recovery curve, often called the knee. This definition can be misleading for complex phase behaviors with no MMP, such as displacement of heavy oil with CO 2 at low temperature. For three-phase displacements, the recovery curve can bend gradually or abruptly with pressure or gas enrichment (Bhambri and Mohanty 2008, Okuno et al. 2011, Pedersen et al. 2012). Slim-tube experiments 18

38 are generally reliable because they use real fluids that can capture the complex interactions between flow and phase behavior in porous media such as those that occur in condensing and vaporizing (CV) drives (Zick 1986, Stalkup 1987, Johns et al. 1993). Slim-tube experiments, however, take significant time to conduct, and are expensive. Also the packing material, asphaletene precipitation (Elsharkwy et al. 1996) and high dispersion (Johns et al. 2002) might affect experiment results. Thus, only a few MMPs can be obtained this way in practice. Other experimental methods like the rising-bubble apparatus (Christiansen and Haines 1987) and the vanishing-interfacial tension test (Rao 1997) have been developed to limit slim-tube experiments use for MMP calculations, but these experiments fail to capture the interaction of phase behavior and flow that occurs in porous media (Zhou and Orr 1998, Orr and Jessen 2007). The MMPs from the rising-bubble apparatus and vanishing-interfacial tension test are accurate for simple binary displacements, but become less accurate as the number of components increase (Jessen and Orr 2008). In addition, slim-tube results have been used for fluid characterization. Due to lack of analytical theories for the rising-bubble apparatus and vanishing-interfacial tension test, the results from these tests cannot be used to improve fluid characterization. Therefore these tests should not be used in practice. Zick (1986) showed that single cell multicontact experiments reliably predict MMP for condensing gas drives, yet the results are not acceptable for condensing/vaporizing drives Computational methods for estimating MMP Computational methods are rapid and convenient ways to complement the otherwise slow and expensive experimental procedures. There are currently three computational methods to determine MMP: 1-D simulation of slim-tube displacements, analytical methods by the MOC, and multiple mixing-cell methods. The main limitation in computational methods is that they rely 19

39 on accurate fluid characterizations using an equation of state (EOS). Thus, the MMP from an EOS model should agree with the slim-tube MMP values (Jaubert et al. 2002, Egwuenu et al. 2008). Once a reliable EOS is developed computational methods can be accurate, fast and robust. The slim-tube experiments can be modeled with 1D flow PDE s. The model can be further simplified using assumptions such as incompressible flow, no volume change on mixing, and no effect of pressure on phase behavior (Dindoruk et al. 1992). The simplified model of slimtube experiments can be solved by numerical or analytical methods. MMP correlations (Yuan et al. 2005) are helpful especially for screening of reservoirs or quality checks of experimental data and fluid characterization. Slim-tube simulation Determination of the MMP by 1-D compositional simulation attempts to mimic the flow in porous media that occurs in slim-tube experiments (Yellig and Metcalfe 1980). Cook et al. (1969) simulated oil vaporization during gas cycling using a 20 cell simplified simulator. Later Metcalfe et al. (1973) applied the method to examine miscibility development. Fine-grid compositional simulations, however, can suffer from numerical-dispersion effects causing the MMP to be in error (Stalkup 1987, Johns et al. 2002). Stalkup (1987) plotted recovery vs. 1/ N where N is the number of grid blocks and extrapolated the recoveries to zero dispersion. Later Stalkup (1990) and Stalkup et al. (1990) studied the effect of numerical dispersion on gas flooding recoveries. Johns et al. showed the effect of dispersion on MME at different levels of dispersion and for different displacement mechanisms. Jessen et al. (2004) examined the effect of dispersion on gas flooding composition paths. Use of higher-order methods can reduce, but not eliminate, the effect of dispersion (Mallison et al. 2005). Yan et al. (2012) developed a parallel algorithm for MMP estimation from 1-D simulations, but used only one simulation at each pressure without varying 20

40 the number of grid blocks. 1-D slim-tube simulations are more cumbersome and time consuming than other computational methods because they require numerous inputs including relative permeability. The simulation time for slim-tube simulation is mainly consumed by phase equilibrium calculations. The phase behavior is simplified by ignoring effects of pressure because the pressure drop in slim tubes is very small. Three hydrocarbon phases can also form in gas floods. Slim-tube simulation can be used to examine recovery factors at different pressures (Okuno et al. 2010b). Three-phase compositional simulations typically use approximate relative three-phase permeability models that commonly do not fit the experimental data well (Delshad et al. 1989). Guler et al. (2001) performed three-phase simulations and showed that relative permeability curves affect oil production time but not ultimate recoveries. One task in compositional simulation to model threehydrocarbon phases is to define the threshold phase density to identify and label the phases since the relative permeability models depend on the phase type (Perschke et al. 1989). Phase mislabeling could occur when the unique threshold density fails to label flow correctly, which can cause discontinuities in the simulation results and subsequent failure (Okuno et al. 2010b). A trial and error technique is often applied to identify the best threshold density between the second liquid (L 2) and vapor (V) phases at relatively high pressure, but this does not always fix the problem. Phase labeling is also important for the reliability of simulations since it affects the relative permeability model and capillary pressure. It is well known that relative permeability curves should be a continuous function of composition (Jerauld 1997, Blunt 2000). Yuan and Pope (2012) used Gibbs energy to incorporate the effect of phase compositions on relative permeability and showed that doing so eliminated discontinuities in the two-phase simulation displacements near a critical point. Ruben and Patzek (2004) studied the consistency of relative permeability models and defined conditions to make the relative permeability model more physical. Two-phase approximations of three-phase flow are usually used in commercial 21

41 simulators that do not allow for three-hydrocarbon phases, however, this can cause errors in recoveries (Wang and Strycker, 2000) and can increase simulation discontinuities and instabilities (Lins et al. 2011, Okuno et al. 2010b). Many of these problems can be avoided if the algorithm used in simulation is independent of phase type, but no current simulator has been formulated this way. Mixing cell model Mixing-cell methods estimate the MMP based on repeated contacts between oil and gas. There are a variety of published mixing-cell methods, but many do not correctly predict the MMP for CV drives. Ahmadi and Johns (2011) published a simple but accurate multiple mixing-cell model to estimate the MMP for any drive mechanism. Although the mixing cell approach is similar to single-point upstream finite difference schemes, the flux calculation is trivial. Hence computational time is less than simulation, yet the results of mixing cell are reliable even for complex phase behavior (Ahmadi et al. 2011, Mogensen et al. 2009, PennPVT toolkit manual 2013, Rezaveisi et al. 2015). Nevertheless, it would still be useful to have other computational methods, such as those based on MOC, as another check of the MMP. Ideally, for large gas floods the MMP from slim-tube experiments, 1-D compositional simulation, multiple mixing cell, and MOC should agree before relying on detailed field scale compositional simulation. In the mixing cell approach, all contacts between equilibrium fluids are retained whether they are forward, backwards or contacts in between. Typical MMP calculations take on the order of seconds so that many MMPs can be done for a variety of initial oil and gas compositions. The only caveat, similar to simulation and analytical solution, is that the MMP from a mixing cell is reliable only for good fluid characterizations since it is based on cubic EOS. The results of the 22

42 mixing cell results indicates that MMP is not a function of fractional flow and we used two phase mixing cell results to check MMPs. Fluid characterization can be improved by matching the MMP calculated using mixing cell and experimental MMPs (Egwuenu et al. 2008) prior to performing compositional simulation. Rezaveisi et al. (2014) used the multiple mixing-cell method to determine tie lines for improvement in computational time and robustness of two-phase flash calculations in compositional simulation. They demonstrated that the computational efficiency of several phasebehavior-calculation methods based on the multiple mixing-cell tie lines is comparable to that of other state-of-the-art techniques in an IMPEC-type reservoir simulator (Rezaveisi et al. 2015). Li (2013) and Li et al. (2015b) applied the mixing cell approach to three-phase displacements by combining three phases to two pseudo phases. The new multiple three-phase mixing cell method was used to determine the pressure for miscibility or more importantly the pressure for high displacement efficiency. The procedure that moves fluid from cell to cell is robust because it is independent of phase labeling (i.e. vapor or liquid), has a robust way to provide good initial guesses for three-phase flash calculations, and is also not dependent on threephase relative permeability (fractional flow). These three aspects give the mixing cell approach significant advantages over using compositional simulation to estimate MMP or to understand miscibility development. The approach can be integrated with previously developed two-phase multiple mixing cell models because it uses the tie-line lengths from the boundaries of tie triangles to recognize when the MMP or pressure for high displacement efficiency is obtained. Application of the mixing cell algorithm shows that unlike most two-phase displacements the dispersion-free MMP may not exist for three-phase displacements, but rather a pressure is reached where the dispersion-free displacement efficiency is maximized. This was the first paper to examine a multiple mixing cell model where two- and three-hydrocarbon phases occur and to calculate the MMP and/or pressure required for high displacement efficiency for such systems. 23

43 Analytical solutions Analytical methods for MMP estimation are based on the analytical solution of dispersion-free 1-D flow (Buckley and Leverett 1942, Helfferich and Klein 1970, Helfferich 1981, Pope 1980, Dindoruk 1992, Johns 1992, Dumore et al. 1984, Orr 2007, Lake et al. 2014). Monroe et al. (1990) first examined the analytical theory for quaternary displacements and showed that there exists a third key tie line in the displacement route, which they called the crossover tie line. Orr et al. (1993) and Johns et al. (1993) confirmed the existence of the crossover tie line for CV drives and presented a simple geometric construction to find the key tie lines (gas, oil, and N c 3 crossover tie lines) when successive tie lines were connected by a shock. They demonstrated that the MMP occurs when one of the key tie lines first intersects a critical point (becomes zero length) as pressure is increased. Johns et al. (1993) further showed that the crossover tie line controls the development of miscibility in CV drives, and that the estimated MMP is below the MMP of either a pure condensing or pure vaporizing drive. Johns and Orr (1996) gave a procedure to calculate the MMP for more than four components, and extended their geometric construction to calculate the first multicomponent displacement of 10- component oil by CO 2. Johns (1992) showed that tie line ruled surfaces formed by a wave are almost flat planes. Therefore nontie-line rarefactions can be estimated by a shock. Wang and Orr (1997) applied Johns shock-jump estimation to calculate MMP of real fluids using the Newton-Raphson method. Jessen et al. (1998) improved the formulation and the solution procedure. Current MOC methods for MMP prediction assume that shocks occur from one key tie line to the next along these surfaces and that there are only N c 1 key tie lines (Wang and Orr, 1997). These methods are referred here to as shock-jump MOC. When shocks are assumed from one key tie line to the next, the MMP is determined when one of these intersecting key tie 24

44 lines becomes zero length. The shock only assumption was made because of the general observation that the composition route traverses a series of nearly planar pseudoternary ruled surfaces (Johns 1992). For many displacements examined, this approach resulted in very small calculation errors of the key tie lines and the associated MMP (Wang and Orr 1997, Jessen et al. 1998, Yuan and Johns 2005, Ahmadi and Johns 2011). Jessen et al. (2001) used shocked-jump MOC to develop a fast approach to estimate the nontie-line rarefactions. However, the shockjump approach can be significantly in error and has other limitations associated with it because it only solves for a selected few tie lines in composition space (Ahmadi et al. 2011) and does not check for the velocity condition. Yuan and Johns (2005) decreased the size of the problem by decreasing the number of equations and unknown parameters. They also discussed the robustness of the method and the effect of initial guesses of the unknown parameters on the convergence of the method. They showed the possibility to converge to incorrect tie-lines. Convergence problems limit the application of the MOC solution to displacements with pure gas injection. One approach to correct the MOC limitation is to determine the exact dispersion-free composition route by avoiding the assumption that shocks exist from one key tie line to the next. This more-accurate MOC approach could estimate the MMP by constructing the composition routes for varying pressure, as was done by Johns and Orr (1996) for 10-component oil displaced by CO 2. However the MOC solution for real gas floods with bifurcating phase behavior or a multicomponent injection gas can be more complicated. Thus, in practice a new approach is needed to simplify the construction of the entire composition route. Mogensen et al. (2009) studied the MMP for fluid samples from the Al-shaheen reservoir where oil density changes spatially. The comparison of MMP calculation with different techniques showed that MMP calculated using MOC for oils heavier than 20 o API is around 5000 psia higher than the MMP calculated by other techniques. Furthermore, MMP predictions by different methods agree well for oil samples lighter than 30 o API. Ahmadi et al. (2011) used a 25

45 pseudo ternary system with similar bifurcatoing phase behavior and showed that the assumptions of MOC solution with only shocks are not valid. The complete solution for Ahmadi et al. (2011) is constructed in Chapter Three-hydrocarbon-phase displacements Low-temperature oil displacements by CO 2 involve complex phase behavior, where three hydrocarbon phases can coexist. Reliable design of miscible gas flooding requires knowledge of the minimum miscibility pressure (MMP), which is the pressure required for 100% recovery in the absence of dispersion or as defined by slim-tube experiments as the knee in the recovery curve with pressure where displacement efficiency is greater than around 90%. There are currently no analytical methods to estimate the MMP for multicomponent mixtures exhibiting three hydrocarbon phases. Also, using compositional simulators to estimate MMP is not always reliable. These challenges include robustness issues of three-phase equilibrium calculations, inaccurate three-phase relative permeability models, and phase identification and labeling problems that can cause significant discontinuities and failures in the simulation results. How miscibility is developed, or not developed, for a three-phase displacement is not well known. Slim-tube measurements show that oil displacement by CO 2 involving three-hydrocarbon phases can achieve greater than 90% displacement efficiency at temperatures typically below 120 F (Yellig and Metcalfe 1980, Gardner et al. 1981, Orr et al. 1983). The one-dimensional displacement simulations by Li and Nghiem (1986) showed high oil recovery without a miscible bank. Simulation results of West Texas oil displacement by CO 2 (Khan et al. 1982) gave high displacement efficiency of more than 90% in the presence of immiscible three-hydrocarbon-phase flow. 26

46 Okuno et al. (2011) explained the mechanism for high displacement efficiency as the result of the composition path approaching critical end points (CEPs). Okuno and Xu (2014) examined further the development of multicontact-miscibility in compositional simulation by introducing new distance parameters based on interphase mass transfer near CEPs. CEPs are states where two of the three coexisting phases merge to a critical point and become identical. There are generally two types of CEPs, the first CEP is where the liquid (L 1 and L 2) phases merge in the presence of the vapor (V) phase, and the other is where the second-liquid (L 2) and V phases merge in the presence of the L 1 phase. A CEP is not a point as it would seem, but is rather a tie line in composition space where the three-phase region (tie triangle) becomes a two-phase tie line as one phase vanishes. A tri-critical point is where three phases simultaneously become identical (Widom 1973, Griffiths 1974). This is a true critical point. Oil displacements by CO 2 involving L 1-L 2-V equilibrium can achieve greater than 95% displacement efficiency even if the L 1 and V phases by themselves would be significantly immiscible. As explained by Okuno et al. (2011) high displacement efficiency is possible because the L 2 phase serves as a buffer between the L 1 and V phases. 2.3 Displacement mechanism of low salinity polymer flooding Polymer flooding can significantly improve sweep efficiency and therefore enhance oil recovery (EOR) (Sheng 2010, Sheng et al., 2015). A combination of polymer flooding with other EOR methods such as gas, alkaline and surfactant flooding has demonstrated synergistic effects that can lead to improved oil recovery (Li et al. 2014, Luo et al. 2015, Sheng 2014a). The efficiency of polymer flooding greatly depends on the salinity of the aqueous phase contacted (Sorbie 2013, Vermolen et al. 2011) because high concentrations of monovalent and divalent ions reduce the polymer viscosity, and thus decrease the sweep efficiency. In practice, a reservoir is 27

47 pre-flushed using low salinity water before polymer flooding to avoid the mixing between the high salinity formation water and the polymer slug. Recently, low salinity water flooding (LSW) is reported to improve oil recovery. In coreflooding experiments, the chemical composition of the injection water is found to have a significant effect on oil recovery. LSW can improve displacement efficiency by changing the wettability of a sandstone reservoir from oil wet to more water wet (Morrow and Buckley 2011). Different mechanisms are proposed including mineral dissolution, fine migration, surface potential change and multi-component ionic exchange (MIE), among which the MIE mechanism (Lager et al. 2007) is the most supported with experimental data and theoretical analysis (Sheng 2014b, Myint and Firoozabadi 2015). In this mechanism, cation exchange between Na +, Ca 2+ and Mg 2+ is considered and how much Na + is adsorbed by the clay surface determines the wettability (Lager et al. 2007). As low salinity brine is injected, Na + is released from the surface and this process alters the surface affinity towards more water wet. The wettability alteration leads to improved oil recovery, as measured in many coreflooding experiments (Austad et al. 2012). Velderr et al. (2010) explained the connate water banking for field scale low salinity floods as evidence for wettability alteration. The divalent cation concentrations usually reach levels below the injection fluid, which indicate the presence of cation exchange reactions. Oil recovery steadily increases in low salinity floods in sandstones even after many pore volumes of low salinity water injection (RezaeiDoust et al. 2011, Shaker Shiran and Skauge 2013, Kozaki 2012), which indicates a slow moving wettability alteration front. Since low salinity water is used to pre-flush the reservoir for polymer, these two processes can work together where wettability alteration and a viscosity increase can improve both sweep efficiency and microscopic displacement efficiency. Mohammadi and Jerauld (2012) developed a mechanistic model for low salinity polymer flooding with relative permeabilities as a function of water salinity. Experimental studies of the combined LSW and polymer flooding lead 28

48 to very high oil recovery (Shaker Shiran and Skauge 2013). However, to the best of our knowledge, there is not a conclusive mechanism that explains how low salinity and polymer interact with each other and what are the mechanisms that leads to such a high recovery. Significant advances have been made in recent years to predict wettability alteration and oil recovery (Jerauld et al. 2008, Dang et al. 2013, Qiao et al. 2015a, Qiao et al. 2016). Jerauld et al. (2008) proposed a fully compositional model that included the transport of salts in the aqueous phase as an additional single-lumped component. They determined the relationship between the relative permeability and residual oil saturation, but from linear interpolation of the wetting state based on total salinity without tracking individual species. Korrani et al. (2014) coupled the UTCOMP reservoir simulator (Chang 1990) and PHREEQC (Parkhurst and Appelo 1999) to model geochemical reactions. Dang et al. (2013) developed a fully coupled geochemical and compositional flow model for low salinity waterflooding in sandstones where cation exchange is believed to be the mechanism for improved oil recovery. However, there is no discussion on how different species controls the process. Qiao et al. (2015a) developed a reservoir simulator for low salinity waterflooding in carbonates by considering geochemical reactions and a mechanistic model for wettability alteration. There is currently a lack of a detailed representation of the surface-geochemical reactions and the corresponding wettability alterations in multiphase-flow models, and there is no simulation study that has considered both wettability alteration caused by cation exchange reaction in low salinity waterflooding in sandstones and the increased viscosity of polymer. Seccombe et al. (2008) coreflood experiments show a bank of low salinity water and no recovery for small slugs of low salinity. They explained that the low recovery was due to dispersion of the small slugs, but no consideration was given to the potential of interacting shocks since a mathematical model of this process is lacking. 29

49 Analytical solutions for cation exchange reactions have been developed for single phase transport. Helfferich and Klein (1970) applied coherence theory to chromatographic separation. Pope et al. (1978) constructed solutions for monovalent-divalent exchange. Appelo et al. (1993) calculated intermediate concentrations for ion exchange transport by assuming that variations consist of shocks only. Venkatraman et al. (2014) has developed a Riemann solver for single phase transport with cation exchange reactions. The analytical solutions for complex LSP processes cannot be easily constructed using conventional MOC. The splitting of the physical equations has shown great performance to simplify the problem. We use splitting in Chapter 4 and 5 to construct the solution for gas floods. De Paula and Pires (2015) and Borazjani et al. (2016) used the splitting approach to develop a front tracking algorithm for polymer injection with variable salinity. They did not consider, however, ion adsorption or wettability alteration. The analytical solution for low salinity polymer flooding is presented in Chapter 7 using the splitting technique. 2.4 Hyperbolic system of equations First order partial differential equations arise in modeling of convective transport by neglecting dispersive effects of heat conduction, diffusion or viscosity. The physical processes are usually coupled and the first order differential equation can only be obtained by idealization or by taking the limiting case (Rhee et al. 2001a). The Euler equation in gas dynamics is a particular important example of hyperbolic equations (Whitham 1999) and many of the theories of hyperbolic equations are developed by gas dynamic scientists. Other applications of hyperbolic equations are in traffic flow, shallow water, chromatographic separation, and enhanced oil recovery models. Hyperbolic equations can be used to examine enhanced oil recovery techniques with dominant convection of one or more phases and negligible effect of dispersive mixing on 30

50 flow. Purely convective flow models for EOR are usually developed assuming negligible compressibility, dispersion, diffusion, and capillary dissipation (Orr 2007). These assumptions are reasonable for 1D displacements and slim-tube experiments. Hyperbolic systems of equations are used in modeling of EOR techniques such as polymer flooding, gas flooding, and surfactant flooding. In addition, 2D models of SWAG at steady state conditions can be modeled with hyperbolic equations (Rossen and Duijn 2004). The following section discuss the procedure to construct the solutions for hyperbolic system of equations for EOR problems. The comprehensive description of MOC can be found in Rhee et al. (2001a, b), Bressan (2013), LeVeque (1992), and Holden and Risebro (2013). Riemann problem A Riemann problem is the conservation law (Eq 2.2) together with piecewise constant initial data with a single discontinuity. C td + F c (C)C XD = 0, (2.2) where C is the volume fraction of components and F is the flux of components. Equations (2.2) are strictly hyperbolic when F C has distinct and real eigenvalues. A Riemann solver constructs the solution for a Riemann problem. For EOR problems a discontinuity occurs at the injection well at time zero and the fronts always move to a production well. The negative portion of the x- axis is always ignored in modeling of EOR processes because the wave velocities are always positive. In petroleum literature, the initial data are referred to as injection and initial condition. Figure 2-2 represents the Buckley-Leverett problem and the solution in x-t space. However, the 31

51 gas flooding solutions in Lagrangian coordinates, as discussed in later sections, can have fronts with negative velocities. The Cauchy problem is the same as a Riemann problem but initial data can be variable. EOR processes can have variable injection condition (i.e. slug injection) which is compatible with the definition of a Cauchy problem, however we can define the initial data along the x and t axes. Wave interaction and front tracking The construction of the Riemann solver is closely related to that of a scalar conservation law with discontinuous coefficients. Additional difficulties arise from the lack of strict hyperbolicity. The ultimate goal of solving Riemann problems is to find a complete and automatic Riemann solver for a specific type of problem, such that the global Riemann solver can construct the solution for any initial and injection conditions. For gas floods, this means finding a Riemann solver for all possible phase behavior, initial and injection compositions, and number of components. Besides being able to solve for MMP automatically, the Riemann solver could be used in front tracking methods (Holden and Risbero 2013) to solve for complex water-alternating-gas (WAG) displacements or other displacements with nonuniform initial and/or injection conditions. Such a solver and front tracking scheme was developed by Issacson (1989) for polymer floods. Later, Johansen and Winther (1989) included multicomponent adsorption in their polymer model. Juanes and Lie (2008) developed a Riemann solver and front tracking method for first-contact miscible water alternating gas floods, while Johns (1992) developed a front tracking algorithm for twocomponent partially miscible gas floods, where components can transfer between phases. 32

52 2.4.2 Method of characteristics MOC is commonly used to construct the solution for hyperbolic equations by converting first order PDE s to a family of ODE s. The ODE s define characteristics curves in x t space and the value of independent variables along the characteristics. The characteristic velocities are determined by eigenvalues of F C in Eq. (2.2) and the eigenvectors of F C indicate the feasible changes of C for the smooth solution. The solution can be constructed by integrating the characteristic curves starting from initial data. The solution is completed by covering the physical domain of the x t plane with characteristics such that only one characteristic passes through each point and composition varies along eigenvectors as well. This procedure may lead to multivalued solutions. To prevent multivalued solutions, the slope of characteristics should increase from the injection condition to the initial condition. Sometimes shocks should be used in the solution to remove multivalued solutions. However, there are usually multiple possible shocks from one composition to the next. The correct physical shock can be determined based on the entropy condition. The MOC constraints are discussed in the following. Velocity constraint Velocity of the characteristics should increase from the left to the right in the solution of a Riemann problem. This condition ensures that the solution is single valued at every x t. 33

53 Entropy condition The system of hyperbolic equations can have discontinuous solutions. The discontinuous solution should satisfy the integral form of Eqs. (2.2) which results in the Rankine-Hugoniot jump conditions, F i U F i D C i U C i D = Λ i = 1, 2, (2.3) where Λ is the dimensionless shock velocity. These constraints are referred to as entropy (or admissible) conditions, and the corresponding shocks as admissible shocks. The entropy condition in gas dynamics is equivalent to the solution of the second law of thermodynamic such that entropy of the gas should increase as it passes through a shock. However, entropy and entropy flux cannot be defined for many conservation laws. Therefore mathematical entropy conditions are developed to ensure uniqueness of the solution. Entropy conditions ensure that a shock continues to propagate in the presence of dispersion, i.e. it is self sharpening. Well-known conditions include the Kruzhkov condition (Kruzhkov 1979) for scalar conservation laws, Lax condition (Lax 1957) for genuinely nonlinear systems, Oleinik (1957) for strictly hyperbolic equations, Liu condition (Liu 1976) which also allows certain local linear degeneracies, and the vanishing viscosity approach by Bianchini and Bressan (2005), and Bressan (2000) for scalar equations and for strictly hyperbolic systems. These conditions are equivalent for the same system where ever the conditions are applicable. For non-hyperbolic systems such as the gas flooding equations, there has not been a unified entropy condition. A generalized Lax entropy condition was proposed by Keyfitz and Kranzer (1980) for a model of elasticity. In connection with scalar conservation laws with discontinuous flux function, Gimse and Risebro (1991 and 1992) introduced the shortest-path 34

54 criterion, and proved its equivalence to the vanishing viscosity limit. We remark that these two entropy conditions are different for certain cases of Riemann problems, and would give very different entropy weak solutions. Lax (1957) stated that a shock of the i-th family for a strictly convex or concave flux function should satisfy the following conditions to be admissible: λ i U > Λ > λ i D i = 1, 2. (2.4) Based on the Lax entropy conditions, a necessary condition for shock admissibility is: λ 2 U > Λ > λ 1 D, (2.5) where λ 2 is always greater than λ 1, and λ 2 is either the tie line or nontie-line eigenvalue. If Eq. (2.5) is not satisfied the shock does not satisfy the Lax entropy conditions and the shock is not admissible. The Liu condition (Liu 1976) is another way to identify if a shock is admissible. This condition states that if a shock is divided into two shocks, any hypothetical upstream shock should be faster than the original shock. The mathematical statement of the Liu condition is, F D i F i C D i C F i U D F i i C U D i C i = 1,, N c 1, i (2.6) in which C i is a point on the shock locus between fixed upstream and downstream compositions. The index i is arbitrary because C i is on the shock locus. Another test for shock admissibility is the vanishing viscosity approach (Bianchini and Bressan 2005). With dispersion Eq. (3.10) becomes, 35

55 ε C i + F i = ε 2 ε C i t D x 2 D x i = 1,, N c 1, D (2.7) where is related to the inverse of the Peclet number and the amount of dispersion is assumed to be the same for each component. The analytical solution with and without a small amount of dispersion should not change significantly. That is, the solution of the above set of equations should converge to the dispersion free solution as the value of ε approaches zero: N c 1 1 lim C i C ε i 2 dx D = 0. ε 0 i=1 0 (2.8) Lantz (1971) showed that numerical dispersion for an explicit solution of two-phase flow is inversely proportional to the number of simulation blocks. Therefore, the concept of vanishing viscosity implies that the numerical simulation results should converge to the dispersion free analytical solution by increasing the number of grid blocks. We use the vanishing viscosity condition to verify the complete composition route from the oil to gas compositions. Appendix C also uses this approach to show that the shock route or path in composition space is not necessarily a straight line for shocks within the two-phase region. As mentioned earlier in this section, the system of equations for a gas flooding problem is reducible. Therefore, the construction of the solution route reduces to connecting the injection composition to the initial composition by a composition route that satisfies the following conditions. 1- Follow eigenvectors (paths) that solve the strong form. 2- Take shocks when necessary a. Shock jump condition for mass balance. b. Enter or exit two-phase zone along a tie-line extension (Larson 1979). 36

56 c. Check entropy condition. 3- Composition velocities should increase from injection composition to initial composition Finite difference estimation of solution The explicit upwind scheme is used to solve Eq. (2.2) in this research (Johns 1992) for gas floods. PennSim (Qiao 2015, PennSim 2013) is used for numerical simulation of low salinity polymer floods. The following equation represents the simulator mathematical formulation. C n+1 i,k = C n i,k Λ(F n i,k n F i,k 1 ), (2.9) where Λ = Δt D /Δx D. The size of the grid blocks and value of Λ affects the amount of numerical dispersion. The induced numerical dispersion can be estimated using Taylor series expansion of Eq. (2.2). The resulting equations can be simplified by assuming constant eigenvectors and eigenvalues for the range of variations in one time step. That is, D = λ 2 Δx D(1 λλ), (2.10) where λ is an estimate of the rarefaction or shock velocity. The explicit scheme is unstable for negative values of D. The limit for positive values of D, Eq. (2.10), is used to define the maximum time-step size. The smaller time-step size improves the stability of the numerical scheme but the computational time and numerical dispersion increases for smaller time steps. Δt Δx λ. (2.11) 37

57 A different number of grid blocks are used to examine the effect of numerical dispersion. The values of Λ are typically in the range of 0.2 to 0.7. The shock and rarefaction velocities converge to 1.0 close to the MMP, therefore larger Λ can be used near the MMP Fractional flow theory The analytical solution for a multicomponent, multiphase flow is essential to examine EOR mechanisms and benchmark simulators. Riemann problems for this type of non-strictly hyperbolic systems arising in simulation of multiphase flow in porous media have been studied by many authors. The complexity of the analytical solutions for transport in porous media, depends on the number of mobile phases and intensity of mass transfer between phases. The solution for these systems can be very complex or not developed yet. Buckley and Leverett (1942) first developed the scalar conservation law for water flooding, which is a two-phase flow problem without mass transfer between phases. Later, Helfferich (1980), Hirasaki (1981) and Pope (1980) extended the models to more complicated processes such as polymer and gas flooding. Helfferich (1981) identified paths for connecting waves of different families for such complex systems, allowing an elegant but heuristic construction for solutions of Riemann problem. The Hellfferich approach has been applied to different EOR problems, such as surfactant flooding (Hirasaki 1981), three-component gas flooding (Dumore et al. 1984) and quaternary displacements (Monroe et al. 1990). However, an exact global Riemann solver is more complicated than what Helfferich (1981) predicted. The two-component displacements can be modeled with scalar conservation laws and Johns (1992) developed a front tracking algorithm for two component partially miscible two-phase flow. The global Riemann solver for threecomponent miscible fluids is complicated. Many researchers have solved the Riemann problems 38

58 for three-component displacements specific boundary conditions (Johns 1992, LaForce and Johns 2005, Seto and Orr 2008). The solution structure varies for different boundary conditions and fluid phase behavior. For example, the complexity of composition routes increases significantly for three-phase displacements (LaForce and Johns 2005). Gas flooding displacements are usually modeled with more than three components (Egwuenu et al. 2008) and the solutions of Riemann problems of such systems are very complicated (Johns and Orr 1996, Orr et al. 1993). The solution can be constructed as several consecutive three-component systems (Johns and Orr 1996), however the solution is still complex. The other approach to simplify the solutions is to use the decoupled nature of thermodynamics in the gas flooding problem such that the solution can be constructed by calculating intersecting tie lines (Johns and Orr 1996, Wang and Orr 1997). However, the assumptions of such solutions are invalid for some fluids (Ahmadi et al. 2011) and solutions of intersecting tie lines can be non-unique (Yuan and Johns 2005). A broad understanding of ternary displacements is the key to finding a Riemann solver for gas floods. This is because ternary systems are the building blocks for multicomponent displacements. That is, composition routes follow a series of successive pseudoternary ruled surfaces (Johns and Orr 1996). Analytical solutions for ternary displacements of various types have been studied by different researchers (Dindoruk 1992, Johns 1992, LaForce and Johns 2005, Seto and Orr 2009), but these solutions are complex to construct and have not led to a global Riemann solver. A simpler approach is needed for these more complex displacements. EOR processes usually are modeled with Riemann problems using constant boundary conditions. However, the global Riemann solvers and front tracking algorithms have significant applications to developing more efficient reservoir simulators and simulation of slug injection EOR processes. Front tracking algorithms are developed for some EOR problems. For the 39

59 polymer flooding models, Johansen, Tveito and Winther (1988, 1989a, 1989b) constructed global Riemann solvers for an adsorption model with various assumptions, and conducted numerical simulations with front tracking. Isaacson and Temple (1986) studied the Riemann problem of a non-adsorptive polymer flooding model, and constructed approximate solutions using Glimm s Random Choice. Using the generalized Langmuir isotherm for the adsorption functions in multicomponent chromatography, Riemann solutions were constructed by Rhee, Aris and Amundson (1970), taking advantage of the fact that the system is Temple class, i.e., with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants (Temple 1983). Dahl, Johansen, Tveito and Winther (1992) constructed Riemann solutions for a model of multicomponent displacements for two-phase flow without mass transfer between phases. Juanes and Lie (2008) applied the Riemann solver of Isaacson and Temple (1986) to three-component water alternating gas floods without mass transfer between phases. Chapter 4 presents a global Riemann solver for three-component systems by extending the splitting approach developed in (Entov and Zazovsky 1997, Pires et al. 2006). The splitting of hydrodynamics from tie lines greatly simplifies the solution to gas flood problems Limitations of current MOC solutions for gas flooding Current MOC methods for MMP prediction assume that shocks occur from one key tie line to the next along these surfaces and that there are only N c 1 key tie lines (Wang and Orr, 1997). We refer to this as the shock-jump MOC method for MMP prediction in this dissertation. When shocks are assumed from one key tie line to the next, the MMP is determined when one of these intersecting key tie lines becomes zero length. The shock only assumption was made because of the general observation that the composition route traverses a series of nearly planar pseudoternary ruled surfaces (Johns 1992). For many displacements examined, this approach 40

60 resulted in very small calculation errors of the key tie lines and the associated MMPs (Wang and Orr 1997, Jessen et al. 1998, Yuan and Johns 2005, Ahmadi and Johns 2011). Jessen et al. (2001) used shock-jump MOC to develop a fast approach to estimate the nontie-line rarefactions. However, the shock-jump MOC approach can be significantly in error and has other limitations associated with it because it only solves for a selected few tie lines in composition space (Ahmadi et al. 2011, Khorsandi et al. 2014). Yuan and Johns (2005) showed that there are multiple sets of intersecting tie lines that could satisfy the shock-jump MOC approach. Further, two-phase regions can bifurcate into separate two-phase regions (Orr and Jensen 1984) so that multiple critical points can exist between the intersecting key tie lines. Mogensen et al. (2009) compared the MMPs predicted by various computational methods for the Al-Shaheen oil displaced by CO 2 and noted a significant difference of thousands of psi between the MMP predicted by the shock-only MOC and other MMP methods for the heavier reservoir fluids. Ahmadi et al. (2011) explained these differences using a simple pseudoternary diagram and their mixing cell method. They showed that the two-phase region splits into two separate two-phase regions (L 1-L 2 and L 1-V regions), and that this bifurcation causes the shock jump MOC method to fail because the key tie lines no longer control miscibility. Ahmadi et al. (2011) gave an approximate fix for bifurcating phase behavior by checking the length of the tie lines between each of the key tie lines to identify if a critical point is present or is forming between them. One approach to correct the MOC limitation is to determine the exact dispersion-free composition route by avoiding the assumption that shocks exist from one key tie line to the next. This more-accurate MOC approach could estimate the MMP by constructing the composition routes for varying pressure, as was done by Johns and Orr (1996) for 10-component oil displaced by CO 2. However the MOC solution for real gas floods with bifurcating phase behavior or a multicomponent injection gas can be more complicated. Thus, in practice a new approach is needed to simplify the construction of the entire composition route. 41

61 Splitting The gas flooding Riemann problem can be simplified significantly by splitting phase behavior from flow. The idea that fractional flow has no effect on a tie-line route and MMP has been developed over the years by many authors, although initially the view was the opposite. Metcalfe et al. (1973), for example, believed that the MMP is dependent on fractional flow. Their conclusions were based, however, on coarse slim-tube simulations that were significantly impacted by a large level of dispersion. Such a high level of dispersion does not exist in slim-tube experiments. In theory, the MMP should be determined with no dispersion present (Johns et al. 2002b). Stalkup (1987) showed using 1-D simulations corrected for dispersion that the MMP is not affected by relative permeability for condensing/vaporizing drives. Jaubert et al. (1998) stated that rock type did not impact experimental MMP measurements for fifty example fluids. Zhao et al. (2006) compared simulation results to their mixing-cell algorithm, and concluded that MMP is not affected by fractional flow. Johns (1992) and Dindoruk (1992) provided evidence that MMPs are likely independent of fractional flow. Johns (1992) demonstrated that tie lines connected by a shock must intersect at a composition outside of the two-phase region, and in many cases outside of positive composition space. Dindoruk (1992) showed that the nontie-line eigenvectors are tangent to the ruled surfaces formed from the intersection of the tie-line extensions at their envelope curves. Bedrikovetsky and Chumak (1992) proposed an auxiliary system of gas flood equations similar to the one that Issacson (1989) derived for polymer flooding. They described the tie-line route for a fourcomponent displacement with constant K-values using their auxiliary system of equations. Entov (2000) further suggested that potential coordinate transformations could be used such that N c 2 of the eigenvalues would become independent of fractional flow, where N c is the number of components. Pires et al. (2006) expanded on this idea and developed Lagrangian coordinates to 42

62 split the equations into two parts; a set of equations dependent only on phase behavior, and one additional equation based on fractional flow. Dutra et al. (2009) used the splitting approach to construct a tie-line route for a four-component displacement, but incorrectly showed an elliptic region in tie-line space. In this research we apply a splitting technique to multicomponent gas displacements to separate the tie-line solution from fractional flow. Our approach does not suffer from the singularities present in Pires et al. (2006) and Dutra et al. (2009). The solution in tie-line space is constructed for a variety of fluid models including pseudoternary displacements with bifurcating phase behavior, and four- and five-component displacements in Chapter 6. The approach developed offers the potential for finding a complete Riemann solver for any initial and injection condition. Finally the MMP is calculated for several fluids using the analytical solution based solely on solving the tie-line problem, where tie-line rarefactions and shocks can exist in tie-line space. Thus, we eliminate the need for the shock jump assumption in determining the MMP. A similar technique is used to solve the Euler equation in gas dynamics where one of the eignevectors has only one non-zero element. In addition, we use the splitting technique to develop the first analytical solutions for the complex coupled process of low salinity-polymer (LSP) slug injection in sandstones that identifies the key parameters that impact oil recovery for LSP, and also improves our understanding of the synergistic process, where cation exchange reactions change the surface wettability. 2.5 Summary The simplified hyperbolic system of equations provides a means to analyze and understand the displacement mechanism of enhanced oil recovery techniques. The solutions for 43

63 multiphase and multiphysics displacements can be very difficult to construct. In addition, new complex EOR techniques emerge every day and more complex mathematical tools are required to analyze these EOR techniques and benchmark numerical simulations. We develop the analytical solutions for complex gas flooding and low salinity polymer floods by splitting the problems into multiple simpler problems. Figure 2-1: Scanning electron microscope (SEM) image of Berea sandstone core (Schembre and Kovscek 2005). Figure 2-2: (Top) Initial data for Buckley-Leverett problem. (Bottom) Characteristics for Buckley-Leverett solution. The characteristic line for the shock is shown in red line. 44

64 Chapter 3 Gas flooding mathematical model In this chapter the multicomponent multiphase flow equations for gas flooding are described. First, we simplify the gas flood model in the form of the standard Riemann problem. The assumptions are discussed and validated. Next, the method of characteristics and the basics of developing an analytical solution are explained. Finally, MOC solutions for ternary displacements with bifurcating phase behavior are illustrated. 3.1 Conservation law Eqs. (3.1) describes multicomponent multiphase flow in one-dimensional (1-D) porous media neglecting the effect of dispersion, diffusion and capillary pressure (Lake et al. 2014). n p t x ijρ j S j + x [v φ x ijρ j f j ] = 0 i = 1,, N c. j=1 n p j=1 (3.1) where x ij is the mole fraction of component i in phase j, ρ j is phase density and S j is phase saturation, φ is the porosity, and v is the total velocity. The fractional flow of phase j is defined by Eq. (3.2). k rj f j = μ j n p j=1 k rj μ j. (3.2) 45

65 The fractional flow curve is a function of the relative permeabilities and viscosities. Fractional flow affects the front velocities but usually have no effect on the structure of the solution for two-phase displacements. Eq. (3.1) can be simplified further using the assumption of no volume change on mixing and incompressible fluids (Helfferich, 1981). Dindoruk (1992) solved the equations with volume change on mixing, where the solution route traversed the same tie lines as those with ideal mixing. The concentration of component i in phase j is defined by Eq. (3.3). x ij c ij = ρ ci n c x kj k=1 ρ ck, (3.3) in which, ρ ck is the constant molar density of the component i. Similarly the phase density can be calculated by the following relationship. N 1 c ρ j = ( x ij ) ρ ci i=1. (3.4) The following equation can be used to replace mole fractions by concentrations in Eq. (3.1) under the assumption of ideal mixing, ρ ci c ij = ρ j x ij. (3.5) Overall concentration and fractional flow of component i are calculated by Eqs. (3.6) and (3.7). 46

66 n p C i = x ij S j, j=1 (3.6) n p F i = x ij f j. j=1 (3.7) The dimensionless form of the equations is more convenient for both simulation and analytical solution. The dimensionless time and location are defined by Eqs. (3.8) and (3.9). t D = vdt φl = PVI, (3.8) x D = x L. (3.9) Pore volume injected (PVI) is usually used in EOR methods as the dimensionless time scale. Eq. (3.10) shows the dimensionless form of the flow equations. C i t D + F i x D = 0 i = 1,, N c 1. (3.10) The flow equations can be shown in matrix form as Eq. (3.11). C td + F c (C)C XD = 0. (3.11) The boundary conditions complete the problem definition. We assume that the reservoir contains only oil at the start of the displacement and the injection composition is constant for the entire displacement. There is no mobile water, so the analytical solution only considers the pore volume of hydrocarbons. The boundary conditions of the Riemann problem are shown as Eq. (3.12). 47

67 C = C inj at x = 0, C = C ini at t = 0. (3.12) The fractional flow has no effect on MMP as discussed in Chapter 6. Therefore, for MMP calculation, we assumed that the all components have the same density. That is c ij is equal to x ij. Although this assumption has no effect on the tie-line route, the assumption should not be used for calculation of the recoveries and front velocities. Figure 3-1 shows the comparison of simulation result of UTCOMP (Chang, 1990) and our explicit single point upstream (EXSPU) scheme for a quaternary displacement with fluid properties as shown in Table 3.1 (Orr et al. 1993). UTCOMP is a comprehensive compositional reservoir simulator while EXSPU is the simulator developed in this research that solves Eq. (3.11) with all MOC assumptions satisfied except that it contains numerical dispersion. Although the front velocities and compositions simulated by the two simulator does not match closely on Figure 3-1, the simulation results match exactly in tie-line space (Figure 3-2). 3.2 Tie lines The phase behavior significantly impacts the analytical solutions for gas flooding. The best way to consider phase behavior is to incorporate it into the flow equations using tie-line definitions. A tie line is a line in composition space that connects the compositions of two phases in equilibrium. Any composition on that line will split into the same two phases. Whitson and Michelsen (1989) introduced the concept of negative flash by extrapolating the tie lines in the single-phase region. The compositions on the extension of a tie line are physically single phase but they can be split mathematically into two phases, one with negative mole fraction and one with mole fraction greater than one. 48

68 There are important features of analytic solutions in terms of tie lines. (1) One of the eigenvectors is always in the direction of a tie line. (2) Miscibility occurs when one of the tie-line lengths of the solution path goes to zero. (3) Tie lines form ruled surfaces in composition space. The nontie-line eigenvectors are tangent to these surfaces. (4) The composition path enters and exits two-phase regions along a tie-line extension (Larson 1979). Chapter 5 demonstrates that the ruled surface route can be determined solely based on phase behavior and independent of fractional flow. A tie-line can be defined with Eq. (3.13). C i = α i 1 (Γ)C 1 + β i 1 (Γ), i = 2,, N c 1. (3.13) where Γ is a N c 2 vector that parameterizes the tie-line space. Johns (1992) used c 11 as the tieline parameter. The extension of compositions space parametrization to more complex fluids is discussed in Chapter 5. The tie-line equation is shown in matrix form by Eq. (3.14). C = ΑC 1 + B. (3.14) Tie-line length (Eq. (3.15)) is defined as the compositional distance between equilibrium compositions in the composition space. Tie-line length is zero at a critical point, while longer tie lines represent more immiscibility. N c TL = (x i1 x i2 ) 2. i=1 (3.15) concentrations. Overall fractional flow of components along a tie line has the same linear relationship as 49

69 F i = α i 1 (Γ)F 1 + β i 1 (Γ) i = 2,, N C 1. (3.16) Substitution of Eqs. (3.13) and (3.16) into Eq. (3.10) for i greater than one will result in following set of equations. C 1 t D + F 1 x D = 0, (3.17) C 1 α i t D + β i t D + F 1 α i x D + β i x D = 0 i = 2,, N C 1. (3.18) A broad understanding of ternary displacements is the key in finding a Riemann solver for gas floods. This is because ternary systems are the building blocks for multicomponent displacements. That is, composition routes follow a series of successive pseudoternary ruled surfaces (Johns and Orr 1996). Analytical solutions for ternary displacements of various types have been studied by different researchers (Dindoruk 1992, Johns 1992, LaForce and Johns 2005, Seto and Orr 2009), but these solutions are complex to construct and have not led to a global Riemann solver. The rest of this chapter and Chapter 4 discusses the analytical solutions for ternary systems. Equations (3.17) and (3.18) can be rewritten for N C = 3 as, C 1 t D c 11 ( t D ) F 1 C 1 F 1 β C 1 x D + = 0, F 1 + h β 0 ( C 1 + h) ( x D ) (3.19) where parameter h is defined as, 50

70 h = dβ dα. (3.20) The indexes for α and β are dropped for simplicity. The physical interpretation of h is the intersection point of two adjacent tie lines defined by Eq. (3.21). C i 1 can be calculated using the tie-line equation. The intersection points form the envelope curve where tie lines are tangent to the envelope curve. C 1 e = h. (3.21) 3.3 MOC solution for gas flooding Equation (3.19) can be solved by MOC, which converts the set of PDE s into an equivalent set of ODE s. The set of ODE s can be categorized into two groups. The first group describes the characteristic curves, while the changes of composition along the characteristics are described by another set of ODE s. The characteristic curves can be calculated for constant compositions as follows. dc i = C i t D dt D + C i x D dx D = 0 i = 1,, N C 1. (3.22) The following equation will result by rearranging Eq. (3.22), C i t D = λ C i x D i = 1,, N C 1, (3.23) in which λ = dx D /dt D defines the characteristic lines in x D t D space. Substitution of Eq. (3.23) converts the flow Eqs. (3.19) to a set of ODE s. 51

71 ( F 1 C 1 λ 0 F 1 C 1 β x D = 0. F 1 + h C 1 + h λ β ) ( x D ) (3.24) Equations (3.24) have non-trivial solutions only for λ equal to eigenvalues of the coefficients matrix of Eqs. (3.24). The eigenvalues are as λ t = F 1 C 1 and λ nt = F 1 + h C 1 + h. (3.25) λ t is the eigenvalue corresponding to the eigenvector along the tie line therefore is called the tie-line eigenvalue and λ nt is the nontie-line eigenvalue. Gas flooding equations are not strictly hyperbolic and the order of eigenvalues changes. Therefore the common indexing of eigenvalues and eigenvectors as used for strictly hyperbolic equations is not used in petroleum engineering literature. A geometric representation of the eigenvalues aids in determining the composition route. The tie-line eigenvalue is equal to the slope of the overall fractional flow plot as a function of overall composition. The nontie-line eigenvalue at a composition on a tie line is equal to the velocity of the line from the envelope composition (-h) for that tie line to the particular two-phase composition. Figure 3-3 shows such a geometric construction for the nontieline eigenvalue. When the line segment is tangent to the overall fractional flow curve, the tie-line and nontie-line eigenvalues must be equal. This occurs for two compositions on the curve, which are known as the umbilic points. The geometric construction also shows that the tie-line eigenvalue is greater than the nontie-line eigenvalue for compositions between the umbilic points. The reverse is true outside the umbilic points. Dindoruk (1992) showed similar results for multicomponent displacements as well. 52

72 Eq. (3.24). The corresponding solutions of Eq. (3.24) are eigenvectors of the coefficient matrix of 1 1 e t = ( ), e nt = λ nt λ t. 0 F 1 ( β ) (3.26) Integration of eigenvectors will result in tie-line and non-tieline paths. The eigenvalues and eigenvectors of Eq. (3.24) are independent of x D and t D. Such systems of equations are called reducible (Rhee et al. 2001b). Therefore the solution for a gas-injection displacement consists of a sequence of compositions that connects injection gas to initial oil and satisfies the material balance, velocity, entropy, and continuity conditions. The eigenvectors of Eq. (3.24) are always real but the system of equations is not strictly hyperbolic because two of eigenvalues are equal at umbilic points. The unique solution for a gas injection displacement is complete with a sequence of compositions that connect the injection composition to the initial composition. The sequence of compositions, called the composition route must follow the eigenvector paths unless the solution becomes multivalued. In that case, shocks or weak solutions of the PDE must be introduced to remove or jump over the multivalued solution. There are many possible solutions that can follow the eigenvectors and shock routes and the trick is to find the unique composition route that satisfies all constraints. 3.4 Ternary compositional routes for complex phase behavior The composition routes for two-phase ternary displacements are discussed by many researchers, yet there are no complete solutions for displacements with bifurcating phase 53

73 behavior. We discuss the bifurcating phase behavior briefly followed by the composition route construction for CO 2 injection. The composition route is not constructed for fluids with more components, however we construct the tie-line route for real fluids with bifurcating phase behavior in Chapter Bifurcating phase behavior At a range of low temperatures, a mixture of CO 2 and a hydrocarbon component like hexadecane form three phases at one pressure. A mixture of CO 2 and two hydrocarbon components like propane and hexadecane can form three phases for a range of pressures and compositions. Sahimi et al. (1985) presented some examples of CO 2/hydrocarbon systems with three phases and they discussed the calculation of phase behavior of such systems using an equation of state (EOS). Orr and Jensen (1984) studied behavior of mixtures of hydrocarbons and CO 2. Their results showed that the phase diagram for a mixture of CO 2, propane and hexadecane has two separate two-phase regions at very high pressures. Mogensen et al. (2009) studied MMPs for real fluids and their results showed large differences between MMP calculated by MOC and other methods. Ahmadi et al. (2011) showed a similar behavior for a mixture of CO 2/C 1/C except that there is no three-phase region in their results. The fluid will form at most two phases at the specified temperature, but current two phase MOC solutions predict an incorrect MMP for the considered fluid. For their fluid system, MOC results predict an MMP value much higher than the results from experimental and the mixing cell method. Their research showed that the tie-line length does not change monotonically between two successive key tie-lines and the shortest tie-line is not a key tie-line. Non-monotone tie-line length explains the change in the order of K-values. Johns (1992) showed that if the K-values remain strictly ordered, the eigenvalues will change monotonically between two successive key 54

74 tie-lines. Therefore the shortest tie line will be a crossover tie line. Consequently tracking the length of crossover tie lines with pressure is enough to find the MMP. Finally, the ruled surface of nontie-line waves can be estimated with flat surfaces or shocks. These two assumptions are the basics of shock jump MOC for MMP calculation. The assumption of strictly ordered K-values is essential for shock-jump MOC solutions. Ahmadi et al. (2011) showed that the order of the K- values changes by composition for their fluid system. As a result, the non-tieline eigenvalues do not change monotonically between two tie-lines. Thus, the non-tieline path could be a combination of shocks and expansion waves (Johns, 1992). They improved the MMP estimation results by searching for the shortest tie-line between successive key tie-lines over the mixing line connecting two key tie-lines. The ternary fluid system presented by Ahmadi et al. (2011) is used in this research. The component properties are shown in Table 3.2. PR EOS (Peng and Robinson, 1976) is used for phase behavior calculations. Figure 3-4 shows the phase behavior using parameters in Table 3.2 at different pressures and 133 o F. The phase behavior at lower pressures is very similar to using constant K-values with composition. As the pressure increases, however, the middle tie lines become smaller, and around 2000 psia, two separate two-phase regions forms. The CO 2 rich two phase region is more like L 1- L 2 behavior, while the C 1N 2 rich two-phase region similar to L 1-V phase behavior. L 1-V region disappears at higher pressures, while there is a L 1-L 2 two-phase region at higher pressures according to PR EOS. The bifurcation of the two-phase region significantly changes the analytical solution. The considered temperature is well above the critical temperature of C 1N 2 and CO 2. The mixture of C 1N 2 and CO 2 has a three-phase region at lower temperature (100 o F) as shown in Figure 3-5. The ternary system has a three phase region because the pressure and temperature of 55

75 the mixture are inside the critical locus of the C 1N 2 and CO 2 mixture (point A in Figure 3-6). These results are consistent with those in Orr and Jensen (1984). Figure 3-7 shows K-values for tie-line extension along the C 1N 2-CO 2 side of the phase diagram. The order of K-values changes as CO 2 mole fraction changes. All K-values are equal to 1.0 at the critical point, which occurs at the center of phase diagram. Conventional MOC solution methods only search for key tie lines therefore miss the critical tie line in between. The nonmonotonic variation of K-values causes tie lines extension to intersect inside the phase diagram within the single phase region. Figure 3-8 shows the structure of tie lines. There is a region bounded by intersection of the envelope curve where three tie lines extensions intersect. Two tie lines intersect along the envelope curve. The integral curves of eigenvectors at psia are shown in Figure 3-9. The nontieline paths merge through a specific composition called watershed (WS) point, which follows the same terminology used in three-phase displacements. The watershed points are two specific umbilic points with F 1 / c 11 = 0 and consequently arbitrary eigenvalues. The only possible path to connect a tie line with CO 2 rich phase to a tie line with rich C 1N 2 vapor phase is shown by the red line Composition-route construction In this section we discuss the analytical solution for the pseudo-ternary system of Figure Consider the composition path for displacement of Oil 1 by pure CO 2 in Figure The two-phase region is shown by the solid line and the envelope curve (dashed line) bounds the region of multiple tie-line extensions. Larson (1979) showed that the composition path should enter or exit the two-phase region by a shock along a tie-line extension. Oil 1 is outside the region of multiple tie-line extensions therefore there is only one possible oil tie line. In addition the 56

76 composition path enters the two phase region along the only possible tie line extension. The solution will be complete by a path connecting the oil and gas tie line. The red lie on Figure 3-9 shows the only possible rarefaction wave that connects oil and gas tie lines. The eigenvalues changes non-monotonically along that path (Figure 3-11). The nontie-line path connects two tie lines if the eigenvalue increases from the gas tie line towards the oil tie line. The shock connects the two tie lines in the case of decreasing eigenvalues. We take a wave starting from the gas tie line followed by a shock to the oil tie line. The only trick is to find the upstream point of shock. We first give a brief description of the entire composition route before describing how to locate the key composition points. The composition route consists of the following parts as shown in Figure Gas to point C: There is a shock from the injection gas composition into the two-phase region along the gas tie line. Point C is the required downstream shock composition because the next segment of the path must take a nontie-line path to the upstream composition α of the nontieline shock. Point C is before the tangent point from the gas composition and also before the first umbilic point. Thus, a constant state occurs at point C because there are two velocities associated with it: the tie-line shock velocity and the nontie-line eigenvalue. C to α: The eigenvalues increase form C to point α so that we can take a nontie-line path, i.e. a rarefaction wave given by the nontie-line eigenvalue. Point α is determined by the oil tie line and the watershed point (WS). α to B: A nontie-line shock within the two-phase region connects point α to point B. Point B lies on the oil tie line. The shock locus of point α must go through the watershed point and its velocity is equal to the nontie-line eigenvalue at point α (the nontie-line shock is a tangent shock at α). 57

77 B to Oil: A shock connects point B to the oil in the single-phase region. The route must shock immediately to the oil because the tie-line eigenvalue cannot be taken owing to the velocity constraint. Point B lies between the tangent point to the oil and the bubble point the gas composition along the oil tie line. There is a constant state at point B since the velocity associated with the nontie-line shock into point B is different than the shock velocity to the oil. The composition route is complex because there is a shock and wave along the nontieline portion that connects the oil and gas tie lines. The watershed point must be determined first in developing this composition route. The watershed point is found by finding the tie line that goes through the tip of the envelope curve shown in Figure 3-10 (see the red tie line in the figure). The watershed point is the umbilic point, which is also the tangent point for the shock from the tip of the envelope curve (see Figure 3-10). The tie line through the watershed point is the smallest tie line in the two-phase diagram. The nontie-line velocity at the watershed point is the maximum velocity along the nontie-line path through the watershed point, i.e. the peak in the curve shown in Figure Next, we must find point α. We describe two methods. The composition at α must lie on the shock locus of the watershed point based on the jump conditions in Eq. (2.3). This is because all shocks must go through the watershed point if they have upstream and downstream compositions on tie lines on opposite sides of the watershed point. The watershed point serves as a funnel of all shocks that cross its tie line. Thus, to find point α we first extend the oil tie line through Oil 1 to the envelope curve to obtain composition A. Point A is the first point along the oil tie line that has two tie-line extensions that intersect it. Then, the tie line through A is found that goes on the other side of the watershed point from the oil tie line. Point α lies on this tie line and is one of only two points on that tie line that is on the shock locus from the watershed point 58

78 as determined by Eq. (2.3). One of these compositions (the closest to point A) can easily be eliminated since it would violate the entropy conditions. Thus, the correct point α is the composition that lies past the second umbilic point on the tie line through composition A. Alternatively, one may find point α and composition B using Newton-Raphson iteration and the switch condition derived in Appendix B. That is, the velocity of the shock from point α to B on the oil tie line must be equal to the nontie-line eigenvalue at point α. The nontie-line shock within the two-phase region is therefore a tangent shock to point α, which just satisfies both the entropy and velocity conditions as shown in Appendix B. The downstream nontie-line shock composition, point B, is found by recognizing that there is a triangle of equal shock velocities that occurs as the oil composition moves along the oil tie line (tie line 2 in Figure 3-10) to composition A on the envelope curve. This triangle of equal shocks is similar to what occurs for condensing gas drives in conventional ternary systems (Johns 1992). The proof that such a triangle of equal shocks exists is based on the shock jump conditions, which show that from an upstream two-phase composition (point α) there are only two downstream compositions that one can shock to on a tie line within the two-phase region. One of those downstream compositions on tie-line 2 can be eliminated as unphysical using the entropy condition. Figure 3-3 shows that a shock from any point on the envelope curve along a tie line is equal to the nontie-line eigenvalue at the composition within the two-phase region. Thus, the velocity of a shock from point A on the envelope curve to point α along the tie line into the two-phase region (tie line 1 in Figure 3-10) is equal to the nontie-line eigenvalue at point α. Further, the velocity of a shock from A to point B, and from B to composition α, and from A to α must give the same velocity, which is equal to the nontie-line velocity. Point B is determined by this construction of equal shock velocities. From point α the route switches to the nontie-line path and that path is taken to the gas tie line. There is only one nontie-line path that goes through α and this path is found by integration along the nontie-line path starting by taking small steps in composition space that point along the 59

79 nontie-line eigenvector until point C on the gas tie line is reached. Once C is reached a shock occurs from that point to the gas composition and the composition route is complete. The compositions for the displacement of Oil 1 by CO 2 as a function of the dimensionless velocity are shown in Figure Numerical simulation with 20,000 grid blocks agrees well with the dispersion-free MOC composition profile shown in Figure Figure 3-13 shows the numerical solution for various numbers of grid blocks for the displacement of Oil 2. The numerical solution is highly sensitive to the level of numerical dispersion (number of grid blocks) and the sensitivity increases as the oil approaches the region with multiple tie-line extension intersections. For example 40,000 grid blocks are required to match the analytical dispersion-free solution for Oil 2. The high degree of sensitivity to numerical dispersion is likely the result of the changing tie-line lengths, where the smallest tie line is at the watershed point and the closeness to the region of multiple tie-line intersections. That is, a path with dispersion may intersect tie lines that go through the region of multiple intersections causing a significant bend in the numerical composition route. Figure 3-13 also shows that the nontie-line shock path is curved. That is, as the number of grid blocks is increased, the shock path on the ternary diagram approaches a curved path, not a straight line path. Figure 3-13 shows the shock path for MOC assuming that the composition path for the shock is a line that goes through the upstream and downstream compositions, but this is not correct (see Appendix C). The shock path must also go through the watershed point Features of displacements with bifurcating phase behavior The composition route and profiles for displacement of Oil 1 show features of both a condensing and vaporizing displacement (C/V), which was not thought to be possible for a ternary displacement. This occurs because the route from Oil 1 to point B, and then to α is like a 60

80 conventional condensing ternary drive, where CO 2 is condensed into the equilibrium liquid phase that forms. The tie-line lengths along this portion of the path become shorter from the oil tie line to the tie line at point α. CO 2 is the intermediate component in this portion of the route because it has a volatility (K-value) between that of the methane-like component (C 1N 2) and the heavy oil pseudocomponent. Along the portion of the route from point α to composition C, however, and then to the injection gas composition (pure CO 2), the displacement is vaporizing in that the methane-like component is vaporized from the equilibrium liquid into the equilibrium vapor. C 1N 2 is the intermediate component in this portion of the ternary displacement because its K- value is less than CO 2. The tie-line lengths increase towards the injection gas in this portion of the displacement. This is similar to a combined condensing and vaporizing displacement in a quaternary system where the condensing region occurs as the tie-line lengths decrease from upstream to downstream, but then increases in the vaporizing portion. Figure 3-14 shows how the K-values for C 1N 2 and CO 2 change order along the pseudoternary axis for the bifurcated phase behavior at 21,000 psia. Another unique feature of this type of displacement is that the analytical MOC solution is not continuous as the oil composition changes further along the oil tie line (tie line 2 in Figure 3-15). This would seem to violate uniqueness criterion for a mathematical solution, but it occurs because the composition route moves through the watershed point, which separates these two types of displacements. For the oil at composition A in Figure 3-15, the route should be only a vaporizing drive not a combined C/V drive since it can take the nontie-line path from tie-line 1 to the gas tie line (a nontie-line shock is no longer needed). The discontinuity occurs along the envelope curve where composition A is displaced by CO 2. That is, there are two dispersion-free composition routes that can be constructed for an oil at point A. The first route is similar to the one already determined in that it would shock along tie line 2 in Figure 3-15 to point B and then to point α. Because the shock velocity from Oil A to B, and then from B to α is the same, this is 61

81 equivalent to a shock from point A to α directly at the same velocity. That velocity is equal to the nontie-line velocity. This type of route, however, introduces a shock that would violate the entropy condition and so this route is not stable in the presence of dispersion. Instead, there must be a tangent shock from A along tie-line 1. The tangent point is between point α and the bubble point on tie-line 1. Thus, a tie-line path is taken to the umbilic point, which is the same as the tangent point from A. Point α is never reached resulting in a discontinuity in the composition routes as is shown in Figure 3-16 and Figure The discontinuity at A is observed in both the analytical and numerical solutions. To demonstrate this, we consider several oil compositions near A along the oil 1 tie line. Point A- is slightly heavier (away from Oil 1), while points A+ and A++ are closer to oil 1 along the same tie line. The plus and minus refers to a change in composition from A by a small value of in composition. For oil compositions slightly nearer Oil 1 along the oil tie line from point A the composition route goes through point α and the simulation results will be close to the unstable route (points A+ and A++). The simulation results for Oil A and slightly heavier than Oil A (points A and A-), however, have composition profiles closer to the stable route as is shown Figure Figure 3-16 also shows that changing from point A- to A results in a very small change in the solution (location of front position), while from A to A+ results in a large relative shift in the leading shock location. From A+ to A++ there is only a small change in the shock location. Thus, the large relative shift from A to A+ confirms the solution discontinuity. Figure 3-17 shows that the recoveries of the heavy component from the simulations also show the same discontinuity, further verifying the dispersion-free discontinuity in the composition route. The discontinuity can be understood better by considering the displacement of oil at 21,000 psia in Figure 3-4. Oil compositions can have two tie lines that extend through them, such as an oil that is pure C (heavy pseudocomponent). One of the tie lines extends from the L 1-L 2 region through this oil, while another extends from the L 1-V region. It is clear that the tie line that 62

82 extends from the L 1-V region is not applicable to this displacement and can be discarded. Instead, the correct oil tie line is the one closest to the gas tie line. If one constructed the solution based on both oil tie lines a discontinuity would result since the composition route to the L 1-V region would have to go through a critical point (that path would be miscible), while the other route (the correct one) would not be miscible. Another unique feature of the displacements for the phase behavior shown in Figure 3-8 is that three tie lines can intersect a single-phase oil composition (this is true for all compositions away from the envelope curve within the tie-line intersection region). Figure 3-18 shows such a case. One of the tie lines (tie-line 1) has a different order of component K-values than the injection tie line (C 1N 2 is more volatile than CO 2). This tie line, which in Figure 3-18 is the upper tie line, can be discarded. Further, tie-line 2 in the figure is unphysical because the shock velocity to the initial composition along it will be smaller than the non-tie line eigenvalue, which violates the velocity condition. Thus, the correct tie line is tie-line 3, which gives a vaporizing gas drive. Numerical simulations of this displacement will converge to this route as well. We now consider the composition route for several oil compositions along the tie line that extends through Oil 1 as shown in Figure 3-19 to Figure Once the oil composition lies within the region of multiple tie-line extensions the displacement is purely vaporizing and the tieline extension through Oil 1 is no longer the key oil tie line. That is, the initial oil tie line changes to the tie-line extension nearest to the injection gas tie line. This type of displacement is exactly the same as would be expected for a conventional ternary vaporizing gas drive. That is, the composition route first takes a tangent shock along the new oil tie line. The route then takes the tie-line eigenvalue and switches at the umbilic point to a nontie-line path. From there, the composition route follows the nontie-line path (wave) until the injection gas tie line is reached. As long as the composition reached is between the tangent point to the gas, there is an immediate 63

83 shock to the gas, resulting in a constant state at that composition. Otherwise, the route takes a tieline path (wave) to the tangent point, from which a tangent shock to the gas composition occurs. Figure 3-20 gives the recoveries for Oils 1 and 4 at 1.2 pore volumes injected. Oil 1 can develop miscibility because it must shock into the L 1-V region of the phase behavior. As pressure increases to about 21,000 psia (the MMP), the recovery goes to 100% as is expected since this displacement must pass through a critical point. That is, the watershed point becomes the critical point at this pressure. The simulation results are highly affected by dispersion, therefore, the numerical recoveries approach 100% at pressures much higher than MMP. The MMP of about 19,617 psia agrees well with the mixing-cell MMP, which is 19,570 +/- 4 psia (PennPVT (2013), Ahmadi and Johns (2011)). Oil 4, however, cannot develop miscibility in this system because the L 1-L 2 region does not vanish as pressure is increased. Thus, Oil 4 always has a tie line that extends through it. The recovery for this oil flattens out albeit at a very large value, but it is not 100%. As for quaternary C/V drives, the shortest tie-line length between the condensing and vaporizing regions controls miscibility. In this pseudoternary case, the shortest tie line corresponds to the tie line through the watershed point, not the gas or oil tie lines. This is why the shock jump MOC method, which finds only the key tie lines (oil or gas tie lines in a ternary displacement), fails to accurately estimate the MMP. The bend in the recovery curve for Oil 4 is not due to numerical dispersion. For example, Pederson et al. (2012) observed similar trends in experimental results for liquid-liquid displacements. A larger methane volume fraction in the initial oil composition can move the oil composition out of the region of liquid-liquid equilibrium and the MMP can occur at lower pressures. Thus, the amount of methane in a displacement like the one studied in this paper can significantly affect the MMP. 64

84 3.5 Summary The hyperbolic equations for gas floods with the zero dispersion assumption is commonly used to examine the displacement mechanism of compositional displacements. The solution for these problems can be constructed using the method of characteristics. However, the solution for EOR problems are complex because the equations are not strictly hyperbolic. The solution for displacements with bifurcating phase behavior is constructed using the method of characteristics. The solution was used to explain the different features of displacements with this type of phase behavior. Table 3.1 Component properties of the fluid system of Orr et al. (1993) Properties Binary Interaction Parameters T c ( o F) P c (psia) ω CO 2 C 1 nc 4 C 10 CO C nc C Table 3.2 Component properties of the fluid system of Ahmadi et al. (2011) Properties Binary Interaction Parameters T c ( o F) P c (psia) ω CO 2 C 1 C CO C C

85 Figure 3-1: Comparison of composition route of UTCOMP and our simulator with MOC assumptions. Figure 3-2: Comparison of tie lines of UTCOMP and our simulator with MOC assumptions. 66

86 Figure 3-3: Geometric construction of the nontie-line and tie-line eigenvalues. The tie-line eigenvalue is equal to the slope of the curve at any point in the two-phase region, while the nontie-line eigenvalue is equal to the slope of the line from h to a twophase composition. These two eigenvalues are equal at the two umbilic points, where the line from h is tangent to the overall fractional flow curve. 67

87 Figure 3-4: Phase behavior of pseudoternary system showing the split of the two-phase region with pressure at 133 o F. 68

88 Figure 3-5: Three phase behavior at 40 o F and 1000 psia. Figure 3-6: Critical locus of C 1N 2 and CO 2. 69

89 Figure 3-7: K-values at psia, 133ᵒF along the line where C concentration is zero. 70

90 Figure 3-8: a) Region of tie-line extensions that intersect within the single-phase liquid region at 16,000 psia, and b) values of tie-line parameters in Eq. (3.13) for all tie lines in positive composition space. The dashed line in figure b) represents a composition in figure a) as shown (see Eq. (3.13)). The intersection of the lines with the solid curve in figure b) shows the tie lines that pass through that composition. Point 2 lies on one of the envelope curves where successive tie-lines intersect. 71

91 Figure 3-9: Non-tie line paths and watershed points. 72

92 Figure 3-10: Triangle of shocks and continuity of solution. Figure 3-11: Tie-line eigenvalue from the oil to gas tie lines along the red non-tie line path shown in Figure

93 Figure 3-12: Analytical and numerical composition profile using 20,000 grid blocks for displacement of oil 1 in Figure 3-10 by pure CO 2 at 16,000 psia. Figure 3-13: Effect of numerical dispersion on the composition route for oil 2 at 16,000 psia. 74

94 Figure 3-14: K-values for the two-phase regions at 21,000 psia (see Figure 3-4) along the C 1N 2- CO 2 axis of the ternary diagram. Figure 3-15: Discontinuous dispersion-free composition routes for point A. 75

95 Figure 3-16: Analytical composition profiles showing a discontinuity in the dispersion-free displacement of oil A. The discontinuity is verified by simulation for various initial compositions near point A with 400,000 grid blocks. The unstable solution shown in the figures for composition A does not satisfy the entropy condition. 76

96 Figure 3-17: Discontinuity in recovery calculated by simulation. Figure 3-18: Example of three intersecting tie lines through an oil composition. 77

97 Figure 3-19: Analytical composition routes for oils 1-4 at 16,000 psia. Figure 3-20: Analytical composition profiles for the heavy pseudocomponent for four oil compositions displaced by pure CO 2 at 16,000 psia. 78

98 Figure 3-21: Analytical composition profiles for the light pseudocomponent for four oil compositions displaced by pure CO 2 at 16,000 psia. Figure 3-22: Analytical composition profiles for the CO 2 component for four oil compositions displaced by pure CO 2 at 16,000 psia. 79

99 Figure 3-23: Recoveries for oil 1 and 4 from numerical simulation with 20,000 grid blocks. The displacement is both condensing and vaporizing for oil 1, but only vaporizing for oil 4. Oil 4 has no MMP at because the displacement remains liquid-liquid at very high pressure. 80

100 Chapter 4 Three-component global Riemann solver using splitting of equations We study a 2 2 system of non-strictly hyperbolic conservation laws arising in three component gas flooding for enhanced oil recovery. The system is not strictly hyperbolic. Along a curve in the domain one family is linearly degenerate, and along two other curves the system is parabolic degenerate. We construct global solutions for the Riemann problem, utilizing the splitting property of thermo-dynamics from the hydro-dynamics. Front tracking simulations are presented using the global Riemann Solver. We consider a simplified compositional displacement model for a three-component system at constant temperature and pressure (Helfferich 1980), (C 1 ) t + (F 1 (C 1, C 2 )) x = 0, (C 2 ) t + (F 2 (C 1, C 2 )) x = 0, (4.1) associated with initial data C 1 (0, x) = C 1(x), C 2 (0, x) = C 2(x). (4.2) The independent variables (t, x) are normalized such that the overall velocity is 1. Here C i is the overall i th component volume fraction, and F i is the overall i th component flux. For the third component, we trivially have C 3 = 1 C 1 C 2, F 3 = 1 F 1 F 2. 81

101 The couplet (C 1, C 2 ) takes values in a triangular domain D = {(C 1, C 2 ) C 1 0, C 2 0,1 C 1 C 2 > 0}. For the phase behaviors that are considered in this paper, there exists a subset D 2 D, referred to as the two-phase region, where the fluid splits into two phases, the liquid and the gaseous phases. In the single phase regiond 1 = D\D 2, we trivially have F 1 (C 1, C 2 ) = C 1, F 2 (C 1, C 2 ) = C 2. We briefly derive the equations in the two-phase region. We denote by c il and c ig the composition of component i in the liquid and gaseous phases, respectively. For (C 1, C 2 ) D 2 the compositions c il and c ig, together with the liquid phase saturation S, satisfy the following equations, C i = c il S + c ig (1 S), F i = c il f + c ig (1 f) i = 1,2, 3 3 c il = c ig = 1. (4.3) i=1 i=1 Here f = f(s, C 1, C 2 ) is the fractional flow of liquid, and S takes values between 0 and 1 in the two-phase region. Typically, for given (C 1, C 2 ), the mapping S f is S-shaped with an inflection point. The K-values, defined as K i = c ig c il i = 1, 2, 3, (4.4) are determined by a phase behavior model and can either be taken as constant or a function of (C 1, C 2 ) (e.g. Michelsen 1982). For given (C 1, C 2 ) and K i, one can calculate c il, c ig and S by 82

102 simultaneous solution of Eqs. (4.3) and (4.4). This simultaneous solution of equations is called a flash calculation in the engineering literature and can be complicated for the systems with more than three components (Johns et al. 1993). In case of composition dependent K-values, the equilibrium compositions are determined by an iterative procedure (Michelsen 1982). Next, the results of flash calculations are used to calculate f and F i. For fixed (c il, c ig ) for i = 1,2, the values (C 1, C 2 ) are linear functions of S. In the phase coordinate (C 1, C 2 ), as S varies from 0 to 1, the trajectory of the couplet (C 1, C 2 ) is the straight line connecting the equilibrium points (c 1g, c 2g ) and (c 1l, c 2l ). When S = 0, we have (C 1, C 2 ) = (c 1l, c 2l ), and when S = 1, we have (C 1, C 2 ) = (c 1g, c 2g ). These lines are called tie-lines. The curves of the end-points of these tie-lines, namely the points (c 1g, c 2g ) and(c 1l, c 2l ), form the boundaries of the two-phase region. One may artificially extend the tie-lines into single-phase region. We assume that the tie-lines do not intersect in the domain D, such that any point (C 1, C 2 ) D lies on one unique tie-line. See Figure 4-1 (left) for a plot of the two-phase region and the tie-lines. It is well-known that the system of conservation laws, Eq. (4.1), is not hyperbolic. There exist two curves in D 2 where the two eigenvalues as well as the two eigenvectors of the Jacobian matrix of the flux function coincide, and the system is singular. On the other hand, the system (Eq. 4.1) has many interesting properties. Indeed, one family of integral curves of the Jacobian matrix are straight lines, which coincide exactly with the tie-lines. This motivates a parametrization of the tie-lines and a variable change of the unknowns. Without loss of generality, we retain the equation for C 1 in Eq. (4.1) and write C = C 1, F = F 1, C 2 = αc + β, F 2 = αf + β, (4.5) where α and β are defined as 83

103 α = c 2l c 2g c 1l c 1g, β = c 2g αc 1g. (4.6) Here α indicates the slope of a tie line, and β its interception point with the linec 1 = 0. Under the assumption that the tie lines do not intersect with each other in the domain D, one may parametrize the tie lines with β (Johansen and Winther 1990, Johns 1992), and consider α = α(β). Treating (C, β) as the unknowns, the system Eq. (4.1) becomes C t + F(C, β) x = 0, C(α(β)) t + β t + F(C, β)(α(β)) x + β x = 0, (4.7) associated with the initial data C(0, x) = C (x), β(0, x) = β (x). (4.8) The tie lines are now horizontal lines in the (C, β)-phase plane, illustrated in Figure 4-1 (right). Construction of solutions of the Riemann problems can be challenging for threecomponent systems as shown in Chapter 3. In (Pires et al. 2006), the following Lagrangian coordinates (φ, ψ) was introduced, φ x = C, φ t = F, and ψ = x t. (4.9) Straight computation leads to the following system ( C F C ) ( 1 φ F C ) = 0, (4.10) ψ β φ + α(β) ψ = 0. (4.11) 84

104 The thermodynamics process described in Eq. (4.11) is decoupled from the fractional flow in Eq. (4.10) (also known as the hydro-dynamics). Solutions of Riemann problems could be rather simply constructed if this coordinate change were well-defined in the whole domain D. In fact, given left and right states (C L, β L ) and (C R, β R ), one could first solve Eq. (4.11) for β, then substitute the solution into Eq. (4.10), and solve a scalar conservation law with possibly discontinuous coefficients. Unfortunately, Eqs. (4.10) and (4.11) does not offer this possibility, since the quantities C and 1 do not allow a single-valued function between them. Furthermore, the coordinate F C F C change is only valid in the set when F > C. Indeed, let J be the Jacobian matrix for this coordinate change, (φ, ψ) J (t, x) = ( F C ), so det(j) = F C. 1 1 Thus det(j) = 0 when F = C, and the coordinate change is not valid there. Furthermore, det(j) < 0 when F < C, so the resulting conservation laws are not equivalent to the original ones. See Wagner (1987) for a discussion on the equivalence between the Eulerian and Lagrangian coordinates for the Euler s equations of gas dynamics. If F < C, we define different Lagrangian coordinates, φ x = C, φ t = F, and ψ = x t. (4.12) The Jacobian matrix J for this coordinate change is J (φ,ψ ) = ( F C ), so det(j) = C F > 0. (t,x)

105 Formal computation leads to the following system: ( C F C ) + ( 1 φ F C ) = 0, (4.13) ψ β φ + α(β) ψ = 0. (4.14) Nevertheless, the splitting nature can still be utilized in both numerical computation and theoretical analysis. In this paper we construct solutions for global Riemann problems, taking advantage of the splitting property. Given left and right states (C L, β L ) and (C R, β R ), we would first solve for β, using either Eq. (4.11) if F > C, and Eq. (4.14) if F < C. This gives us a-priori information on waves connecting different tie-lines. The global Riemann solver for Eq. (4.7) can be constructed based on this information. The Riemann solver is then used to generate piecewise constant front tracking approximate solutions. The rest of the chapter is organized as follows. In Section 4.1 we give some basic analysis, the precise assumptions on the model, along with the main results. Wave behaviors of both families are analyzed in detail in Section 4.2. In Section 4.3 we connect various waves and construct global existence of solutions for Riemann Problems. Some numerical simulation using wave front tracking algorithm is performed and the results presented Section 4.4, to solve the three-component slug injection problem with mass transfer between phases. 4.1 Basic analysis, precise assumptions, and the main results We assume that in the phase plan (C 1, C 2 ), no two tie-lines intersect in the domain D. Using Eqs. (4.6) and (4.3), we have 86

106 α(β) = β(1 K 2 )(K 1 K 3 ) β(k 1 1)(K 2 K 3 ) + (K 2 K 1 )(1 K 3 ). (4.15) Computation shows that the intersection point of any two tie-lines is outside the domain D if the K-values satisfy one of the following conditions K 3 < K 2 < 1 < K 1, or K 1 < 1 < K 2 < K 3. (4.16) Such conditions are called strictly ordered K-values in the petroleum engineering literature (Orr et al. 1993). This labeling of components can be different from the conventional ordering of components based on molecular weight. Under the assumption Eq. (4.16), every couplet (C 1, C 2 ) D corresponds to a unique couplet (C, β). Defining the unknown vector u (C, β) t, (4.17) the Eqs. (4.7) can be written into the quasi-linear form FC F u t + A(u)u x = 0, where Au ( ) F '( ) 1. (4.18) 0 C '( ) 1 The matrix A(u) has the following eigenvalues and right-eigenvectors λ C = F C, C r 1 0, λβ = F+[α (β)] 1 C+[α (β)] 1, r F FC. (4.19) 87

107 Here the labeling of the two families are not with respect to wave speed. We referred to λ C and λ β as the eigenvalues for the tie-line and non tie-line families, respectively. Sample integral curves for the β-eigenvectors (nontie-line paths) are plotted in Figure 4-2. The values (C = [α (β)] 1, β) gives the envelope curve of the tie lines. A computation on the directional derivative of λ β in the direction r β gives λ β. r β = 1 (Cα (β) + 1) 2 α (β)(f C λ β )(F C). (4.20) This indicates that along the curve F = C, the eigenvalue λ β remains constant. This curve lies between the two groups of integral curves (see the green curve in Figure 4-2), and is a βintegral curve (see the proof of Lemma 4.2), along which the β-family is linearly degenerate. This curve is referred to as the equi-velocity curve, and we will use the abbreviation EVC throughout this paper. Furthermore, Eq. (4.20) also indicates that along a β-integral curve, the derivative of λ β changes sign at the point where F C λ β = 0. The S-shape of the map C F(C, β) for any fixed β gives rise to exactly two such points in the two-phase region. At these points we also have λ C = F C = F + [α (β)] 1 C + [α (β)] 1 = λβ, r C = r β = (1,0) t, i.e., the two eigenvalues as well as the two eigenvectors coincide, so the system is parabolic degenerate. These points are referred to as the umbilical points. As β varies, we have two curves in the two-phase region, one on each side of the EVC, where the system is degenerate. For the convenience of our analysis, we introduce a new functional. For fixed β and parameter a, we define a function F(C; β, a) as 88

108 F(C; β, a) F(C, β) + a C + a. (4.21) This function takes the value of the slope between the point ( a, a) and (C, F), see Figure 4-3 plots (a) and (b) for an illustration. For a = [α (β)] 1 the function takes the values of λ β. Note that for fixed β and a, the function C F reaches its minimum and maximum values at C min and C max respectively, where the lines ( a, a) (C min, F(C min )) and ( a, a) (C max, F(C max )) are tangent to the graph of F(C, β) in plot (a). We now state the precise assumptions on the functions F(C, β) and α(β) as follows. A1. The map β α is C 2 either strictly concave α < 0 or strictly convex α > 0. A2. The function F(C, β) is C 2. For any fixed β, the map C F is an S-shaped function with a unique inflection point. In the two-pause region, the map C F is strictly convex F CC > 0 on the left of the inflection point, and strictly concave F CC < 0 on the right of the inflection point. A3. The length of tie-lines in the two-phase region is a monotone function in β, such that the followings hold. Between any two tie-lines, say with β 1 and β 2, either everything point on the line β = β 1 can be connected to some point on the line β = β 2 by at least one βintegral curve, or every point on the line β = β 2 can be connected to some point on the line β = β 1 by at least one β-integral curve. We remark that, the explicit expression for integral curves of the systems with constant K-values shows the same behavior as (A3) (Dindoruk 1992). However, for phase behavior with composition dependent K-values, if the order of K-values changes, (A3) may not hold (Khorsandi et al. 2014). Below is the main result of this chapter. 89

109 Theorem 4.1 The Riemann problem for Eq. (4.7) has a unique global solution for any Riemann data u L and u R. Furthermore, in the phase plane (C, β), the path of the β-wave lies on the same side of the EVC as the left state u L. 4.2 Basic wave behavior The C-waves We first recall the Liu admissibility condition (Liu 1976) for shocks. Let u + = S β (σ)(u L ) for some σ R be a point on the β-shock curve through the left state u L. We say that the shock with left and right state (u L, u + ) satisfies the Liu admissibility condition provided that its speed is less or equal to the speed of every smaller shock, joining u L with an intermediate state u = S β (σ)(u L ), s [o, σ]. When β is a constant, then two equations in Eq. (4.7) are the same. This scalar conservation law, where C is the unknown, has a Buckley-Leverett (1942) type flux function. Solutions of Riemann problems are well-understood, see for example (Smoler, 1969). We referred the waves there as C-waves. Let (C L, β) and (C R, β) be the left and right states, the solution of the Riemann problem is constructed such that all shocks satisfy the Liu admissibility condition, and it could consist of composite waves. To construct these wave, if C L > C R, we make the concave upper envelope of the flux function, while if C L < C R, we make the lower convex envelope, and the C-waves are constructed accordingly. See Figure 4-4 for an illustration. All C-shocks satisfy the Liu admissible condition. 90

110 4.2.2 The β-waves β-waves. The waves that connect two different tie lines, i.e., two different β values, are referred as The β-shocks We recall the Lax admissible condition for shocks. Along a shock curve of the i th family in the (x, t) plan, the nearby characteristics of the same family must merge into the shock. For scalar conservation law with general flux function, Lax condition is necessary but not sufficient. However, if the flux is strictly convex or concave, these two conditions are equivalent. In our model, the system is degenerate along two curves, therefore it is difficult to define admissible shock loci across these degenerate curves. Indeed, shock locus might be discontinuous, thus it is unclear how to apply the Liu condition. Since the β-family is strictly convex or concave, we apply instead the Lax admissible condition. We remark that the Lax condition, combined with the minimum jump condition (Gimse and Risebro 1992) will eventually yield the unique solution for Riemann problems, proved in Section 4.3. For a β-shock, the C value is not constant across the shock. We first show that the Lax admissibility condition for β-shocks for the Eqs. (4.7) is equivalent to the same condition for the scalar Eqs. (4.11) or (4.14), for F > C or F < C respectively. Lemma 4.2 Let (C, β) be piecewise continuous solution of Eqs. (4.7), and let (C L, β L ) and (C R, β R ) be the left and right state of a β-shock that satisfies the Rankine-hugoniot condition. Then, we have sign (F(C L, β L ) C L ) = sign (F(C R, β R ) C R ). (4.22) Furthermore, the followings hold. 91

111 If F(C L, β L ) = C L, then F(C R, β R ) = C R, and this shock is a contact discontinuity. If F(C L, β L ) > C L and F(C R, β R ) > C R, then the shock (C L, β L ) (C R, β R ) satisfies the Lax condition if and only if (β L, β R ) is a shock for Eq. (4.11) that satisfies the Lax condition. If F(C L, β L ) < C L and F(C R, β R ) < C R, then the shock (C L, β L ) (C R, β R ) satisfies the Lax condition if and only if (β L, β R ) is a shock for Eq. (4.14) that satisfies the Lax condition. Proof. Let (C L, β L ) and (C R, β R ) be the left and right state of a β-shock, respectively, and let σ β be the shock speed. The Rankine-Hugoniot condition requires σ β (C L C R ) = F L F R, (4.23) σ β (α L C L + β L α R C R β R ) = α L F L + β L α R F R β R. (4.24) Here we used the short hands F L = F(C L, β L ), F R = F(C R, β R ), α L = α(β L ), α R = α(β R ). We can eliminate C R or C L by multiplying Eq. (4.23) with suitable factor and subtract the remaining equation from Eq. (4.24). Simple calculation gives σ β = FL + σ β 1 C L + = FR 1 + σ β σ β 1 C R +, σ β 1 where σ β = αl α R β L. (4.25) βr Note that σ β is the Rankine-Hugoniot speed for Eq. (4.11) in the Lagrangian coordinate. In the phase plane (C 1, C 2 ), the two tie-lines associated with β L and β R intersect at the point where C 1 = σ β 1. Under our assumption, this point lies outside the domain D, either to the left or to the right of D. Assuming it is on the left such that σ β 1 < 0, we illustrate the geometric 92

112 meaning of Eq. (4.25), in Figure 4-5 for an illustration. This clearly implies Eq. (4.22). The case where the intersection point is on the right of D is completely similar. For the rest of the proof we only consider the case σ β 1 < 0. If F L = C L, i.e., the left state is on the EVC, then by Eq. (4.25) we have σ β = 1, and we must have F R = C R for every state (C R, β R ) that could be connected to (C L, β L ) with a β-shock. Thus the right state must also lie on the EVC. Along such a shock curve, the second eigenvalue λ β 1, and the β-family is linearly degenerate. This discontinuity is actually a contact discontinuity, proved later in Lemma 4.2. Otherwise if F L > C L, by Eq. (4.25) we have σ β > 1, and therefore F R > C R. In order to show the equivalence of the two Lax conditions, i.e., α (β L ) > σ β > α (β R ) λ β (C L, β L ) > σ β > λ β (C R, β R ), it suffices to show that the mapping s F + s 1 C + s 1 is strictly increasing for any fixed F and C with F > C. This fact can be easily verified. The proof for the case F L < C L is completely similar. The same results can be shown similarly for the case where the intersection point of the two tie-lines is on the right of D β-rarefactions. A β-rarefaction wave will connect (C L, β L ) to (C R, β R ) along the integral curves of the β-field. Similar to Lemma 4.2, we have the following Lemma. 93

113 Lemma 4.3 Consider piecewise continuous solutions of Eqs. (4.7), and let (C L, β L ) and (C R, β R ) be the left and right states of a β-rarefaction wave in the two phase region. Then, we have (i) If F(C L, β L ) = C L, then F(C R, β R ) = C R, and this wave is a contact discontinuity. (ii) If F(C L, β L ) > C L then F(C R, β R ) > C R, and (β L, β R ) is a rarefaction wave for Eq. (4.11). (iii) If F(C L, β L ) < C L, then F(C R, β R ) < C R, and (β L, β R ) is a rarefaction wave for Eq. (4.14). Proof. In the phase plane (C, β), the β-rarefaction curves are the integral curves of the second eigenvector of the Jacobian matrix of the flux function for Eq. (4.6), given in Eqs. (4.19). Let s R(s)(C L, β L ) denote a β-rarefaction curve initiated at (C L, β L ) where s is the parametrization of the curve such that R(0)(C L, β L ) = (C L, β L ). We first show that the EVC is an integral curve. It suffices to show that (C s, β s ) t is parallel to the eigenvector r β. Indeed, taking partial derivative in s of the equation F(C, β) = C, we get F C C s + F β C s = 0, i. e., C C 1 s F. 0. F s If F = C, we have λ β = 1 and so r β = ( F β, F C 1) t. Thus (C s, β s ) t is parallel to r β, as claimed. This proves (i). By the uniqueness of the integral curves, (ii) and (iii) follows, completing the proof. 94

114 4.3 Global solutions of Riemann problems The solution of a Riemann problem is the key building block in a front tracking algorithm. In this section we construct solutions for Riemann problems with any Riemann data, taking advantage of the splitting property in the Lagrangian coordinates Connecting C-waves with β-shock Connecting C-waves with a β-shock results in the Riemann problem for a scalar conservation law with discontinuous coefficient function. Let u L = (C L, β L ) t and u R = (C R, β R ) t be the left and right states of the Riemann data, and assume that β L β R is connected by a single β-shock. We consider an implicit Riemann problem for a scalar conservation law with discontinuous flux function, C t + F (C, x) x = 0, F (C, x) = { FL (C) = F(C, β L ), x > σ β t, F R (C) = F(C, β R ), x < σ β t, (4.26) with initial Riemann data C(0, x) = { CL, x > 0, C R, x < 0. (4.27) Note that the wave speed σ β is unknown, and it will be determined after the Riemann problem is solved. This feature makes the Riemann problem solver implicit. In order to remove the implicit feature, we recall the definition of the function F(C; β, a) in Eq. (4.21). Given β L and β R, we define the F functions 95

115 F L = F(C; β L, σ β), F R = F(C; β R, σ β), where σ β = β L β R α(β L ) α(β R ). (4.28) Note that relation between the graphs of F L and F R are topologically identical to that of the graphs of F L and F R. Riemann problem for a scalar conservation law with (F L, F R ) as the flux function, will generate the same types of waves if using (F L, F R ) as the flux functions, although with different wave speeds. The advantage of using F L and F R lies in the fact that βwaves will be stationary. This makes the construction of Riemann solution clearer. For the Riemann data of Eq. (4.27), we are now consider the following scalar equation C t + F(C, x) x = 0, where F(C, x) = { FL (C), x 0, F R (C), x > 0. Existence and uniqueness of Riemann solution for scalar conservation law with flux function with spacial discontinuity was established by Gimse and Risebro (1991), using the minimum jump condition, under the assumption that the flux functions f(u, x) are smooth in u. Our flux functions F(C, x) are only continuous and piecewise smooth in C. Nevertheless, the construction of the Riemann solution remains rather similar. We denote (u 1, u 2 ; f) the Riemann problem for a scalar conservation law u t + f(u) x = 0 with u 1, u 2 as the left and right states. The construction follows a three-step algorithm. S1: Given F L (C) and C L, we identify the set I L (C L, F L ) {C m ; (C L, C m ; F L ) is solved by waves of non positive speed} {C L }. S2: Given F R (C) and C R, we identify the set I R (C R, F R ) {C M ; (C M, C R ; F R ) is solved by waves of non negative speed} {C R }. 96

116 S3: Find the β-wave position (C m, β L ) (C M, β R ) by minimizing C M C m in the set {C m I L, C M I R, F L (C m ) = F R (C M )}. Next Theorem guarantees the existence and uniqueness of the Riemann solution. Theorem 4.4 Consider the Riemann problem with u L = (C L, β L ) and u R = (C R, β R ) as the left and right states, where β L and β R is connected with a single β shock. There exists a unique solution for this Riemann problem. Proof. We first observed that it suffices to prove the existence and uniqueness of the path for the β-shock. Once this path is located, the solution for the Riemann problem is uniquely determined. We define the set for the values of the flux function on the set I L and I R as J L (C L, F L ) {F L (C); C I L }, J R (C R, F R ) {F R (C); C I R }. (4.29) We first claim that the intersection of these two sets are not empty, J L (C L, F L ) J R (C R, F R ). (4.30) Indeed, due to the properties of our flux function, it is convenient to list all the cases. Given F L, let (C 0, F L 0 ) and (C 2, F L 2 ) be the minimum and maximum points, respectively. Also we let C1 be the unique point such that C 0 < C 1 < C 2 and F L (C 1 ) = 1. See Figure 4-6 for an illustration. There are 4 cases. If C L C 0, then we have I L = (0, C 0 ], J L = [F L (C 0 ), 1]. If C 0 < C L < C 1, then let C L be the unique point such that C L < C 0 and F L (C L) = F L (C L ). We have 97

117 I L = (, C L] {C L }, J L = [F L (C L ), 1]. If C 1 < C L < C 2, then let C L be the unique point such that C L > C 2 and F L (C L) = F L (C L ). We have I L = {C L } [C L, 1), J L = [1, F L (C L )]. If C L C 2, then we have We note that 1 J L in all cases. I L = [C 2, 1), J L = [1, F L (C 2 )]. Now, given F R, let (C 3, F R 3 ) and (C 4, F R 4 ) be the minimum and maximum points for F R respectively. There are 3 cases, illustrated in Figure 4-6. If C R < C 3, then let C R be the unique point such that C R > C 3 and F R (C R) = F R (C R ). We have I R = {C R } [C R, C 4 ], J R = [F R (C R), F 4 R ]. This includes the case where C R lies in the single phase region on the left of D 2. If C 3 C R C 4, then we have I R = [C 3, C 4 ], J R = [F 3 R, F 4 R ]. If C R > C 4, then let C R be the unique point such that C R < C 4 and F R (C R) = F R (C R ). We have I R = [C 3, C R] {C R }, J R = [F 3 R, F R (C R)]. This includes the case where C R lies in the one phase region on the right of D 2. 98

118 We note that 1 J R. Thus J L J R is non-empty, proving (4.10). To see that there is a unique solution to the minimizing problem, we first exclude the possible isolated points in the sets I L, I R, and denote the sets by I L o, I R o. On the set I L o, the function F L (C) is strictly decreasing, while on the set I R o, the function F R (C) is strictly increasing. Given F J L J R, let C M I R and F R (C M ) = F, and let C m I L and F L (C m ) = F. Denote also D F C M C m. Then, the function F D F is strictly increasing, and there exist a unique minimum for the map F D F. Finally, if F L (C L ) and/or F R (C R ) are/is in J L J R, there could be multiple minimum paths. In this case, we will select the path with the more isolated points. This yields a unique path for the β-shock. We have an immediate Corollary on the location of the β-shock. Corollary 4.5 In the setting of Theorem 4.2, the path of the β-shock lies on the same side of EVC as the left state of the Riemann data. Sample Riemann problems connecting single phase and two-phase regions. Let l t be the tieline that is tangent to the two-phase region, and let (C t, β t ) be the tangent point. This tie-line lies in the single phase region, and the flux function (C, β t ) = C. Consider another tie-line l 2 through the two-phase region with the flux (C, β 2 ). The solutions for the Riemann problems with left and right states on each of these tie-lines are illustrated in Figure 4-7, where we plotted the functions F(C, ). Case 1. If the left state is l t, then it will be connected to the point M with a C-contact discontinuity that travels with speed 1. Note that M is on the EVC. In fact it is the endpoint of EVC as it reaches the single-phase region. From M one can connect to any R on the tie-line l 2 on 99

119 the red curve by solving a Riemann problem of a scalar equation, which will yield a shock of speed 0. Case 2. (a) If the right state is on l t, and the left state is on the the right side of the EVC on the tie-line l 2, then the wave path L-M-R will go through the upper point for M. (b) On the other hand, if the left state is on the left of the EVC on the tie-linel 2, the wave path L-M-R will go through the lower point for M. Finally, if the left or right state are in the single-phase region along a tie-line extension, the single-phase region and two-phase region is connected by a C-wave (Hellferich, 1980). These discussions indicate that there are two ways that a wave path can connect states in the single-phase and two-phase regions: (i) through tie lines, and (ii) through the point M in Case 1. This point M is referred to as the Plait Point Connecting β-rarefaction wave to C-waves Definition 4.6 In the (C, β)-plane, given a β-integral curve C, a curve C is called the critical curve of C, if for every fixed β the β-eigenvalue λ β (C, β) = F(C; β, α (β) 1 ) has the same values on the curves C and C, and the curves C and C are separated by the degenerate curve. Due to the S-shape of the flux function C F, the existence and uniqueness of the critical curve is clear. Next Lemma provides its relative location to the β-integral curves. Lemma 4.7 Let C 1 C 2 be a β-integral curve, separated by the degenerate curve with C 1 on the left and C 2 on the right, lying on the same side of EVC. LetC 3 and C 4 be the 100

120 corresponding critical curves forc 1 and C 2 respectively. Then, eitherc 3 is on the left of C 2 andc 4 is on the left of C 1, or C 3 is on the right of C 2 and C 4 is on the right of C 1. Proof. We parametrize all these curve with β, i.e., C 1 is the graph of the function β C 1 (β) etc. We first observe that λ β (C 1, β) = F(C 1, β)α + 1 C 1 α + 1 = F(C 3, β)α + 1 C 3 α + 1 = λ β (C 3, β) implies F(C 1, β) C 1 = F(C 1, β)α + 1 F(C 3, β) C 3 F(C 3, β)α + 1 = C 1α + 1 C 3 α + 1. (4.31) Along C 1, using (2.6), the directional derivative of the β-eigenvalue is λ β. r β = α (β)(f(c 1, β) C 1 ) (α (β) + 1) 2. (4.32) must have Along C 3, the directional derivative of the β-eigenvalue is the same as in Eq. (4.32). We λ C β (C 3, β)c 3 (β) + λ β β (C 3, β) = α (β)(f(c 1, β) C 1 ) (α (β) + 1) 2. (4.33) Note that λ β (C 1, β) = λ β (C 3, β), and we will simply write λ β. Also, since α(β) is a function of β, we will drop the independent variable and simply write α, α, α. Using the partial derivatives β (F C λ β )α λ C = Cα + 1, λ β F β α + Fα λ β Cα β = Cα,

121 and the identities (4.31), we can solve (4.33) with respect to C 3 and obtain C 3 (β) = 1 (F C (C 3, β) λ β )α [(C 3α + 1) 2 (C 1 α + 1) 2. α (F(C 1, β) C 1 ) (C 3 α + 1) F β (C 3, β)α F(C 3, β)α + λ β C 3 α ] 1 (F(C 3, β) C 3 ) = (F C (C 3, β) λ β )α [α (C 1 α F + 1) β (C 3, β)α (F(C 3, β) + λ β C 3 )α ]. Fix a point on C 3, denoted as (C 3, β), let C 5 denote the β-integral curve through (C 3, β), parametrize it in β. We have C 5 (β) = F β(c 3, β) F C (C 3, β) λ β. Direct computation gives C 3 (β) C 5 = α (F C (C 3, β) λ β )α [F(C 3, β) C 3 C 1 α F(C + 1 3, β) + λ β C 3 ] = = α (F C (C 3, β) λ β )α [F(C 3, β) C 3 C 1 α F(C + 1 3, β) + F(C 1, β)α C 1 α + 1 C 3] α F C (C 3, β) λ β [F(C 1, β)c 3 F(C 3, β)c 1 ] α C 1 C 3 = F C (C 3, β) λ β [F(C 1, β) C 1 F(C 3, β) C 3 ]. The factor F C (C 3, β) λ β changes sign crossing the degenerate curves, and the term F(C 1, β)/c 1 F(C 3, β)/c 3 changes from positive to negative as it crosses EVC. We always have C 1 0, C 2 0. We have the following conclusion: Case 1. If α < 0, then on the left of EVC, we have 102

122 F C (C 3, β) λ β > 0, F(C 1, β)/c_1 F(C 3, β)/c 3 < 0, C 3 > C 5. By the uniqueness of the β-integral curve, C 3 lies on the right of C 2. Similarly, C 4 lies on the right of C 1. If these curves lie on the right of EVC, then we have F C (C 3, β) λ β < 0, F(C 1, β)/c_1 F(C 3, β)/c 3 > 0, C 3 > C 5. Then, C 3 lies on the right of C 2, and similarly C 4 lies on the right of C 1. Case 2. If α > 0, a completely similar argument shows that C 3 lies on the left of C 2, and C 4 lies on the left of C 1. These two cases are illustrated in Figure 4-8, on the left of EVC. Next Theorem establishes the existence and uniqueness of solutions for a Riemann problem which contains β-rarefaction waves. Theorem 4.8 Consider the Riemann problem withu L = (C L, β L ) andu R = (C R, β R ) as the left and right state, whereβ L andβ R is connected with a single β rarefaction wave. There exists a unique solution for this Riemann problem. Proof. Under our assumptions, given β L and β R, then either (i) every point on β = β R can be connected to β L through a β-integral curve, or (ii) every point on β = β L can be connected to β R through a β-integral curve. To fix the idea, we consider case (i), while case (ii) can be treated in a completely similar way. Recall the definition of the function F(C; β, a) in (4.21). We denote now F L (C) = F(C; β L, α (β L ) 1 ), F R (C) = F(C; β R, α (β R ) 1 ). 103

123 Let C, C be the two values where F R reaches its min and max value. Then, there exists two integral curves through each of C and C that connect toβ L. We denote these curves as C 1, C 2, C 1, C 2. On the line β = β L, we denote by I the set of C values that can not be connected to the right with a β-integral curve. Clearly, this set includes the C values between the curves C 1 and C 2, and those between the curves C 1 and C 2. Given u L, we let I L denote the set of C values on the line β = β L such that the Riemann problem (C L, C; F L ) is solved with non-positive speed, and the point C can be connected to β = β R along a β-integral curve. Recall the sets I L and I R used in the proof of Theorem 4.1. We have I L = I L \I. Furthermore, let I L denote the set of the corresponding C values on the line β = β R that can be connected to the set I L through a β-rarefaction curve. We will only consider the case where the β-rarefaction path lies on the left of the EVC, while the other case can be treated similarly. We consider the two Cases in Figure 4-8 separately. Case 1. We assume first that C L lies on the left side of EVC, and we identify the set I L for all cases of C L locations. In Figure 4-9 we show three different situations. If C L < C 2, then contains the interval on the left of C 1 and I L contains the interval on the left of C. The set I L I R includes exactly one point. If C 2 < C L < C 3, then I L contains an addition pointc L, and I L contains an additional point which can be connected to C L through a β-integral curve. The set I L I R includes either one point or two points. If it includes two points, one of then must be the isolated point in I L, which will be selected. 104

124 If C 3 < C L < C 0, we denote the integral curve through C L by C 4 and it corresponding critical curve by C 5. Then the set I L includes the point C L plus the interval on the left of the critical curve C 5. The set I L consists of the point on C 4 and the interval on the left of the point that can be connected to I L with integral curve, where the right endpoint lies on the left of C 5. Thus, the set I L I R includes exactly one point. Case 2. The proof is very similar, except in the case when C 3 < C L < C 2, where there exists composite paths, see Figure In the plot on the left, C L can be connected to C as follows: FromC L, the path follows a β-integral curve, until it intersects with the critical curve C 3 at a. Then it takes a horizontal path, through a C-shock, until it reaches the curve C 1 at b. From there it follows C 1 to reach C. In the plot on the right, we show another path. In fact, at any point a before reaching a, one could take a horizontal path to reach the critical curve of the integral curve through C L at b, then take the β-integral curve from there to reach the line β = β R at a point to the left of C. Thus, we redefine the set I L to include the points on the line β = β R that can be connected to the set I L through a composite path. Clearly, I L includes all C C. Following a same argument as for Case 1, we conclude the uniqueness of the path. Similar to Corollary, we immediately have the next result on the position of the βrarefaction wave. Corollary 4.9 In the setting of Theorem, the path of the β-rarefaction lies on the same side of the EVC as the left stateu L Global existence and uniqueness of solutions for Riemann problems Proof. (Of Theorem 4.1.) We now complete a constructive proof for the main Theorem. Given a left and right state u L = (C L, β L ) and u R = (C R, β R ), the solution of the Riemann 105

125 problem is constructed in two steps. We first solve the β-wave using information beaded on (β L, β R ) and the equation (4.11). This determines the type of β-wave that will connect to the possible C-waves on the left and right. Thanks to Theorem 4.4 and Theorem 4.8, there exists a unique path for the location of the β-wave. Then, the C-waves are constructed by solving the scalar conservation laws, possibly for both left and right equations with β = β L and β = β R. The uniqueness of these C-waves follows from standard theory for scalar conservation laws. Thus, combining with Corollary 4.5 and Corollary 4.11 we complete the proof of Theorem 4.1. We have two immediate Corollaries. Corollary 4.10 The two-phase region is invariant for Riemann problems. Furthermore, the EVC cuts the region into two sub-regions, where each one is invariant for Riemann problems. For example, if both u L and u R lie on the left (or on the right) of the EVC with F L C L and F R C R, then the solution remains on the same side of the EVC and F C. Combining Corollary and Corollary, the next Corollary follows. Corollary 4.11 Let u L = (C L, β L ) and u R = (C R, β R ) be the left and right states of the Riemann problem, where (β L, β R ) is connected with a single β-wave, i.e., either a β-shock or a β-rarefaction wave. Then, the path of β-wave wave and the left state lie on the same side of the EVC. Furthermore, the solution path in the phase plane (C, β) crosses the EVC exactly once. 4.4 Numerical Simulations with Front Tracking The Riemann solver as described in Section 4.3 is implemented in a front tracking algorithm. The results of the front tracking is demonstrated for several examples and are compared with finite difference simulation results. 106

126 Let ε > 0 be the parameter for the front tracking algorithm. We discretize the space for β values, and let B ε = {β n } denote the set of the discrete values for β, with β n > β n 1, β n β n 1 ε n = 1,2,, N 1. (4.34) We let α ε (β) denote the piecewise affine approximation to α(β), with α ε (β n ) = α(β n ) for every n. Next, we need to discretize C along each tie line. Unfortunately, the C grid is not constant and depend on the β-wave. Therefore, we need to update the C ε = {C n,m } after calculating a new β-wave. The set of C ε = {C n,m } denote the discritized values for C, with C n,m > C n,m 1, C n,m C n,m 1 ε C, n = 1,2,, N 1, m = 1,2,, M 1. (4.35) Then we estimate f(s) with piecewise linear f ε (S). The parameter ε C is ε devided by a constant. The discrete initial data is piecewise constant u ε (0, x) = (C ε (0, x), β ε (0, x)), where β ε takes only the values in B ε. Let x i be the points of discontinuities in the discrete initial data. We denote the cell values as β ε (0, x) = β i, C ε (0, x) = C i, x i 1 x < x i. At t = 0, a set of Riemann problems shall be solved at every point x i where the initial discrete data has a jump. The rarefaction fronts are approximated by jumps of size less than or equal to ε (Figure 4-11, Left). One can use the result of the Theorem 4.5 to calculate the intermediate points, where both approaches result to the same solution (Figure 4-11, right). Each front is labeled to be either a C- or β-front, and it travels with Rankine-Hugonoit velocity. At a later time t > 0 where two fronts meet, a new Riemann problem is solved. The process continues 107

127 until the final time T is reached. In the case of a variation in injection conditions, the initialization process should be repeated. The β and C values calculated by front tracking has a significantly different behavior. The definition of α ε (β n ) constrains the values of β in the solution to B ε, unless the initial data contain values out of B ε. However the C values of the solution are not necessarily in C ε even if all the initial data are in C ε. Therefore, to control the number of fronts, C-waves with the same velocity should be merged into one C-wave, and C-waves smaller than a threshold should be eliminated. We used a three component system with properties shown in tables 4.1 and 4.2 at 2650 psia and 160 o F to demonstrate front tracking algorithm use of the global Riemann solver. Peng Robinson (1976) equation of state is used to calculate phase compositions. Slug injection is commonly used in gas flooding where the boundary condition at x = 0 is changed at different times (or cycles). Furthermore, the finite difference simulation with single point upwind flux estimation is used to simulate gas flooding. We compared the simulation results of the front tracking algorithm with the finite difference simulations. Example 2 has initial oil shown by R in Figure 4-12 and slug composition by L 1, which changes to L 2 at t = 0.2. Figure 4-12 compares the compositions at t = 0.8 and Figure 4-13 shows the comparison of composition profiles at t = 0.8. Example 3 is simulation of a problem with variable initial condition. In addition, the composition at x = 0 is varied at different times to mimic the slug injection process. Figure 4-14 shows the fronts of the example, and Figure 4-15 shows the profiles at different times. 108

128 4.5 Initial and injection tie-line selection There is a potential problem with the solution of the tie-line route that requires additional explanation. Larson (1979) proved that shocks in and out of the two-phase region must shock along a tie-line extension. However, multiple tie lines can extend through the initial and injection compositions if they are single-phase compositions, especially for more complicated phase behavior that includes bifurcation of a two-phase region. Thus, the correct initial and injection tie lines for use as boundary conditions in Eqs. (4.11) and (4.14) must be carefully selected. In general, we first assume that the tie line found from a negative flash calculation that extends through the initial or injection composition is the correct tie line. This may or may not be true, and a method is needed to check its validity. One possibility is to first solve the tie-line route based on these boundary conditions and then determine the entire composition route. If the composition route is physical (solution is single-valued, satisfies the mass balances and entropy conditions) the correct tie lines as boundary conditions are selected. Alternatively, a simple test could be done without the need to calculate the entire composition route. We determine that test next. We first solve for the entire tie-line route to identify rarefactions and shock, and their corresponding velocities in tie-line space. Next, we transform the auxiliary relations for the boundary conditions in x D t D coordinates to the Lagrangian coordinates of Eqs. (4.9) or (4.12), where only single-phase initial and injection compositions are considered (a two-phase composition has a unique tie line and no test is needed). The transformed boundary conditions for a single-phase composition that is the intersection of multiple tie lines with constant Γ gives, φ = C 1,k ψ, Γ = Γ k, k = inj, ini. (4.36) 109

129 We then check if the solved tie-line route satisfies the transformed boundary conditions by comparing the eigenvalues at the initial and injection compositions to the eigenvalue of any rarefaction or shock associated with the tie line that extends through that composition. For example, the eigenvalue of the tie line through the initial composition (or the shock velocity from that tie line to another tie line) must be less than the eigenvalue associated with the initial composition as given by Eq. (4.36). For the injection composition, the reverse is true. Equation (4.36) defines the single-phase eigenvalues ( d / d ) associated with the boundary conditions in Lagrangian space as 1/C 1,ini and 1/C 1,inj (slopes of Eq. (4.36)). For C 1,k = 0, however, the slope is infinite so that special care must be taken. To consider this singularity, we relate the rarefaction and shock velocities to their intersection compositions as defined by C x 1 = 1/λ β for rarefactions and C x 1 = 1/Λ for shocks and give an equation for the validity of the tie-line selected as, sgn(c x 1,k C 1,k )sgn(c x 1,k EVC(Γ k )) > 0, k = ini, inj. (4.37) That is, Eq. (4.37) determines if the proposed tie-line route with the specified initial or injection compositions will violate the velocity condition (single-valued solution). Equation (4.37) is useful as a simple check of the validity of a tie line found by a negative flash. However, if the tie line is found not to satisfy Eq. (4.37) one must search for another tie line that extends through the initial or injection composition. Li and Johns (2007) and later Li et al. (2012) developed a constant K-value flash to calculate tie lines without calculating saturations. Their method can be used to calculate all tie lines that pass through a composition. Juanes (2008) also developed a similar method to find all tie lines that extend through a single-phase composition when K-values are constant, but that method is not practical as the number of components increase. Both methods are not satisfactory for our purpose because tie lines that 110

130 intersect at a single-phase composition (initial or injection composition) can also have different K-values. In this paper, we develop a new and more robust approach to determine tie lines with different K-values that passes through the initial or injection composition Consider the case that the tie-line route using the initial tie line Γ ini1 does not satisfy Eq. (4.37) and therefore, we should find another tie line, Γ ini2, that passes through the initial composition. We know that Γ ini1 and Γ ini2 intersect at the initial composition, therefore these two tie lines, if hypothetically connected by a nontie-line shock, would have a shock velocity of Λ = 1/C 1,ini. The problem therefore reduces to finding a hypothetical nontie-line shock with the velocity determined by the intersection point. Thus, we use Eqs. (4.25) to find Γ ini. That is, we perform a line search along 2 each Hugoniot locus given by Eqs. (4.25) and test if Λ = 1/C 1,ini. 4.6 Example ternary displacements We first construct solutions in tie-line space for ternary displacements to illustarate the advantages of the new MOC approach. The first example is a simple case where K-values are independent of composition. The second example is complex case with bifurcating phase behavior as shown in Ahmadi et al. (2011) and Chapter 3. One of the key steps in the solution for ternary displacements in tie-line space is to show whether the injection and initial tie lines are connected with a shock or a rarefaction wave or a combination of both. In addition, multiple tie lines may pass through the initial or injection compositions. Thus, we must check if the tie line from a negative flash converged to the appropriate tie line using Eqs. (4.37). The two-phase region with constant K-values (0.05, 1.2, and 2.5) is shown in Figure 4-16 by the solid lines. Three tie lines are also shown in Figure 4-16, labeled TL 1, TL 2 and TL 3. The 111

131 composition path for displacements with this type of phase behavior is well studied (Johns 1992, Orr 2007, Seto and Orr 2009). First we consider displacement of I 1 with J 1, where only one tie line extends through the initial or injection composition. In composition space, the nontie-line path takes the rarefaction wave indicated by R 1 in Figure 4-16 that goes from the initial tie line (TL 3 ) to the injection tie line (TL 1 ). A rarefaction wave occurs here because the velocity of the nontie-line eigenvalue given by the eigenvalue problem formed from Eqs. (4.1) increases from the injection tie line to the initial tie line. Details of the composition route can be found in Orr (2007). In tie-line space, we solve directly for the tie lines using Eqs. (4.11) or (4.14) without considering composition space. The tie-line coefficients (Eqs. (4.6)) for all tie lines between TL 1 and TL 3 are shown in Figure Because C 1,J1 < EVC(Γ J1 ), the eigenvalues in tie-line space are the slope of the curve of α versus β. Therefore, the single eigenvalue in Lagrangian coordinates ( dα/dβ) increases from TL 1 to TL 3 and the two tie lines are connected with a rarefaction wave as they were in the composition space. Figure 4-18 shows graphically the scalar equation solved, which is similar to how Buckley-Leverett (1942) displacements are presented. Thus, displacements in tie-line space for ternary systems are easier than Buckley-Leverett solutions because the flux function in Figure 4-17 is strictly concave. The eigenvalue is given along the x- axis in Figure 4-18, where dψ/dφ = dα/dβ (Eq. (4.14)). Other solutions in both composition and tie-line space are given in Figure 4-16 to Figure If we displace I 2 by J 2, TL 3 is connected to TL 1 by the shock S 1 shown in Figure 4-16 by the dashed line and in Figure 4-18 by the dotted line. A shock must occur here because the eigenvalue decreases from the injection to initial tie line. This shock is a tangent shock (like a Welge tangent shock) that satisfies the entropy condition because it does not cut through the flux function in Figure

132 Next, we construct the solution for displacement of I 3 with J 1, where two tie lines, TL 2 and TL 3, now pass through I 2. A negative flash calculation will typically yield TL 3. Once a tieline route is constructed we apply the test of Eq. (4.37) for that tie line to see if it is correct. x Equation (4.37) is only satisfied, however, for TL 2 because the value of C 1 is equal to C 1 at a as shown in Figure Thus, TL 3 is discarded and we search for another tie line using the procedure outlined previously. It is easy to see in Figure 4-16 that the only remaining tie line is TL 2, so a tie-line route is constructed with that tie line. The correct (physical) tie-line route exhibits a rarefaction wave from TL 2 to TL 1 (see R 2 in Figure 4-16 and in Figure 4-18). In the last example for constant K-values, we discuss the displacement of I 4 with J 3 in Figure 4-16, where now C 1,J3 > EVC(Γ J3 ) and the positive sign is used in Eqs. (4.11). The single eigenvalue is therefore equal to the slope of the curve of α vs β in Figure Thus, the eigenvalue increases from TL 2 to TL 1, giving a rarefaction wave in tie-line space (Figure 4-18). In composition space, the nontie-line rarefaction wave in Figure 4-16 is labelled by R 3. The composition route is clearly different from R 2, while the tie-line route follows the same tie lines, but in reverse order. Now, we construct the solution in tie-line space for three-component displacements with bifurcating phase behavior to demonstrate the application and simplicity of using MOC in tie-line space (see Figure 4-19). The corresponding MOC solutions using Eqs. (4.1) in composition space were complex to construct and can be found in Khorsandi et al. (2014). The solutions are identical, but in tie-line space the problem is as simple as the Buckley-Leverett problem with an S-shaped flux function. The tie-line coefficients for the tie lines in Figure 4-19 now give an S- shaped flux function because of the bifurcating behavior (see Figure 4-20). The initial and injection compositions considered along with component properties for the PR EOS are given in Table

133 Two displacements for Figure 4-19 are examined here with the same injection composition J 1. Because C 1,J1 < EVC(Γ J1 ), we use the negative sign in Eqs. (4.14). The eigenvalue shown in Figure 4-21 is equal to the slope of the curve α vs. β. For the displacement of I 2 with J 1, there is only one tie line that passes through I 2. The solution for displacement of I 2 by J 1 in tie-line space, therefore, is a rarefaction wave to TL 3 followed by a tangent shock from TL 3 to TL 5 (see Figure 4-20 and Figure 4-21). There are three possible tie lines for initial composition I 1, however. We can construct the solution in tie-line space for each one and then x check if the solution satisfies Eq. (4.37). For TL 5, C 1 is equal to C 1 at point a in Figure 4-19, x which violates Eq. (4.37). Similarly, for the tie-line path of TL 4, C 1 is equal to C 1 at point b, x which again violates Eq. (4.37). For TL 2, C 1 is outside of the phase diagram and satisfies Eq. (4.37). Therefore TL 2 is the correct tie line and the rest of the solution route is easily generated as a rarefaction from TL 1 to TL 2. Figure 4-22 gives the tie-line solutions plotted against the Lagrangian coordinate. 4.7 Summary We demonstrated the existence and uniqueness for solution of global Riemann problem for a two phase flow model with three components. The construction of a Riemann solution was then used in a front tracking algorithm, allowing constructing solutions for slug injection problems. A more interesting and challenging problem is the existence of entropy weak solutions for the Cauchy problem, established as the convergence limit of the front tracking approximate solutions. Towards this goal, one needs to establish proper a-priori estimates on the approximate solutions, in particular, some bounds on the total variation in certain form for compactness. The 114

134 key step in these analysis is the wave interaction estimates. In the literature among models on reservoir simulation, the existence of entropy weak solutions is only available for non-adsorptive models for two phase polymer flooding, under specific assumptions. For the gas flood problems, due to the various degeneracies and the nonlinear resonance, this remains an open problem. Table 4.1 Fluid characterization for the ternary system T o C F P C (psi) ω C CO C Table 4.2 Binary interaction coefficients for the ternary system CO 2 C 1 C CO Table 4.3 Initial condition for example problem 3 < x x < C β Table 4.4 Injection condition for example problem 3 < t t < C β Table 4.5 Component properties and compositions for bifurcating phase behavior developed based on Ahmadi et al. (2011) Properties Compositions T c ( o F) P c (psia) ω I 1 I 2 I 3 J 1 BIP C 1N C 1N 2 CO 2 nc 4 CO , nc C

135 Figure 4-1: Illustration of three-component phase diagram with constant K-values (K 1, K 2, K 3 ) = (0.05,1.5,2.5). Left plot uses the C 1, C 2 ) coordinate, which the right plot uses the (C, β) coordinate. The two red curves are the boundary of the two-phase region, and green lines are tie lines. Figure 4-2: Integral curves for the β-family in the phase plane (C, β), corresponding to the case in Figure 4-1. Here, the red curves are the boundary of the two-phase region and are called binodal curves. 116

136 (a) Figure 4-3: Functions F(C, β) and F(C; β, a). a=0.2 on the left plot. (b) Figure 4-4: Solutions to Riemann problems for C-waves. Left: If C L < C R, the lower convex envelope L M 1 M 2 R gives a shock L M 1, a rarefaction fan M 1 M 2 and a shock M 2 R. Right: If C L > C R, the upper concave envelope L M R gives a rarefaction fan L M and then a shock M R. 117

137 Figure 4-5: Illustration for β shock. Here the red and blue curves are graphs for F L and F R, and is the point ( σ β 1, σ β 1 ). The green line has slope σ β. Then, C L and C R must be selected from the corresponding graphs of F L and F R that intersect with the green line. Figure 4-6: The set I L and J L are the x and y coordinates for the thick curves in (L1)-(L4). The set I R and J R are the x and y coordinates for the thick curves in (R1)-(R3). 118

138 Figure 4-7: Riemann solver for the special case, where a tie-line is tangent to the two phase region, plots of the functions C F(., β L ) and C F(., β R ), where blue curve is for the left state, and red curve is for the right state. Case 1 Case 2 Figure 4-8: Two possible relations between the curves C 1, C 2, C 3 and C 4. Figure 4-9: Three situations for different locations of C L and the corresponding sets of Ĩ L (with thick three line on L) and Î L (with thick red line on R). 119

139 Figure 4-10: Case 2, when C 3 < C L < C 2, the β-wave path consists of two β-rarefaction waves with a C-contact in between. Figure 4-11: Estimation of large β-rarefaction with smaller waves (Left) and convergence of results to the correct solution (Right). 120

140 Figure 4-12: Comparison of the composition path calculated by finite difference simulation and front tracking. Figure 4-13: Comparison of the composition profiles calculated by finite difference simulation using 10,000 grid blocks and front tracking with ε =

141 Figure 4-14: Fronts for variation of initial condition where two slugs are injected. 122

142 Figure 4-15: Composition profiles at t = 0.0 (bottom),0.1,0.2,0.3,0.4 and 0.5 (top). 123

143 Figure 4-16: Analytical solution for ternary displacement in composition space. Point a is on the envelope curve and C 1 a = 1/λ β. Figure 4-17: Tie-line coefficients for ternary phase behavior of Figure 1 in tie-line space. 124

144 Figure 4-18: Analytical solution for three-component displacements showing shocks and rarefactions in Lagrangian coordinates. Figure 4-19: Injection and initial compositions considered for bifurcating phase behavior (Ahmadi et al. 2011). Three tie lines extend through composition I 1. Points a and b lie on the envelope curve (see Khorsandi et al. 2014). 125

145 Figure 4-20: Tie-line coefficients for three-component bifurcating phase behavior in tie-line space, where negative sign in Eqs. (3) is used. Figure 4-21: Eigenvalue for three-component displacements with bifurcating phase behavior. Dashed line is a shock. 126

146 Figure 4-22: Analytical solution showing shocks and rarefactions in Lagrangian coordinates. 127

147 Chapter 5 Tie-line routes for multicomponent displacements In this section we extend the splitting technique developed in Chapter 4 to multicomponent gas displacements. First, we describe the mathematical model in tie-line space for multicomponent displacements. The models rely on a concrete parametrization of tie-line space which are discussed next. Finally, the tie-line route is constructed for multiple displacements. 5.1 Mathematical model The calculation of MMP relies on the accurate development of the dispersion-free composition route using the method of characteristics. The multicomponent 1-D dispersion-free displacements can be modeled with (Helfferich 1981) C i t D + F i x D = 0 i = 1,, N c 1, (5.1) where C i and F i are overall volume fraction and overall fractional flow of component i, t D is dimensionless time or pore volume injected (PVI), x D is dimensionless length, and N c is the number of components. Equations (5.1) can be split into two parts; equations that depend only on tie lines, and an equation that depends on flow. Splitting is achieved using the following Lagrangian coordinate transformation developed in Chapter 4, φ xd = C 1, φ td = ±F 1 and ψ = x D t D. (5.2) 128

148 The signs in Eqs. (5.2) are determined based on the location of the injection composition relative to the equal velocity curve (EVC), where the EVC curve is defined as the curve in the two-phase region where F 1 equals C 1. The upper sign of Eqs. (5.2) is used when C 1inj > C 1EVC (Γ inj ), while the lower sign is used otherwise. Γ is a N c 2 vector that has elements with unique values for each tie line. The condition for defining the tie-line space by vector Γ is described in Appendix A. Substitution of Eqs. (5.2) and the tie-line equations in Γ space (shown in Eqs. (5.5) and (5.6)) gives the following transformed set of equations that depends solely on phase behavior. β i φ ± α i ψ = 0 i = 1,, N c 2. (5.3) where α i and β i are tie-line coefficients, and α i = f(β 1,, β Nc 2) i = 1,, N c 2. (5.4) The tie lines and their coefficients are defined by, C i+1 = α i C 1 + β i i = 1,, N c 2, α i = x i+1 1 (1 K i+1 ) x 1 1 (1 K 1 ), β i = x i+1 1 (K 1 K i+1 ) K 1 1 i = 1,, N c 2, (5.5) (5.6) where x 1 i is mole fraction of component i in phase 1 and K i is the K-value of component i. The sign for the second term of Eqs. (5.3) is positive if C 1inj > C 1EVC (Γ inj ) and negative otherwise. The eigenvectors are independent of the sign in Eqs. (5.3), but the sign of the eigenvalues change correspondingly. We describe the solution for Eqs. (5.3) with positive sign because the solution for negative sign is very similar. 129

149 fractional flow. The final equation that results from the transformation of Eqs. (5.1) is dependent on φ ( C 1 F 1 C 1 ) ± ψ ( 1 F 1 C 1 ) = 0. (5.7) Ideally, Eq. (5.7) could be used to calculate the composition route once the tie-line route is determined. However, this equation is singular in the single-phase region and at the EVC. In addition the flux, 1 C, is not a single-valued function of the conserved quantity 1. Thus, Eq. F 1 C 1 F 1 C 1 (5.7) is not used further in this paper, and instead, Eqs. (5.3) are first solved for the tie lines using the specified boundary conditions, and then in a separate step Eqs. (5.1) are solved by mapping the fractional flow onto the fixed tie lines (similar to Gimse and Risbero, 1992). The characteristic equation using Eqs. (5.3) are (R λ i ) dβ dψ = 0 i = 1,, N c 2, (5.8) where R is a N c 2 by N c 2 matrix with elements R ij = α i β j. The elements of matrix R can be calculated analytically as shown in Appendix A. The eigenvalues of matrix R are related to the envelope curve of the tie lines. Each tie line is tangent to N c 2 envelope curves (Dindoruk 1992) at the corresponding composition, C e 1,k, so that for the k th eigenvalue, e C 1,k = 1 k = 1,, N λ c 2. k (5.9) A series of tie lines tangent to the same envelope curve form a ruled surface and the ruled surfaces can be calculated by integrating the eigenvectors of Eqs. (5.8), given by dβ. These ruled dψ 130

150 surfaces are planar when K-values are constant (independent of composition) as shown in Appendix C. The eigenvalues of the characteristic equation (Eqs. (5.8)) give the allowable rates for the overall Lagrangian flux that can be taken to solve the strong form of Eqs. (5.3), while the eigenvectors give allowable N c 2 ruled surfaces on which the composition route must lie as described by Johns (1992) and Johns and Orr (1996). The eigenvalues for the tie lines in tie-line space can be both positive and negative, but are always real. The solution consists of following the eigenvector paths (the strong form) from the initial to the injection condition, where the solution in tie-line space should be single valued. A single-valued solution means the path taken in tie-line space should have a corresponding monontonic, but increasing eigenvalue from the injection tie line to the initial tie line. If the tie-line solution becomes multivalued based solely on the strong form, the weak form (shocks) must be introduced. Shocks in tie-line space must be taken that avoid multivalued solutions, but also must satisfy the entropy conditions. The Lax (1957) and Liu (1976) entropy conditions or the vanishing viscosity condition (Bianchini and Bressan 2005) can be used to check shock admissibility. The eigenvalues are easily written for simplified systems using Eqs. (5.8). For threecomponent gas floods, there is only one eigenvalue compared with two in the traditional method that solves Eqs. (5.1). This makes the solution of three-component problems simpler, where the sole eigenvalue is equal to dα and its eigenvector is trivial (arbitrary scalar). For four-component dβ displacements, there are two eigenvalues given by, (5.10) 131

151 where the eigenvalues can be calculated as, λ = α 1 + α 2 ± ( α 1 + α 2 2) 4 ( α 1 α 2 α 1 α 2 ) β 1 β 2 β 1 β 2 β 1 β 2 β 2 β 1 2. (5.11) Each term in Eq. (5.11) can be calculated analytically as shown in Appendix A. Once an eigenvalue is determined, its corresponding eigenvector can be calculated from Eqs. (5.10) or more generally for any number of components from the matrix R given in Eqs. (5.8). Shocks must be conservative. Thus, the upstream and downstream values must satisfy the weak solution (Rankine-Hugoniot condition) for Eqs. (5.3), Λ = Δα i Δβ i i = 1,, N c 2, (5.12) where Λ is the shock rate of change of the overall Lagrangian flux. Although Λ is not a velocity, we use the term shock velocity loosely as shorthand for Λ (and for λ) in this paper. Eqs. (5.12) are satisfied if the two tie lines intersect at C 1 = 1/Λ. (5.13) Johns and Orr (1996) showed a similar result to Eq. (5.13), where the velocity of the shock was calculated by a triangular geometric construction in composition space. In tie-line space, however, we do not need to apply the geometric construction to find the tie-line routes, although, if desired it can be used to find the upstream and downstream compositions of the shock once the tie-line route is known. 132

152 5.2 Tie-line space There must be a one-to-one relationship between each tie line in tie-line space and composition space for the transformation to be useful. First we discuss the thermodynamic definition of the tie-line space. Next we use the inverse function theorem to define necessary conditions for one-to-one correspondence in tie-line space. Last, we discuss a simple transformation to show that parameters that describe tie-line space are conservative. Unfortunately the Gibbs phase rule cannot be used directly to define the tie-line space. The Gibbs phase rule is the fundamental for thermodynamic phase equilibrium calculation. Gibbs (1875) demonstrated that a system of r coexistent phases, each of which has the same n independently variable components is capable of n + 2 r variation of phase. However, the Gibbs phase rule is usually inaccurately interpreted (Prausnitz et al. 1998, Smith et al. 2001) as the state of the system can be determined by specifying F = n + 2 r state variables. This extension of Gibbs phase rule is not always true. A contradictory example is the thermodynamic state of two-phase water. Based on the extended phase rule, the state of a two-phase water system is determined by specifying one intensive property such as pressure. However there are four two phase states at atmospheric pressure as shown by red squares in Figure 5-1. The correct interpretation of Gibbs phase rule is that the change of any intensive property of the system can be calculated by specifying the change in one of the intensive properties of the system at a twophase state, assuming that the system remains two-phase after the change. 133

153 5.2.1 Thermodynamic definition Equilibrium is achieved for equality of the component fugacities at the same temperature and pressure. Fugacity is a function of temperature, pressure and phase composition. Therefore, at equilibrium the tie lines satisfy, f i 1 = f i 2, i = 1,, N c, Σx i 1 = 1, Σx i 2 = 1. (5.14) where the superscript indicates the phase. There are 2N c unknown compositional variables and N c + 2 equations in Eqs. (5.14). The system of equations is completed by adding N c - 2 definitions of tie-line space parameters, γ i, g i (x 1 1,, x 1 Nc, x 2 1,, x 2 Nc ) = γ i i = 1,, N c 2. (5.15) A simple form of g i can be used such as γ i = 0.5(x i 1 + x i 2 ) (Voskov and Tchelepi, 2008), or γ i = x i 1, or γ i = x i 2, or γ i = β i (see Eqs. (5.6)). In a well-defined tie-line space, each tie line is determined by a unique Γ. Γ = [γ 1,, γ Nc 2]. (5.16) The corresponding tie line for a given Γ can be calculated by solving Eqs. (5.14) and (5.16). C 1 should also be specified to calculate the moles of a phase. That is, n 1 = C 1 x 1 2 x 1 1 x 1 2. (5.17) Therefore, we can calculate molar properties of the reservoir fluid by specifying the vector v. 134

154 v = [C 1, γ 1,, γ Nc 2]. (5.18) Typically in reservoir engineering, overall composition C is used to calculate molar properties of a fluid at constant temperature and pressure. C = [C 1,, C Nc 1]. (5.19) We need to show that there is one-to-one relationship between all physical values of C and v and also that Γ is constant along a tie line. If these conditions are true, we can conclude that Γ is well defined. To demonstrate the above conditions, we define an implicit function f that maps composition space to tie-line space. The exact form of f is not necessary to determine if the transformation is one-to-one, but we just need to show that f is invertible. Thus, the mapping is expressed by, C = f(v). (5.20) A tie-line definition is valid, if (1) f, defined below, is a one-to-one function, (2) dγ is zero along each tie line and (3) x 1 y 1 for all tie lines. Based on the inverse function theorem, an implicit function is one-to-one if df is locally invertible everywhere in the domain of the function. C 1 C 1 C 1 C 1 γ 1 γ Nc 2 df =. C Nc 1 C Nc 1 C N c 1 [ C 1 γ 1 γ Nc 2] (5.21) 135

155 We can remove the first row and column of df because those elements are either 1.0 or 0.0, which means df = dγ where dγ is defined as C 2 C 2 γ 1 γ Nc 2 dγ = = A C 1 + B, C Nc 1 C N c 1 [ γ 1 γ Nc 2] (5.22) where A and B are derivatives of the tie-line coefficients in tie-line space as calculated in Appendix A. The condition number of dγ can be used to determine if dγ is invertible (condition number must be finite for invertibility). The inverse function theorem requires checking the condition number of dγ for all tie lines, which is not practical. However, we can check the criterion at a limited number of tie lines. For example, we can generate a series of tie lines using the mixing cell method of Ahmadi et al. (2011) with only a few contacts, and find the best choice of Γ, i.e. the choice for the tie-simplexes in Eqs. (5.16) that give the smallest average condition number for dγ. Tie-line space as defined above can be used to research phase behavior. One more condition, however, should be checked before using the tie-line space definition for analytical solutions using the method of characteristics because the weak solution of hyperbolic systems can change for a nonlinear change of variables (Gelfan 1963). We define a conservative tie-line space such that the weak solution in tie-line space is the same as the weak solution of Eqs. (5.1). The transformation should satisfy Δg(β) = g(δβ), where the β i are the conserved quantity in Eqs. (5.3). We define tie-line space in such a way that β i provides a one-to-one mapping from composition space to tie-line space. The β values are unique for each tie line if the envelope curves of the tie lines do not intersect the hyperplane of C 1 = 0. Therefore, we can define a new composition C i such that the hyperplane of C 1 = 0 lies inside the two-phase region. This would 136

156 ensure that the envelope curves do not intersect the hyperplane because intersection of tie lines inside the two-phase region is unphysical. The details of transformation are described in next section Composition space parametrization We would like to transform the composition coordinates and define a new composition space such that β can directly be used to parametrize tie lines without any of the problems discussed in previous section. Furthermore the flat ruled surface will appear as lines in the new tie-line space. We can transfer the composition space linearly and define a new set of conserved quantities by Eqs. (5.23), C = QC + IC r, (5.23) such that for all tie lines x 1 < 0 and y 1 > 0. C is a composition and C is the same composition in the new coordinates. All elements of C r beside the first one are zero. The following are the steps required to define the composition space. 1. Generate a random set of compositions and calculate the tie lines for these compositions. We used 5N c as the size of data set. An estimate of the composition route can be used as the data set as well. The estimate can be generated by simulation or mixing-cell. 2. Fit a plane to the middle points of the tie lines using the least square algorithm. The middle points are defined by z i = x i+y i, i = 1,, N 2 c The normal vector of the fitted plane defines the first base vector for the new composition space and the intercept of the plane is the value of C r,1. 137

157 4. The set of orthonormal bases are calculated using Gram-Schmidt process (Dukes et al. 2005). 5. The base vectors calculated in step 4 are rows of Q. The inverse of Q can be calculated for inverse conversion form C to C. We used this algorithm to define the new composition space for many different fluids. The error of least squares in step 2 is usually very low values, even for fluids with complex phase behavior. Figure 5-2 demonstrates the new coordinates with red arrows. The new coordinate helps to eliminate negative flash calculations as well as described in next section. Flash calculation in new composition coordinate The steps required to calculate phase compositions of a tie line are described as follow. 1. The composition corresponding to the tie line is, C 1 = 0 and C i+1 = γ i, i = 1,, N C 2, where γ i = β i. 2. Calculate C using inverse of Eqs. (5.23). This composition is guaranteed to be inside the two phase region. 3. Perform flash using C as feed and calculate phase compositions. 5.3 Example ruled surface routes Constant K-value displacement We show for any N c that the ruled surfaces defined by the eigenvectors are planar if K- values are not composition dependent. Dindoruk (1992) and Johansen et al. (2005) showed the 138

158 same result for four-component displacements. First we show that shock loci are planar surfaces in hyperspace, then we demonstrate that ruled surfaces generated from the eigenvalues coincide with the shock loci and are therefore planar as well. The shock loci are given by (Rankine Hugoniot) condition. Λ = Δα i Δβ i = ( x 1Δx i x i Δx 1 x 1 (x 1 + Δx 1 )Δx i ) ( 1 K i K i K 1 ) i = 2,, N c 1. (5.24) The term x 1 (x 1 + Δx 1 ) is common in Eqs. (5.24). Therefore we define the modified shock velocity Λ = x 1 (x 1 + Δx 1 )Λ and rearrange Eqs. (5.24) in a form similar to an eigenvalue problem, where x i are solved. N c 1 K Nc K j [ (x 1 + x i ) ( 1 K i ) Λ ] Δx K 1 K Nc K i K i x i ( 1 K i ) ( K N c K j ) 1 K i K 1 K 1 K Nc = 0 i = 2,, N c 1. j=2 j i Δx j (5.25) The coefficient matrix in Eqs. (5.25) is not a function of Δx i, which means that shock loci are straight lines in tie-line space. In addition, the binodal curves are planes for phase behaviors with constant K-values. The tie lines of a shock locus intersect two straight lines, the upstream tie line of the shock and the shock locus on the binodal surface. Therefore, the shock loci are planes in composition space. The eigenvectors can be calculated by the limit of Eqs. (5.25) as Δx i goes to zero. Because shock loci lie in a hyperplane and an eigenvector is always tangent to the shock locus, the eigenvector is constant along that shock locus. Therefore, the ruled surfaces given by the eigenvectors are also planar and coincide with the planes of the shock loci. 139

159 5.3.2 Example four-component displacements In this section, we give tie-line routes for various four-component displacements. The first example considered is one where K-values are constant and there are multiple crossover tie lines that satisfy the conditions for shock-jump MOC (Yuan and Johns 2005). The second case considered has K-values that change with composition based on the PR EOS. This second case is the same phase behavior and displacement as the one considered in Dutra et al. 2009, but here we show that there is no complex eigenvalue as they reported. The third example is a four component displacement with bifurcating phase behavior. This case is too complex to be solved in composition space using the eigenvalue problem based on Eqs. (5.1), but here we show it is not difficult to solve it in tie-line space using Eqs. (5.3). Johns and Orr (1996) showed that the multicomponent solution consists of N c 2 pseudo-ternary displacements in composition space. Each of these pseudo-ternary displacements are along one ruled surface of the route for one family of nontie-line eigenvalues. In tie-line space, the four-component solutions have one less eigenvalue and only the eigenvalues corresponding to the ruled surfaces remain. That is, there are N c 2 eigenvectors in tie-line space, so for four components there are two dependent parameters to describe tie-line space and ternary diagrams can be used to represent the solutions. The eigenvalue in tie-line space that is the largest is associated the eigenvector path we term the fast path. The slow path is for the smallest eigenvalue. Again, these eigenvalues are not velocities like they are for Eqs. (5.1), but for simplicity we keep that terminology. Physical solution routes must give single-valued solutions (velocity conditions must be satisfied) and any shocks present must satisfy the entropy conditions. Yuan and Johns (2005) demonstrated the possibility of multiple solutions for the crossover tie line in the shock-jump MOC method. They considered relatively simple four- 140

160 component phase behavior using K-values independent of composition (see K-values and compositions in Table 5.1). We consider the same case here to show that tie-line route is easily found using tie-line space. The initial and injection tie lines, and multiple crossover tie line solutions for the shock-only MOC approach are given in Figure 5-3. Yuan and Johns (2005) only gave two of these three crossover tie lines, while we found three. Only one tie line extends through the initial or injection compositions. The ruled surfaces that a composition route must follow in tie-line space are shown in Figure 5-4. The ruled surfaces in tie-line space are not surfaces, but are curves, or in this case lines because K-values are constant. The shock loci coincide exactly with these lines as well (see Appendix C). Figure 9 shows several fast and slow paths for two different representations, one using a ternary diagram of β i where 3 i=1 β i = 1, and another a Cartesian plot of the equilibrium phase composition of one of the phases (x i ). Thus, with four components the tie-line route is given only by two parameters. Unlike the composition route solution using Eqs. (1) where there are 2(N c 2) umbilic points for each tie line, there is only one equal eigenvalue point (umbilic point) in tie-line space for this problem. The solution route can switch from one eigenvector path to another at any of these umblic points in composition space. This fact greatly simplifies the construction in tie-line space since there is only one point where a switch can be made. Figure 5-5 shows the eigenvalues in tie-line space and the single umbilic point along the β 1 β 3 axis. No other umbilic points exist in tie-line space. The three possible crossover tie lines according to the shock jump MOC method are also plotted in Figure 5-4. The only physical route that can be constructed is the one with crossover tie line 2 (solution 2 in Figure 5-4). Figure 5-4 shows that the correct solution consists of two shocks S 1 and S 2 in tie-line space. Crossover tie lines 1 and 3 are not possible because they would give a multivalued solution (violate the velocity condition). For example, for crossover tie-line solution 1, the route would have to traverse from the injection point in tie-line space along the slow path 141

161 that corresponds to the β 1 β 3 axis and then continue along the fast path portion of the same axis until a path to the initial tie line is found. A switch, however, at the corresponding slow path through the initial tie line is not allowed (violates velocity condition). Other combinations of shocks and rarefaction waves can also be eliminated for the same reason. The tie-line route does not change if we reverse the initial and injection tie lines. This is easily shown because the slow paths just become fast paths and vice versa. The composition route, however, will change as we showed for the ternary displacements in Figure For the reversed displacement, the sign changes for the second term in Eqs. (5.3). This result is important because it may explain why complicated reservoir flow simulations are bounded by the mixingcell tie lines (see Rezaveisi et al. 2015). Dutra et al. (2009) studied a four-component displacement using similar, but slightly different transformation parameters. Their solution with input parameters given in Table 5.2 exhibited a region with complex eigenvalues. Figure 5-7 shows tie-line space for this problem at 2900 psia and 170 o F using the equilibrium phase mole fractions. There is only one tie line that extends through the initial and injection compositions, but as discussed previously there are multiple possible crossover tie lines using the shock jump MOC method. Here, we just show the correct solution, which consists of a shock S from tie line J 1 to TL 1 that nearly coincides with the slow path, followed by a rarefaction wave from TL 1 along the fast path to tie-line I 1. The eigenvalues are always real in our calculation unlike that of Dutra et al. (2009). The ruled surface formed by the Rankine-Hugoniot conditions (Eqs. (5.12)) is not planar in this case and does not coincide exactly with the ruled surface associated with the slow eigenvalue. These two surfaces (in tie-line space) are tangent at the tie line point at which the shock is to occur. Thus, the shock velocity at this point is equal to the eigenvalue at that point. Last, we consider a four-component fluid with bifurcating phase behavior is created by adding nc 4 to the three-component system of Ahmadi et al. (2011) (Table 4.5). We construct the 142

162 tie line route for displacement of I 3 in Table 1 by pure CO 2. Figure 5-8 demonstrate the composition path for this system in composition space. Figure 5-9 demonstrate the ruled surfaces along with the tie-line route for the displacement of I 3 by pure CO 2. The solution in tie-line space follows the path for the slower eigenvalue with rarefaction R 1 form TL J1 to TL 1. Then the solution follows the fast eigenvalue with the rarefaction R 2 to TL 2 and then the tangent shock S 1 to TL I3. Bifurcation occurs along the part of the tie-line route for faster eigenvalue. Figure 5-8 shows that the tie-line length does not change monotonically along the fast path. Contrary to the assumptions of current MOC methods, the shortest tie line is not the crossover, injection or initial tie lines. However, TL 3 is the shortest tie line in simulation results, because the shock is estimated with a shock layer in simulation (see appendix C). TL 2 approaches to TL 3 as pressure increases, and become the same critical tie line at MMP. 5.4 Summary We developed a transformation to split the flow equations without singularities. The splitting technique is applied to calculate all tie lines in multicomponent displacements where both shocks and rarefactions exist. The tie-line path is constructed for complex phase behaviors that have not been solved before. The solution in tie-line space is robust and does not have the nonuniqness problem of shock-jump MOC solutions. Furthermore, the examination of phase behavior in tie-line space revealed additional possible solutions using the shock-jump MOC method. We defined the conditions for a one-to-one mapping of composition space to tie-line space. Analytical calculation of derivatives of tie lines helped to calculate eigenvalues and eigenvectors accurately even very close to critical point. We demonstrate that ruled surfaces coincide with shock loci when K-values are independent of composition. 143

163 Table 5.1 The compositions and K-values for example case (Yuan et al. 2005) x 1 x 2 x 3 x 4 K-values Oil Gas Table 5.2 Component properties and compositions for displacement by Dutra et al. (2009) Properties Compositions T c ( o F) P c (psia) ω I 1 J 1 BIP N N 2 C 3 C 6 C C C Figure 5-1: Phase diagram of water. There are four possible two-phase states at atmospheric pressure as shown by red squares (from Chaplin 2003). 144

164 Figure 5-2: The new coordinates are demonstrated with red arrows. 145

165 Figure 5-3: Quaternary displacement with three possible crossover solutions based solely on shock-jump MOC (Yuan and Johns, 2005). The two-phase region for each ternary face is outlined by the blue and purple dashed lines. Figure 5-4: Phase diagram in tie-line space and ruled surfaces for constant K-values. 146

166 Figure 5-5: Eigenvalues along the line β 2 = 0. Figure 5-6: Parametrization of the tie-line path for automatic construction of tie-line path for quaternary phase behavior of Table

167 Figure 5-7: Phase diagram in tie-line space and ruled surfaces for four-component displacements in Table 5.2 at 2900 psia and 160 o F. Figure 5-8: Quaternary phase diagram with bifurcating phase behavior generated based on Ahmadi et al. (2011) at 8000 psia and 133 o F. 148

168 Figure 5-9: Four-component displacement of Figure 5-8 in tie-line space at 8000 psia and 133 o F. 149

169 Chapter 6 Robust and accurate MMP calculation using an equation-of-state In this chapter, the approximate Riemann solver in tie-line space is used to construct the ruled-surface routes. Next, the Riemann solvers are used for MMP calculation of multicomponent displacement. Finally, several challenging MMP calculation examples are provided. 6.1 Riemann solver in tie-line space We construct the solution of multicomponent dispaclements similar to the approach of Johns and Orr (1996) by solving N C 2 pseudoternary displacements sequentially as discussed in Chapter 5. The Riemann solver is optimized for MMP calculation algorithm build ruled surface route from injection to initial tie line by constructing the pseudo ternary displacements sequentially Pseudoternary ruled surfaces The total length traversed along each pseudo-ternary ruled surface is parameterized in tieline space so that the initial and injection tie lines become connected. The parameter l i is the eigenvector length taken along ruled surface i. The value of l i for N C 2 ruled surfaces can be positive or negative depending on the eigenvector (or shock) direction taken. We allow for the added complexity of a combination of n total rarefactions and shocks along each ruled surface, which is possible for two-phase regions that bifurcate. We define the parameter l i to describe each pseudo-ternary displacement as, 150

170 n j l j = ± l j,k k=1 j = 1,, N c 2. (6.1) The parameter l i,k corresponds to the length taken for one rarefaction or shock along a portion of a ruled surface, and as defined below is always positive, l j,k = N (dγ j dγ 1 ) 2 c 2 j=2 dγ 1 Rarefaction N ( γ { j ) 2 c 2 j=2 Shock j = 1,, N c 2, k = 1, n. (6.2) For four components, we have only two values of l i, but each of these has n possible segments depending on the change in the eigenvalues along the ruled surface. Johns (1992) showed that a pseudo-ternary displacement has only one nontie-line shock or rarefaction along it if K-values are strictly ordered (n = 1). The ternary displacement with bifurcating phase behavior, however as shown in Chapter 3, can have both shocks and rarefactions along the same ruled surface. When both exist along a given ruled surface, the shock velocity is the same as the eigenvalue at which the rarefaction starts or ends in order to satisfy both the velocity (singlevalued) and entropy conditions. Although written more generally, we assume here that the route along a ruled surface can consist of a maximum of only two segments (one rarefaction and shock so that n = 2) because it is unlikely that for crude oil displacements the two-phase region will split into three two-phase regions as pressure is increased. The sign in Eq. (6.1) is positive if the ith eigenvalue increases from the left tie line (upstream) to the right tie line (downstream). Integration of rarefaction paths can be done numerically to the accuracy desired. Therefore the right tie line can be defined as a function of the left tie line and l j as 151

171 Γ j R = f(γ j L, l j ). (6.3) The i-th ruled surface can be a combination of a shock and rarefaction for bifurcating phase behavior. The bifurcation of the two-phase region is easily checked from the change in the eigenvalues at Γ L i and Γ R i. If the eigenvalues are monotonic from Γ L i and Γ R i then only a shock or rarefaction exists along that ruled surface depending on the sign, but not both Estimation of ruled surfaces The solution construction for Eqs. (5.3) and associated initial and injection boundary conditions with any number of components and complex phase behavior is challenging. A robust Riemann solver is needed especially for MMP calculations, which require convergence near critical regions. At is simplest, the Riemann solver must integrate in small steps along each ruled surface for rarefactions and solve the Rankine-Hugoniot (RH) equations for shocks. Newton iterations are required to find the unique solution of Eqs. (5.3), but it is possible that integration along a ruled surface could result in negative phase compositions, or could step into the supercritical region. Therefore the Riemann problem is usually replaced with an approximate Riemann problem. However, approximate Riemann solvers such as the ones by Roe (1981) and Toro (2009) are specifically designed to be used in numerical simulators with a Godunov scheme. The Riemann problems for such cases usually consist of only one wave, but here we have multiple waves (and shocks). There are many possibilities for a Riemann solver algorithm. The approach presented here is to approximate eigenvector paths with a second order polynomial, while calculating shocks exactly. We can use an estimation of Eqs. (6.4) to improve robustness and computational speed as it avoids complete integration along a ruled surface when a rarefaction exists. Consider 152

172 the vectors e R = Γ ini Γ inj for the displacement and e i = Γ i R Γ i L i = 1,, N c 2 for each pseudo-ternary displacement. Assuming that there is a good estimate for ẽ i, the ruled surface route can be calculated by solving e R = l ie i i = 1,, N c 2, (6.4) which is a linear system of equations. The eigenvectors of the injection tie line are used as an initial estimate of e i for the first iteration and estimates of e i are improved in each iteration using the average of the i-th eigenvector of Γ R i and Γ L i. This estimation is equivalent to assuming that the integral curves are second order polynomials. In addition, the shocks are calculated accurately using the RH condition Riemann solver in tie-line space Riemann solver in tie-line space: The solution in tie-line space allows for the development of an automatic Riemann solver using Eqs. (5.3) for arbitrary initial and injection tie lines. As note earlier, there are N c 2 eigenvectors in tie-line space, each one corresponding to a pseudo-ternary ruled surface. The ruled-surface route for the Riemann problem of Eqs. (5.3) with the tie lines Γ inj and Γ ini as the initial data, consist of N c 2 ruled surfaces, such that a ruled surface corresponding to k th L family of eigenvectors of Eqs. (5.3) can be identified with its left and right tie lines, Γ k and Γ R k. The ruled-surface route is complete when Γ inj = Γ 1 L, R Γ ini = Γ Nc 2, Γ R L k = Γ k+1 k = 1,, N c 3. (6.5) 153

173 N c 2 of the above equations have trivial solutions. Therefore the solution can be constructed by solving only the last remaining equation. We select the equations that correspond to the shortest tie line. This selection significantly increases the robustness of the algorithm near the supercritical region. We illustrate the solution procedure for the parameters in Table 5.1 as shown in Figure 6-1 where the solution can be constructed by solving Γ R 1 = Γ L 2. The similar procedure can be used for multicomponent systems. The construction of tie-line route starts from injection tie line by constructing the first pseudo-ternary displacement as determined by l 1. The process should be repeated for each family of eigenvalues starting from the right tie line of the previous pseudo-ternary displacement. Such that we start from Γ inj and construct m ruled surfaces to Γ R m. Next we start from Γ ini and construct the N c 2 m pseudo-ternary L displacement to Γ m+1. Then the Newton method can be used to solve Γ R L m = Γ m+1. m should be R selected such that Γ m is the shortest tie line among all Γ R i for i = 1,, N c 2. m is equal to zero when injection tie line is the shortest one. The computation error decreases using this approach since we are not required to start a pseudo-ternary displacement from a critical tie line Validation of estimate Riemann solver The ruled-surface routes calculated by approximate Riemann solver are compared to the numerical simulation results. First example is the four component displacement of Table 6.1. The simulation results are projected onto the plane of x C1 x C10. The ruled surface route is a shock from injection tie-line to crossover tie-line followed by a shock to initial tie-line. Both Riemann solvers are used for this problem, and the result have negligible difference. Next example, is a four component displacement with bifurcating phase behavior. The component properties are from Table 4.5 and the phase diagram is shown in Figure 5-9. The ruled surface route is a 154

174 rarefaction from injection tie-line to crossover tie line along the integral curve corresponding to the slow eigenvalue. The ruled surface corresponding to fast eigenvalue consists of a rarefaction and a tangent shock. Figure 6-3 demonstrates the simulation and analytical results projected to the x C26 35 x C1 surface. The crossover tie line and the shortest tie line along the RHL are shown in Figure 6-3 as well, which closely matches with the simulation results. Next example is the six component displacement in Table 6.2. The tie-line space for a six-component displacement is in 4D space, therefore we compare the simulation and analytical results using a 2D projection of results to the x C1 x C20 surface as shown in Figure MMP calculation for N c displacements MMP calculations are based on constructing the ruled surface routes at different pressures and track the shortest tie-line length. MMP calculation algorithms rely on tracking the shortest tie line at different pressures and find the pressure in which the shortest tie-line length become zero. Johns et al. (1996) proved that a four-component displacement is MCM when the crossover tie-line length become zero. We showed in Figure 4-7 that for a ternary displacement with critical initial or injection data, the composition route lies on EVC or the binodal curve. The results immediately can be extended to multicomponent systems that the composition route for a displacement with a critical tie line lies on EVC or bimodal curves. Therefore the composition route is one shock with velocity one and the displacement is piston like. In this section we use the tie-line space Riemann solvers to calculate the ruled surface routes at different pressures and estimate the MMP values. Compare to previous MMP calculation algorithms based on shockjump MOC, the number of variables in our algorithm are significantly reduced. For example, to construct the solution for an displacement of an eleven-component oil by a five-component gas, Wang and Orr (2002) uses 99 variables, Jessen et al. (1998) uses 286 variables and Yuan and 155

175 Johns (2002) uses 19 variables while our approach requires only 9 variables. The smaller number of variables, significantly increases the robustness of the solution. In addition, our algorithm is the only one that checks for velocity and entropy condition, therefore the Riemann solver will not converge to wrong solutions of the shock-jump MOC. Furthermore, the approximate Riemann solver iterations are faster than shock-jump MOC. The results of the approximate Riemann solver are used as initial estimate for comprehensive Riemann solver when more accurate results are desired. The MMP is estimated for three different examples. We compare the analytical solutions to numerical simulation results. The 1-D slim-tube simulation formulation given in Johns (1992) is used with 10,000 grid blocks for simulations to compare with analytical solution. For MMP estimations from the simulation, we extrapolated the shortest resulting tie-line lengths to an infinite number of grid blocks (zero dispersion). The extrapolation is done using only one simulation up to 0.5 PVI at each pressure. The shortest tie-line length at each time is linearly extrapolated versus 1/time (Yan et al. 2012) Four-component displacements The MMP is estimated for the displacement in Table 5.2. The tie line route parameters are shown in Figure 6-5. The tie-line route parameters are almost constant even close to MMP. The analytically calculated key tie-line lengths along with the extrapolated shortest tie-line length from simulation are shown in Figure 6-6. The shortest tie line is a cross-over tie line and displacement mechanism is C/V. We estimated MMP for the four component displacement of Figure 5-8 and Figure 5-9 which has bifurcating phase behavior. At lower pressures (1000 psia) the bifurcating does not appear in analytical solution and the fast path is just a shock (Figure 6-13). However, the shortest 156

176 tie line calculated by simulation is not a crossover, injection or initial tie line, as discussed earlier. At higher pressures the bifurcating tie line approaches the shortest tie line calculated by simulation. The two tie lines become the same at MMP. Shock-Jump MOC misses the bifurcating tie line and for this example, the shortest tie line would be the initial tie line. Therefore, the shock-jump MOC over predict the MMP for this displacement by 4000 psia Five-component displacement MMP is calculated for five-component displacements with gas mixture injection of Table 6.3. Figure 6-8 shows the key tie-line length for the displacement in Table 6.3 along with the shortest tie-line length calculated by simulation. Figure 6-9 demonstrates the structure of the solution. The tie-line route is a shock from injection tie line to first cross-over tie line and another shock to the second cross-over tie line. The tie line routes is completed with a rarefaction to the initial tie line Twelve-component displacement by Zick (1986) Zick (1986) examined a real fluid and demonstrated the existence of the condensing/vaporizing drive mechanism. This fluid model has examined by many researchers. We used the approximate Riemann solver to construct the ruled surface route and estimate MMP to be at 3097 psia (Figure 6-10). The shortest tie-line length is compared against slim-tube simulation results. Zick (1986) reported an MMP of 3125 psia from slim-tube experiment. Jessen et al. reported an MMP of 3095 psia using analytical solution. Ahmadi and Johns used mixing cell to calculate MMP as 3104 psia. Figure 6-11 clearly demonstrate the condensing/vaporizing displacement mechanism as the shortest tie line is the seventh crossover tie line. 157

177 6.2.4 Eleven-component displacement by Metcalfe and Yarborough (1979) In this section, we calculate the MMP for displacement of an eleven-component synthetic oil by pure CO 2. The slim-tube experiments results are provided by Metcalfe and Yarborough (1979). Later Turek et al. (1984) examined the phase behavior and demonstrate existence of a three phase region for a range of pressures and temperatures. Johns and Orr (1996) construct the entire composition route for the eleven-component displacement at different pressures and estimated the MMP. We used the Riemann solver in tie-line space to construct the solution at different pressures and estimated MMP (Figure 6-12). The slim-tube simulator is used to calculate MMP as well Bifurcating phase behavior Mogensen et al. (2009) compared the MMPs predicted by various computational methods for the Al-Shaheen oil displaced by CO 2 and noted a significant difference of thousands of psi between the MMP predicted by the shock-only MOC and other MMP methods for the heavier reservoir fluids (Figure 6-13). Ahmadi et al. (2011) explained these differences using a simple pseudoternary diagram and their mixing cell method and we presented the analytical solution for such ternary displacements in Section 3.4. Ahmadi et al. (2011) gave an approximate fix for bifurcating phase behavior by checking the length of the tie lines between each of the key tie lines to identify if a critical point is present or is forming between them. They used a linear interpolation between key tie-lines, however ruled surfaces are curved for bifurcating pseudoternary displacements. As a result the predicted MMP by improved shock only MOC deviates from MMPs calculated by mixing cell for heavier oil samples (Figure 6-13). The approximate 158

178 Riemann solver can find the shortest tie-line accurately and result in a better estimation of the MMP. 6.3 Summary An approximate Riemann solver is developed to construct the tie-line surface route for multicomponent displacements. The new Riemann solver number of variables is significantly less than used in previous shock-jump MOC solutions. Therefore the solver is more robust. The Riemann solver is used to develop an MMP calculation algorithm. The new MMP calculation algorithm is tested with multiple challenging compositional displacements. The approximate Riemann solver can be improved for robustness and accuracy. Table 6.1 The compositions for four-component example with component properties in Table 6.3 C 1 CO 2 nc 4 C 10 Injection Initial Table 6.2 Input properties for six-component MMP calculations from Johns (1992) Properties Compositions T c ( o F) P c (psia) ω I 1 J 1 BIP C C 1 CO 2 nc 4 C 10 C 14 CO nc C C C

179 Table 6.3 Input properties for five-component MMP calculations Properties Compositions T c ( o F) P c (psia) Ω I 1 J 1 BIP C C 1 CO 2 nc 4 C 6 CO , nc C C Table 6.4 The compositions for 12-component displacement from Zick (1986) CO 2 C 1 C 2 C 3 C 4 C 5 C 6 1+ C 7 2+ C 7 3+ C 7 4+ C 7 5+ C 7 Injection Initial Figure 6-1: Parametrization of ruled surfaces. The solution can be constructed by solving Γ 1 L = Γ 2 R. 160

180 Figure 6-2: Projection of tie-line routes to C 1 C 10 plane for four component displacement of Table 6.1. Figure 6-3: Projection of tie-line routes to x C1 x C26 35 plane for the four component displacement of I 3 by J 1 in Table 4.5 with bifurcating phase behavior at 8000 psia and 133 o F. 161

181 Figure 6-4: Projection of tie-line routes to x C1 x C20 plane for six component displacement in Table 6.2 at 1000 psia and 160 o F. Figure 6-5: The tie line route parameters for the displacement of I 1 by J 1 in Table

182 Tie-line length Figure 6-6: Key tie-line lengths calculated by analytical solution and shortest tie-line length calculated by simulation for the displacement in Table Injection Crossover Bifurcating Initial Simulation Pressure, psia Figure 6-7: Four-component displacements with bifurcating phase behavior. The shock-jump MOC over predict MMP by almost 4000 psi. 163

183 Figure 6-8: The key tie line length for five-component displacement. The shortest tie-line length is calculated by simulation. Figure 6-9: The tie-line route at different pressures for five-component displacements. 164

184 Figure 6-10: Tie-line length variation with pressure calculated using approximate Riemann solver for displacement in Table 6.4 at 185 o F. MMP is estimated to be at 3097 psia. Figure 6-11: Tie-line length variation form injection to initial tie-line at different pressures and 185 o F with compositions in Table

185 Figure 6-12: Shortest tie-line length variation with pressure for displacement from Johns and Orr (1996). Figure 6-13: MMP calculaiton for complex phase behavior of Mogensen et al. (2008). The shockonly MOC, improved shock-only MOC and mixing cell results are copied form Ahamdi et al. (2011). 166

186 Chapter 7 Application of splitting technique to low salinity polymer flooding In this chapter, we develop the first analytical solutions for the complex coupled process of low salinity-polymer (LSP) slug injection in sandstones that identifies the key parameters that impact oil recovery for LSP, and also improves our understanding of the synergistic process, where cation exchange reactions change the surface wettability. Both secondary and tertiary LSP is considered. Section 7.1 presents the mathematical models. Then, splitting of the analytical equations is developed along with analytical solutions for different scenarios in Section 7.2. The developed analytical solutions are validated against experimental results and numerical simulation in Section 7.3. The velocity of the different ions and saturation fronts are compared to demonstrate the insight of the new analytical solutions. 7.1 Mathematical model We use the in-house general purpose compositional simulator, PennSim (PennSim 2013, Qiao 2015), to make all numerical simulation calculations. The basic equations needed to model LSP flooding are outlined here. Mass conservation of oil, water, polymer and aqueous ionic species are included along with cation exchange reactions, adsorption of salts and polymer, inaccessible pore volume, and wettability alteration. A mechanistic approach that includes the cation exchange of Ca 2+ and Na + is used to model the wettability alteration. The viscosity is a function of polymer and ionic species concentration. 167

187 7.1.1 Immiscible oil/water flow The mass conservation equations for immiscible oil and water phases are as follows: t (φs αρ α ) + (ρ α u α ) = 0 α = o, w. (7.1) Darcy s law governs the flow rate of each phase, u α = k rα μ α K (P α ρ α gz) α = o, w. (7.2) The subscript w refers to the water phase, while o to the oil phase. Capillary pressure relates the pressure of oil and water phases, P cow = P o P w. (7.3) The saturation relation completes the set of equations S o + S w = 1. (7.4) The primary unknowns for the multiphase flow system are P o and S w Cation Exchange Reaction Network The cation exchange between clay and the aqueous phase is assumed to be the main mechanism for wettability alteration. The primary cations include Na +, Ca 2+ and Mg 2+. With Na- X, Ca-X 2 and Mg-X 2 representing the surface sites occupied by sodium, calcium and magnesium, the cation exchange reactions can be written as 168

188 Ca Na X Ca X 2 + 2Na +, Mg Na X Mg X 2 + 2Na +, K eq,ca K eq,mg where K eq,ca and K eq,mg are the reaction equilibrium constants for Ca 2+ and Mg 2+ exchange reactions, respectively. Since the surface reactions occur very fast, it is usually assumed that the cation exchange reactions are in equilibrium. For the above reactions, the mass action law is written as K eq,ca = (Ca X 2)(Na + ) 2 (Ca 2+ )(Na X) 2, K eq,ca = (Mg X 2)(Na + ) 2 (Mg 2+ )(Na X) 2, where () represents thermodynamic activities. Here, for the convenience of the analytical solutions, we assume dilute aqueous solution and the activities for surface species are (Na X) = [Na X], CEC (Ca X 2 ) = 2[Ca X 2], CEC (Mg X 2 ) = 2[Mg X 2], CEC where [] denote the concentration in mol/g solid and CEC represents the total surface site concentration in mol/g as CEC = [Na X] + 2[Ca X 2 ] + 2[Mg X 2 ]. 169

189 7.1.3 Reactive Transport Model The mass conservation equations for the primary species p is N sec t (M p + ν qp M q ) + (F p + ν qp F q ) = 0 p = 1,, N pri, (7.5) q=1 N sec q=1 where the first term is the accumulation of total moles and the second term is the total molar flux of the primary component p. The equations are based on the stoichiometric relationship among the species participating in reactions. The derivation of the general reactive transport equations can be found in Qiao et al. (2015b). For the cases considered in this paper, the reactive transport equations for Na + Ca 2+ and Mg 2+ are written as t [φs wρ w C Na + + (1 φ )ρ s C Na X ] + (u w C Na +) = 0, t [φs wρ w C Ca 2+ + (1 φ )ρ s C Ca X2 ] + (u w C Ca 2+) = 0, t [φs wρ w C Mg 2+ + (1 φ )ρ s C Mg X2 ] + (u w C Mg 2+) = Wettability alteration We use a linear interpolation model as follows: k rw = (1 θ) k rw,ww + θk rw,ow, (7.6) where k rw,ow, and k rw,ww are water relative permeabilities at the end-point oil-wet (ow) and endpoint water-wet states (ww). The same linear interpolation is used for other coefficients of the relative permeability model. These end-point states do not have to be at the complete oil-wet or 170

190 water-wet states, but ideally should be measured at initial reservoir conditions (mixed wet state), and at the most water-wet state possible (state achieved during LSW). The Brooks-Corey model is used (Brooks and Corey 1966), k rw = k rw (S ) n w, k ro = k ro (1 S ) n o, where the normalized water saturation S is calculated by S = S w S wr 1 S wr S or. Here S wr is the residual water saturation and S or is the residual oil saturation that depends on wettability: S or = (1 θ)s ww or + θs ow or. We further assume that the wettability alteration is controlled by the surface concentration of adsorbed Na +, namely θ = [Na X] [Na X] ww [Na X] ow [Na X] ww. (7.7) Polymer Flooding Model Polymer is dissolved and well mixed in the aqueous phase. The mass conservation equation for polymer is t [φ(s w φ IPV )C p + (1 φ)c p] + (C p u w ) = 0, (7.8) 171

191 where φ IPV is the inaccessible pore volume and C p is the polymer that is adsorbed on the rock surface. The viscosity of polymer solution is a function of polymer, Na + and Ca 2+ concentrations as (Delshad et al. 1996) μ p = μ w [1 + (a 1 C p + a 2 C 2 p + a 3 C 3 sp p )C se ], C se = C Na + + (β p 1)C Ca 2+. The adsorbed polymer concentration C p is a function of aqueous polymer concentration C p, which is determined from a table-lookup function. The shear rate dependence and viscoelastic effects of the polymer are not considered. The residual oil saturation decreases during polymer flooding as a function of the trapping number (Delshad and Pope, 1989) Numerical solution We used a finite volume method to discretize the PDEs. For each control volume k, the pressure P j,k, water saturation S w,k and molar concentrations C i,k are assumed to be at the geometric center. The volumetric flow rate is evaluated at the interface between two control volumes using a central finite difference scheme and upstream weighing. The temporal discretization uses a generalized non-iterative IMPEC solution, which treats the pressure variable using the backward Euler method and the total moles of primary species using the forward Euler method. A speciation calculation is performed after pressure and mole numbers are calculated. The last step is to update the properties that include the effects of surface reactions on porous media properties such as changing wettability. A more detailed solution procedure can be found in Qiao (2015). 172

192 7.2 Analytical solution The mathematical model is simplified assuming 1-D incompressible dispersion-free flow. In addition, reaction kinetics are ignored so that chemical reactions are always in equilibrium. Mass conservation of oil, salt components and polymer (Eqs. 7.1, 7.5 and 7.7) are then given by, S o t D + f o x D = 0, (7.9) t D [(S w φ IPV φ ) C p + C p] + x D C p f w = 0, (7.10) (S t w C i + C i) + C D x i f w = 0, i = 1,, N c, (7.11) D where f w = (1 + μ 1 w(c p,c s ) k ro (S w,c s ) ) and f k rw (S w,c s ) μ o = 1 f w, x D and t D are dimensionless distance o and time, and N c is the number of cations. The oil conservation equation can be rewritten using S o = 1 S w as S w t D + f w x D = 0. (7.12) The conservation equations for polymer and ions can be expanded using the chain rule and simplified using the above equation. That is, (S w + D p ) C p t D + f w C p x D = 0, (7.13) N c C i C j C i S w + C ij + f t D t w = 0 i = 1,, N D x C, (7.14) D j=1 173

193 where C ij = C i C j. In the next section, the analytical solutions are developed by splitting the flow equation into three sub problems Decoupled system of equations We define the new coordinates (Pires et al. 2006) as x D,t D φ = (f w dt D S w dx D ), 0,0 (7.15) ψ = x D. (7.16) The elements of Eqs. (7.12) are transformed therefore to the new coordinates as f w f w = S x w D t φ + f w D ψ ψ, φ S w S w = f t w D x φ. D ψ (7.17) The same calculation can be repeated for Eqs. (7.13) and (7.14). The final result of the transformation to the new coordinates after some manipulation is ψ ( 1 ) f w φ (S w ) = 0, (7.18) f w C p ψ + φ (C p + φ IPV φ C p) = 0, (7.19) C i ψ + C i φ = 0, j = 1,, N c. (7.20) 174

194 The hyperbolic equations (Eqs ) have the same form as the conservation equations when ψ is the time, φ is the location, f w 1, C p and C i are the conserved quantities, and S w /f w, C p and C i + φ IPV C φ p are the flux functions. Therefore we can use the method of characteristics (MOC) to solve the equations by defining the characteristic velocity, σ, in the new coordinates as σ = 1 λ = dφ dψ, (7.21) where λ is the eigenvalue of the characteristics matrix. Furthermore, λ represents the front retardation for two-phase flow. Fronts with larger retardation appear later; hence, the solution is single valued when λ decreases from the injection composition to initial composition. When ψ = x D, the eigenvalues increase from the injection composition to initial composition. We can convert the PDEs of Eqs. ( ) to ordinary differential equations using the definition of λ. The characteristics equations of Eqs. ( ) are (S w f w f λ ) df 1 w w dφ + f w / f w dc p C p dφ + f w / f w dc i C S w,c i dφ i S w,c p,c j i (D p + φ IPV φ N c dc j C ij dφ j=1 N c i=1 = 0, (7.22) λ ) dc p dφ = 0, (7.23) λ dc i dφ = 0, i = 1,, N c, (7.24) where f w = f w S w C p,c i. The following equations demonstrate the characteristic matrix for low salinity polymer floods with two cations and one anion. Two independent ion concentrations are necessary to calculate the equilibrium composition of the water phase and solid surface. 175

195 [ S w f w f w λ f w / f w C p S w,c i f w / f w C 1 S w,c p,c 2 f w / f w C 2 S w,c p,c 1 0 D p + φ IPV φ λ C 11 λ C C 21 C 22 λ ] [ df w 1 dφ dc p dφ dc 1 dφ dc 2 dφ ] = 0. (7.25) The eigenvalues of this system are λ 1,2 = C 11 + C 22 ± C 11 + C 22 2C 11 C C 12 C 21 2, (7.26) λ 3 = D p + φ IPV φ, (7.27) λ 4 = S w f w f w. (7.28) The system of equations is not strictly hyperbolic and the eigenvalues are not ordered based on their values. The corresponding eigenvectors are as follows e 1,2 = [ e 1s 0 1 λ 1,2 C 11 C 12 ], e 3 = e 3s 1 0 [ 0 ], e 4 = [ 0], (7.29) where, e 1s = f w / f w C1 Sw,Cp,C2 + λ 1,2 C 11 f w / f w C 12 C2 Sw,Cp,C1 S w f w f w λ 1,2 and e 3s = 176 f w / f w Cp Sw,C i S w f w f w λ 3 The eigenvectors and eigenvalues, Eqs. ( ), have two important features. First, λ 4 is the only eigenvalue that is function of fractional flow and saturation. Second, composition is constant along e 4. Therefore, we can solve the reaction and polymer transport independent of.

196 fractional flow and the path along e 4. In addition, the polymer and reaction systems are uncoupled. The solution for the uncoupled conservation equations can be constructed independently even when the system of equations are not strictly hyperbolic. (For example uncoupled advection equations, Leveque 2002). Therefore, we first solve for concentrations based solely on reaction and polymer transport. Then we map fractional flow onto these concentrations. In this paper we assumed that adsorption of polymer is independent of salinity. In contrast, the adsorption of polymer can be considered as a function of salinity, therefore the polymer transport equation will be dependent on the reactive transport solution. Yet, the reactive transport solution will be independent of polymer concentration. The weak solution for Eqs. ( ) in Lagrangian coordinates is equivalent to the weak solution of the first equation (Wagner 1987). Therfore, we can calculate the shock velocities and determine the front types based on the solution in Lagrangian coordinates Reactive transport solution Equation (7.20) is similar to the single phase transport with cation exchange. Equation (7.30) is the characteristic matrix for the single phase reactive transport with two cations (Venkatraman et al. 2014), 1 + C 11 λ dc 1 C 12 [ C C 22 λ ] [ ] = 0. (7.30) dc 2 follows The eigenvalues for single-phase transport are related to the eigenvalues of Eqs. (7.24) as λ = λ + 1, 177

197 such that a front with speed of 1.0 in the single-phase region has λ = 1 and λ = 0, which implies no retardation in the Lagrangian coordinates. However the eigenvectors of Eqs. (7.24) are the same as single-phase reactive transport. The Riemann solver for the single phase reactive transport developed by Venkatraman et al. (2014) can be used to construct solutions for low salinity cation exchange reactions in low salinity flooding. The important features of the solutions of cation exchange reactions are as follows. We assumed the anion is not adsorbed, therefore the anion front is a contact discontinuity with characteristic speed of 1.0. Therefore, the retardation of the anion, λ, is always zero in Lagrangian coordinates and the anion front moves ahead of the cation exchange front. In addition, as a result of the constant CEC assumption, the anion shock has no effect on the surface composition. The cation fronts for low salinity injection are always shocks and the front for high salinity injection are always rarefactions with negligible retardation. Therefore, the anion shock has no effect on surface concentrations and as a result wettability is not altered with the anion shock Polymer transport solution The polymer eigenvalue, λ 3, is the retardation factor for the polymer front. The eigenvalue is calculated based on the slope of the adsorption isotherm. As mentioned earlier, although the polymer viscosity is a function of salinity, the MOC solutions for polymer in Lagrangian coordinates are independent of salinity solutions. The salinity of water can affect the adsorption of the polymer. In that case, the polymer concentration will change along with the reactive transport eigenvectors. However the reactive transport system remains independent of polymer transport. The shock velocity can be calculated using the Rankine-Hugoniot condition as 178

198 Λ = ΔC p ΔC p + φ IPV φ. The adsorption isotherms are usually concave so that the solutions for polymer injection with polymer adsorption always have a shock in the polymer concentration. In contrast, the solutions for injection of chase fluid in a polymer flood exhibit a rarefaction wave for polymer concentration due to gradual desorption of polymer from the rock surface. Polymer adsorption and porosity degradation is commonly considered as an irreversible process, therefore the polymer will not desorb from the rock surface and the solution for chase fluid injection will have a shock in the polymer concentration. Furthermore, non-newtonian behavior of polymer can be incorporated in the current solution (Rossen et al. 2010) Maping fractional flow The last step of constructing the solution is to map fractional flow onto concentration solutions. Our solution can be considered as an extension of fractional flow theory to multicomponent systems. Alternatively, we can construct the complete solution in Lagrangian coordinates using Eqs. (7.22), then transform the solution to x D t D coordinates. The solution construction for compositional shocks are sufficient to construct the low salinity polymer injection solution because rarefactions only occur for injection of a slug of polymer and low salinity. The slug injection problem can be solved using front tracking algorithms where rarefactions are estimated with several shocks. The shock velocity in x D t D coordinate is Λ = Δx D /Δt D then Λ = Λ on the S w axis. f w S w +Λ. This means the extension of a C-shock should pass through the point 179

199 Now, we demonstrate the steps to construct the solutions for multiple cases with only one C-shock and Λ = 0.5. Figure 7-1 (left) demonstrates two hypothetical fractional flows for upstream and downstream compositions of the shock. The shock between these two water composition states has Λ = 0.5. The change in fractional flow properties is a result of a change in polymer concentration and/or wettability alteration. The solutions for one upstream composition, L, and three downstream compositions, R 1, R 2 and R 3 are shown in Figure 7-1. The C-shock is independent of the downstream composition. The solutions consist of a rarefaction form L to a, then a tangent shock to b. The final part of the solution can be constructed as a Buckley leverett problem with b as injection composition and R i as initial composition. The profiles for these three problems are shown in Figure 7-1 (right). Figure 7-2 shows the C-shock for different upstream compositions. The C-shock is always a function of the downstream composition. For example the solution for L 1 R 1 is a shock to a then a jump between two fractional flows that is tangent at b and a rarefaction to R 1. The second solution is a shock from L 2 to c followed by a Buckley-Leverett shock to R 1. Finally, the last composition path is a shock from L 3 to d followed by saturation shock to R 2. The examples in Figures 7-1 and 7-2 demonstrate that the C-shock is always a function of the upstream composition; therefore, the solution for multiple C-shocks can be constructed sequentially from the injection composition to initial composition without trial and error Front tracking algorithm The analytical solutions for different combinations of injection and initial conditions are described in the previous section. These solutions can be used to calculate the interaction of fronts for complex slug injection problems and for varying initial conditions. The basic procedure is as 180

200 follows. First, the fractional flow is estimated with a piecewise linear function as shown in Figure 7-3. The solution for a simple water flood based on the smooth piecewise fractional flow curves is shown in Figure 7-4. That is, the rarefaction is converted to a series of shocks. The leading front velocity is slightly different between the two solutions. The mass is, however, conserved in both cases. Furthermore, the accuracy of solution can be increased by approximating the fractional flow curve with more linear pieces. The second step is to estimate the initial and injection compositions with piecewise constant values. Figure 7-5 (left) shows initial water saturation for a reservoir with stepwise initial water saturation. The front tracking algorithms start by constructing solutions for the initial condition jumps at time zero. These jumps are shown by red dots along the horizontal axis of Figure 7-5 (right). Each line in Figure 7-5 (right) represents a shock and saturations have constant values between the lines. The shocks may intersect depending on their velocity. That is, the upstream faster shock could catch up with the downstream slower shock as shown by point a in Figure 7-5 (right). When they intersect, a new Riemann problem forms. The solution should be constructed for the upstream saturation (S w = 0.59) and downstream saturation (S w = 0.1). The algorithm is finalized when there are no additional shock intersections. More details of the front tacking algorithm can be found in Holden and Risebro (2013) Matching wettability front retardation independent of reactions We simplified the analytical solution by assuming that only one of the cation shocks alters the wettability, which we call the wettability front. This assumption helps to reduce the cation exchange reaction model to a single retardation coefficient for the wettability front. The retardation coefficient and the produced water chemistry can be matched using a Riemann solver for single-phase reactive transport or by trial and error using numerical simulators to calculate the 181

201 reaction model parameters. The retardation coefficient is a function of the low salinity water composition, CEC and reaction equilibrium coefficients. Although the wettability front changes the surface composition significantly, the change in water composition is smaller and the front is usually smeared out in the production data. Furthermore, when the high salinity water is injected the fronts are rarefactions that move very fast, so that we can ignore the retardation effect for high salinity injection in a low salinity flooded reservoir. Therefore the low salinity experiments can be matched by these steps. 1. Measure viscosity, relative permeabilities, capillary pressures, polymer viscosities and polymer adsorption isotherms for high and low salinity mixtures. 2. Match the recovery curves by adjusting the retardation coefficient for the wettability front. 3. Convert produced water composition to single phase data as described in next section. Then match the compositions and retardation coefficient Matching reactions independent of fractional flow The single phase reactive transport simulation codes are commonly used to match the geochemical reactions in low salinity floods. This estimation is valid because the oil saturation is usually very small and close to residual saturation, therefore flow can be assumed to be single phase transport. We use the splitting approach to eliminate the effect of fractional flow on experimental results. The Lagrangian coordinate φ can be calculated using the recovery curves. φ(t, 1) = 0,1 0,0 S w dx t,1 + f w dx 0,1 t,1 = S wi + f w dx 0,1. (7.31) 182

202 The parameter φ + 1 is the equivalent single phase flow time. Then we plot ion concentrations as a function of φ + 1, which makes the results independent of fractional flow curve. The reactions, therefore, can be matched independent of fractional flow. 7.3 Results First we validate the analytical solutions for two-phase flow with cation exchange reactions, where wettability alteration is initially neglected. Next, the effect of CE and wettability alteration on the analytical solutions is demonstrated. Finally a low salinity polymer experiment and a series of low salinity slug injection displacements are matched with the analytical solutions Two phase CEC without wettability alteration In this case all eigenvalues are independent of each other and we have three completely decoupled systems. The front velocities for salinities can be calculated as the slope of the line tangent to the curve drawn from S w = Λ as shown in Figure 7-6. We used the injection and initial composition in Voegelin et al. (2000) as shown in Table 7.1. Figure 7-7 demonstrates the analytical solutions and numerical simulation results for the single- and two-phase transport with cation exchange reactions. The fronts for the two phase case moves faster than the single-phase case because a portion of the pore volume is filled with oil. The solution consists of an anion shock with zero retardation and two cation shocks, which are retarded for 7.5 and 16.3 PVI. The concentration of [Na X]at x D = 1 is shown in Figure 7-8. The surface composition is significantly changed by the first cation shock. Since the surface wettability is considered a function of [Na X], the first cation shock alters the surface wettability as it moves through the reservoir. 183

203 7.3.2 Low salinity waterflooding The analytical solutions for examples in the previous section are constructed with wettability alteration caused by cation exchange reactions as shown in Figure 7-9. The flow models are described in Table 7.2. The sensitivity of the results to CEC is demonstrated in Figure 7-9. Figure 7-10 demonstrates the analytical solutions using fractional flow curves. The analytical solution using no CEC over predicts the wettability front velocity Low salinity polymer experiment Shaker Shiran and Skauge (2013) low salinity polymer experiments are matched with our analytical solutions. The input parameters for the model are shown in Table 7.2 and 7.3. The experiments are matched by tuning the CEC, high salinity residual oil saturation, and the residual oil saturation reduction by polymer. The matched CEC value is shown as CEC2 in Table 7.2. The experiments were conducted with a low salinity slug followed by low salinity polymer buffer. The compositions of the water in both cases are the same. We first demonstrate the analytical solution for low salinity polymer injection, then we used the front tracking algorithm to match polymer slug injection. The solution is presented using a Walsh diagram (Walsh and Lake 1989 and Lake et al. 2014) in Figure The solution consists of a small rarefaction from J to a along the water-wet polymer fractional flow curve (WWP) (Figure 7-11 top, left) followed by a wettability front from a to b. Then, there is a rarefaction along the oil-wet polymer curve (OWP) to c followed by a tangent polymer shock to d. The solution is completed by a leading saturation shock along the oil-wet fractional flow curve (OW) from d to I. Figure 7-11 (top, right) shows recovery is poor because of the slow moving wettability front (shock ba). As shown in Fig. 8, only the first cation 184

204 exchange front changes surface wettability, therefore we only considered one wettability front in the analytical solution of Figure The oil recovery is continued to 17 PVI for a low salinity flood. The reaction model parameters are shown in Table 7.2. The cation exchange shock velocity is matched by adjusting the CEC value shown as CEC2 in Table 7.2. Figure 7-12 demonstrates the match between analytical solutions and experimental data for the low salinity polymer flooding experiments by Shaker Shiran and Skauge (2013). Figure 7-13 gives a Walsh diagram for injection of a low salinity water slug followed by polymer. The low salinity front moves very slow in the reservoir and the polymer shock interacts with the low salinity shock even after a long period of low salinity injection as shown in Figure 7-13 (bottom right). The front tracking algorithm is used to calculate the analytical solution after the polymer injection. A preflush of the reservoir with low salinity water is commonly used to improve polymer flood performance. Figure 7-13 shows that the distance between polymer and high salinity water increases in the reservoir even for a small slug of preflush. Therefore, the optimum low salinity preflush can be determined based on the dispersion level in the reservoir Low salinity slug injection with varying slug size Seccombe et al. (2008) performed low salinity slug injection experiments with varying slug sizes. Their results showed no oil recovery for small slugs, which was explained as the result of mixing. We used the relative permeability data provided in the paper, and tuned the CEC to match their results with our analytical solutions. The analytical solution is not affected by dispersion, but the oil is still not produced. The analytical solutions demonstrate that the zero oil production for small slugs in the experiments is a result of intersecting shocks, not dispersive mixing. 185

205 Figure 7-15 (left) demonstrates the fronts for a 0.2 pore volume low salinity slug. The high salinity slug catches up to the wettability front at point b on Figure 7-15 (left). Therefore no more oil is added to the oil bank and the oil bank spreads in the core, significantly increasing the breakthrough time. Figure 7-15 (right) demonstrates the fronts for 0.6 pore volume injection where the wettability front breaks through before the high salinity chase water catches up, hence the oil bank is produced. Figure 7-16 shows the saturation profiles at 15 PVI. The oil bank moves slowly and spreads out for 0.1 and 0.2 PVI injection. For the 0.3 PV low salinity slug experiment, the oil bank breaks through, but after a long time. The reduction in residual oil saturation is matched for different slug sizes by adjusting the CEC value as shown in Figure A relatively small CEC value (CEC3 in Table 7.2) was used to match the results, and the simulation results were sensitive to dispersion so that a large number of grid blocks were required to match the analytical solutions. The PennSim results using 100 grid blocks are shown in Figure Seccombe et al. (2008) concluded that the 0.2 PV low salinity slug is ineffective because of mixing, however our analysis shows that the interaction of high salinity and wettability fronts can explain this phenomena. The produced water chemistry is required to match the reaction parameter models more precisely. Lager et al. (2011) examined the produced water geochemistry of the same reservoir and concluded that the cation exchange reactions are possibly different from the aquifer freshening model (Valocchi et al. 1981). 7.4 Summary Analytical solutions were constructed for low salinity polymer flood in sandstones considering a mechanistic model of wettability alteration based on cation exchange reactions. The solutions were developed by splitting the equations into reaction, polymer, and fractional flow parts. Numerical simulation and analytical solutions predicated the same results. In addition, the 186

206 recovery mechanism of low salinity flood is determined to be based on a slow moving wettability front in reservoir. The simple model of wettability front was used to match low salinity flood and slug injections. Table 7.1 Water composition for the single- and two-phase displacements. Voegelin et al. (2000) Shaker Shiran, and Seccombe et al. Ion (mol/l) Skauge (2013) (2008) Injection Initial Injection Initial Injection Initial Na Mg Ca Cl Table 7.2 Reaction parameters. CEC2 and CEC3 are calculated by matching experiments. CEC1 (mol/l) CEC2 CEC3 Parameter K eq,ca K eq,mg (Venkatraman et al. 2014) (mol/l) (mol/l) Value Table 7.3 The Corey relative permeability parameters for the experiments. Shaker Shiran, and Seccombe et al. Skauge (2013) (2008) Parameters Water Water wet Oil wet Oil wet wet S wr S or n w n o kr w kr o μ w μ o μ p

207 Figure 7-1: Mapping of fractional flow curve to the composition solution. Left figure uses the standard approach as is solved for the fractional flow problem for polymer flooding. Right figure demonstrates the wave velocities for the three different Riemann problems. Figure 7-2: Mapping of fractional flow curve to composition solution. Left figure uses the same approach as fractional flow for polymer flooding. Right figure demonstrate the wave velocities for the three different Riemann problems. 188

208 Figure 7-3: Piecewise linear approximation of fractional flow is commonly used in front tracking algorithms. Figure 7-4: The piecewise estimate of the fractional flow curve converts the rarefactions to small shocks. The error of approximation decreases as the number of the linear pieces of fractional flow is increased. 189

209 Figure 7-5: The interaction of shocks in a water flooding displacement with variable initial condition. The initial condition should be approximated with a piecewise constant function. Figure 7-6: The single phase CE reactions are converted to two-phase transport. The slope of dashed lines are equal to cation front velocities. 190

210 Figure 7-7: Comparison of single- and two-phase transport of Mg ++. Wettability alteration is not included in this model. The simulation results are shown with dotted lines. Figure 7-8: Comparison of single- and two-phase adsorbed concentration of Na at x D = 1 for the floods of Figure 7-7. The surface composition is not affected by the anion shock. The simulation results are shown with dotted lines. 191

211 Figure 7-9: Comparison of analytical solution results (solid line) and simulation results (dotted line) for high salinity and low salinity injection considering the effect of wettability alteration. The analytical solution with no CEC over predicts the effect of low salinity injection. Figure 7-10: Solutions for low salinity water flooding. Left figure shows the analytical solution with original CEC and the right figure shows the analytical solution without CEC. The wettability front velocity is over estimated in the right figure. 192

212 Figure 7-11: Walsh diagram for low salinity polymer injection. Fractional flows are shown for oil wet (OW), oil wet with polymer (OWP) and water wet with polymer (WWP). The anion and polymer shocks have the same velocity. The wettability front is very slow. 193

213 Figure 7-12: Analytical solution and simulation results matched experimental data (Shaker Shiran and Skauge 2013). CEC and oil wet S or were not provided for the experimental data and they are the only two fitting parameters used to match the low salinity flood. 194

214 Figure 7-13: Walsh diagram for low salinity flood followed by polymer injeciton. The fractional flows are shown for oil wet (OW), water wet (WW), and water wet polymer (WWP). The solution is not self similar and the results are calculated by the front tracking algorithm. 195

215 Figure 7-14: Low salinity pre-flush. The yellow area shows the high salinity water and blue area represents the polymer flooded region. Figure 7-15: Saturation fronts for 1D low salinity slug injection. The low salinity slug size is 0.2 PV for left figure and 0.6 PV for the right figure. The Na + significantly reduces at the front shown by the red line so that wettability alteration occurs across this line. The shaded region represents the water with very low salinity. 196