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

ii

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

iii

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

iv

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.

v

Table of Contents

List of Figures .......................................................................................................................... viii

List of Tables ........................................................................................................................... xv

Nomenclature ........................................................................................................................... xvi

Acknowledgments.................................................................................................................... xix

Chapter 1 Introduction ............................................................................................................. 1

1.1 Description of problem .............................................................................................. 1 1.2 Research objectives .................................................................................................... 5 1.3 Structure of the dissertation ....................................................................................... 5

Chapter 2 Background and literature review ........................................................................... 7

2.1 Enhanced oil recovery ................................................................................................ 7 2.1.1 Phase behavior and fluid properties ................................................................ 9 2.1.2 Petrophysics .................................................................................................... 13 2.1.3 Volumetric sweep efficiency ........................................................................... 13 2.1.4 Local displacement efficiency ......................................................................... 16

2.2 Displacement mechanisms of gas floods ................................................................... 17 2.2.1 Experimental methods for estimating MMP ................................................... 18 2.2.2 Computational methods for estimating MMP ................................................. 19 2.2.3 Three-hydrocarbon-phase displacements ........................................................ 26

2.3 Displacement mechanism of low salinity polymer flooding ...................................... 27 2.4 Hyperbolic system of equations ................................................................................. 30

2.4.2 Method of characteristics ................................................................................ 33 2.4.3 Finite difference estimation of solution ........................................................... 37 2.4.4 Fractional flow theory ..................................................................................... 38 2.4.5 Limitations of current MOC solutions for gas flooding .................................. 40

2.5 Summary .................................................................................................................... 43

Chapter 3 Gas flooding mathematical model ........................................................................... 45

3.1 Conservation law ........................................................................................................ 45 3.2 Tie lines ...................................................................................................................... 48 3.3 MOC solution for gas flooding .................................................................................. 51 3.4 Ternary compositional routes for complex phase behavior ....................................... 53

3.4.1 Bifurcating phase behavior .............................................................................. 54 3.4.2 Composition-route construction ...................................................................... 56 3.4.3 Features of displacements with bifurcating phase behavior ............................ 60

3.5 Summary .................................................................................................................... 65

Chapter 4 Three-component global Riemann solver using splitting of equations ................... 81

vi

4.1 Basic analysis, precise assumptions, and the main results ......................................... 86 4.2 Basic wave behavior .................................................................................................. 90

4.2.1 The C-waves .................................................................................................... 90 4.2.2 The β-waves .................................................................................................... 91

4.3 Global solutions of Riemann problems ...................................................................... 95 4.3.1 Connecting C-waves with β-shock .................................................................. 95 4.3.2 Connecting β-rarefaction wave to C-waves .................................................... 100 4.3.3 Global existence and uniqueness of solutions for Riemann problems ............ 105

4.4 Numerical Simulations with Front Tracking .............................................................. 106 4.5 Initial and injection tie-line selection ......................................................................... 109 4.6 Example ternary displacements .................................................................................. 111 4.7 Summary .................................................................................................................... 114

Chapter 5 Tie-line routes for multicomponent displacements ................................................. 128

5.1 Mathematical model ................................................................................................... 128 5.2 Tie-line space ............................................................................................................. 133

5.2.1 Thermodynamic definition .............................................................................. 134 5.2.2 Composition space parametrization ................................................................ 137

5.3 Example ruled surface routes ..................................................................................... 138 5.3.1 Constant K-value displacement ....................................................................... 138 5.3.2 Example four-component displacements ........................................................ 140

5.4 Summary .................................................................................................................... 143

Chapter 6 Robust and accurate MMP calculation using an equation-of-state ......................... 150

6.1 Riemann solver in tie-line space ................................................................................ 150 6.1.1 Pseudoternary ruled surfaces ........................................................................... 150 6.1.2 Estimation of ruled surfaces ............................................................................ 152 6.1.3 Riemann solver in tie-line space ..................................................................... 153 6.1.4 Validation of estimate Riemann solver ........................................................... 154

6.2 MMP calculation for 𝑵𝒄 displacements ..................................................................... 155 6.2.1 Four-component displacements ....................................................................... 156 6.2.2 Five-component displacement ......................................................................... 157 6.2.3 Twelve-component displacement by Zick (1986) ........................................... 157 6.2.4 Eleven-component displacement by Metcalfe and Yarborough (1979) .......... 158 6.2.5 Bifurcating phase behavior .............................................................................. 158

6.3 Summary .................................................................................................................... 159

Chapter 7 Application of splitting technique to low salinity polymer flooding....................... 167

7.1 Mathematical model ................................................................................................... 167 7.1.1 Immiscible oil/water flow ............................................................................... 168 7.1.2 Cation Exchange Reaction Network ............................................................... 168 7.1.3 Reactive Transport Model ............................................................................... 170 7.1.4 Wettability alteration ....................................................................................... 170 7.1.5 Polymer Flooding Model ................................................................................ 171 7.1.6 Numerical solution .......................................................................................... 172

7.2 Analytical solution ..................................................................................................... 173

vii

7.2.1 Decoupled system of equations ....................................................................... 174 7.2.2 Reactive transport solution .............................................................................. 177 7.2.3 Polymer transport solution .............................................................................. 178 7.2.4 Maping fractional flow .................................................................................... 179 7.2.5 Front tracking algorithm .................................................................................. 180 7.2.6 Matching wettability front retardation independent of reactions .................... 181 7.2.7 Matching reactions independent of fractional flow ......................................... 182

7.3 Results ........................................................................................................................ 183 7.3.1 Two phase CEC without wettability alteration ............................................... 183 7.3.2 Low salinity waterflooding ............................................................................. 184 7.3.3 Low salinity polymer experiment .................................................................... 184 7.3.4 Low salinity slug injection with varying slug size .......................................... 185

7.4 Summary .................................................................................................................... 186

Chapter 8 Conclusions ............................................................................................................. 198

8.1 Summary and conclusions.......................................................................................... 198 8.2 Future research ........................................................................................................... 201

8.2.1 Application of splitting to compositional simulation ...................................... 201 8.2.2 WAG injection and hysteresis ......................................................................... 202 8.2.3 Fluid characterization ...................................................................................... 202 8.2.4 Displacement mechanism for combined EOR techniques .............................. 202

References ................................................................................................................................ 203

Appendix A Tie-line derivatives .............................................................................................. 225

Appendix B Switch condition .................................................................................................. 230

Appendix C Shock composition paths ..................................................................................... 235

Appendix D Reservoir simulation in tie-line space ................................................................. 238

Appendix E Object oriented design of phase equilibrium calculation ..................................... 239

viii

List of Figures

Figure 2-1: Scanning electron microscope (SEM) image of Berea sandstone core (Schembre and Kovscek

2005). .................................................................................................................................................... 44

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

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

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

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........... 67

Figure 3-4: Phase behavior of pseudoternary system showing the split of the two-phase region with

pressure at 133oF. ................................................................................................................................. 68

Figure 3-5: Three phase behavior at 40 oF and 1000 psia. ........................................................................... 69

Figure 3-6: Critical locus of C1N2 and CO2. ................................................................................................. 69

Figure 3-7: K-values at 16000 psia, 133ᵒF along the line where C26-35 concentration is zero. ..................... 70

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

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

Figure 3-10: Triangle of shocks and continuity of solution........................................................................... 73

ix

Figure 3-11: Tie-line eigenvalue from the oil to gas tie lines along the red non-tie line path shown in Figure

3-10. ...................................................................................................................................................... 73

Figure 3-12: Analytical and numerical composition profile using 20,000 grid blocks for displacement of oil

1 in Figure 3-10 by pure CO2 at 16,000 psia. ....................................................................................... 74

Figure 3-13: Effect of numerical dispersion on the composition route for oil 2 at 16,000 psia. ................... 74

Figure 3-14: K-values for the two-phase regions at 21,000 psia (see Figure 3-4) along the C1N2-CO2 axis of

the ternary diagram. .............................................................................................................................. 75

Figure 3-15: Discontinuous dispersion-free composition routes for point A. ............................................... 75

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

Figure 3-17: Discontinuity in recovery calculated by simulation. ................................................................. 77

Figure 3-18: Example of three intersecting tie lines through an oil composition. ......................................... 77

Figure 3-19: Analytical composition routes for oils 1 - 4 at 16,000 psia. ..................................................... 78

Figure 3-20: Analytical composition profiles for the heavy pseudocomponent for four oil compositions

displaced by pure CO2 at 16,000 psia. .................................................................................................. 78

Figure 3-21: Analytical composition profiles for the light pseudocomponent for four oil compositions

displaced by pure CO2 at 16,000 psia. .................................................................................................. 79

Figure 3-22: Analytical composition profiles for the CO2 component for four oil compositions displaced by

pure CO2 at 16,000 psia. ....................................................................................................................... 79

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

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)

x

coordinate. The two red curves are the boundary of the two-phase region, and green lines are tie lines.

............................................................................................................................................................ 116

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

Figure 4-3: Functions 𝑭(𝑪, 𝜷) and 𝓕(𝑪; 𝜷, 𝒂). a=0.2 on the left plot. ...................................................... 117

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. ................................................................................................................................................ 117

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. .............................................. 118

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). ...................................................... 118

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. ............................................................................................................................ 119

Figure 4-8: Two possible relations between the curves 𝑪𝟏, 𝑪𝟐, 𝑪𝟑 and 𝑪𝟒. ............................................ 119

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). ....................................................................................... 119

Figure 4-10: Case 2, when 𝑪𝟑 < 𝑪𝑳 < 𝑪𝟐, the 𝜷-wave path consists of two β-rarefaction waves with a 𝑪-

contact in between. ............................................................................................................................. 120

Figure 4-11: Estimation of large β-rarefaction with smaller waves (Left) and convergence of results to the

correct solution (Right). ...................................................................................................................... 120

Figure 4-12: Comparison of the composition path calculated by finite difference simulation and front

tracking. .............................................................................................................................................. 121

xi

Figure 4-13: Comparison of the composition profiles calculated by finite difference simulation using

10,000 grid blocks and front tracking with 𝜺 = 𝟎. 𝟎𝟓. ....................................................................... 121

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

Figure 4-15: Composition profiles at 𝒕 = 𝟎. 𝟎 (bottom),0.1,0.2,0.3,0.4 and 0.5 (top). ............................... 123

Figure 4-16: Analytical solution for ternary displacement in composition space. Point a is on the envelope

curve and 𝑪𝟏𝒂 = −𝟏/𝝀𝜷 . ................................................................................................................. 124

Figure 4-17: Tie-line coefficients for ternary phase behavior of Figure 1 in tie-line space. ....................... 124

Figure 4-18: Analytical solution for three-component displacements showing shocks and rarefactions in

Lagrangian coordinates. ...................................................................................................................... 125

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). ....................................................................................................................... 125

Figure 4-20: Tie-line coefficients for three-component bifurcating phase behavior in tie-line space, where

negative sign in Eqs. (3) is used. ........................................................................................................ 126

Figure 4-21: Eigenvalue for three-component displacements with bifurcating phase behavior. Dashed line is

a shock. ............................................................................................................................................... 126

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

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

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

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. ............................................................................................................................ 146

Figure 5-4: Phase diagram in tie-line space and ruled surfaces for constant K-values. ............................. 146

Figure 5-5: Eigenvalues along the line 𝜷𝟐 = 𝟎. ....................................................................................... 147

Figure 5-6: Parametrization of the tie-line path for automatic construction of tie-line path for quaternary

phase behavior of Table 5.1. ............................................................................................................... 147

xii

Figure 5-7: Phase diagram in tie-line space and ruled surfaces for four-component displacements in Table

5.2 at 2900 psia and 160oF. ................................................................................................................. 148

Figure 5-8: Quaternary phase diagram with bifurcating phase behavior generated based on Ahmadi et al.

(2011) at 8000 psia and 133oF. ........................................................................................................... 148

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

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 6.1.

............................................................................................................................................................ 161

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 oF. ........................... 161

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 oF. ...................................................................................................................... 162

Figure 6-5: The tie line route parameters for the displacement of I1 by J1 in Table 5.2. ........................... 162

Figure 6-6: Key tie-line lengths calculated by analytical solution and shortest tie-line length calculated by

simulation for the displacement in Table 5.2 . .................................................................................... 163

Figure 6-7: Four-component displacements with bifurcating phase behavior. The shock-Jump MOC over

predict MMP by almost 4000 psi. ....................................................................................................... 163

Figure 6-8: The key tie line length for five-component displacement. The shortest tie-line length is

calculated by simulation. .................................................................................................................... 164

Figure 6-9: The tie-line route at different pressures for five-component displacements. ............................ 164

Figure 6-10: Tie-line length variation with pressure calculated using approximate Riemann solver for

displacement in Table 6.4 at 185 oF. MMP is estimated to be at 3097 psia. ....................................... 165

Figure 6-11: Tie-line length variation form injection to initial tie-line at different pressures and 185 oF with

compositions in Table 6.4 . ................................................................................................................. 165

Figure 6-12: Shortest tie-line length variation with pressure for displacement from Johns and Orr (1996).

............................................................................................................................................................ 166

xiii

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). ... 166

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. .......................................................... 188

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

Figure 7-3: Piecewise linear approximation of fractional flow is commonly used in front tracking

algorithms. .......................................................................................................................................... 189

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

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. ................................... 190

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

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. ...................................... 191

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. ..................................................................................................................... 191

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. ............................................ 192

xiv

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

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

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. ...................................................... 194

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

Figure 7-14: Low salinity pre-flush. The yellow area shows the high salinity water and blue area represents

the polymer flooded region. ................................................................................................................ 196

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

Figure 7-16: Water saturation profiles for different low salinity slug sizes after 15 PVI calculated by MOC

with cation exchange reaction. ........................................................................................................... 197

Figure 7-17: Comparison of Sor 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. .......................................................................................................................... 197

xv

List of Tables

Table 3.1 – Component properties of the fluid system of Orr et al. (1993) .................................................. 65

Table 3.2 – Component properties of the fluid system of Ahmadi et al. (2011) ........................................... 65

Table 4.1 – Fluid characterization for the ternary system ........................................................................... 115

Table 4.2 – Binary interaction coefficients for the ternary system .............................................................. 115

Table 4.3 – Initial condition for example problem 3 ................................................................................... 115

Table 4.4 – Injection condition for example problem 3 .............................................................................. 115

Table 4.5 – Component properties and compositions for bifurcating phase behavior developed based on

Ahmadi et al. (2011) ........................................................................................................................... 115

Table 5.1 – The compositions and K-values for example case (Yuan et al. 2005) ...................................... 144

Table 5.2 – Component properties and compositions for displacement by Dutra et al. (2009) ................... 144

Table 6.1 – The compositions for four-component example with component properties in Table 6.3 ....... 159

Table 6.2 – Input properties for six-component MMP calculations from Johns (1992) .............................. 159

Table 6.3 – Input properties for five-component MMP calculations ........................................................... 160

Table 6.4 – The compositions for 12-component displacement from Zick (1986) ..................................... 160

Table 7.1 – Water composition for the single- and two-phase displacements. ............................................ 187

Table 7.2 – Reaction parameters. CEC2 and CEC3 are calculated by matching experiments. ................... 187

Table 7.3 – The Corey relative permeability parameters for the experiments. ............................................ 187

xvi

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

𝐶𝑠 The concentration of a species

��𝑠 The adsorbed concentration of a species

𝐷 Composition domain

𝐷 Dispersion coefficient

F Overall fractional flow

𝐹𝑝 Molar rate of the primary species p

𝐹𝑞 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

𝐾𝑒𝑞,𝑟 Equilibrium constant of reaction 𝑟

𝐾𝑖𝑗 Derivatives of K-value respect to phase mole fraction

𝐿 Horizontal distance between injection and production wells

𝐿𝑗 Liquid phase 𝑗 for three phase fluids

𝑀𝑝 Molar density of the primary species p

𝑀𝑞 Molar density of the secondary species 𝑞

𝑁𝑝 The number of the primary species

𝑁𝑠𝑒𝑐 The number of secondary reactions

𝑄𝑝 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

xvii

Superscripts

A Upstream shock

B Downstream shock

D Downstream

I Independent

L Left

R Right

U Upstream

𝑘𝑟𝛼 Relative permeability of phase 𝛼

𝑘𝑟𝛼∗ Endpoint relative permeability of phase 𝛼

𝑘𝑟𝛼,𝑤𝑤∗ Water wet endpoint relative permeability of phase 𝛼

𝑘𝑟𝛼,𝑜𝑤∗ Oil wet endpoint relative permeability of phase 𝛼

nc Number of components

𝑛𝛼 Exponent in Corey’s model for phase α

𝑛𝑝 Number of phases

Pe Peclet number

t time

x length

x Mole fraction in phase

z Mixture mole fraction

∅𝑖 Fugacity coefficient of component i

𝜌𝑐𝑖 Constant density of component i

𝜌𝑗 Molar or volumetric density of phase j

α Tie-line coefficient

β Tie-line coefficient

ε Dispersion coefficient

λ Eigenvalue

Λ Shock velocity

𝚨 Tie-line coefficient matrix

𝚩 Tie-line coefficient matrix

𝛤 Tie line

𝛾 Tie line space parameter

𝜇 Viscosity

𝜉 Shock layer moving coordinate

𝜑 Lagrangian coordinate

𝜓 Lagrangian coordinate

𝜙 Porosity

∅𝑖 Fugacity coefficient of component i

𝜌𝑐𝑖 Constant density of component i

𝜌𝑗 Molar or volumetric density of phase j

xviii

e Envelope curve

ini Initial

inj Injeciton

l Liquid

v Vapor

Subscripts

D Dimensionless

g Gas

i Component

i Initial

i Tie-line parameter index

ij Component i in phase j

inj Injection

j Phase

nt Non-tie line

o Oil

P Production

r Residual

t Tie line

w 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

xix

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.

1

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

2

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

3

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

4

not exist for some oil and gas compositions. In those cases, L1-L2 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

5

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 two-

phase 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

6

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.

.

7

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 CO2

storage (Li et al. 2015a). Injection of CO2 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, CO2 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

8

storage of CO2 in saline formations (Hitzman 2013, Zoback and Gorelick 2012, National

Academy of Sciences 2012). The incremental oil recovery from CO2 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 CO2 EOR and storage a potentially profitable process. Effective use of CO2

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 = 𝐸𝑣𝐸𝐷 . (2.1)

Volumetric sweep efficiency, 𝐸𝑣, is defined as the volume of oil contacted over the

amount of oil in place while local displacement efficiency, 𝐸𝐷, 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.

9

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 CO2 injection has

been used widely in west Texas (Mizenko 1992, Orr and Taber 1984, Tanner 1992, Stein 1992).

CO2 is widely used as a solvent in gas injection because miscibility can be achieved at lower

pressures, and CO2 has a liquid like density. The phase behavior of a mixture of CO2 and

hydrocarbon components is more complicated compared to hydrocarbon mixtures. The mixture of

CO2 and reservoir oil can form a second hydrocarbon liquid at reservoir conditions. Although

CO2 floods usually have complex phase behavior, CO2 flooding has very positive features. For

example, underground CO2 storage can be used to mitigate the CO2 effect on global warming.

Also the second hydrocarbon liquid in three-phase CO2 floods acts as an extraction agent to

10

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

11

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.

12

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.

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 Nc 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).

13

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.

2.1.3 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

14

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,

15

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 CO2 storage capacity based on the assumption that the volume

occupied by oil will become available for CO2. Wood et al. (2008) developed dimensionless

groups for continuous gravity-stable CO2 flooding of homogeneous reservoirs with high vertical

permeabilities and large dip angles. They estimated oil recovery and CO2 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 CO2 EOR and

storage using horizontal wells. The scaling groups are as follows.

𝐿

𝐻√𝐾𝑧𝐾𝑥

Effective aspect ratio

𝑉𝐷𝑃 Dykstra-Parsons coefficient

𝜆𝑥𝐷 Correlation length coefficient in x-direction

16

𝜆𝑧𝐷 Correlation length coefficient in z-direction

𝜇𝑜𝜇𝑔

Mobility ratio (CO2-oil)

𝜌𝑜 − 𝜌𝑔

𝜌𝑤 − 𝜌𝑜 Buoyancy ratio

𝐻(𝜌𝑜 − 𝜌𝑤)𝑔

𝑃𝑝 − 𝑃𝑖𝑛𝑗 Buoyancy number

1 − 𝑆ℎ𝑟 − 𝑆𝑤𝑖1 − 𝑆ℎ𝑟 − 𝑆𝑤𝑟

Normalized initial oil saturation

𝜇𝑜𝜇𝑤

𝑘𝑟𝑤𝑜

𝑘𝑟𝑜𝑜 Mobility ratio (water-oil)

𝑆ℎ𝑟 Residual hydrocarbon saturation

𝑆𝑤𝑟 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.

2.1.4 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)

17

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

18

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 CO2

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.

2.2.1 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 CO2 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

19

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.

2.2.2 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

20

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 slim-

tube 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

21

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 three-

hydrocarbon 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 (L2) 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

22

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

23

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 phase-

behavior-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 three-

phase 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.

24

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 𝑁𝑐 − 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 CO2.

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 𝑁𝑐 − 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

25

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 shock-

jump 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 CO2. 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 oAPI 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 oAPI. Ahmadi et al. (2011) used a

26

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 3.

2.2.3 Three-hydrocarbon-phase displacements

Low-temperature oil displacements by CO2 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 CO2 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 CO2 (Khan et al. 1982) gave high

displacement efficiency of more than 90% in the presence of immiscible three-hydrocarbon-phase

flow.

27

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 (L1 and L2) phases merge

in the presence of the vapor (V) phase, and the other is where the second-liquid (L2) and V phases

merge in the presence of the L1 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 CO2 involving

L1-L2-V equilibrium can achieve greater than 95% displacement efficiency even if the L1 and V

phases by themselves would be significantly immiscible. As explained by Okuno et al. (2011)

high displacement efficiency is possible because the L2 phase serves as a buffer between the L1

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

28

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+, Ca2+ and

Mg2+ 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

29

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.

30

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

31

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.

𝑪𝑡𝐷 + 𝑭𝒄(𝑪)𝑪𝑋𝐷 = 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 𝑭𝑪 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

32

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 two-

component partially miscible gas floods, where components can transfer between phases.

33

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 𝑭𝑪 in Eq. (2.2) and the eigenvectors of 𝑭𝐶 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.

34

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,

𝐹𝑖𝑈 − 𝐹𝑖

𝐷

𝐶𝑖𝑈 − 𝐶𝑖

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

35

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:

𝜆𝑖𝑈 > 𝛬 > 𝜆𝑖

𝐷 𝑖 = 1, 2 . (2.4)

Based on the Lax entropy conditions, a necessary condition for shock admissibility is:

𝜆2𝑈 > 𝛬 > 𝜆1

𝐷 , (2.5)

where λ2 is always greater than λ1, and λ2 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,

𝐹𝑖∗ − 𝐹𝑖

𝐷

𝐶𝑖∗ − 𝐶𝑖

𝐷 ≥𝐹𝑖𝑈 − 𝐹𝑖

𝐷

𝐶𝑖𝑈 − 𝐶𝑖

𝐷 𝑖 = 1,… , 𝑁𝑐 − 1 , (2.6)

in which 𝐶𝑖∗ is a point on the shock locus between fixed upstream and downstream compositions.

The index 𝑖 is arbitrary because 𝐶𝑖∗ 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,

36

𝜕𝐶𝑖𝜀

𝜕𝑡𝐷+𝜕𝐹𝑖𝜕𝑥𝐷

= 휀𝜕2𝐶𝑖

𝜀

𝜕𝑥𝐷2 𝑖 = 1,… ,𝑁𝑐 − 1 , (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:

𝑙𝑖𝑚𝜀→0

∑ ∫ ‖𝐶𝑖 − 𝐶𝑖𝜀‖2

1

0

𝑑𝑥𝐷

𝑁𝑐−1

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).

37

c. Check entropy condition.

3- Composition velocities should increase from injection composition to initial composition

2.4.3 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.

𝐶𝑖,𝑘𝑛+1 = 𝐶𝑖,𝑘

𝑛 − Λ(𝐹𝑖,𝑘𝑛 − 𝐹𝑖,𝑘−1

𝑛 ) , (2.9)

where 𝛬 = 𝛥𝑡𝐷/𝛥𝑥𝐷. 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

2𝛥𝑥𝐷(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)

38

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.

2.4.4 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 three-

component miscible fluids is complicated. Many researchers have solved the Riemann problems

39

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

40

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 multi-

component 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 multi-

component 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.

2.4.5 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 𝑁𝑐 − 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

41

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 CO2 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 (L1-L2 and L1-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 CO2. 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.

42

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 four-

component displacement with constant K-values using their auxiliary system of equations. Entov

(2000) further suggested that potential coordinate transformations could be used such that 𝑁𝑐 − 2

of the eigenvalues would become independent of fractional flow, where 𝑁𝑐 is the number of

components. Pires et al. (2006) expanded on this idea and developed Lagrangian coordinates to

43

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

44

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.

45

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).

𝜕

𝜕𝑡∑𝑥𝑖𝑗𝜌𝑗𝑆𝑗

𝑛𝑝

𝑗=1

+𝜕

𝜕𝑥[𝑣

𝜙∑𝑥𝑖𝑗𝜌𝑗𝑓𝑗

𝑛𝑝

𝑗=1

] = 0 𝑖 = 1,… ,𝑁𝑐 . (3.1)

where 𝑥𝑖𝑗 is the mole fraction of component i in phase 𝑗, 𝜌𝑗 is phase density and 𝑆𝑗 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).

𝑓𝑗 =

𝑘𝑟𝑗𝜇𝑗

∑𝑘𝑟𝑗𝜇𝑗

𝑛𝑝𝑗=1

. (3.2)

46

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).

𝑐𝑖𝑗 =

𝑥𝑖𝑗𝜌𝑐𝑖

∑𝑥𝑘𝑗𝜌𝑐𝑘

𝑛𝑐𝑘=1

, (3.3)

in which, 𝜌𝑐𝑘 is the constant molar density of the component i. Similarly the phase density can be

calculated by the following relationship.

𝜌𝑗 = (∑𝑥𝑖𝑗

𝜌𝑐𝑖

𝑁𝑐

𝑖=1

)

−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,

𝜌𝑐𝑖𝑐𝑖𝑗 = 𝜌𝑗𝑥𝑖𝑗 . (3.5)

Overall concentration and fractional flow of component i are calculated by Eqs. (3.6) and

(3.7).

47

𝐶𝑖 =∑𝑥𝑖𝑗𝑆𝑗

𝑛𝑝

𝑗=1

, (3.6)

𝐹𝑖 =∑𝑥𝑖𝑗𝑓𝑗

𝑛𝑝

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).

𝑡𝐷 =∫𝑣𝑑𝑡

𝜙𝐿= 𝑃𝑉𝐼 ,

(3.8)

𝑥𝐷 =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.

𝜕𝐶𝑖𝜕𝑡𝐷

+𝜕𝐹𝑖𝜕𝑥𝐷

= 0 𝑖 = 1,… ,𝑁𝑐 − 1 . (3.10)

The flow equations can be shown in matrix form as Eq. (3.11).

𝑪𝑡𝐷 + 𝑭𝒄(𝑪)𝑪𝑋𝐷 = 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).

48

𝐶 = 𝐶𝑖𝑛𝑗 𝑎𝑡 𝑥 = 0 ,

𝐶 = 𝐶𝑖𝑛𝑖 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 cij is equal to xij.

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.

49

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).

𝐶𝑖 = 𝛼𝑖−1(𝚪)𝐶1 + 𝛽𝑖−1(𝚪), 𝑖 = 2,… , 𝑁𝑐 − 1 . (3.13)

where 𝚪 is a 𝑁𝑐 − 2 vector that parameterizes the tie-line space. Johns (1992) used 𝑐11 as the tie-

line 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 = 𝜜𝐶1 +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.

𝑇𝐿 = √∑(𝑥𝑖1 − 𝑥𝑖2)2

𝑁𝑐

𝑖=1

. (3.15)

Overall fractional flow of components along a tie line has the same linear relationship as

concentrations.

50

𝐹𝑖 = 𝛼𝑖−1(𝚪)𝐹1 + 𝛽𝑖−1(𝚪) 𝑖 = 2,… ,𝑁𝐶 − 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.

𝜕𝐶1𝜕𝑡𝐷

+𝜕𝐹1𝜕𝑥𝐷

= 0 ,

(3.17)

𝐶1𝜕𝛼𝑖𝜕𝑡𝐷

+𝜕𝛽𝑖𝜕𝑡𝐷

+ 𝐹1𝜕𝛼𝑖𝜕𝑥𝐷

+𝜕𝛽𝑖𝜕𝑥𝐷

= 0 𝑖 = 2,… ,𝑁𝐶 − 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 𝑁𝐶 = 3 as,

(

𝜕𝐶1𝜕𝑡𝐷

𝜕𝑐11𝜕𝑡𝐷 )

+

(

𝜕𝐹1𝜕𝐶1

𝜕𝐹1𝜕𝛽

0𝐹1 + ℎ

𝐶1 + ℎ)

(

𝜕𝐶1𝜕𝑥𝐷

𝜕𝛽

𝜕𝑥𝐷)

= 0 ,

(3.19)

where parameter h is defined as,

51

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). 𝐶𝑖≠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.

𝐶1𝑒 = −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.

𝑑𝐶𝑖 =𝜕𝐶𝑖𝜕𝑡𝐷

𝑑𝑡𝐷 +𝜕𝐶𝑖𝜕𝑥𝐷

𝑑𝑥𝐷 = 0 𝑖 = 1,… ,𝑁𝐶 − 1 . (3.22)

The following equation will result by rearranging Eq. (3.22),

𝜕𝐶𝑖𝜕𝑡𝐷

= −𝜆𝜕𝐶𝑖𝜕𝑥𝐷

𝑖 = 1,… ,𝑁𝐶 − 1 , (3.23)

in which 𝜆 = 𝑑𝑥𝐷/𝑑𝑡𝐷 defines the characteristic lines in 𝑥𝐷 − 𝑡𝐷 space. Substitution of Eq.

(3.23) converts the flow Eqs. (3.19) to a set of ODE’s.

52

(

𝜕𝐹1𝜕𝐶1

− 𝜆𝜕𝐹1𝜕𝛽

0𝐹1 + ℎ

𝐶1 + ℎ− 𝜆

)

(

𝜕𝐶1𝜕𝑥𝐷

𝜕𝛽

𝜕𝑥𝐷)

= 0 .

(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

𝜆𝑡 =𝜕𝐹1𝜕𝐶1

and 𝜆𝑛𝑡 =𝐹1 + ℎ

𝐶1 + ℎ .

(3.25)

𝜆𝑡 is the eigenvalue corresponding to the eigenvector along the tie line therefore is called

the tie-line eigenvalue and 𝜆𝑛𝑡 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 nontie-

line 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.

53

The corresponding solutions of Eq. (3.24) are eigenvectors of the coefficient matrix of

Eq. (3.24).

𝑒𝑡 = (1

0), 𝑒𝑛𝑡 =

(

1

𝜆𝑛𝑡 − 𝜆𝑡𝜕𝐹1𝜕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 𝑥𝐷 and 𝑡𝐷. 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

54

behavior. We discuss the bifurcating phase behavior briefly followed by the composition route

construction for CO2 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 6.

3.4.1 Bifurcating phase behavior

At a range of low temperatures, a mixture of CO2 and a hydrocarbon component like

hexadecane form three phases at one pressure. A mixture of CO2 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 CO2/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

CO2. Their results showed that the phase diagram for a mixture of CO2, 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 CO2/C1/C26-35 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

55

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 oF. 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 CO2 rich two phase region is more like L1-

L2 behavior, while the C1N2 rich two-phase region similar to L1-V phase behavior. L1-V region

disappears at higher pressures, while there is a L1-L2 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 C1N2 and CO2. The

mixture of C1N2 and CO2 has a three-phase region at lower temperature (100 oF) as shown in

Figure 3-5. The ternary system has a three phase region because the pressure and temperature of

56

the mixture are inside the critical locus of the C1N2 and CO2 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 C1N2-CO2 side of the phase

diagram. The order of K-values changes as CO2 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 non-

monotonic 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 16000 psia are shown in Figure 3-9. The nontie-

line 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 ∂F1/ ∂c11 = 0 and consequently arbitrary eigenvalues. The only possible

path to connect a tie line with CO2 rich phase to a tie line with rich C1N2 vapor phase is shown by

the red line.

3.4.2 Composition-route construction

In this section we discuss the analytical solution for the pseudo-ternary system of Figure

3-10. Consider the composition path for displacement of Oil 1 by pure CO2 in Figure 3-10. 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

57

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 3-10.

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 nontie-

line 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).

58

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 nontie-

line 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 3-11.

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

59

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

60

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 CO2 as a function of the dimensionless

velocity are shown in Figure 3-12. Numerical simulation with 20,000 grid blocks agrees well with

the dispersion-free MOC composition profile shown in Figure 3-12.

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.

3.4.3 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

61

conventional condensing ternary drive, where CO2 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. CO2 is the intermediate component in this portion of the route because it

has a volatility (K-value) between that of the methane-like component (C1N2) 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 CO2), the displacement is vaporizing in that the

methane-like component is vaporized from the equilibrium liquid into the equilibrium vapor.

C1N2 is the intermediate component in this portion of the ternary displacement because its K-

value is less than CO2. 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 C1N2 and CO2 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 CO2. 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

62

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 3-17.

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 0.001 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 3-16. 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 C26-36 (heavy pseudocomponent). One of the tie lines extends from the L1-L2

region through this oil, while another extends from the L1-V region. It is clear that the tie line that

63

extends from the L1-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 L1-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 (C1N2 is more volatile than CO2). 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 3-22. Once the oil composition lies

within the region of multiple tie-line extensions the displacement is purely vaporizing and the tie-

line 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

64

shock to the gas, resulting in a constant state at that composition. Otherwise, the route takes a tie-

line 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 L1-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

L1-L2 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.

65

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

Tc (oF) Pc (psia) ω CO2 C1 nC4 C10

CO2 87.90 1071.0 0.2250 - - - -

C1 -116.63 667.8 0.0104 0.1000 - - -

nC4 305.65 550.7 0.2010 0.1257 0.0270 - -

C10 652.10 305.7 0.4900 0.0942 0.0420 0.0080 0.0080

Table 3.2 – Component properties of the fluid system of Ahmadi et al. (2011)

Properties Binary Interaction Parameters

Tc (oF) Pc (psia) ω CO2 C1 C26-35

CO2 87.89 1069.87 0.225 - - -

C1 -118.67 664.06 .00857 0.1186 - -

C26-35 1258.83 224.98 1.0908 0.0847 0.008 -

66

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.

67

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.

68

Figure 3-4: Phase behavior of pseudoternary system showing the split of the two-phase region

with pressure at 133oF.

69

Figure 3-5: Three phase behavior at 40 oF and 1000 psia.

Figure 3-6: Critical locus of C1N2 and CO2.

70

Figure 3-7: K-values at 16000 psia, 133ᵒF along the line where C26-35 concentration is zero.

71

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.

72

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

73

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 3-10.

74

Figure 3-12: Analytical and numerical composition profile using 20,000 grid blocks for

displacement of oil 1 in Figure 3-10 by pure CO2 at 16,000 psia.

Figure 3-13: Effect of numerical dispersion on the composition route for oil 2 at 16,000 psia.

75

Figure 3-14: K-values for the two-phase regions at 21,000 psia (see Figure 3-4) along the C1N2-

CO2 axis of the ternary diagram.

Figure 3-15: Discontinuous dispersion-free composition routes for point A.

76

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.

77

Figure 3-17: Discontinuity in recovery calculated by simulation.

Figure 3-18: Example of three intersecting tie lines through an oil composition.

78

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 CO2 at 16,000 psia.

79

Figure 3-21: Analytical composition profiles for the light pseudocomponent for four oil

compositions displaced by pure CO2 at 16,000 psia.

Figure 3-22: Analytical composition profiles for the CO2 component for four oil compositions

displaced by pure CO2 at 16,000 psia.

80

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.

81

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),

(𝐶1)𝑡 + (𝐹1(𝐶1, 𝐶2))𝑥 = 0 , (𝐶2)𝑡 + (𝐹2(𝐶1, 𝐶2))𝑥 = 0 ,

(4.1)

associated with initial data

𝐶1(0, 𝑥) = 𝐶1(𝑥) , 𝐶2(0, 𝑥) = 𝐶2(𝑥) . (4.2)

The independent variables (𝑡, 𝑥) are normalized such that the overall velocity is 1. Here

𝐶𝑖 is the overall ith

component volume fraction, and 𝐹𝑖 is the overall 𝑖𝑡ℎ component flux. For the

third component, we trivially have

𝐶3 = 1 − 𝐶1 − 𝐶2 , 𝐹3 = 1 − 𝐹1 − 𝐹2 .

82

The couplet (𝐶1, 𝐶2) takes values in a triangular domain

𝐷 = {(𝐶1, 𝐶2)|𝐶1 ≥ 0, 𝐶2 ≥ 0,1 − 𝐶1 − 𝐶2 > 0} .

For the phase behaviors that are considered in this paper, there exists a subset 𝐷2 ⊂ 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 region𝐷1 = 𝐷\𝐷2, we trivially have

𝐹1(𝐶1, 𝐶2) = 𝐶1 , 𝐹2(𝐶1, 𝐶2) = 𝐶2 .

We briefly derive the equations in the two-phase region. We denote by 𝑐𝑖𝑙 and 𝑐𝑖𝑔 the

composition of component i in the liquid and gaseous phases, respectively. For (𝐶1, 𝐶2) ∈ 𝐷2 the

compositions 𝑐𝑖𝑙 and 𝑐𝑖𝑔, together with the liquid phase saturation 𝑆, satisfy the following

equations,

𝐶𝑖 = 𝑐𝑖𝑙𝑆 + 𝑐𝑖𝑔(1 − 𝑆), 𝐹𝑖 = 𝑐𝑖𝑙𝑓 + 𝑐𝑖𝑔(1 − 𝑓) 𝑖 = 1,2 , ∑𝑐𝑖𝑙

3

𝑖=1

=∑𝑐𝑖𝑔

3

𝑖=1

= 1. (4.3)

Here 𝑓 = 𝑓(𝑆, 𝐶1, 𝐶2) is the fractional flow of liquid, and S takes values between 0 and 1

in the two-phase region. Typically, for given (𝐶1, 𝐶2), the mapping 𝑆 ↦ 𝑓 is S-shaped with an

inflection point. The K-values, defined as

𝐾𝑖 =𝑐𝑖𝑔

𝑐𝑖𝑙 𝑖 = 1, 2, 3 , (4.4)

are determined by a phase behavior model and can either be taken as constant or a function of

(𝐶1, 𝐶2) (e.g. Michelsen 1982). For given (𝐶1, 𝐶2) and 𝐾𝑖, one can calculate 𝑐𝑖𝑙, 𝑐𝑖𝑔 and 𝑆 by

83

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 𝐹𝑖.

For fixed (𝑐𝑖𝑙 , 𝑐𝑖𝑔) for 𝑖 = 1,2, the values (𝐶1, 𝐶2) are linear functions of S. In the phase

coordinate (𝐶1, 𝐶2), as 𝑆 varies from 0 to 1, the trajectory of the couplet (𝐶1, 𝐶2) is the straight

line connecting the equilibrium points (𝑐1𝑔, 𝑐2𝑔) and (𝑐1𝑙 , 𝑐2𝑙). When 𝑆 = 0, we have (𝐶1, 𝐶2) =

(𝑐1𝑙 , 𝑐2𝑙), and when 𝑆 = 1, we have (𝐶1, 𝐶2) = (𝑐1𝑔, 𝑐2𝑔). These lines are called tie-lines. The

curves of the end-points of these tie-lines, namely the points (𝑐1𝑔, 𝑐2𝑔) and(𝑐1𝑙 , 𝑐2𝑙), 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

(𝐶1, 𝐶2) ∈ 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 𝐷2 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 𝐶1 in Eq. (4.1) and write

𝐶 = 𝐶1, 𝐹 = 𝐹1, 𝐶2 = 𝛼𝐶 + 𝛽, 𝐹2 = 𝛼𝐹 + 𝛽 , (4.5)

where 𝛼 and 𝛽 are defined as

84

𝛼 =𝑐2𝑙 − 𝑐2𝑔

𝑐1𝑙 − 𝑐1𝑔, 𝛽 = 𝑐2𝑔 − 𝛼𝑐1𝑔 .

(4.6)

Here 𝛼 indicates the slope of a tie line, and 𝛽 its interception point with the line𝐶1 = 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

𝐶𝑡 + 𝐹(𝐶, 𝛽)𝑥 = 0, 𝐶(𝛼(𝛽))𝑡 + 𝛽𝑡 + 𝐹(𝐶, 𝛽)(𝛼(𝛽))

𝑥+ 𝛽𝑥 = 0 , (4.7)

associated with the initial data

𝐶(0, 𝑥) = 𝐶(𝑥), 𝛽(0, 𝑥) = ��(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 three-

component systems as shown in Chapter 3. In (Pires et al. 2006), the following Lagrangian

coordinates (𝜑, 𝜓) was introduced,

𝜑𝑥 = −𝐶, 𝜑𝑡 = 𝐹, 𝑎𝑛𝑑 𝜓 = 𝑥 − 𝑡 . (4.9)

Straight computation leads to the following system

(𝐶

𝐹 − 𝐶)𝜑− (

1

𝐹 − 𝐶)𝜓= 0 , (4.10)

𝛽𝜑 + 𝛼(𝛽)𝜓 = 0 . (4.11)

85

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 (𝐶𝐿 , 𝛽𝐿) and (𝐶𝑅 , 𝛽𝑅), 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

𝐹−𝐶 and

1

𝐹−𝐶 do not allow a single-valued function between them. Furthermore, the coordinate

change is only valid in the set when 𝐹 > 𝐶. Indeed, let 𝐽 be the Jacobian matrix for this

coordinate change,

𝐽 ≐𝜕(𝜑,𝜓)

𝜕(𝑡, 𝑥)= (

𝐹 −𝐶−1 1

) , 𝑠𝑜 det(𝐽) = 𝐹 − 𝐶 .

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,

��𝑥 = 𝐶, ��𝑡 = −𝐹, 𝑎𝑛𝑑 �� = 𝑥 − 𝑡. (4.12)

The Jacobian matrix 𝐽 for this coordinate change is

𝐽 ≐𝜕(��,��)

𝜕(𝑡,𝑥)= (

−𝐹 𝐶−1 1

) , 𝑠𝑜 det(𝐽) = 𝐶 − 𝐹 > 0 .

86

Formal computation leads to the following system:

(𝐶

𝐹 − 𝐶)��+ (

1

𝐹 − 𝐶)��= 0 , (4.13)

𝛽�� + 𝛼(𝛽)�� = 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 (𝐶𝐿 , 𝛽𝐿) and (𝐶𝑅 , 𝛽𝑅), 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 (𝐶1, 𝐶2), no two tie-lines intersect in the domain 𝐷.

Using Eqs. (4.6) and (4.3), we have

87

𝛼(𝛽) =𝛽(1 − 𝐾2)(𝐾1 − 𝐾3)

𝛽(𝐾1 − 1)(𝐾2 − 𝐾3) + (𝐾2 − 𝐾1)(1 − 𝐾3) . (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

𝐾3 < 𝐾2 < 1 < 𝐾1, 𝑜𝑟 𝐾1 < 1 < 𝐾2 < 𝐾3 . (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 (𝐶1, 𝐶2) ∈ 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

𝑢𝑡 + 𝐴(𝑢)𝑢𝑥 = 0, 𝑤ℎ𝑒𝑟𝑒 ( ) '( ) 10

'( ) 1

CF F

A u F

C

. (4.18)

The matrix 𝐴(𝑢) has the following eigenvalues and right-eigenvectors

𝜆𝐶 = 𝐹𝐶 , 1

0

Cr

, 𝜆𝛽 =𝐹+[𝛼′(𝛽)]

−1

𝐶+[𝛼′(𝛽)]−1,

C

Fr

F

. (4.19)

88

Here the labeling of the two families are not with respect to wave speed. We referred to

𝜆𝐶 and 𝜆𝛽 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 𝜆𝛽in the direction 𝑟𝛽 gives

∇𝜆𝛽. 𝑟𝛽 =1

(𝐶𝛼′(𝛽) + 1)2𝛼′′(𝛽)(𝐹𝐶 − 𝜆

𝛽)(𝐹 − 𝐶) . (4.20)

This indicates that along the curve 𝐹 = 𝐶, the eigenvalue 𝜆𝛽 remains constant. This curve

lies between the two groups of integral curves (see the green curve in Figure 4-2), and is a 𝛽-

integral 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 𝜆𝛽

changes sign at the point where 𝐹𝐶 − 𝜆𝛽 = 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

𝜆𝐶 =𝜕𝐹

𝜕𝐶=𝐹 + [𝛼′(𝛽)]−1

𝐶 + [𝛼′(𝛽)]−1= 𝜆𝛽 , 𝑟𝐶 = 𝑟𝛽 = (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

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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

𝜆𝛽. Note that for fixed 𝛽 and 𝑎, the function 𝐶 ↦ ℱ reaches its minimum and maximum values at

𝐶𝑚𝑖𝑛 and 𝐶𝑚𝑎𝑥 respectively, where the lines (−𝑎,−𝑎) − (𝐶𝑚𝑖𝑛, 𝐹(𝐶min)) and (−𝑎,−𝑎) −

(𝐶𝑚𝑎𝑥, 𝐹(𝐶max)) 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 ℂ2 either strictly concave 𝛼′′ < 0 or strictly convex 𝛼′′ > 0.

A2. The function 𝐹(𝐶, 𝛽) is ℂ2. 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

𝐹𝐶𝐶 > 0 on the left of the inflection point, and strictly concave 𝐹𝐶𝐶 < 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 𝛽1 and 𝛽2, either everything point

on the line 𝛽 = 𝛽1 can be connected to some point on the line 𝛽 = 𝛽2 by at least one 𝛽-

integral curve, or every point on the line 𝛽 = 𝛽2 can be connected to some point on the

line 𝛽 = 𝛽1 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.

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Theorem 4.1 The Riemann problem for Eq. (4.7) has a unique global solution for any

Riemann data 𝑢𝐿 and 𝑢𝑅. 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 𝑢𝐿.

4.2 Basic wave behavior

4.2.1 The C-waves

We first recall the Liu admissibility condition (Liu 1976) for shocks. Let 𝑢+ =

𝑆𝛽(𝜎)(𝑢𝐿) for some 𝜎 ∈ ℝ be a point on the 𝛽-shock curve through the left state 𝑢𝐿. We say that

the shock with left and right state (𝑢𝐿 , 𝑢+) satisfies the Liu admissibility condition provided that

its speed is less or equal to the speed of every smaller shock, joining 𝑢𝐿 with an intermediate state

𝑢∗ = 𝑆𝛽(𝜎)(𝑢𝐿), 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 (𝐶𝐿 , 𝛽) and (𝐶𝑅 , 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 𝐶𝐿 > 𝐶𝑅, we

make the concave upper envelope of the flux function, while if 𝐶𝐿 < 𝐶𝑅, 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.

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4.2.2 The β-waves

The waves that connect two different tie lines, i.e., two different 𝛽 values, are referred as

𝛽-waves.

3.2.1 The β-shocks

We recall the Lax admissible condition for shocks. Along a shock curve of the ith

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 (𝐶𝐿 , 𝛽𝐿)

and (𝐶𝑅 , 𝛽𝑅) be the left and right state of a 𝛽-shock that satisfies the Rankine-hugoniot

condition. Then, we have

sign (𝐹(𝐶𝐿 , 𝛽𝐿) − 𝐶𝐿) = sign (𝐹(𝐶𝑅 , 𝛽𝑅) − 𝐶𝑅) . (4.22)

Furthermore, the followings hold.

92

• If 𝐹(𝐶𝐿 , 𝛽𝐿) = 𝐶𝐿, then 𝐹(𝐶𝑅 , 𝛽𝑅) = 𝐶𝑅, and this shock is a contact discontinuity.

• If 𝐹(𝐶𝐿 , 𝛽𝐿) > 𝐶𝐿 and 𝐹(𝐶𝑅 , 𝛽𝑅) > 𝐶𝑅, then the shock (𝐶𝐿 , 𝛽𝐿) − (𝐶𝑅 , 𝛽𝑅) satisfies the

Lax condition if and only if (𝛽𝐿 , 𝛽𝑅) is a shock for Eq. (4.11) that satisfies the Lax

condition.

• If 𝐹(𝐶𝐿 , 𝛽𝐿) < 𝐶𝐿 and 𝐹(𝐶𝑅 , 𝛽𝑅) < 𝐶𝑅, then the shock (𝐶𝐿 , 𝛽𝐿) − (𝐶𝑅 , 𝛽𝑅) satisfies the

Lax condition if and only if (𝛽𝐿 , 𝛽𝑅) is a shock for Eq. (4.14) that satisfies the Lax

condition.

Proof. Let (𝐶𝐿 , 𝛽𝐿) and (𝐶𝑅 , 𝛽𝑅) be the left and right state of a 𝛽-shock, respectively,

and let 𝜎𝛽 be the shock speed. The Rankine-Hugoniot condition requires

𝜎𝛽(𝐶𝐿 − 𝐶𝑅) = 𝐹𝐿 − 𝐹𝑅 , (4.23)

𝜎𝛽(𝛼𝐿𝐶𝐿 + 𝛽𝐿 − 𝛼𝑅𝐶𝑅 − 𝛽𝑅) = 𝛼𝐿𝐹𝐿 + 𝛽𝐿 − 𝛼𝑅𝐹𝑅 − 𝛽𝑅 . (4.24)

Here we used the short hands

𝐹𝐿 = 𝐹(𝐶𝐿 , 𝛽𝐿), 𝐹𝑅 = 𝐹(𝐶𝑅 , 𝛽𝑅), 𝛼𝐿 = 𝛼(𝛽𝐿), 𝛼𝑅 = 𝛼(𝛽𝑅) .

We can eliminate 𝐶𝑅 or 𝐶𝐿 by multiplying Eq. (4.23) with suitable factor and subtract the

remaining equation from Eq. (4.24). Simple calculation gives

𝜎𝛽 =𝐹𝐿 + ��𝛽

−1

𝐶𝐿 + ��𝛽−1 =

𝐹𝑅 + ��𝛽−1

𝐶𝑅 + ��𝛽−1 , where ��𝛽 =

𝛼𝐿 − 𝛼𝑅

𝛽𝐿 − 𝛽𝑅 . (4.25)

Note that ��𝛽 is the Rankine-Hugoniot speed for Eq. (4.11) in the Lagrangian coordinate.

In the phase plane (𝐶1, 𝐶2), the two tie-lines associated with 𝛽𝐿 and 𝛽𝑅 intersect at the

point where 𝐶1 = −��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 −��𝛽−1 < 0, we illustrate the geometric

93

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 −��𝛽−1 < 0. If 𝐹𝐿 = 𝐶𝐿, i.e., the left

state is on the EVC, then by Eq. (4.25) we have 𝜎𝛽 = 1, and we must have 𝐹𝑅 = 𝐶𝑅 for every

state (𝐶𝑅 , 𝛽𝑅) that could be connected to (𝐶𝐿 , 𝛽𝐿) with a 𝛽-shock. Thus the right state must also

lie on the EVC. Along such a shock curve, the second eigenvalue 𝜆𝛽 ≡ 1, and the 𝛽-family is

linearly degenerate. This discontinuity is actually a contact discontinuity, proved later in Lemma

4.2.

Otherwise if 𝐹𝐿 > 𝐶𝐿, by Eq. (4.25) we have 𝜎𝛽 > 1, and therefore 𝐹𝑅 > 𝐶𝑅. In order to

show the equivalence of the two Lax conditions, i.e.,

𝛼′(𝛽𝐿) > ��𝛽 > 𝛼′(𝛽𝑅) ⟺ 𝜆𝛽(𝐶𝐿 , 𝛽𝐿) > 𝜎𝛽 > 𝜆𝛽(𝐶𝑅 , 𝛽𝑅) ,

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 𝐹𝐿 < 𝐶𝐿 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.

3.2.2 β-rarefactions.

A β-rarefaction wave will connect (𝐶𝐿 , 𝛽𝐿) to (𝐶𝑅 , 𝛽𝑅) along the integral curves of the

𝛽-field. Similar to Lemma 4.2, we have the following Lemma.

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Lemma 4.3 Consider piecewise continuous solutions of Eqs. (4.7), and let (𝐶𝐿 , 𝛽𝐿) and

(𝐶𝑅 , 𝛽𝑅) be the left and right states of a 𝛽-rarefaction wave in the two phase region. Then, we

have

(i) If 𝐹(𝐶𝐿, 𝛽𝐿) = 𝐶𝐿, then 𝐹(𝐶𝑅, 𝛽𝑅) = 𝐶𝑅, and this wave is a contact discontinuity.

(ii) If 𝐹(𝐶𝐿, 𝛽𝐿) > 𝐶𝐿 then 𝐹(𝐶𝑅, 𝛽𝑅) > 𝐶𝑅, and (𝛽𝐿 , 𝛽𝑅) is a rarefaction wave for Eq. (4.11).

(iii) If 𝐹(𝐶𝐿, 𝛽𝐿) < 𝐶𝐿, then 𝐹(𝐶𝑅, 𝛽𝑅) < 𝐶𝑅, and (𝛽𝐿, 𝛽𝑅) 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)(𝐶𝐿 , 𝛽𝐿) denote a 𝛽-rarefaction curve initiated at (𝐶𝐿 , 𝛽𝐿) where 𝑠 is the

parametrization of the curve such that 𝑅(0)(𝐶𝐿 , 𝛽𝐿) = (𝐶𝐿 , 𝛽𝐿). We first show that the EVC is an

integral curve. It suffices to show that (𝐶𝑠, 𝛽𝑠)𝑡 is parallel to the eigenvector 𝑟𝛽. Indeed, taking

partial derivative in 𝑠 of the equation 𝐹(𝐶, 𝛽) = 𝐶, we get

𝐹𝐶𝐶𝑠 + 𝐹𝛽 − 𝐶𝑠 = 0, 𝑖. 𝑒., 1

. 0Cs

s

FC

F

.

If 𝐹 = 𝐶, we have 𝜆𝛽 = 1 and so 𝑟𝛽 = (−𝐹𝛽 , 𝐹𝐶 − 1)𝑡. Thus (𝐶𝑠, 𝛽𝑠)

𝑡 is parallel to 𝑟𝛽,

as claimed. This proves (i). By the uniqueness of the integral curves, (ii) and (iii) follows,

completing the proof.

95

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.

4.3.1 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 𝑢𝐿 = (𝐶𝐿 , 𝛽𝐿)𝑡 and 𝑢𝑅 =

(𝐶𝑅 , 𝛽𝑅)𝑡 be the left and right states of the Riemann data, and assume that 𝛽𝐿 − 𝛽𝑅 is connected

by a single 𝛽-shock. We consider an implicit Riemann problem for a scalar conservation law with

discontinuous flux function,

𝐶𝑡 + ��(𝐶, 𝑥)𝑥 = 0, ��(𝐶, 𝑥) = {𝐹𝐿(𝐶) = 𝐹(𝐶, 𝛽𝐿), 𝑥 > 𝜎𝛽𝑡,

𝐹𝑅(𝐶) = 𝐹(𝐶, 𝛽𝑅), 𝑥 < 𝜎𝛽𝑡, (4.26)

with initial Riemann data

𝐶(0, 𝑥) = {𝐶𝐿 , 𝑥 > 0,

𝐶𝑅 , 𝑥 < 0. (4.27)

Note that the wave speed 𝜎𝛽 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 𝛽𝐿 and 𝛽𝑅, we define the ℱ functions

96

ℱ𝐿 = ℱ(𝐶; 𝛽𝐿, ��𝛽), ℱ𝑅 = ℱ(𝐶; 𝛽𝑅 , ��𝛽), where ��𝛽 =

𝛽𝐿 − 𝛽𝑅

𝛼(𝛽𝐿) − 𝛼(𝛽𝑅). (4.28)

Note that relation between the graphs of ℱ𝐿 and ℱ𝑅 are topologically identical to that of

the graphs of 𝐹𝐿 and 𝐹𝑅. Riemann problem for a scalar conservation law with (𝐹𝐿 , 𝐹𝑅) as the

flux function, will generate the same types of waves if using (ℱ𝐿, ℱ𝑅) as the flux functions,

although with different wave speeds. The advantage of using ℱ𝐿 and ℱ𝑅 lies in the fact that 𝛽-

waves 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

𝐶𝑡 + ℱ(𝐶, 𝑥)𝑥 = 0, where ℱ(𝐶, 𝑥) = {ℱ𝐿(𝐶), 𝑥 ≤ 0 ,

ℱ𝑅(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 (𝑢1, 𝑢2; 𝑓) the Riemann problem for a scalar conservation law 𝑢𝑡 + 𝑓(𝑢)𝑥 =

0 with 𝑢1, 𝑢2 as the left and right states. The construction follows a three-step algorithm.

S1: Given 𝐹𝐿(𝐶) and 𝐶𝐿, we identify the set

𝐼𝐿(𝐶𝐿, ℱ𝐿) ≐ {𝐶𝑚; (𝐶𝐿 , 𝐶𝑚; ℱ𝐿) 𝑖𝑠 𝑠𝑜𝑙𝑣𝑒𝑑 𝑏𝑦 𝑤𝑎𝑣𝑒𝑠 𝑜𝑓 𝑛𝑜𝑛 − 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑠𝑝𝑒𝑒𝑑} ∪ {𝐶𝐿}.

S2: Given ℱ𝑅(𝐶) and 𝐶𝑅, we identify the set

𝐼𝑅(𝐶𝑅 , ℱ𝑅) ≐ {𝐶𝑀; (𝐶𝑀 , 𝐶𝑅; ℱ𝑅) 𝑖𝑠 𝑠𝑜𝑙𝑣𝑒𝑑 𝑏𝑦 𝑤𝑎𝑣𝑒𝑠 𝑜𝑓 𝑛𝑜𝑛 − 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑠𝑝𝑒𝑒𝑑} ∪ {𝐶𝑅}.

97

S3: Find the 𝛽-wave position (𝐶𝑚, 𝛽𝐿) − (𝐶𝑀 , 𝛽𝑅) by

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑖𝑛𝑔 |𝐶𝑀 − 𝐶𝑚| 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑡 {𝐶𝑚 ∈ 𝐼𝐿 , 𝐶𝑀 ∈ 𝐼𝑅 , ℱ𝐿(𝐶𝑚) = ℱ𝑅(𝐶𝑀)}.

Next Theorem guarantees the existence and uniqueness of the Riemann solution.

Theorem 4.4 Consider the Riemann problem with 𝑢𝐿 = (𝐶𝐿 , 𝛽𝐿) and 𝑢𝑅 = (𝐶𝑅 , 𝛽𝑅) as the left

and right states, where 𝛽𝐿 and 𝛽𝑅 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 𝐼𝐿 and 𝐼𝑅 as

𝐽𝐿(𝐶𝐿 , ℱ𝐿) ≐ {ℱ𝐿(𝐶); 𝐶 ∈ 𝐼𝐿}, 𝐽𝑅(𝐶𝑅 , ℱ𝑅) ≐ {ℱ𝑅(𝐶); 𝐶 ∈ 𝐼𝑅}. (4.29)

We first claim that the intersection of these two sets are not empty,

𝐽𝐿(𝐶𝐿, ℱ𝐿) ∩ 𝐽𝑅(𝐶𝑅 , ℱ𝑅) ≠ ∅. (4.30)

Indeed, due to the properties of our flux function, it is convenient to list all the cases.

Given ℱ𝐿, let (𝐶0, ℱ0𝐿) and (𝐶2, ℱ2

𝐿) be the minimum and maximum points, respectively. Also we

let 𝐶1 be the unique point such that 𝐶0 < 𝐶1 < 𝐶2 and ℱ𝐿(𝐶1) = 1. See Figure 4-6 for an

illustration. There are 4 cases.

If 𝐶𝐿 ≤ 𝐶0, then we have

𝐼𝐿 = (0, 𝐶0], 𝐽𝐿 = [ℱ𝐿(𝐶0), 1].

If 𝐶0 < 𝐶𝐿 < 𝐶1, then let ��𝐿 be the unique point such that ��𝐿 < 𝐶0 and ℱ𝐿(��𝐿) =

ℱ𝐿(𝐶𝐿). We have

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𝐼𝐿 = (−∞, ��𝐿] ∪ {𝐶𝐿}, 𝐽𝐿 = [ℱ𝐿(𝐶𝐿), 1].

If 𝐶1 < 𝐶𝐿 < 𝐶2, then let ��𝐿 be the unique point such that ��𝐿 > 𝐶2 and ℱ𝐿(��𝐿) =

ℱ𝐿(𝐶𝐿). We have

𝐼𝐿 = {𝐶𝐿} ∪ [��𝐿, 1), 𝐽𝐿 = [1, ℱ𝐿(𝐶𝐿)].

If 𝐶𝐿 ≥ 𝐶2, then we have

𝐼𝐿 = [𝐶2, 1), 𝐽𝐿 = [1, ℱ𝐿(𝐶2)].

We note that 1 ∈ 𝐽𝐿 in all cases.

Now, given ℱ𝑅, let (𝐶3, ℱ3𝑅) and (𝐶4, ℱ4

𝑅) be the minimum and maximum points for

ℱ𝑅 respectively. There are 3 cases, illustrated in Figure 4-6.

If 𝐶𝑅 < 𝐶3 , then let ��𝑅 be the unique point such that ��𝑅 > 𝐶3 and ℱ𝑅(��𝑅) = ℱ𝑅(𝐶𝑅).

We have

𝐼𝑅 = {𝐶𝑅} ∪ [��𝑅, 𝐶4], 𝐽𝑅 = [ℱ𝑅(��𝑅), ℱ4

𝑅].

This includes the case where 𝐶𝑅 lies in the single phase region on the left of 𝐷2.

If 𝐶3 ≤ 𝐶𝑅 ≤ 𝐶4, then we have

𝐼𝑅 = [𝐶3, 𝐶4], 𝐽𝑅 = [ℱ3

𝑅 , ℱ4𝑅].

If 𝐶𝑅 > 𝐶4, then let ��𝑅 be the unique point such that ��𝑅 < 𝐶4 and ℱ𝑅(��𝑅) = ℱ𝑅(𝐶𝑅).

We have

𝐼𝑅 = [𝐶3, ��𝑅] ∪ {𝐶𝑅}, 𝐽𝑅 = [ℱ3

𝑅, ℱ𝑅(��𝑅)].

This includes the case where 𝐶𝑅 lies in the one phase region on the right of 𝐷2.

99

We note that 1 ∈ 𝐽𝑅. Thus 𝐽𝐿 ∩ 𝐽𝑅 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 𝐼𝐿 , 𝐼𝑅 , and denote the sets by 𝐼𝑜𝐿 , 𝐼𝑜

𝑅. On the set 𝐼𝑜𝐿, the function

ℱ𝐿(𝐶) is strictly decreasing, while on the set 𝐼𝑜𝑅, the function ℱ𝑅(𝐶) is strictly increasing. Given

ℱ ∈ 𝐽𝐿 ∩ 𝐽𝑅, let 𝐶𝑀 ∈ 𝐼𝑅 and ℱ𝑅(𝐶𝑀) = ℱ, and let 𝐶𝑚 ∈ 𝐼𝐿 and ℱ𝐿(𝐶𝑚) = ℱ. Denote also

𝐷ℱ ≐ 𝐶𝑀 − 𝐶𝑚. Then, the function ℱ ↦ 𝐷ℱ is strictly increasing, and there exist a unique

minimum for the map ℱ ↦ |𝐷ℱ|.

Finally, if ℱ𝐿(𝐶𝐿) and/or ℱ𝑅(𝐶𝑅) are/is in 𝐽𝐿 ∩ 𝐽𝑅, 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 𝑙𝑡 be the tie-

line that is tangent to the two-phase region, and let (𝐶𝑡, 𝛽𝑡) be the tangent point. This tie-line lies

in the single phase region, and the flux function (𝐶, 𝛽𝑡) = 𝐶. Consider another tie-line 𝑙2 through

the two-phase region with the flux (𝐶, 𝛽2). 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 lt, 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 𝑙2 on

100

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 lt, and the left state is on the the right side of the EVC

on the tie-line 𝑙2, 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-line𝑙2, 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.

4.3.2 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

𝜆𝛽(𝐶, 𝛽) = ℱ(𝐶; 𝛽, 𝛼′(𝛽)−1)

has the same values on the curves 𝐶 and ��, and the curves 𝐶 and �� 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 𝐶1 ∪ 𝐶2 be a 𝛽-integral curve, separated by the degenerate curve with

𝐶1 on the left and 𝐶2 on the right, lying on the same side of EVC. Let𝐶3 and 𝐶4 be the

101

corresponding critical curves for𝐶1 and 𝐶2 respectively. Then, either𝐶3 is on the left of 𝐶2 and𝐶4

is on the left of 𝐶1, or 𝐶3 is on the right of 𝐶2 and 𝐶4 is on the right of 𝐶1.

Proof. We parametrize all these curve with 𝛽, i.e., 𝐶1 is the graph of the function 𝛽 ↦

𝐶1(𝛽) etc. We first observe that

𝜆𝛽(𝐶1, 𝛽) =𝐹(𝐶1, 𝛽)𝛼

′ + 1

𝐶1𝛼′ + 1

=𝐹(𝐶3, 𝛽)𝛼

′ + 1

𝐶3𝛼′ + 1

= 𝜆𝛽(𝐶3, 𝛽)

implies

𝐹(𝐶1, 𝛽) − 𝐶1𝐹(𝐶3, 𝛽) − 𝐶3

=𝐹(𝐶1, 𝛽)𝛼

′ + 1

𝐹(𝐶3, 𝛽)𝛼′ + 1

=𝐶1𝛼

′ + 1

𝐶3𝛼′ + 1

. (4.31)

Along 𝐶1, using (2.6), the directional derivative of the 𝛽-eigenvalue is

∇λβ. 𝑟𝛽 =𝛼′′(𝛽)(𝐹(𝐶1, 𝛽) − 𝐶1)

(𝛼′(𝛽) + 1)2 . (4.32)

Along 𝐶3, the directional derivative of the 𝛽-eigenvalue is the same as in Eq. (4.32). We

must have

𝜆𝐶𝛽(𝐶3, 𝛽)𝐶3

′(𝛽) + 𝜆𝛽𝛽(𝐶3, 𝛽) =

𝛼′′(𝛽)(𝐹(𝐶1, 𝛽) − 𝐶1)

(𝛼′(𝛽) + 1)2 . (4.33)

Note that 𝜆𝛽(𝐶1, 𝛽) = 𝜆𝛽(𝐶3, 𝛽), and we will simply write 𝜆𝛽. 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

𝜆𝐶𝛽=(𝐹𝐶 − 𝜆

𝛽)𝛼′

𝐶𝛼′ + 1, 𝜆𝛽

𝛽=𝐹𝛽𝛼

′ + 𝐹𝛼′′ − 𝜆𝛽𝐶𝛼′′

𝐶𝛼′ + 1,

102

and the identities (4.31), we can solve (4.33) with respect to 𝐶3′ and obtain

𝐶3′(𝛽) =

1

(𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽)𝛼′

[(𝐶3𝛼

′ + 1)2

(𝐶1𝛼′ + 1)2

.𝛼′′(𝐹(𝐶1, 𝛽) − 𝐶1)

(𝐶3𝛼′ + 1)

− 𝐹𝛽(𝐶3, 𝛽)𝛼′ − 𝐹(𝐶3, 𝛽)𝛼

′′ + 𝜆𝛽𝐶3𝛼′′]

=1

(𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽)𝛼′

[𝛼′′(𝐹(𝐶3, 𝛽) − 𝐶3)

(𝐶1𝛼′ + 1)

− 𝐹𝛽(𝐶3, 𝛽)𝛼′ − (𝐹(𝐶3, 𝛽) + 𝜆

𝛽𝐶3)𝛼′′].

Fix a point on 𝐶3, denoted as (𝐶3, 𝛽), let 𝐶5 denote the 𝛽-integral curve through (𝐶3, 𝛽),

parametrize it in 𝛽. We have

𝐶5′(𝛽) = −

𝐹𝛽(𝐶3, 𝛽)

𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽 .

Direct computation gives

𝐶3′(𝛽) − 𝐶5

′ =𝛼′′

(𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽)𝛼′

[𝐹(𝐶3, 𝛽) − 𝐶3𝐶1𝛼

′ + 1− 𝐹(𝐶3, 𝛽) + 𝜆

𝛽𝐶3]

=𝛼′′

(𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽)𝛼′

[𝐹(𝐶3, 𝛽) − 𝐶3𝐶1𝛼

′ + 1− 𝐹(𝐶3, 𝛽) +

𝐹(𝐶1, 𝛽)𝛼′

𝐶1𝛼′ + 1

𝐶3]

=𝛼′′

𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽[𝐹(𝐶1, 𝛽)𝐶3 − 𝐹(𝐶3, 𝛽)𝐶1]

=𝛼′′𝐶1𝐶3

𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽[𝐹(𝐶1, 𝛽)

𝐶1−𝐹(𝐶3, 𝛽)

𝐶3] .

The factor 𝐹𝐶(𝐶3, 𝛽) − 𝜆𝛽 changes sign crossing the degenerate curves, and the term

𝐹(𝐶1, 𝛽)/𝐶1 − 𝐹(𝐶3, 𝛽)/𝐶3 changes from positive to negative as it crosses EVC. We always have

𝐶1 ≥ 0, 𝐶2 ≥ 0. We have the following conclusion:

Case 1. If 𝛼′′ < 0, then on the left of EVC, we have

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𝐹𝐶 (𝐶3, 𝛽) − 𝜆𝛽 > 0, 𝐹(𝐶1, 𝛽)/𝐶_1 − 𝐹(𝐶3, 𝛽)/𝐶3 < 0, → 𝐶3

′ > 𝐶5′ .

By the uniqueness of the 𝛽-integral curve, 𝐶3 lies on the right of 𝐶2. Similarly, 𝐶4 lies

on the right of 𝐶1.

If these curves lie on the right of EVC, then we have

𝐹𝐶 (𝐶3, 𝛽) − 𝜆𝛽 < 0, 𝐹(𝐶1, 𝛽)/𝐶_1 − 𝐹(𝐶3, 𝛽)/𝐶3 > 0, → 𝐶3

′ > 𝐶5′ .

Then, 𝐶3lies on the right of 𝐶2, and similarly 𝐶4 lies on the right of 𝐶1.

Case 2. If 𝛼′′ > 0, a completely similar argument shows that 𝐶3 lies on the left of 𝐶2, and

𝐶4 lies on the left of 𝐶1.

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 with𝑢𝐿 = (𝐶𝐿 , 𝛽𝐿) and𝑢𝑅 = (𝐶𝑅 , 𝛽𝑅) as

the left and right state, where𝛽𝐿 and𝛽𝑅 is connected with a single 𝛽 rarefaction wave. There

exists a unique solution for this Riemann problem.

Proof. Under our assumptions, given 𝛽𝐿 and 𝛽𝑅, then either (i) every point on 𝛽 = 𝛽𝑅

can be connected to 𝛽𝐿 through a 𝛽-integral curve, or (ii) every point on 𝛽 = 𝛽𝐿 can be connected

to 𝛽𝑅 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

ℱ𝐿(𝐶) = ℱ(𝐶; 𝛽𝐿, 𝛼′(𝛽𝐿)−1), ℱ𝑅(𝐶) = ℱ(𝐶; 𝛽𝑅 , 𝛼′(𝛽𝑅)−1).

104

Let 𝐶, �� be the two values where ℱ𝑅 reaches its min and max value. Then, there exists two

integral curves through each of 𝐶 and �� that connect to𝛽𝐿. We denote these curves as

𝐶1, 𝐶2, ��1, ��2.

On the line 𝛽 = 𝛽𝐿, 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 𝐶1 and 𝐶2,

and those between the curves ��1 and ��2.

Given 𝑢𝐿, we let 𝐼𝐿 denote the set of 𝐶 values on the line 𝛽 = 𝛽𝐿 such that the Riemann

problem (𝐶𝐿 , 𝐶; 𝐹𝐿) is solved with non-positive speed, and the point 𝐶 can be connected to 𝛽 =

𝛽𝑅 along a 𝛽-integral curve. Recall the sets 𝐼𝐿and 𝐼𝑅 used in the proof of Theorem 4.1.

We have

𝐼𝐿 = 𝐼𝐿\𝐼′.

Furthermore, let 𝐼𝐿 denote the set of the corresponding 𝐶 values on the line 𝛽 = 𝛽𝑅 that

can be connected to the set 𝐼𝐿 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 𝐶𝐿 lies on the left side of EVC, and we identify the set 𝐼𝐿 for

all cases of 𝐶𝐿 locations. In Figure 4-9 we show three different situations.

If 𝐶𝐿 < 𝐶2, then contains the interval on the left of 𝐶1 and 𝐼𝐿 contains the interval on the

left of 𝐶. The set 𝐼𝐿 ∩ 𝐼𝑅 includes exactly one point.

If 𝐶2 < 𝐶𝐿 < 𝐶3, then 𝐼𝐿 contains an addition point𝐶𝐿, and 𝐼𝐿 contains an additional point

which can be connected to 𝐶𝐿 through a 𝛽-integral curve. The set 𝐼𝐿 ∩ 𝐼𝑅 includes either

one point or two points. If it includes two points, one of then must be the isolated point in

𝐼𝐿, which will be selected.

105

If 𝐶3 < 𝐶𝐿 < 𝐶0, we denote the integral curve through 𝐶𝐿 by 𝐶4 and it corresponding

critical curve by 𝐶5. Then the set 𝐼𝐿 includes the point 𝐶𝐿plus the interval on the left of

the critical curve 𝐶5. The set 𝐼𝐿 consists of the point on 𝐶4 and the interval on the left of

the point that can be connected to 𝐼𝐿 with integral curve, where the right endpoint lies on

the left of 𝐶5. Thus, the set 𝐼𝐿 ∩ 𝐼𝑅 includes exactly one point.

Case 2. The proof is very similar, except in the case when 𝐶3 < 𝐶𝐿 < 𝐶2, where there

exists composite paths, see Figure 4-10. In the plot on the left, 𝐶𝐿 can be connected to 𝐶 as

follows: From𝐶𝐿, the path follows a 𝛽-integral curve, until it intersects with the critical curve 𝐶3

at 𝑎. Then it takes a horizontal path, through a 𝐶-shock, until it reaches the curve 𝐶1at 𝑏. From

there it follows 𝐶1 to reach 𝐶. In the plot on the right, we show another path. In fact, at any point

�� before reaching 𝑎, one could take a horizontal path to reach the critical curve of the integral

curve through 𝐶𝐿 at ��, then take the 𝛽-integral curve from there to reach the line 𝛽 = 𝛽𝑅 at a

point to the left of 𝐶. Thus, we redefine the set 𝐼𝐿 to include the points on the line 𝛽 = 𝛽𝑅 that

can be connected to the set 𝐼𝐿 through a composite path. Clearly, 𝐼𝐿 includes all 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 𝛽-

rarefaction 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 state𝑢𝐿.

4.3.3 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 𝑢𝐿 = (𝐶𝐿 , 𝛽𝐿) and 𝑢𝑅 = (𝐶𝑅 , 𝛽𝑅), the solution of the Riemann

106

problem is constructed in two steps. We first solve the 𝛽-wave using information beaded on

(𝛽𝐿, 𝛽𝑅) 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 = 𝛽𝐿 and 𝛽 = 𝛽𝑅. 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 𝑢𝐿 and 𝑢𝑅 lie on the left (or on the right) of the EVC with 𝐹𝐿 ≥ 𝐶𝐿

and 𝐹𝑅 ≥ 𝐶𝑅, 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 𝑢𝐿 = (𝐶𝐿 , 𝛽𝐿) and 𝑢𝑅 = (𝐶𝑅 , 𝛽𝑅) be the left and right states of the

Riemann problem, where (𝛽𝐿, 𝛽𝑅) 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.

107

Let 휀 > 0 be the parameter for the front tracking algorithm. We discretize the space for 𝛽

values, and let ℬ𝜀 = {𝛽𝑛} denote the set of the discrete values for 𝛽, with

𝛽𝑛 > 𝛽𝑛−1, |𝛽𝑛 − 𝛽𝑛−1| ≤ 휀 𝑛 = 1,2,… , 𝑁 − 1 . (4.34)

We let 𝛼𝜀(𝛽) denote the piecewise affine approximation to 𝛼(𝛽), with 𝛼𝜀(𝛽𝑛) = 𝛼(𝛽𝑛)

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 𝐶𝜀 = {𝐶𝑛,𝑚} after calculating a new

𝛽-wave. The set of 𝐶𝜀 = {𝐶𝑛,𝑚} denote the discritized values for 𝐶, with

𝐶𝑛,𝑚 > 𝐶𝑛,𝑚−1, |𝐶𝑛,𝑚 − 𝐶𝑛,𝑚−1| ≤ 휀𝐶 , 𝑛 = 1,2, … ,𝑁 − 1, 𝑚 = 1,2,… ,𝑀 − 1. (4.35)

Then we estimate 𝑓(𝑆) with piecewise linear 𝑓𝜀(𝑆). The parameter 휀𝐶 is 휀 devided by a

constant.

The discrete initial data is piecewise constant 𝑢𝜀(0, 𝑥) = (𝐶𝜀(0, 𝑥), 𝛽𝜀(0, 𝑥)), where 𝛽𝜀

takes only the values in ℬ𝜀. Let 𝑥𝑖 be the points of discontinuities in the discrete initial data. We

denote the cell values as

𝛽𝜀(0, 𝑥) = 𝛽𝑖, 𝐶𝜀(0, 𝑥) = 𝐶𝑖, 𝑥𝑖−1 ≤ 𝑥 < 𝑥𝑖 .

At 𝑡 = 0, a set of Riemann problems shall be solved at every point 𝑥𝑖 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

108

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 𝛼𝜀(𝛽𝑛) constrains the values of 𝛽 in the solution to ℬ𝜀, unless the initial data

contain values out of ℬ𝜀. However the 𝐶 values of the solution are not necessarily in 𝐶𝜀 even if

all the initial data are in 𝐶𝜀. 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 oF 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 𝐿1, which changes to 𝐿2 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.

109

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 𝑥𝐷 − 𝑡𝐷 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 = −𝐶1,𝑘𝜓 , 𝜞 = 𝜞𝑘 , 𝑘 = 𝑖𝑛𝑗, 𝑖𝑛𝑖 . (4.36)

110

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/𝐶1,𝑖𝑛𝑖 and −1/𝐶1,𝑖𝑛𝑗 (slopes of Eq. (4.36)). For 𝐶1,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 𝐶1𝑥 = −1/𝜆𝛽 for

rarefactions and 𝐶1𝑥 = −1/Λ for shocks and give an equation for the validity of the tie-line

selected as,

𝑠𝑔𝑛(𝐶1,𝑘𝑥 − 𝐶1,𝑘)𝑠𝑔𝑛(𝐶1,𝑘

𝑥 − 𝐸𝑉𝐶(𝜞𝑘)) > 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

111

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 𝜞𝑖𝑛𝑖1does not satisfy Eq. (4.37) and therefore, we should

find another tie line, 𝜞𝑖𝑛𝑖2, that passes through the initial composition. We know that 𝜞𝑖𝑛𝑖1 and

𝜞𝑖𝑛𝑖2 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/𝐶1,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 𝜞𝑖𝑛𝑖2. That is, we perform a line search along

each Hugoniot locus given by Eqs. (4.25) and test if Λ = −1/𝐶1,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 𝑇𝐿1, 𝑇𝐿2 and 𝑇𝐿3. The

112

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 𝐼1 with 𝐽1, 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 𝑅1 in Figure 4-16 that goes from the initial tie line (𝑇𝐿3) to the injection tie

line (𝑇𝐿1). 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 TL1

and TL3 are shown in Figure 4-17. Because 𝐶1,𝐽1 < 𝐸𝑉𝐶(Γ𝐽1), 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 𝑇𝐿1 to 𝑇𝐿3 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

4-18. If we displace 𝐼2 by 𝐽2, 𝑇𝐿3 is connected to 𝑇𝐿1 by the shock 𝑆1 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 4-17.

113

Next, we construct the solution for displacement of 𝐼3 with 𝐽1, where two tie lines, 𝑇𝐿2

and 𝑇𝐿3, now pass through 𝐼2. A negative flash calculation will typically yield TL3. Once a tie-

line route is constructed we apply the test of Eq. (4.37) for that tie line to see if it is correct.

Equation (4.37) is only satisfied, however, for 𝑇𝐿2 because the value of 𝐶1𝑥 is equal to 𝐶1 at 𝑎 as

shown in Figure 4-16. Thus, TL3 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

𝑇𝐿2, so a tie-line route is constructed with that tie line. The correct (physical) tie-line route

exhibits a rarefaction wave from 𝑇𝐿2 to 𝑇𝐿1 (see R2 in Figure 4-16 and in Figure 4-18).

In the last example for constant K-values, we discuss the displacement of 𝐼4 with 𝐽3 in

Figure 4-16, where now 𝐶1,𝐽3 > 𝐸𝑉𝐶(Γ𝐽3) 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 4-17. Thus, the

eigenvalue increases from 𝑇𝐿2 to 𝑇𝐿1, 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 𝑅3. The

composition route is clearly different from 𝑅2, 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 4.5.

114

Two displacements for Figure 4-19 are examined here with the same injection

composition 𝐽1. Because 𝐶1,𝐽1 < 𝐸𝑉𝐶(𝜞𝐽1), 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 𝐼2 with 𝐽1, there is only one tie line that passes through 𝐼2. The solution for displacement of 𝐼2

by 𝐽1 in tie-line space, therefore, is a rarefaction wave to TL3 followed by a tangent shock from

TL3 to TL5 (see Figure 4-20 and Figure 4-21). There are three possible tie lines for initial

composition 𝐼1, however. We can construct the solution in tie-line space for each one and then

check if the solution satisfies Eq. (4.37). For 𝑇𝐿5, 𝐶1𝑥 is equal to 𝐶1 at point 𝑎 in Figure 4-19,

which violates Eq. (4.37). Similarly, for the tie-line path of 𝑇𝐿4, 𝐶1𝑥 is equal to 𝐶1 at point 𝑏,

which again violates Eq. (4.37). For 𝑇𝐿2, 𝐶1𝑥 is outside of the phase diagram and satisfies Eq.

(4.37). Therefore 𝑇𝐿2 is the correct tie line and the rest of the solution route is easily generated as

a rarefaction from 𝑇𝐿1 to 𝑇𝐿2. 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

115

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

TC oF PC (psi) ω

C10

611.161 305.76 0.5764

CO2 87.89 1071 0.225

C1 -116.59 667.8 0.008

Table 4.2 – Binary interaction coefficients for the ternary system

CO2 C

1

C10

0.0942 0.0420

CO2 - 0.1

Table 4.3 – Initial condition for example problem 3

< 𝑥 𝑥 < 𝐶 𝛽

0 0.2 0.52 0

0.2 0.4 0.3 0

0.4 ∞ 0.318 0.5

Table 4.4 – Injection condition for example problem 3

< 𝑡 𝑡 < 𝐶 𝛽

0 0.1 0.05 0.5

0.1 0.2 0.5 0

0.2 ∞ 0.01 0

Table 4.5 – Component properties and compositions for bifurcating phase behavior developed

based on Ahmadi et al. (2011)

Properties Compositions

BIP Tc (oF) Pc (psia) ω I1 I2 I3 J1

C1N2 -116.59 667.8 0.008 0.1 0.75 0.455 0 C1N2 CO2 nC4

CO2 87.89 1,071 0.225 0.1 0.1 0.000 1 0.119 0.000

nC4 305.69 550.7 0.193 0.0 0.0 0.272 0 0.013 0.126 0.000

C26-35 1258.83 244.98 1.090 0.8 0.15 0.273 0 0.000 0.094 0.000

116

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) 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.

117

(a) (b)

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.

118

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).

119

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.

Case 1 Case 2

Figure 4-8: Two possible relations between the curves 𝐶1, 𝐶2, 𝐶3 and 𝐶4.

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).

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Figure 4-10: Case 2, when 𝐶3 < 𝐶

𝐿 < 𝐶2, 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).

121

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 휀 = 0.05.

122

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

123

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

124

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.

125

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).

126

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.

127

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

128

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)

𝜕𝐶𝑖𝜕𝑡𝐷

+𝜕𝐹𝑖𝜕𝑥𝐷

= 0 𝑖 = 1,… ,𝑁𝑐 − 1 , (5.1)

where 𝐶𝑖 and 𝐹𝑖 are overall volume fraction and overall fractional flow of component 𝑖, 𝑡𝐷 is

dimensionless time or pore volume injected (PVI), 𝑥𝐷 is dimensionless length, and 𝑁𝑐 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,

𝜑𝑥𝐷 = ∓𝐶1, 𝜑𝑡𝐷 = ±𝐹1 and 𝜓 = 𝑥𝐷 − 𝑡𝐷 . (5.2)

129

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 𝐹1 equals 𝐶1 . The upper sign of Eqs. (5.2) is used when 𝐶1𝑖𝑛𝑗 >

𝐶1𝐸𝑉𝐶(𝜞𝑖𝑛𝑗), while the lower sign is used otherwise. 𝜞 is a 𝑁𝑐 − 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.

𝜕𝛽𝑖𝜕𝜑

±𝜕𝛼𝑖𝜕𝜓

= 0 𝑖 = 1,… ,𝑁𝑐 − 2 . (5.3)

where 𝛼𝑖 and 𝛽𝑖 are tie-line coefficients, and

𝛼𝑖 = 𝑓(𝛽1, … , 𝛽𝑁𝑐−2) 𝑖 = 1,… ,𝑁𝑐 − 2 . (5.4)

The tie lines and their coefficients are defined by,

𝐶𝑖+1 = 𝛼𝑖𝐶1 + 𝛽𝑖 𝑖 = 1,… ,𝑁𝑐 − 2, (5.5)

𝛼𝑖 =𝑥𝑖+11 (1 − 𝐾𝑖+1)

𝑥11(1 − 𝐾1)

, 𝛽𝑖 =𝑥𝑖+11 (𝐾1 − 𝐾𝑖+1)

𝐾1 − 1 𝑖 = 1,… ,𝑁𝑐 − 2, (5.6)

where 𝑥𝑖1 is mole fraction of component i in phase 1 and 𝐾𝑖 is the K-value of component i. The

sign for the second term of Eqs. (5.3) is positive if 𝐶1𝑖𝑛𝑗 > 𝐶1𝐸𝑉𝐶(𝜞𝑖𝑛𝑗) 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.

130

The final equation that results from the transformation of Eqs. (5.1) is dependent on

fractional flow.

𝜕

𝜕𝜑(

𝐶1𝐹1 − 𝐶1

) ±𝜕

𝜕𝜓(

1

𝐹1 − 𝐶1) = 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

𝐹1−𝐶1, is not a single-valued function of the conserved quantity

𝐶1

𝐹1−𝐶1. Thus, Eq.

(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 − 𝜆𝑖)𝑑𝜷

𝑑𝜓= 0 𝑖 = 1,… ,𝑁𝑐 − 2, (5.8)

where 𝑹 is a 𝑁𝑐 − 2 by 𝑁𝑐 − 2 matrix with elements 𝑹𝑖𝑗 =𝜕𝛼𝑖

𝜕𝛽𝑗 . 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 𝑁𝑐 − 2 envelope curves (Dindoruk

1992) at the corresponding composition, 𝐶1,𝑘𝑒 , so that for the kth eigenvalue,

𝐶1,𝑘𝑒 = −

1

𝜆𝑘 𝑘 = 1,… ,𝑁𝑐 − 2 . (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

𝑑𝜓. These ruled

131

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 𝑁𝑐 − 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 three-

component 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

𝑑𝛽 and its eigenvector is trivial (arbitrary scalar). For four-component

displacements, there are two eigenvalues given by,

(5.10)

132

where the eigenvalues can be calculated as,

𝜆 =

𝜕𝛼1𝜕𝛽1

+𝜕𝛼2𝜕𝛽2

±√(𝜕𝛼1𝜕𝛽1

+𝜕𝛼2𝜕𝛽2

)2

− 4(𝜕𝛼1𝜕𝛽1

𝜕𝛼2𝜕𝛽2

−𝜕𝛼1𝜕𝛽2

𝜕𝛼2𝜕𝛽1

)

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 =Δ𝛼𝑖Δ𝛽𝑖

𝑖 = 1,… , 𝑁𝑐 − 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

𝐶1 = −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.

133

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 two-

phase state, assuming that the system remains two-phase after the change.

134

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,

𝑓𝑖1 = 𝑓𝑖

2, 𝑖 = 1,… ,𝑁𝑐 ,

Σ𝑥𝑖1 = 1, Σ𝑥𝑖

2 = 1.

(5.14)

where the superscript indicates the phase. There are 2𝑁𝑐 unknown compositional variables and

𝑁𝑐 + 2 equations in Eqs. (5.14). The system of equations is completed by adding 𝑁𝑐 - 2

definitions of tie-line space parameters, 𝛾𝑖,

𝑔𝑖(𝑥11, … , 𝑥𝑁𝑐

1 , 𝑥12, … , 𝑥𝑁𝑐

2 ) = 𝛾𝑖 𝑖 = 1,… ,𝑁𝑐 − 2 . (5.15)

A simple form of 𝑔𝑖 can be used such as 𝛾𝑖 = 0.5(𝑥𝑖1 + 𝑥𝑖

2) (Voskov and Tchelepi,

2008), or 𝛾𝑖 = 𝑥𝑖1, or 𝛾𝑖 = 𝑥𝑖

2, or 𝛾𝑖 = 𝛽𝑖 (see Eqs. (5.6)). In a well-defined tie-line space, each tie

line is determined by a unique 𝜞.

𝜞 = [𝛾1, … , 𝛾𝑁𝑐−2]. (5.16)

The corresponding tie line for a given 𝜞 can be calculated by solving Eqs. (5.14) and

(5.16). 𝐶1 should also be specified to calculate the moles of a phase. That is,

𝑛1 =𝐶1 − 𝑥1

2

𝑥11 − 𝑥1

2 . (5.17)

Therefore, we can calculate molar properties of the reservoir fluid by specifying the

vector 𝒗.

135

𝒗 = [𝐶1, 𝛾1, … , 𝛾𝑁𝑐−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 = [𝐶1, … , 𝐶𝑁𝑐−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) 𝑥1 ≠ 𝑦1 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.

𝑑𝑓 =

[ 𝜕𝐶1𝜕𝐶1

𝜕𝐶1𝜕𝛾1

…𝜕𝐶1𝜕𝛾𝑁𝑐−2

⋮ ⋮ ⋱ ⋮𝜕𝐶𝑁𝑐−1

𝜕𝐶1

𝜕𝐶𝑁𝑐−1

𝜕𝛾1…

𝜕𝐶𝑁𝑐−1

𝜕𝛾𝑁𝑐−2]

. (5.21)

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We can remove the first row and column of 𝑑𝑓 because those elements are either 1.0 or

0.0, which means |𝑑𝑓| = |𝑑��| where 𝑑�� is defined as

𝑑�� =

[ 𝜕𝐶2𝜕𝛾1

…𝜕𝐶2𝜕𝛾𝑁𝑐−2

⋮ ⋱ ⋮𝜕𝐶𝑁𝑐−1

𝜕𝛾1…

𝜕𝐶𝑁𝑐−1

𝜕𝛾𝑁𝑐−2]

= 𝑨′𝐶1 +𝑩′,

(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 𝑑�� can be used to determine if 𝑑�� is invertible (condition

number must be finite for invertibility). The inverse function theorem requires checking the

condition number of 𝑑�� 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 𝑑��.

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 𝛽𝑖 are the conserved quantity in Eqs.

(5.3). We define tie-line space in such a way that 𝛽𝑖 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 ��1 = 0. Therefore, we can define a new

composition ��𝑖 such that the hyperplane of ��1 = 0 lies inside the two-phase region. This would

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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.

5.2.2 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),

�� = 𝑸𝑪 + 𝑰𝐶𝑟 , (5.23)

such that for all tie lines ��1 < 0 𝑎𝑛𝑑 ��1 > 0. 𝑪 is a composition and �� is the same composition in

the new coordinates. All elements of 𝑪𝒓 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 5𝑁𝑐 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 𝑧𝑖 =𝑥𝑖+𝑦𝑖

2, 𝑖 = 1,… ,𝑁𝑐 − 1.

3. 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 𝐶𝑟,1.

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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 �� 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, ��1 = 0 and ��𝑖+1 = 𝛾𝑖 , 𝑖 =

1,… ,𝑁𝐶 − 2, where 𝛾𝑖 = ��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

5.3.1 Constant K-value displacement

We show for any 𝑁𝑐 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

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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 =𝛥𝛼𝑖𝛥𝛽𝑖

= (𝑥1𝛥𝑥𝑖 − 𝑥𝑖𝛥𝑥1𝑥1(𝑥1 + 𝛥𝑥1)𝛥𝑥𝑖

) (1 − 𝐾𝑖𝐾𝑖 − 𝐾1

) 𝑖 = 2,… ,𝑁𝑐 − 1. (5.24)

The term 𝑥1(𝑥1 + Δ𝑥1) is common in Eqs. (5.24). Therefore we define the modified

“shock velocity” Λ = 𝑥1(𝑥1 + Δ𝑥1)Λ and rearrange Eqs. (5.24) in a form similar to an

eigenvalue problem, where ix are solved.

[−(𝑥1 + 𝑥𝑖𝐾𝑁𝑐 −𝐾𝑗

𝐾1 − 𝐾𝑁𝑐)(

1 − 𝐾𝑖𝐾𝑖 − 𝐾1

) − Λ] Δ𝑥𝑖 − 𝑥𝑖 (1 − 𝐾𝑖𝐾𝑖 − 𝐾1

) ∑ (𝐾𝑁𝑐 − 𝐾𝑗

𝐾1 −𝐾𝑁𝑐)

𝑁𝑐−1

𝑗=2𝑗≠𝑖

Δ𝑥𝑗

= 0 𝑖 = 2,… ,𝑁𝑐 − 1 .

(5.25)

The coefficient matrix in Eqs. (5.25) is not a function of Δ𝑥𝑖, 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 Δ𝑥𝑖 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.

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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 𝑁𝑐 − 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 𝑁𝑐 − 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-

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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 𝛽𝑖 where ∑ 𝛽𝑖3𝑖=1 = 1, and another a Cartesian plot of the equilibrium

phase composition of one of the phases (𝑥𝑖). 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(𝑁𝑐 − 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 𝛽1 − 𝛽3 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

S1 and S2 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

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that corresponds to the 𝛽1 − 𝛽3 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 4-16. 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 mixing-

cell 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 170oF 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 𝐽1 to 𝑇𝐿1 that nearly coincides with the

slow path, followed by a rarefaction wave from 𝑇𝐿1 along the fast path to tie-line 𝐼1. 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 nC4 to the three-component system of Ahmadi et al. (2011) (Table 4.5). We construct the

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tie line route for displacement of 𝐼3 in Table 1 by pure CO2. 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 𝐼3 by pure CO2. The solution in tie-line space

follows the path for the slower eigenvalue with rarefaction 𝑅1 form 𝑇𝐿𝐽1to 𝑇𝐿1. Then the solution

follows the fast eigenvalue with the rarefaction 𝑅2 to 𝑇𝐿2 and then the tangent shock 𝑆1 to 𝑇𝐿𝐼3 .

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, 𝑇𝐿3 is the shortest tie line in simulation results, because the shock is estimated with a

shock layer in simulation (see appendix C). 𝑇𝐿2 approaches to 𝑇𝐿3 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.

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Table 5.1 – The compositions and K-values for example case (Yuan et al. 2005)

𝑥1 𝑥2 𝑥3 𝑥4

K-values 5.0 2.0 1.2 0.01

Oil 0.0386 0.3485 0.0874 0.5256

Gas 0.1984 0.000 0.000 0.8016

Table 5.2 – Component properties and compositions for displacement by Dutra et al. (2009)

Properties Compositions

BIP Tc (oF) Pc (psia) ω I1 J1

N2 -232.51 492.32 0.040 0.0 0.8 N2 C3 C6

C3 205.97 615.76 0.152 0.0 0.2 0.085 0.000

C6 453.65 430.59 0.296 0.3 0.0 0.150 0.027 0.000

C10 611.16 353.76 0.576 0.7 0.0 0.155 0.020 0.000

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).

145

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

146

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.

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Figure 5-5: Eigenvalues along the line 𝜷𝟐 = 𝟎.

Figure 5-6: Parametrization of the tie-line path for automatic construction of tie-line path for

quaternary phase behavior of Table 5.1.

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Figure 5-7: Phase diagram in tie-line space and ruled surfaces for four-component displacements

in Table 5.2 at 2900 psia and 160oF.

Figure 5-8: Quaternary phase diagram with bifurcating phase behavior generated based on

Ahmadi et al. (2011) at 8000 psia and 133oF.

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

150

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 𝑁𝐶 − 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.

6.1.1 Pseudoternary ruled surfaces

The total length traversed along each pseudo-ternary ruled surface is parameterized in tie-

line space so that the initial and injection tie lines become connected. The parameter 𝑙𝑖 is the

eigenvector length taken along ruled surface i. The value of 𝑙𝑖 for 𝑁𝐶 − 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 𝑙𝑖 to describe

each pseudo-ternary displacement as,

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𝑙𝑗 = ±∑ 𝑙𝑗,𝑘

𝑛𝑗

𝑘=1

𝑗 = 1,… ,𝑁𝑐 − 2 . (6.1)

The parameter 𝑙𝑖,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,

𝑙𝑗,𝑘 =

{

∫√∑ (𝑑𝛾𝑗 𝑑𝛾1⁄ )

2𝑁𝑐−2𝑗=2 𝑑𝛾1 𝑅𝑎𝑟𝑒𝑓𝑎𝑐𝑡𝑖𝑜𝑛

√∑ (∆𝛾𝑗)2𝑁𝑐−2

𝑗=2 𝑆ℎ𝑜𝑐𝑘

𝑗 = 1,… ,𝑁𝑐 − 2,

𝑘 = 1, 𝑛.

(6.2)

For four components, we have only two values of 𝑙𝑖, 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 (single-

valued) 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 𝑙𝑗 as

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𝛤𝑗𝑅 = 𝑓(𝛤𝑗

𝐿 , 𝑙𝑗). (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 𝜞𝑖𝐿 and 𝜞𝑖

𝑅. If the eigenvalues are monotonic from 𝜞𝑖𝐿 and 𝜞𝑖

𝑅 then only a shock or

rarefaction exists along that ruled surface depending on the sign, but not both.

6.1.2 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

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the vectors ��𝑅 = 𝛤𝑖𝑛𝑖 − 𝛤𝑖𝑛𝑗 for the displacement and ��𝑖 = 𝛤𝑖𝑅 − 𝛤𝑖

𝐿 𝑖 = 1,… ,𝑁𝑐 − 2 for each

pseudo-ternary displacement. Assuming that there is a good estimate for ei, the ruled surface

route can be calculated by solving

��𝑅 = 𝑙𝑖��𝑖 𝑖 = 1,… ,𝑁𝑐 − 2 , (6.4)

which is a linear system of equations. The eigenvectors of the injection tie line are used as an

initial estimate of ��𝑖 for the first iteration and estimates of ��𝑖 are improved in each iteration using

the average of the i-th eigenvector of 𝜞𝑖𝑅 and 𝜞𝑖

𝐿. 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.

6.1.3 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 𝑁𝑐 − 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 𝜞𝑖𝑛𝑗 and

Γini as the initial data, consist of 𝑁𝑐 − 2 ruled surfaces, such that a ruled surface corresponding to

kth family of eigenvectors of Eqs. (5.3) can be identified with its left and right tie lines, 𝜞𝑘𝐿 and

𝜞𝑘𝑅.

The ruled-surface route is complete when

𝜞𝑖𝑛𝑗 = 𝜞1𝐿 , 𝜞𝑖𝑛𝑖 = 𝜞𝑁𝑐−2

𝑅 , 𝜞𝑘𝑅 = 𝜞𝑘+1

𝐿 𝑘 = 1,… ,𝑁𝑐 − 3. (6.5)

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𝑁𝑐 − 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 𝜞1𝑅 = 𝜞2

𝐿. 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 𝑙1. 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 𝜞𝑖𝑛𝑗 and construct 𝑚 ruled

surfaces to 𝜞𝑚𝑅 . Next we start from 𝜞𝑖𝑛𝑖 and construct the 𝑁𝑐 − 2 −𝑚 pseudo-ternary

displacement to 𝜞𝑚+1𝐿 . Then the Newton method can be used to solve 𝜞𝑚

𝑅 = 𝜞𝑚+1𝐿 . 𝑚 should be

selected such that 𝜞𝑚𝑅 is the shortest tie line among all 𝜞𝑖

𝑅 for 𝑖 = 1,… ,𝑁𝑐 − 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.

6.1.4 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 𝑥𝐶1 − 𝑥𝐶10. 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

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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

𝑥𝐶26−35 − 𝑥𝐶1 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 𝑥𝐶1 − 𝑥𝐶20 surface as shown in Figure 6-4.

6.2 MMP calculation for 𝑵𝒄 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 shock-

jump 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

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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).

6.2.1 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

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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.

6.2.2 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.

6.2.3 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.

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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 CO2. 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.

6.2.5 Bifurcating phase behavior

Mogensen et al. (2009) compared the MMPs predicted by various computational methods

for the Al-Shaheen oil displaced by CO2 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 pseudo-

ternary 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

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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

C1 CO2 nC4 C10

Injection 0.3 0.0 0.2 0.5

Initial 0.1 0.9 0.0 0.0

Table 6.2 – Input properties for six-component MMP calculations from Johns (1992)

Properties Compositions

BIP

Tc

(oF)

Pc

(psia) ω I1 J1

C1 87.9 1071.34 0.225 0.20 0.00 C1 CO2 nC4 C10 C14

CO2 -116.63 667.8 0.0104 0.05 1.00 0.1000 0.0000

nC4 305.65 550.7 0.201 0.05 0.00 0.1257 0.0270 0.0000

C10 652.1 305.7 0.49 0.40 0.00 0.0942 0.0420 0.0080 0.0000

C14 789.51 235.2 0.679 0.10 0.00 0.1098 0.0725 0.0078 0.0000 0.0000

C20 920.91 161.4 0.907 0.20 0.00 0.0865 0.0540 0.0000 0.0000 0.0000

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Table 6.3 – Input properties for five-component MMP calculations

Properties Compositions

BIP Tc (oF) Pc (psia) Ω I1 J1

C1 -116.59 667.8 0.008 0.2067 0.1200 C1 CO2 nC4 C6

CO2 87.89 1,071 0.225 0.1059 0.7590 0.100 0.000

nC4 305.69 550.7 0.193 0.2347 0.0600 0.027 0.126 0.000

C6 453.65 429.8 0.296 0.1100 0.0600 0.422 0.145 0.017

C10 611.16 305.8 0.575 0.3428 0.0000 0.420 0.094 0.000 0.000

Table 6.4 – The compositions for 12-component displacement from Zick (1986)

CO2 C1 C2 C3 C4 C5 C6 C71+ C7

2+ C73+ C7

4+ C75+

Injection 0.066 0.371 0.054 0.037 0.026 0.019 0.022 0.179 0.091 0.061 0.045 0.030

Initial 0.178 0.388 0.188 0.220 0.027 0 0 0 0 0 0 0

Figure 6-1: Parametrization of ruled surfaces. The solution can be constructed by solving 𝜞1𝐿 =

𝜞2𝑅.

161

Figure 6-2: Projection of tie-line routes to 𝐶1 − 𝐶10 plane for four component displacement of

Table 6.1.

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 oF.

162

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 oF.

Figure 6-5: The tie line route parameters for the displacement of I1 by J1 in Table 5.2.

163

Figure 6-6: Key tie-line lengths calculated by analytical solution and shortest tie-line length

calculated by simulation for the displacement in Table 5.2 .

Figure 6-7: Four-component displacements with bifurcating phase behavior. The shock-Jump

MOC over predict MMP by almost 4000 psi.

0

0.2

0.4

0.6

0.8

1

0 4000 8000 12000 16000

Tie

-lin

e le

ng

th

Pressure, psia

InjectionCrossoverBifurcatingInitialSimulation

164

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.

165

Figure 6-10: Tie-line length variation with pressure calculated using approximate Riemann

solver for displacement in Table 6.4 at 185 oF. 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 oF with compositions in Table 6.4 .

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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 shock-

only MOC, improved shock-only MOC and mixing cell results are copied form

Ahamdi et al. (2011).

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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 Ca2+ and Na+

is used to model the wettability alteration. The viscosity is a function of polymer and ionic

species concentration.

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7.1.1 Immiscible oil/water flow

The mass conservation equations for immiscible oil and water phases are as follows:

𝜕

𝜕𝑡(𝜙𝑆𝛼𝜌𝛼) + 𝛻 ⋅ (𝜌𝛼 ��𝛼) = 0 𝛼 = 𝑜,𝑤. (7.1)

Darcy’s law governs the flow rate of each phase,

��𝛼 =𝑘𝑟𝛼𝜇𝛼

𝐊 ⋅ ∇(𝑃𝛼 − 𝜌𝛼𝑔𝑍) 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,

𝑃𝑐𝑜𝑤 = 𝑃𝑜 − 𝑃𝑤 . (7.3)

The saturation relation completes the set of equations

𝑆𝑜 + 𝑆𝑤 = 1. (7.4)

The primary unknowns for the multiphase flow system are 𝑃𝑜 and 𝑆𝑤.

7.1.2 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+, Ca2+ and Mg2+. With Na-

X, Ca-X2 and Mg-X2 representing the surface sites occupied by sodium, calcium and magnesium,

the cation exchange reactions can be written as

169

𝐶𝑎2+ + 2𝑁𝑎 − 𝑋 ↔ 𝐶𝑎 − 𝑋2 + 2𝑁𝑎+, 𝐾𝑒𝑞,𝐶𝑎

𝑀𝑔2+ + 2𝑁𝑎 − 𝑋 ↔ 𝑀𝑔 − 𝑋2 + 2𝑁𝑎+, 𝐾𝑒𝑞,𝑀𝑔

where 𝐾𝑒𝑞,𝐶𝑎 and 𝐾𝑒𝑞,𝑀𝑔 are the reaction equilibrium constants for Ca2+ and Mg2+ 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

𝐾𝑒𝑞,𝐶𝑎 =(𝐶𝑎 − 𝑋2)(𝑁𝑎

+)2

(𝐶𝑎2+)(𝑁𝑎 − 𝑋)2 ,

𝐾𝑒𝑞,𝐶𝑎 =(𝑀𝑔 − 𝑋2)(𝑁𝑎

+)2

(𝑀𝑔2+)(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 − 𝑋2) =2[𝐶𝑎 − 𝑋2]

𝐶𝐸𝐶 ,

(𝑀𝑔 − 𝑋2) =2[𝑀𝑔 − 𝑋2]

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 − 𝑋2] + 2[𝑀𝑔 − 𝑋2].

170

7.1.3 Reactive Transport Model

The mass conservation equations for the primary species 𝑝 is

𝜕

𝜕𝑡(𝑀𝑝 +∑ 𝜈𝑞𝑝𝑀𝑞

𝑁𝑠𝑒𝑐

𝑞=1

)+ ∇ ⋅ (𝐹𝑝 +∑ 𝜈𝑞𝑝𝐹𝑞

𝑁𝑠𝑒𝑐

𝑞=1

) = 0 𝑝 = 1,… ,𝑁𝑝𝑟𝑖 , (7.5)

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+ Ca2+ and Mg2+ are written as

𝜕

𝜕𝑡[𝜙𝑆𝑤𝜌𝑤𝐶𝑁𝑎+ + (1 − 𝜙 )𝜌𝑠𝐶𝑁𝑎−𝑋] + ∇ ⋅ (��𝑤𝐶𝑁𝑎+) = 0,

𝜕

𝜕𝑡[𝜙𝑆𝑤𝜌𝑤𝐶𝐶𝑎2+ + (1 − 𝜙 )𝜌𝑠𝐶𝐶𝑎−𝑋2] + ∇ ⋅ (��𝑤𝐶𝐶𝑎2+) = 0,

𝜕

𝜕𝑡[𝜙𝑆𝑤𝜌𝑤𝐶𝑀𝑔2+ + (1 − 𝜙 )𝜌𝑠𝐶𝑀𝑔−𝑋2] + ∇ ⋅ (��𝑤𝐶𝑀𝑔2+) = 0.

7.1.4 Wettability alteration

We use a linear interpolation model as follows:

𝑘𝑟𝑤∗ = (1 − 𝜃) 𝑘𝑟𝑤,𝑤𝑤

∗ + 𝜃𝑘𝑟𝑤,𝑜𝑤∗ , (7.6)

where 𝑘𝑟𝑤,𝑜𝑤∗ , and 𝑘𝑟𝑤,𝑤𝑤

∗ are water relative permeabilities at the end-point oil-wet (ow) and end-

point 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

171

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),

𝑘𝑟𝑤 = 𝑘𝑟𝑤∗ (𝑆∗)𝑛𝑤 ,

𝑘𝑟𝑜 = 𝑘𝑟𝑜∗ (1 − 𝑆∗)𝑛𝑜 ,

where the normalized water saturation 𝑆∗ is calculated by

𝑆∗ =𝑆𝑤 − 𝑆𝑤𝑟

1 − 𝑆𝑤𝑟 − 𝑆𝑜𝑟 .

Here 𝑆𝑤𝑟 is the residual water saturation and 𝑆𝑜𝑟 is the residual oil saturation that

depends on wettability:

𝑆𝑜𝑟 = (1 − 𝜃)𝑆𝑜𝑟𝑤𝑤 + 𝜃𝑆𝑜𝑟

𝑜𝑤 .

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)

7.1.5 Polymer Flooding Model

Polymer is dissolved and well mixed in the aqueous phase. The mass conservation

equation for polymer is

𝜕

𝜕𝑡[𝜙(𝑆𝑤 − 𝜙𝐼𝑃𝑉)𝐶𝑝 + (1 − 𝜙)��𝑝] + ∇ ⋅ (𝐶𝑝��𝑤) = 0 , (7.8)

172

where 𝜙𝐼𝑃𝑉 is the inaccessible pore volume and ��𝑝 is the polymer that is adsorbed on the rock

surface. The viscosity of polymer solution is a function of polymer, 𝑁𝑎+ and 𝐶𝑎2+

concentrations as (Delshad et al. 1996)

𝜇𝑝 = 𝜇𝑤[1 + (𝑎1𝐶𝑝 + 𝑎2𝐶𝑝2 + 𝑎3𝐶𝑝

3)𝐶𝑠𝑒𝑠𝑝] ,

𝐶𝑠𝑒 = 𝐶𝑁𝑎+ + (𝛽𝑝 − 1)𝐶𝐶𝑎2+ .

The adsorbed polymer concentration ��𝑝 is a function of aqueous polymer concentration

𝐶𝑝, 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).

7.1.6 Numerical solution

We used a finite volume method to discretize the PDEs. For each control volume k, the

pressure 𝑃𝑗,𝑘, water saturation 𝑆𝑤,𝑘 and molar concentrations 𝐶𝑖,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).

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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,

𝜕𝑆𝑜𝜕𝑡𝐷

+𝜕𝑓𝑜𝜕𝑥𝐷

= 0 , (7.9)

𝜕

𝜕𝑡𝐷[(𝑆𝑤 −

𝜙𝐼𝑃𝑉𝜙)𝐶𝑝 + ��𝑝] +

𝜕

𝜕𝑥𝐷𝐶𝑝𝑓𝑤 = 0 ,

(7.10)

𝜕

𝜕𝑡𝐷(𝑆𝑤𝐶𝑖 + ��𝑖) +

𝜕

𝜕𝑥𝐷𝐶𝑖𝑓𝑤 = 0, 𝑖 = 1,… ,𝑁𝑐 , (7.11)

where 𝑓𝑤 = (1 +𝜇𝑤(𝐶𝑝,𝐶𝑠)

𝑘𝑟𝑤(𝑆𝑤,𝐶𝑠)

𝑘𝑟𝑜(𝑆𝑤,𝐶𝑠)

𝜇𝑜)−1

and 𝑓𝑜 = 1 − 𝑓𝑤, 𝑥𝐷 and 𝑡𝐷 are dimensionless distance

and time, and 𝑁𝑐 is the number of cations. The oil conservation equation can be rewritten using

𝑆𝑜 = 1 − 𝑆𝑤 as

𝜕𝑆𝑤𝜕𝑡𝐷

+𝜕𝑓𝑤𝜕𝑥𝐷

= 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,

(𝑆𝑤 + 𝐷𝑝)𝜕𝐶𝑝

𝜕𝑡𝐷+ 𝑓𝑤

𝜕𝐶𝑝

𝜕𝑥𝐷= 0, (7.13)

𝑆𝑤𝜕𝐶𝑖𝜕𝑡𝐷

+∑��𝑖𝑗𝜕𝐶𝑗

𝜕𝑡𝐷

𝑁𝑐

𝑗=1

+ 𝑓𝑤𝜕𝐶𝑖𝜕𝑥𝐷

= 0 𝑖 = 1,… ,𝑁𝐶 , (7.14)

174

where ��𝑖𝑗 =𝜕��𝑖

𝜕𝐶𝑗. In the next section, the analytical solutions are developed by splitting

the flow equation into three sub problems.

7.2.1 Decoupled system of equations

We define the new coordinates (Pires et al. 2006) as

𝜑 = ∫ (𝑓𝑤𝑑𝑡𝐷 − 𝑆𝑤𝑑𝑥𝐷) 𝑥𝐷,𝑡𝐷

0,0

, (7.15)

𝜓 = 𝑥𝐷 . (7.16)

The elements of Eqs. (7.12) are transformed therefore to the new coordinates as

𝜕𝑓𝑤𝜕𝑥𝐷

|𝑡𝐷

= −𝑆𝑤𝜕𝑓𝑤𝜕𝜑

|𝜓

+𝜕𝑓𝑤𝜕𝜓

|𝜑

,

𝜕𝑆𝑤𝜕𝑡𝐷

|𝑥𝐷

= 𝑓𝑤𝜕𝑆𝑤𝜕𝜑

|

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

𝑓𝑤) −

𝜕

𝜕𝜑(𝑆𝑤𝑓𝑤) = 0 , (7.18)

𝜕𝐶𝑝

𝜕𝜓+

𝜕

𝜕𝜑(��𝑝 +

𝜙𝐼𝑃𝑉𝜙

𝐶𝑝) = 0 , (7.19)

𝜕𝐶𝑖𝜕𝜓

+𝜕��𝑖𝜕𝜑

= 0, 𝑗 = 1,… , 𝑁𝑐 . (7.20)

175

The hyperbolic equations (Eqs. 7.18 – 7.20) have the same form as the conservation

equations when 𝜓 is the time, 𝜑 is the location, 𝑓𝑤−1, 𝐶𝑝 and 𝐶𝑖 are the conserved quantities, and

𝑆𝑤/𝑓𝑤, ��𝑝 and ��𝑖 +𝜙𝐼𝑃𝑉

𝜙𝐶𝑝 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

�� =1

��=𝑑𝜑

𝑑𝜓 ,

(7.21)

where �� is the eigenvalue of the characteristics matrix. Furthermore, �� represents the front

retardation for two-phase flow. Fronts with larger retardation appear later; hence, the solution is

single valued when �� decreases from the injection composition to initial composition. When 𝜓 =

−𝑥𝐷, the eigenvalues increase from the injection composition to initial composition. We can

convert the PDEs of Eqs. (7.18 – 7.20) to ordinary differential equations using the definition of ��.

The characteristics equations of Eqs. (7.18 – 7.20) are

(𝑆𝑤 −𝑓𝑤𝑓𝑤′− ��)

𝑑𝑓𝑤−1

𝑑𝜑+ 𝑓𝑤

′/𝜕𝑓𝑤𝜕𝐶𝑝

|𝑆𝑤,𝐶𝑖

𝑑𝐶𝑝

𝑑𝜑+∑𝑓𝑤

′/𝜕𝑓𝑤𝜕𝐶𝑖

|𝑆𝑤,𝐶𝑝,𝐶𝑗≠𝑖

𝑑𝐶𝑖𝑑𝜑

𝑁𝑐

𝑖=1

= 0 , (7.22)

(𝐷𝑝 +𝜙𝐼𝑃𝑉𝜙

− ��)𝑑𝐶𝑝

𝑑𝜑= 0 , (7.23)

∑��𝑖𝑗𝑑𝐶𝑗

𝑑𝜑

𝑁𝑐

𝑗=1

− ��𝑑𝐶𝑖𝑑𝜑

= 0, 𝑖 = 1,… ,𝑁𝑐 , (7.24)

where 𝑓𝑤′ =

𝜕𝑓𝑤

𝜕𝑆𝑤|𝐶𝑝,𝐶𝑖

. 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.

176

[ 𝑆𝑤 −

𝑓𝑤𝑓𝑤′− �� 𝑓𝑤

′/𝜕𝑓𝑤𝜕𝐶𝑝

|𝑆𝑤,𝐶𝑖

𝑓𝑤′/𝜕𝑓𝑤𝜕𝐶1

|𝑆𝑤,𝐶𝑝,𝐶2

𝑓𝑤′/𝜕𝑓𝑤𝜕𝐶2

|𝑆𝑤,𝐶𝑝,𝐶1

0 𝐷𝑝 +𝜙𝐼𝑃𝑉𝜙

− �� 0 0

0 0 ��11 − �� ��12

0 0 ��21 ��22 − �� ]

[ 𝑑𝑓𝑤

−1

𝑑𝜑𝑑𝐶𝑝

𝑑𝜑𝑑𝐶1𝑑𝜑𝑑𝐶2𝑑𝜑 ]

= 0 . (7.25)

The eigenvalues of this system are

��1,2 =��11 + ��22 ±√��11 + ��22 − 2��11��22 + 4��12��21

2 ,

(7.26)

��3 = 𝐷𝑝 +𝜙𝐼𝑃𝑉𝜙 ,

(7.27)

��4 = 𝑆𝑤 −𝑓𝑤𝑓𝑤′ .

(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

��1,2 =

[

𝑒1𝑠01

��1,2 − ��11

��12 ]

, ��3 =

[ 𝑒3𝑠

1

0

0 ]

, ��4 =

[ 1

0

0

0]

, (7.29)

where, 𝑒1𝑠 =𝑓𝑤′ /𝜕𝑓𝑤𝜕𝐶1

|𝑆𝑤,𝐶𝑝,𝐶2

+��1,2−��11

��12𝑓𝑤′ /𝜕𝑓𝑤𝜕𝐶2

|𝑆𝑤,𝐶𝑝,𝐶1

𝑆𝑤−𝑓𝑤

𝑓𝑤′ −��1,2

and 𝑒3𝑠 =

𝑓𝑤′ /𝜕𝑓𝑤𝜕𝐶𝑝

|𝑆𝑤,𝐶𝑖

𝑆𝑤−𝑓𝑤

𝑓𝑤′ −��3

.

The eigenvectors and eigenvalues, Eqs. (7.26 – 7.29), have two important features. First,

��4 is the only eigenvalue that is function of fractional flow and saturation. Second, composition is

constant along ��4. Therefore, we can solve the reaction and polymer transport independent of

177

fractional flow and the path along ��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. (7.18 – 7.20) 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.

7.2.2 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 + ��11 − �� ��12

��21 1 + ��22 − ��] [

𝑑𝐶1

𝑑𝐶2

] = 0 . (7.30)

The eigenvalues for single-phase transport are related to the eigenvalues of Eqs. (7.24) as

follows

�� = �� + 1,

178

such that a front with speed of 1.0 in the single-phase region has �� = 1 and �� = 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, ��, 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.

7.2.3 Polymer transport solution

The polymer eigenvalue, ��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

179

Λ =Δ��𝑝Δ𝐶𝑝

+𝜙𝐼𝑃𝑉𝜙 .

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).

7.2.4 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 𝑥𝐷 − 𝑡𝐷 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 𝑥𝐷 − 𝑡𝐷 coordinate is Λ =

Δ𝑥𝐷/Δ𝑡𝐷 then Λ =𝑓𝑤

𝑆𝑤+Λ. This means the extension of a C-shock should pass through the point

– Λ on the 𝑆𝑤 axis.

180

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, 𝑅1, 𝑅2 and 𝑅3 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 𝑅𝑖 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 𝐿1 − 𝑅1 is a

shock to 𝑎 then a jump between two fractional flows that is tangent at 𝑏 and a rarefaction to 𝑅1.

The second solution is a shock from 𝐿2 to 𝑐 followed by a Buckley-Leverett shock to 𝑅1. Finally,

the last composition path is a shock from 𝐿3 to 𝑑 followed by saturation shock to 𝑅2. 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.

7.2.5 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

181

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 (𝑆𝑤 =

0.59) and downstream saturation (𝑆𝑤 = 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).

7.2.6 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

182

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.

7.2.7 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) = ∫ −𝑆𝑤𝑑𝑥0,1

0,0+ ∫ 𝑓𝑤𝑑𝑥

𝑡,1

0,1= −𝑆��𝑖 + ∫ 𝑓𝑤𝑑𝑥

𝑡,1

0,1 . (7.31)

183

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.

7.3.1 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 𝑆𝑤 = −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 𝑥𝐷 = 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.

184

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.

7.3.3 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 7-11. 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

185

exchange front changes surface wettability, therefore we only considered one wettability front in

the analytical solution of Figure 7-11. 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.

7.3.4 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.

186

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 7-17. 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 7-17. 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

187

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.

Ion (mol/l)

Voegelin et al. (2000) Shaker Shiran, and

Skauge (2013)

Seccombe et al.

(2008)

Injection Initial Injection Initial Injection Initial

Na+ 0.00166 0.47000 0.04940 0.49400 0.01001 0.45916

Mg++ 0.00062 0.04900 0.00118 0.01178 0.00244 0.11175

Ca++ 0.00310 0.01150 0.00547 0.05473 0.00043 0.01970

Cl− 0.00910 0.59100 0.06270 0.92701 0.01574 0.72206

Table 7.2 – Reaction parameters. CEC2 and CEC3 are calculated by matching experiments.

Parameter Keq,Ca Keq,Mg CEC1 (mol/l)

(Venkatraman et al. 2014)

CEC2

(mol/l)

CEC3

(mol/l)

Value 46.933 67.8390 0.1171 3.1617 0.0033

Table 7.3 – The Corey relative permeability parameters for the experiments.

Parameters

Shaker Shiran, and

Skauge (2013)

Seccombe et al.

(2008)

Water wet Oil wet Water

wet Oil wet

𝑆𝑤𝑟 0.22 0.22 0.15 0.07

𝑆𝑜𝑟 0.10 0.16 0.24 0.29

𝑛𝑤 1.60 1.60 9.46 3.70

𝑛𝑜 2.80 2.80 1.48 6.16

𝑘𝑟𝑤∗ 0.50 0.50 0.40 0.40

𝑘𝑟𝑜∗ 0.93 0.93 1.00 1.00

𝜇𝑤 1.03 1.03 1.00 1.00

𝜇𝑜 2.40 2.40 1.20 1.20

𝜇𝑝 2.60 2.60 - -

188

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.

189

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.

190

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.

191

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.

192

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.

193

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.

194

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.

195

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.

196

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.

197

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 Sor 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.

198

Chapter 8

Conclusions

We developed a general framework for solving multicomponent, multiphysics transport

in porous media. Method of characteristics along with splitting of the phase equilibrium model

from the transport model significantly simplified the solutions for gas floods and low salinity

polymer floods. The mass transfer between the phases is the key recovery mechanism for gas

flooding, and wettability alteration is important for LSW. We developed analytical solutions for

composition routes using the method of characteristics for the case when the two-phase zone

bifurcates at high pressure. We split the equations into two parts; one dependent on phase

behavior only and the other on fractional flow. We used the splitting method to develop a new

MMP calculation algorithm.

8.1 Summary and conclusions

The key conclusions related to development of analytical solutions for bifurcating phase

behavior are

1- The solution for particular oil compositions becomes discontinuous when the watershed

point no longer controls the composition route.

2- A combined C/V drive occurs for the ternary displacements because the order of the K-

values changes between the two lightest components. This is in contrast to other papers

that report that four or more components are needed for a condensing/vaporizing

mechanism to exist.

199

3- The MMP determined by the composition route (not just the key tie lines) agrees well

with the MMP from the mixing-cell method. The shortest tie line controls miscibility and

this tie line is between the condensing and vaporizing regions.

4- The MMP does not exist for some oil compositions in these complex ternary

displacements where L1-L2 regions remain at high pressures. The displacement efficiency

for these cases approaches a constant value of around 95%.

5- The composition route along the nontie-line path consists of a shock and wave. The

composition route must go through a key composition called the “watershed” point.

6- The composition path and numerical solution agree well, although the finite difference

solution is significantly affected by numerical dispersion. For the two to agree closely,

we used over 100,000 grid blocks in the 1-D numerical solution. The simulation results

for 400,000 grid blocks verify the discontinuity in the solution at the envelope curve.

Shocks within the two-phase zone are represented by curves in composition space.

Shocks from single-phase compositions to (or from) the two-phase zone are represented

by straight lines along the tie-line extensions.

7- There exists a single-phase region where three tie-line extensions intersect, which must

be considered in the development of the composition route. We show which tie line is the

correct one to take.

8- A new switch condition is developed that satisfies both the velocity and entropy

conditions simultaneously. The switch condition significantly aids in determining key

composition points.

Conclusion related to analytical solutions in tie-line space are

200

1- The tie lines in a composition route can be solved more simply using the method of

characteristics (MOC) in tie-line space than in composition space. The eigenvectors in

tie-line space describe the ruled surfaces in composition space, but reduce to curves in

tie-line space. Further, there is only one umbilic point in tie-line space for all

displacements examined.

2- Solution routes in tie-line space are constructed for complex displacements that have not

been solved previously, including a four-component bifurcating displacement with a

shock a rarefaction along one of the ruled surfaces.

3- Solutions previously difficult to rule out (such as multiple tie-line extensions through the

initial or injection composition) are now easily eliminated using a simple condition based

on “velocity” constraints of the tie lines.

4- The MMP is more accurately determined when calculated using the tie-line MOC

approach because shorter tie lines between key tie lines (bifurcating tie lines) could

control miscibility. The approximate shock jump method for MMP calculation can have

significant errors on the order of 5000 psia in such cases (cases where the two-phase

region bifurcates).

5- The approximate Riemann solver significantly improved the robustness of the MMP

calculation algorithm. Further investigation of the approximate Riemann solver is needed

to improve MMP calculation accuracy and robustness.

Conclusions related to analytical solutions for low salinity polymer flooding are

1- The analytical solution results match the experimental data indicating that the proposed

model for wettability alteration through cation exchange is the likely mechanism.

201

2- The cation exchange front moves slower than the salinity shock (anion front). Surface

compositions and wettability alteration occurs only along the cation exchange front.

3- Oil recoveries are matched by adjusting parameters that retard the wettability front,

without the need to match the parameters for the cation exchange reaction model. After

recoveries are matched, the reaction model is tuned to match water composition,

independent of fractional flow. This makes the model parameters involved in tuning more

reliable.

4- Oil is recovered gradually over several pore volumes of low salinity water injection. Most

of the oil is recovered once the wettability front breakthrough.

5- Small slugs of low salinity water can be ineffective because the high salinity shock

moves faster than the wettability front, causing wettability alteration to cease at that

point. The low salinity slug should be of sufficient size to propagate the wettability front

to the production wells.

8.2 Future research

8.2.1 Application of splitting to compositional simulation

We used the splitting technique to simplify the MMP calculation algorithm. A similar

approach can be used to improve phase behavior calculations in reservoir simulation. The flash

calculation can be simplified by use of tie-line space parameters instead of overall compositions.

The current compositional reservoir simulators, however, use overall composition or phase

compositions as the independent variables. The front tracking algorithm developed in this

research can be used directly in streamline simulation. In addition, the front tracking algorithm

can be used to reformulate the compositional reservoir simulators based on tie-line parameters.

202

8.2.2 WAG injection and hysteresis

Many of the current analytical solutions for WAG are limited to simultaneous WAG. The

front tracking algorithm in tie-line space can be used to construct analytical solutions for WAG

considering hysteresis, dissolution of gas in water, and vaporization of water. The results can be

used to calculate the optimum WAG ratio

8.2.3 Fluid characterization

The phase diagrams in tie-line space reveal new features of transport properties of fluids.

For example, the eigenvalues in tie-line space can be used to optimize the number of components

based on transport properties of the fluid independent of fractional flow curves. The K-values are

commonly used to determine the optimum number of components. Our preliminary results,

however, demonstrated that the heavy components with vastly different K-values can vaporize on

almost the same front in a immiscible gas flood.

8.2.4 Displacement mechanism for combined EOR techniques

We constructed solutions for low salinity polymer flooding. The solution can be used to

determine the contribution of low salinity and polymer to the final recovery. The combination of

multiple EOR techniques to increase the overall recovery and economics of the process is

becoming very popular. The splitting technique can be used to analyze the synergistic effect of

different chemicals, and displacement mechanisms.

203

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225

Appendix A Tie-line derivatives

The analytical derivatives of tie-line coefficients are necessary for construction of the

solution in tie-line space. These derivatives can assist flash calculation in reservoir simulator and

parameterization of composition space. A tie line can be defined by following equation.

𝐶𝑖 = 𝛼𝑖−1𝐶1 + 𝛽𝑖−1 𝑖 = 2,… , 𝑁𝑐 − 1 , (A-1)

where the tie line coefficients must be calculated using the results of flash calculations and Eq.

A.2.

𝛼𝑖−1 =𝑥𝑖(1 − 𝐾𝑖)

𝑥1(1 − 𝐾1), 𝛽𝑖−1 =

𝑥𝑖(𝐾𝑖 − 𝐾1)

1 − 𝐾1 𝑖 = 2,… ,𝑁𝑐 − 1 .

(A-2)

The derivatives are calculated as following.

𝜕𝛼𝑖−1

𝜕𝑥𝑗𝐼 =

(1 − 𝐾𝑖)𝜕𝑥𝑖𝜕𝑥𝑗

− 𝑥𝑖𝐾𝑖𝑗 − 𝛼𝑖[(1 − 𝐾1)𝛿1𝑗 − 𝑥1𝐾1𝑗]

𝑥1(1 − 𝐾1),

𝜕𝛽𝑖−1

𝜕𝑥𝑗𝐼 =

(𝐾𝑖 −𝐾1)𝜕𝑥𝑖𝜕𝑥𝑗

+ 𝑥𝑖(𝐾𝑖𝑗 − 𝐾1𝑗) + 𝛽𝑖𝐾1𝑗

1 − 𝐾1 𝑖 = 2,… ,𝑁𝑐 − 1,

𝑗 = 1,… ,𝑁𝑐 − 2 ,

(A-3)

where 𝐾𝑖𝑗 = 𝜕𝐾𝑖/𝜕𝑥𝑗𝐼. A set of 𝑁𝑐 – 2 mole fractions should be selected as independent

variables. Any one of the liquid or vapor mole fractions can be used as independent variables. It

should be noted that some combinations of independent variables might not be suitable.

226

K-values derivatives with respect to independent variables are necessary to calculate the

derivatives of tie-line coefficients. The following discussion demonstrates the number of

independent variables required to determine a tie line, along with the relationship between

dependent variables and independent variables. Two phases at thermodynamic equilibrium should

satisfy the following condition for all components.

𝑓𝑖𝑣 = 𝑓𝑖

𝑙 . (A-4)

The change of fugacities of components should be the same for both phases to keep the

phases at equilibrium. That is the phases are at equilibrium before and after the change of

compositions.

𝑑𝑙𝑛𝑓𝑖𝑣 = 𝑑𝑙𝑛𝑓𝑖

𝑙 , (A-5)

where,

𝑑𝑙𝑛𝑓𝑖 =∑𝜕𝑙𝑛𝑓𝑖𝜕𝑛𝑗

𝑑𝑛𝑗

𝑁𝑐

𝑗=1

. (A-6)

Change in fugacity goes to infinity as the mole fraction goes to zero. The following

equation will resolve the problem.

𝑦𝑖𝑑𝑙𝑛𝑓𝑖𝑣 = 𝑦𝑖𝑑𝑙𝑛𝑓𝑖

𝑙 . (A-7)

We are interested to calculate change of tie-line coefficients; therefore we assume that the

overall compositions stay at the middle of tie line. Further we can assume that the total number of

moles remains constant. These assumptions can be shown by following equations.

227

∑𝑑𝑛𝑖𝑙

𝑁𝑐

𝑖=1

= 0, ∑𝑑𝑛𝑖𝑣

𝑁𝑐

𝑖=1

= 0 . (A-8)

The number of equations is 𝑁𝑐 − 2 less than the number of variables. Therefore by

specifying 𝑁𝑐 − 2 component with change in number of moles, all the other number of moles can

be calculated. The independent change in mole fractions can be defined as new variable.

𝑑𝑛𝑖𝐼 = 𝑑𝑛𝑘

𝑙 𝑜𝑟 𝑑𝑛𝑘𝑣 , 𝑘 = 1,… , 𝑁𝑐 − 2, 𝑖 = 1,… ,𝑁𝑐 − 2 . (A-9)

The equations can be rearranged by choosing the dependent and independent variables.

[𝐷𝐷𝑁] = [𝐶𝑀][𝐼𝐷𝑁] , (A-10)

where the matrix elements are defined by Eq A-11.

𝐶𝑀𝑖𝑗 =𝜕𝐷𝑛𝑖

𝜕𝑛𝑗𝐼 . (A-11)

𝐷𝑛 is the number of moles for dependent components and 𝑛𝐼 is the number of moles for

independent components. The derivatives of fugacity can be calculated based on derivatives of

fugacity coefficients as follows,

𝜕𝑙𝑛𝑓𝑖𝜕𝑛𝑗

=𝜕𝑙𝑛∅𝑖𝜕𝑛𝑗

+1

𝑥𝑖

𝜕𝑥𝑖𝜕𝑛𝑗

,

𝜕𝑥𝑖𝜕𝑛𝑗

= {

1 − 𝑥𝑖, 𝑖 = 𝑗 ,

−𝑥𝑖, 𝑖 ≠ 𝑗 .

(A-12)

K-values can be defined as the number of moles of 𝑁𝑐 − 2 component of the liquid phase.

𝑙𝑛𝐾𝑖 = 𝑙𝑛𝜙𝑖𝑙 − 𝑙𝑛𝜙𝑖

𝑣 = 𝑓(𝑛1𝐼 , … , 𝑛𝑁𝑐−2

𝐼 ) 𝑖 = 1,… ,𝑁𝑐 . (A-13)

228

The derivative of K-value respect to composition is shown as follows,

𝜕𝑙𝑛𝐾𝑖

𝜕𝑥𝑗𝐼 = ∑

𝜕𝑙𝑛𝐾𝑖

𝜕𝑛𝑘𝐼

𝑁𝑐−2

𝑘=1

𝜕𝑛𝑘𝐼

𝜕𝑥𝑗𝐼 𝑖 = 1, … , 𝑁𝑐, 𝑗 = 1, … , 𝑁𝑐 − 2 . (A-14)

The derivatives of K-value respect to liquid component moles are calculated as follows,

𝜕𝑙𝑛𝐾𝑖

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

=𝜕𝑙𝑛𝜙𝑖

𝑙

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

−𝜕𝑙𝑛𝜙𝑖

𝑣

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

𝑖 = 1,… ,𝑁𝑐, 𝑗 = 1,… ,𝑁𝑐 − 2 . (A-15)

K-values are defined for two phases at equilibrium but fugacity is defined for any single

phase. The derivatives of liquid phase fugacity coefficients are calculated using the PR EOS. The

following demonstrates the method to calculate the derivatives of fugacity coefficients for two

phases at equilibrium.

𝜕𝑙𝑛𝜙𝑖𝑙

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

=∑𝜕𝑙𝑛𝜙𝑖

𝑙

𝜕𝑛𝑘𝑙 |

𝑛𝑚≠𝑘𝑙

𝜕𝑛𝑘𝑙

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

𝑁𝑐

𝑘=1

, 𝜕𝑙𝑛𝜙𝑖

𝑣

𝜕𝑛𝑗𝐼 =∑

𝜕𝑙𝑛𝜙𝑖𝑣

𝜕𝑛𝑘𝑣 |

𝑛𝑚≠𝑘𝑣

𝜕𝑛𝑘𝑣

𝜕𝑛𝑗𝐼 |

𝑛𝑘≠𝑗𝐼

𝑁𝑐

𝑘=1

𝑖 = 1,… ,𝑁𝑐 , 𝑗 = 1,… ,𝑁𝑐 − 2 .

(A-16)

The constraints of Eq. A-7 imply that 𝑑𝑥𝑖 = 𝑑𝑛𝑖 for both phases. Therefore Eq. A-14 can

be simplified further.

𝜕𝑙𝑛𝐾𝑖

𝜕𝑥𝑗𝐼 =

𝜕𝑙𝑛𝐾𝑖

𝜕𝑛𝑗𝐼 𝑖 = 1,… ,𝑁𝑐 , 𝑗 = 1,… ,𝑁𝑐 − 2 . (A-17)

To summarize, the tie-line derivatives can be calculated in following order.

1- Calculate fugacity coefficients and their derivatives using an equation of state.

2- Eqs. A-6, A-7 and A-8 are used simultaneously to calculate [𝐶𝑀].

229

3- Calculate Kij using Eq. A-17.

4- A’, B’ can be calculated by Eq. A-2.

The above derivatives can be used to generate initial estimates for flash calculations.

First, the change in phase liquid mole fractions by a change of overall mole fraction should be

calculated as follows,

[∆𝑥𝐼] = 𝑓([∆𝑧], 𝐴′, 𝐵′). (A-18)

Eq. A-18 is equivalent to the simultaneous solution of the following four equations,

𝐴𝑖 = 𝐴𝑖𝑟 +∑

𝜕𝐴𝑖

𝜕𝑥𝑖𝐼 ∆𝑥𝑖

𝐼 , (A-19)

𝐵𝑖 = 𝐵𝑖𝑟 +∑

𝜕𝐵𝑖

𝜕𝑥𝑖𝐼 ∆𝑥𝑖

𝐼 , (A-20)

𝑧𝑖 = 𝐴𝑖𝑧1 + 𝐵𝑖 , (A-21)

𝑧𝑖 = 𝑧𝑖𝑟 + ∆𝑧𝑖 . (A-22)

The superscript r indicates the values the composition with calculated tie line. The next

step is to update K-values using Eq. A-17.

230

Appendix B Switch condition

We combined the velocity and entropy conditions to show important new relationships

between eigenvalues and shocks to derive a general switch condition for shocks and waves. We

also show that the nontie-line shock must be tangent to the nontie-line eigenvalue at α for the

nontie-line shock in this pseudoternary displacement.

The entropy conditions for a variety of shock types can be written as,

Lax 1-type:

𝜆2𝑈 ≥ 𝜆1

𝑈 ≥ Λ

𝜆2𝐷 ≥ Λ ≥ 𝜆1

𝐷

(B-1)

Lax 2-type:

𝜆2𝑈 ≥ Λ ≥ 𝜆1

𝑈

Λ ≥ 𝜆2𝐷 ≥ 𝜆1

𝐷

(B-2)

Transitional:

𝜆2𝑈 ≥ Λ ≥ 𝜆1

𝑈

𝜆2𝐷 ≥ Λ ≥ 𝜆1

𝐷

(B-3)

Compressive:

𝜆1𝑈 ≥ Λ ≥ 𝜆1

𝐷

𝜆2𝑈 ≥ Λ ≥ 𝜆2

𝐷

(B-4)

231

where the order of the eigenvalues is assumed, that is, 2 1 everywhere. The Lax conditions

were derived specifically for a strictly hyperbolic system where the order of the eigenvalues

remains the same in the dependent variable space. In Eq. B-1, however, the order can change as a

function of composition. Thus, the fast eigenvalue is always 2 , except at the umbilic point where

the tie-line and nontie-line eigenvalues are equal. For compositions between the umbilic points

2 t and 1 nt (see Figure 3-3). Otherwise,

2 nt and 1 t .

The entropy conditions have physical meaning. The eigenvalues inside a box indicate

those eigenvalues associated with characteristics that impinge on or are parallel to the shock. The

1- and 2-Lax shocks have three characteristics that impinge on or are parallel to the shock on a

time-distance diagram, which makes those shocks self-sharpening and resistant to dispersion. The

transitional shock, however, only has two characteristics that impinge on or are parallel to the

shock, while an overly compressive shock has four characteristics that impinge on or are parallel

to the shock. Generally, a transitional shock is least resistant to dispersion, while a compressive

shock is very resistant (highly self-sharpening).

For all shock types, we can see from the boxes in Eqs. B-1 to B-4 that 2 1

U D is

always true for a valid shock.

Consider the common occurrence in a composition route where a constant state is

separated by two shocks, where shock A is upstream of shock B. That is, the downstream of

shock A is the upstream of shock B, and they are separated only by a constant state. The constant

state is the upstream composition of shock B or downstream composition of shock A. Such a

composition route occurs often both for conventional ternary displacements and for the

pseudoternary displacements in this paper.

We first consider that only a constant state is separated by two shocks and we combine

the entropy and velocity conditions to derive a general switch condition. Later, we apply this

232

switch condition to the nontie-line shock that is connected to a nontie-line path in the

pseudoternary displacements to show that the nontie-line shock must be a tangent shock to the

nontie-line path.

Case 1: Shocks A and B are Lax-1 type:

𝜆2 ≥ Λ𝐴 ≥ 𝜆1, 𝐿𝑎𝑥 1 𝑓𝑜𝑟 𝐴

𝜆2 ≥ 𝜆1 ≥ Λ𝐵, 𝐿𝑎𝑥 1 𝑓𝑜𝑟 𝐵

Λ𝐵 ≥ Λ𝐴, 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 .

(B-5)

A comparison of the above conditions gives, Λ𝐵 = Λ𝐴 = 𝜆1, which means that there

cannot be a constant state between Lax-1 type shocks. These two shocks must merge to form one

shock.

Case 2: Shock A and B are Lax-2 type:

𝛬𝐴 ≥ 𝜆2 ≥ 𝜆1, 𝐿𝑎𝑥 2 𝑓𝑜𝑟 𝐴

𝜆2 ≥ 𝛬𝐵 ≥ 𝜆1, 𝐿𝑎𝑥 2 𝑓𝑜𝑟 𝐵

𝛬𝐵 ≥ 𝛬𝐴, 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 .

(B-6)

A comparison of the above conditions gives, Λ𝐵 = Λ𝐴 = 𝜆2, which means that there

cannot be a constant state between Lax-2 type shocks. These two shocks must again merge to

form one shock.

Case 3: Shock A is Lax-2 type and shock B is Lax-1 type:

Λ𝐴 ≥ 𝜆2 ≥ 𝜆1, 𝐿𝑎𝑥 2 𝑓𝑜𝑟 𝐴

𝜆2 ≥ 𝜆1 ≥ Λ𝐵, 𝐿𝑎𝑥 1 𝑓𝑜𝑟 𝐵

Λ𝐵 ≥ Λ𝐴, 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 .

(B-7)

233

A comparison of the above conditions gives, Λ𝐵 = Λ𝐴 = 𝜆1 = 𝜆2, which means that there

cannot be a constant state between a Lax-2 type shock upstream of a Lax-1 type shock. These two

shocks must again merge to form one shock.

Case 4: Shock A is Lax 1 type and shock B is Lax-2 type:

𝜆2 ≥ Λ𝐴 ≥ 𝜆1, 𝐿𝑎𝑥 1 𝑓𝑜𝑟 𝐴

𝜆2 ≥ Λ𝐵 ≥ 𝜆1, 𝐿𝑎𝑥 2 𝑓𝑜𝑟 𝐵

Λ𝐵 ≥ Λ𝐴, 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 .

(B-8)

A comparison of the above conditions gives, 𝜆2 ≥ Λ𝐵 ≥ Λ𝐴 ≥ 𝜆1. These shocks can exist

separated by a constant state.

A similar analysis can be done for transitional and overly compressive shocks. The

eigenvalue condition for downstream of a transitional shock is the same as for a downstream Lax-

1 shock. Also the eigenvalue condition for upstream of a transitional shock is the same as an

upstream Lax-2 shock. Therefore a transitional shock can be present before or after the constant

state. However, a compressive shock has a similar upstream entropy condition as a Lax-1 shock

and downstream condition as a Lax-2 shock. Therefore we cannot have a compressive shock

before or after a constant state.

Now consider the composition route specific to the pseudoternary displacements in

Section 3.4. For a combination of an upstream nontie-line shock to a constant state followed by a

downstream tie-line shock, the velocity and entropy conditions must apply. We then have for the

entropy conditions from Eq. (2.5),

𝜆2𝛼 ≥ Λ𝑛𝑡 ≥ 𝜆1

𝐵 , (B-9)

𝜆2𝐵 ≥ Λ𝑡 ≥ 1 , (B-10)

234

where composition B is the constant state composition as shown in Figure 3-10 (the value of 1.0

is the eigenvalue at the oil composition). The nontie-line shock is a Lax-1 shock, while the tie-

line shock is Lax-2 type. Thus, combining the two entropy conditions with the velocity constraint

gives:

𝜆2𝐵 ≥ Λ𝑡 ≥ Λ𝑛𝑡 ≥ 𝜆1

𝐵 . (B-11)

When the constant state is between the umbilic points on the oil tie line and point α is

outside of the umbilic points on the tie line through A, and a nontie-line path upstream of the

nontie-line shock exists, we must have,

𝜆𝑡𝐵 ≥ Λ𝑡 ≥ 𝜆𝑛𝑡

𝛼 = Λ𝑛𝑡 ≥ 𝜆𝑛𝑡𝐵 . (B-12)

This forces the nontie-line shock to be a tangent shock to the nontie-line path so that both

the entropy and velocity conditions are satisfied.

235

Appendix C Shock composition paths

A shock consists of a jump from an upstream state to a downstream state in the MOC

solution. Generally it is represented on a ternary diagram as a straight line that connects these two

endpoint states, but this is not necessarily the case.

Rhee and Amundson (1974) investigated the effect of dispersion and mass transfer on a

shock composition path for chromatography using the concept of a shock layer. A shock layer is a

series of compositions that connects the two ending states of the shock, where the end states

move together with the same velocity as the shock velocity. Exactly the same approach can be

used here to construct the composition path for a shock where various levels of dispersion are

present in a gas flood.

Consider the 1-D convection-dispersion equation, which is similar to Eq. (3.10):

1

𝑃𝑒

𝜕2𝐶𝑖

𝜕𝑥𝐷2 =

𝜕𝐶𝑖𝜕𝑡𝐷

+𝜕𝐹𝑖𝜕𝑥𝐷

𝑖 = 1, 2 . (C-1)

We assume there is a shock layer that moves with the same velocity given by ��, and has

the following boundary conditions:

𝐶𝑖 = 𝐶𝑖𝑈 𝑎𝑡 𝑥 = −∞ ,

𝐶𝑖 = 𝐶𝑖𝐷 𝑎𝑡 𝑥 = +∞ ,

(C-2)

where U is the upstream state and D the downstream state. We define a moving coordinate to

follow the shock layer as,

𝜉 = 𝑥𝐷 − ��𝑡𝐷 . (C-3)

236

We also ensure that the solution is bounded so that the boundary conditions become,

𝐶𝑖 = 𝐶𝑖𝑈 ,

𝑑𝐶𝑖𝑑𝜉

= 0 𝑎𝑡 𝑥 = −∞ ,

𝐶𝑖 = 𝐶𝑖𝐷 ,

𝑑𝐶𝑖𝑑𝜉

= 0 𝑎𝑡 𝑥 = +∞ .

(C-4)

Substitution of Eq. (C-3) into Eq. (C-1) converts the PDEs to the following ODEs,

1

𝑃𝑒

𝑑2𝐶𝑖𝑑𝜉2

−𝑑𝐹𝑖𝑑𝜉

+ ��𝑑𝐶𝑖𝑑𝜉

= 0 𝑖 = 1, 2 . (C-5)

Integration of Eq. (C-5) from −∞ to 𝜉 and using the boundary conditions gives,

1

𝑃𝑒

𝑑𝐶𝑖𝑑𝜉

− 𝐹𝑖 + ��𝐶𝑖 = −𝐹𝑖𝑈 + ��𝐶𝑖

𝑈 𝑖 = 1, 2 . (C-6)

Application of the boundary condition at +∞ shows that the shock layer should move

with the velocity of the discontinuous shock. That is, the shock has the same speed as that given

by the Rankine-Hugoniot jump conditions,

�� =𝐹𝑖𝐷 − 𝐹𝑖

𝑈

𝐶𝑖𝐷 − 𝐶𝑖

𝑈 𝑖 = 1, 2 . (C-7)

Integration of Eq. (C-6) with the assumption that the Peclet numbers for each component

are equal gives,

𝑑𝐶1𝑑𝐶2

=𝐹1 − 𝐹1

𝑈 + ��(𝐶1𝑈 − 𝐶1)

𝐹2 − 𝐹2𝑈 + ��(𝐶2

𝑈 − 𝐶2) . (C-8)

237

Eq. (C-8) gives the composition path (or direction) for a shock of any type (a shock

within the two-phase region or a shock in or out of the two-phase region). The shock path or

locus given by Eq. C-8 is a line if the value of 𝑑𝐶1/𝑑𝐶2 = 1/𝛼 for all values of 1

C along the

locus, where α is the slope of the tie line in Eq. 2.13. This is the case for a shock in or out of the

two-phase region along a tie-line extension.

Shocks within the two-phase region, however, do not point along a tie line and therefore

Eq. (C-8) will result in varying values for 𝑑𝐶1/𝑑𝐶2 as we move away from upstream state. Thus,

the shock path is generally curved for a shock within the two-phase region and cannot be

represented by a line between the upstream and downstream endpoint compositions of the shock

on a ternary diagram. The shock path can be determined numerically by integration of Eq. (C-8)

starting from either one of the endpoint compositions of the shock. The velocity of the shock and

its endpoint compositions are fixed in the integration.

238

Appendix D Reservoir simulation in tie-line space

The transport in new composition space, Eqs. (5.23) can be used to improve phase

equilibrium calculation in reservoir simulators. The conventional simulators use Eq. (3.10) for

transport calculation.

𝜕𝐶𝑖𝜕𝑡𝐷

+𝜕𝐹𝑖𝜕𝑥𝐷

= 0 𝑖 = 1,… ,𝑁𝑐 − 1 . (3.10)

We can use the following transport equation for a ternary displacement instead.

𝜕��1𝜕𝑡𝐷

+𝜕��1𝜕𝑥𝐷

= 0 ,

𝜕��

𝜕𝑡𝐷+��2���� + 1

��2���� + 1

𝜕��

𝜕𝑥𝐷= 0 .

(D.1)

The �� can be used directly in flash calculation. In addition K-values at constant pressure

and temperature are only a function of ��. There, an estimation of K-values with a function can be

used to eliminate iteration in flash calculation.

239

Appendix E Object oriented design of phase equilibrium calculation

class CompProp { public: double Tc; double Pc; double Vc; double MW; double omega; double VS; // Volume shift double Par; // Parachor double ConsDen; //Component density for no volume change on mixing std::string Name; int type; }; class FluidProp { public: FluidProp(); ~FluidProp(); FluidProp(int Num_comp); CompProp * CompArray; int NumOfComp; double * BIP; double R; double * Q; double * invQ; double C1T; bool Qinitialized; }; class TieProp { public: int numberOfPhases; int num_comp; int primaryComp; //The primary component for tie line equaiton int * IndIndexes; //The independent components //Index starts from 0 for gas and from nc for liquid double * EigVal; double * EigVec; double * EigAngl; double * Coefs; //This one can handele bothe two phase and three phases double * TieDer; //All A derivatives followed by B derivatives double * CM; //Coefficient matrix dxi/dgi double * EndPoints; //Mole fraction of primary component in gas phase then liquid phase double TieLength;

240

bool InvCheck; //To calculate determinant of matrix double conCM, conGamma, conR; TieProp(); TieProp(int NumberOfComps, int NofPhases); TieProp(const TieProp& Other); void CopyFrom(const TieProp& Other); static double FCSign; }; //double TieProp::FCSign =0; class PhaseProp { public: PhaseProp(); ~PhaseProp(); PhaseProp(int Nc_in); PhaseProp(const PhaseProp& Other); void CopyFrom(const PhaseProp& Other); void Reinitialize(int Nc_in); double * composition; double * compositionP; double * LnFugacityCoef; double * PureLnFC; double T; double P; //double R; double V; //molar volume double nt; //total number of moles double sat; double aveMW; int num_comp; double ExcessGibbs; }; class PhysicalCond : public PhaseProp { public: PhysicalCond(); ~PhysicalCond(); PhysicalCond(int nc_in, int np_in); void Reinitialize(int nc_in, int np_in); PhaseProp * Phases; TieProp * TieLine; double PureFCPressure; double PureFCTemperature; int numberOfPhases; double * kval; PhysicalCond(const PhysicalCond& Other); void CopyFrom(const PhysicalCond& Other); bool InitializedK; int ErrorCode;

241

void set_P(double Pressure); void set_T(double Tempreature); double getGamma(int index); void setGamma(int index, double value); void changeGammas(double * dGamma); }; class EOS // For now we will update everything in this class { public: EOS(FluidProp *inputFluidModel); void Initialize(FluidProp *inputFluidModel); void ChnageOfTemperature(PhaseProp &Condition); void UpdateFlash(PhaseProp &Condition); void UpdateFinal(PhaseProp &Condition); void InitializeKvalue(double *Kvalue, double pressure, double tempreature); //FluidProp *Mixture; PhaseProp *Condition; static int EosUpdateCount; void UpdatePureFC(PhysicalCond &Condition); void GibssSinglePhase(PhysicalCond &Condition); void CalFugCDerivatives(PhaseProp &Condition, double * FugDerivtiv); double R; FluidProp *FluidModel; void ConvertComp(double * z, double * zp); void invConvertComp(double * zp, double * z); private: double * alpha; //change sby temperature double * a; double * aij; double * ac; double * ap; double * b; double * m; double * BufMinFC; double * ZArr; double P, T, oneORT; double at, bt; double * LnPhi; double G1,G2; int num_comp; double * Composition; double * ZRoots(double az, double bz, double cz); double NegPw3(double val); //Variables shared between functions double Adim, Bdim, OneORT,atInv,btInv; static const double sqrt2 ;

242

static const double Pi; static const double OneThird; }; class KvalueModel { public: double * Update(PhysicalCond &Condition); }; class EquilibriumCalculation { public: EOS * FluidEOS; //Should be a pointer bool InitializedKvale; double tolerance, SwitchTol;//Convergence Tolerance and NewtonSwitch double MlFrctnTol; //tolerance for compositon double MaxIter; //Maximum number of iterations double error; double StabTol; }; class StabilityAnalysis : public EquilibriumCalculation { StabilityAnalysis(); //This one is not necessary StabilityAnalysis(int number_components, EOS &EqOfState); }; class TwoPhaseFlash : public EquilibriumCalculation { public: TwoPhaseFlash(); TwoPhaseFlash(int Nc_in); void CalculateProp(PhysicalCond *InputCond); void CalculatePropLJ(PhysicalCond *InputCond); void BubblePoint(PhysicalCond &InputCond); //This one should be defined as new class void Stability(PhysicalCond &InputCond); //Should be a new class (Not necessarily) void CalculateTieline(PhysicalCond *InputCond); void CalculateTielineSS(PhysicalCond *InputCond); void UpdateTieMatrix(PhysicalCond *InputCond); void UpdateGamma(PhysicalCond *InputCond); //Calculate a tie line shock with the length as l = Σ(Δγ)² void CalculateTielineShock(PhysicalCond *CondU,PhysicalCond *CondD, double l ); void SetDebugMode(const bool debug); void SetConstantK(const double * constantK); double * k; double * k2; PhaseProp *Phases; int TotalFlash, TotalIteration, TotalInnerItr;

243

private: void SolveFlashSS(void); void SolveFlashNR(void); void CalculateTielineConsK(PhysicalCond *InputCond); void UpdateTieMatrixConsK(PhysicalCond *InputCond); double CalculateAngle(double eigenvalue, PhysicalCond * Cond); bool DebugMode, ConstantK; int num_comp; double nl; double * z; double * x; double * y; double * FG; double * FL; double * preVal; //Newton Method parameters double * dFugidnjP0; double * dFugidnjP1; double * N_L; double * N_Linv; double * N_D; double * N_A; double * N_B; double * N_betha; double * N_f; double N_step; //Stepsize to prevent divergence //Deirvatives of tie line and eigenvalues calculation double * dlnfugL ; double * dlnfugV; double * kij; double * DerFugLIn; double * DerFugVIn; double * derA ; double * derB ; double * dxnc_1dxj ; double * CXN ; double * A; double * B ; double * Mat1; double * Mat2 ; double * CM ; double * RM; double * xp; double * yp; double * zp; double * betaDer2; double * betaDer; doublereal *Aty; doublereal *bty; integer * UTMipiv; integer * UTMiwork; doublereal * UTMwork; doublereal * UTMwr; doublereal * UTMwi; doublereal * UTMvl; doublereal * UTMvr; }; class ThreePhaseFlash : public EquilibriumCalculation { public: ThreePhaseFlash(); ThreePhaseFlash(int Nc_in); void CalculateProp(PhysicalCond *InputCond); void SetDebugMode(const bool debug); PhaseProp *Phases; int TotalFlash, TotalIteration, TotalInnerItr;

244

private: bool DebugMode; int num_comp; double nl1, nl2; double * z; double * x1; double * x2; double * y; double * FG; double * FL; double * k1; double * k2; double * k1_p; double * k2_p; }; class PropPackData{ public: PropPackData(const int numComp); int Num_comp; std::string * CompName; double * Tc; double * Pc; double * Vc; double * omega; double * Vs; double * MW; double * ConDen; double * BIP; double * composition; double * Kval; double Pressure; double Temperature; double R; double toelrance, MaxIter, SwitchTol, MlFrctnTol; double StabTol; int FlashOpt, StabOpt; bool ConstantK; int PrimaryComp; int * IndepeIndex; bool SetTielineSpace; }; class TwoPhasePropertyPackage { public: TwoPhasePropertyPackage(char * InputFile); TwoPhasePropertyPackage(PropPackData * Input); FluidProp *Mixture; //should be a pointer void CalculateProp(); void CalculateProp(PhysicalCond *InputCondition); void UpdateTieline(PhysicalCond *InputCondition); void ChangeGammaTieLine(PhysicalCond *InputCondition, double * dGamma); PhysicalCond *Condition; //Should be private void SetDebugMode(const bool debug); void CalculateTieLineSpaceParams(); void UpdateC1T(PhysicalCond *InputCondition); void ConvertComp(double * z, double * zp); int FlashOption; private:

245

bool ConstantK; EOS *FluidEOS; TwoPhaseFlash *FlashObj; double * WKST1; bool DebugMode; };

Vita

Saeid Khorsandi was born in 1985 in Esfahan, Iran, the son of Ali Khorsandi and Nahid

Khalilian. After graduating in Math and Physics from Nilfrooshzadeh High School in 2003, he

entered Petroleum University of Technology, Abadan, Iran. He graduated with Bachelors of

Science in Petroleum Engineering in 2007. In the following year, he entered the Reservoir

Engineering program at the Sharif University of Technology where he received the degree of

Master of Science in Petroleum Engineering in December of 2010. He then moved to State

College where he started doctoral studies at Energy and Mineral Engineering Department, The

Pennsylvania State University, University Park in 2011.

This dissertation was typed by Saeid Khorsandi.