relative permeability of near-miscible fluids in ... · relative permeability of near-miscible...

RELATIVE PERMEABILITY OF NEAR-MISCIBLE FLUIDS IN

COMPOSITIONAL SIMULATORS

A THESIS

SUBMITTED TO THE DEPARTMENT OF ENERGY

RESROUCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTERS OF SCIENCE

Ala Nabeel Al-Zayer

September 2015

© Copyright by Ala Nabeel Al-Zayer 2015

All Rights Reserved

ii

I certify that I have read this report and that, in my opinion, it is fully

adequate in scope and quality as a report for the degree of Masters of

Science in Petroleum Engineering.

(Dr. Hamdi Tchelepi) Principal Adviser

I certify that I have read this report and that, in my opinion, it is fully

adequate in scope and quality as a report for the degree of Masters of

Science in Petroleum Engineering.

(Dr. Denis Voskov)

iii

Abstract

The relative permeability functions are a key parameter in Darcy’s law extension

for modeling multiphase flow. They are empirical functions that lump the effects

of complex interactions between flowing fluids and the porous medium, but they

are usually reported as functions of saturation only. The dependence of the relative

permeability on phase identification can lead to significant complications in near-

miscible displacements.

We present an analysis of existing methods that aim to account for miscibility

effects by including compositional dependence in the relative permeability functions.

The solution evolution in compositional space is analyzed, and the impact of compo-

sitional changes in the relative permeabilities on simulation results and performance

is quantified. We show the sensitivity of different methods to the choice of refer-

ence points used, and we provide guidelines to limit the modification of the relative

permeabilities to physically reasonable amounts.

We use the Gibbs free energy based strategy with some modifications. The new

approach was implemented in a general-purpose simulator (AD-GPRS), and tested

on a wide range of compositional displacements. We have found that including any

compositional dependence in the relative permeability near the critical point improves

the nonlinear convergence significantly. Only slight differences are observed in the

final saturation distributions and well production rates. The new approach, which

applies a correction to the area above the critical tie line extension, results in smoother

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transitions between the single and two phase regions.

In summary, we show a clear advantage of incorporating compositional dependence

in the relative permeability in terms of nonlinear performance. This is especially

clear in displacements near the critical point (near-miscible). The differences between

different models are sensitive to the reference points used, which can only be validated

with experimental evidence and a more solid physical foundation. We provide a basic

framework in the AD-GPRS simulator for possible further investigation into this

topic.

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Acknowledgments

I am very grateful for the opportunity of spending the past two years at Stanford, it

has proved to be a truly rewarding and eye opening experience. This of course is only

a result of the amazing individuals that have supported me all the way, and those I

had the pleasure of getting to know and learn from.

I would like to start by thanking my advisor, Hamdi Tchelepi, for giving me the

opportunity to dive into the world of research and reservoir simulation. He has been

a source of support and inspiration these past two years. His insight has set a new

standard for me in the way science is to be taught, and how research should be

approached, which I am sure will benefit me throughout my life.

I would like to express my deep appreciation for the support I have recieved from

my second advisor, Denis Voskov. Denis has provided me with guidance from choosing

my topic to finalizing this thesis, his patience and advice these past two years have

been key in me developing as a researcher and accomplishing this work. His kindness

and ability to work with many students is truly inspiring.

I would like to recognize Chengwu Yuan and Gary Pope, whos work inspired

this investigation. I especially thank Chengwu for his advice and cooperation on

clarifying questions I had. I would also like to thank Huanquan Pan for his help

with AD-GPRS and flash-related questions, Francois Hamon for providing me with

his hysteresis implementation and making AD-GPRS less intimidating.

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I am also very grateful for all the friendships that made the past two years much

more enjoyable, I thank you all for the priceless discussions and memories. I look

forward to continuing the discussion even after Stanford, so please keep in touch even

if I fail to do so.

I wouldn’t be here without the support of my family. I am always humbled by

the amount of support and love from my parents, Amal and Nabeel, no words can

express my appreciation for all that you have done for me. I would like to thank

Moataz who has been the companion in this chapter, and who has always engaged

me in stimulating discussions. Finally, my beloved partner, Sara, thank you for

all your support and patience throughout this demanding journey. Thank you for

reminding me to leave the office and enjoy life.

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Contents

Abstract v

Acknowledgments vii

1 Introduction 3

1.1 Compositional Models . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Relative Permeability Review . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Factors Affecting Relative Permeability . . . . . . . . . . . . . 8

1.2.2 Data Collection and Utilization . . . . . . . . . . . . . . . . . 11

1.3 Study Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . 13

2 Compositional Dependence of Relative Permeability 15

2.1 Capturing Miscibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Numerical Implementation of Coats Model . . . . . . . . . . . . . . . 18

2.2.1 Surface Tension Calculation . . . . . . . . . . . . . . . . . . . 18

2.2.2 Computing Interpolation Parameter . . . . . . . . . . . . . . . 19

2.2.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Compositional Consistency 39

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3.1 Phase Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Compositionally Consistent Methods . . . . . . . . . . . . . . . . . . 42

3.3 Numerical Implementation of Gibbs Based Model . . . . . . . . . . . 46

3.4 Compositionally Consistent Example . . . . . . . . . . . . . . . . . . 48

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Gibbs Free Energy Based Approach 55

4.1 Model Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.1 Miscibility within the Phase Envelope . . . . . . . . . . . . . . 57

4.1.2 Miscibility in the Super Critical Region . . . . . . . . . . . . . 61

4.1.3 Compositional Consistency . . . . . . . . . . . . . . . . . . . . 63

4.2 Phase Misidentification . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Simulation Results 79

5.1 3-Component Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 4-Component Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Hysteresis 99

6.1 Hysteresis Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Compositional Effects on Trapping . . . . . . . . . . . . . . . . . . . 103

6.3 Hysteresis Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Conclusions 113

A Miscible Displacements 117

B Gibbs Free Energy 121

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List of Tables

2.1 Initial conditions and compositions of Metcalfe and Yarborough (1979)

System 1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Relative permeability parameters used for example 3.4 . . . . . . . . 48

4.1 Initial and well control conditions for a 3-component system exhibiting

phase flip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1 Injection schedule for WAG example . . . . . . . . . . . . . . . . . . 109

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List of Figures

1.1 Relative permeability measurments by Wyckoff and Botset (1936). . . 9

1.2 An example of an immiscible (left) and miscible (right) relative per-

meability curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 3-component system (C1-NC4-C10) initialized at 100 bars/400 K (left)

and 200 bars/400◦K (right) with the color bar representing IFT in

[dynes/cm]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 3-component system (C1-NC4-C10) at 150 bars/400 K showing the

dimensionless interpolation parameter Fk value calculated using a ref-

erence value of 6.8 dynes/cm (left) and 2.2 dynes/cm (right). . . . . . 21

2.3 Fk vs. IFT for different reference surface tension values. . . . . . . . . 22

2.4 Plot of Fk vs. IFT for different values of the exponent N. . . . . . . . 22

2.5 Interpolating between immiscible curves and the expected miscible

curves: Using actual expected miscible curves (top), scaling residual

saturation on miscible curves (bottom-left) and scaling residual satu-

rations on both miscible and immiscible curves (bottom-right). . . . . 24

2.6 Different approaches to scaling the end-points of the miscible curves:

No scaling (top), scale with Fk between limits (bottom-left) and full

scaling to the immiscible end-points (bottom-right). . . . . . . . . . . 26

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2.7 Relative permeability curves used were generated using the Corey cor-

relation with Sgr = 0.1, Sor = 0.2, kroep = 0.5, krgep = 0.75 and an

exponent of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Ternary representation of the displacement in scenario 1 showing the

difference between the two cases (left) and the interpolation parameter

Fk values for each grid block - represented by points - for the corrected

case (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.9 Gas saturation distribution at the end of the simulation for the two

cases (left) and the corresponding interpolation paramter Fk values

(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 Cumulative Newton iterations for the two cases showing significant

improvement when applying the correction. . . . . . . . . . . . . . . . 30

2.11 The gas saturation Sg (top-left), the gas relative permeability krg (top-

right) and the cumulative Newton iterations (bottom) for the two cases

using different interpolation methods from Figure 2.5. . . . . . . . . . 31

2.12 Ternary representation of the displacement in scenario 1 showing the

difference between the two cases (left) and the interpolation parameter

Fk values for each grid block - represented by points - for the corrected

case (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.13 Gas saturation distribution at the end of the simulator for the two cases

(left) and the corresponding interpolation paramter Fk values (right). 32

2.14 Cumulative Newton iterations for the two cases (left) and the gas rel-

ative permeability (right). . . . . . . . . . . . . . . . . . . . . . . . . 33

2.15 Cumulative Newton iterations for the two cases (left) and the gas rel-

ative permeability (right). . . . . . . . . . . . . . . . . . . . . . . . . 34

2.16 Phase misidentification case as evident from the oscillations observed

in saturation profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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2.17 Ternary diagram showing the compositional path and the gas satura-

tion values (right) of the grid blocks, highlighting the misidentification

on the dew point line of the phase envelope (left). . . . . . . . . . . . 35

2.18 Ternary representation of the displacement (left) and gas saturation

(right) for scenario 3 with and without corrections. . . . . . . . . . . 36

2.19 Ternary representation of the displacement in scenario 2 compared to

the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Ternary plots of the 3-component system C1−NC4−C10 at a pressure

of 138 bars and 411 K. Shown is gas relative permeability (left), oil

relative permeability (right) and the gas-oil relative permeability in the

single phase region (center). Relative permeabilities generated using a

corey correlation with residual saturation of 0.1, exponents of 2, gas

end-point of 0.8 and oil end-point of 0.5 . . . . . . . . . . . . . . . . 41

3.2 Extereme case of phase misidentification seen on the saturation profile

where an oscillatory behavior is observed (left) with the ternary plot

showing the grid blocks color coded based on gas saturation (right) . 42

3.3 3-component system (C1-C4-C10) at 130.8 bars and 411K showing

normalized gibbs energy of the gas phase g∗g (left) and of the oil phase

g∗o (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 3-component system (C1-C4-C10) at 3 bars and 290 K showing nor-

malized gibbs energy of the gas phase g∗g (left) and of the oil phase g∗o

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 3-component system C1-NC4-C10 at 130.8 bars and 410.95 K showing

compositional path through critical point from Yuan’s example. . . . 49

3.6 Relative permeability values of solution without (left) and with (right)

ensuring compositional consistency using Yuans Gibbs free energy model. 50

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3.7 Gas saturation after 25 days with and without using Yuans Gibbs free

energy model to ensure compositional consistency. . . . . . . . . . . . 50

3.8 Relative permeability values of corrected case (left), and comparison of

saturation fronts (right) using the initial composition as the reference

state for the oil phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.9 ROWS: Top: Normal run without any modifications to relative per-

meabilities Middle: Yuans model with max and min values as ref.

points (Fig. 3.3) Bottom: Yuans model with initial/injection com-

positions as ref. points (Fig. 3.8) COLUMNS: Left: kro and krg in

single phase region Middle: krg Right: kro . . . . . . . . . . . . . . 52

3.10 Relative permeability values of corrected case (left), and comparison of

saturation fronts (right) using the initial composition as the reference

state for the oil phase. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.11 kro and krg in single phase region (left), krg (center) and kro (right)

using reference points close to the critical point. . . . . . . . . . . . . 54

4.1 Normalized Gibbs free energy of the oil (left) and gas (right) phase in

the compositional space for the three component system at 100 bars

and 450 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Interpolation parameter Fk using Coats (1980) IFT approach (left) and

propose Gibbs free energy approach (right). . . . . . . . . . . . . . . 59

4.3 Interpolation parameter Fk using Coats IFT approach (blue) and pro-

posed Gibbs free energy approach (red) showing similar behavior. . . 59

4.4 Sensitivity of interpolation parameter Fk to exponent in proposed Gibbs

free energy approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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4.5 C1-NC4-C10 at 100 bars and 450 K with super critical region (above

extension of critical tie line) is highlighted (left) with Fk parameter

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Phase IDs (left) and Gibbs free energy (right) in the compositional

space for the three component system at 100 bars and 450 K. . . . . 63

4.7 kro and krg in single phase region using an exponent of 20 (top-left)

and 10 (top-right) with corresponding Fk interpolation parameter in

all the compositional space using an exponent of 5 for the two phase

region (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.8 kro and krg in single phase region using critical temperature (left) and

using Gibbs free energy (right) as a criteria for single phase identifica-

tion showing the elimination of the discontinuity when using Gibbs. . 66

4.9 Gas saturation profile (left) and compositional path on ternary diagram

color coded with gas saturation (right) for system in Table 4.1 . . . . 68

4.10 Gas saturation profile (left) and compositional path on ternary dia-

gram color coded with gas saturation (right) for system in Table 4.1

highlighting two-phase misidentification . . . . . . . . . . . . . . . . . 69

4.11 Overall, oil and gas phase compositions for cell #72 when the timestep

is at 18 days in different Newton iterations without using the Gibbs

free energy check starting from Newton iteration #1 in the top-left

corner and increasing to the right. . . . . . . . . . . . . . . . . . . . . 70

4.12 Overall, oil and gas phase compositions for cell #72 when the timestep

is at 18 days in different Newton iterations using the Gibbs free energy

check starting from Newton iteration #1 in the top-left corner and

increasing to the right. . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.13 Converged solution for block #87 at final time step (50 days) with

compositions misidentified - using the Gibbs check corrects this . . . 72

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4.14 Gas saturation (left) and Newton iterations (right) for the normal case

without using the Gibbs free energy check and the corrected case that

does . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.15 Gas saturation profile (left) and compositional path on ternary dia-

gram color coded with gas saturation (right) for system in Table 4.1

highlighting single-phase misidentification. . . . . . . . . . . . . . . . 73

4.16 Gas saturation (left) and Newton iterations (right) for the normal case

vs. the corrected case. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.17 Gas saturation (left) and Newton iterations (right) for the different cases 75

4.18 krg (left) and kro (right) with magnified images on the bottom to high-

light the differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.19 Interpolation parameter Fk at the end of the simulation, a smoother

transition exists between the single and two phase regions in the new

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1 Log(kx) map of 20th layer of SPE10 - block size: 6.1×3.05×0.61m . . 80

5.2 Relative permeability curves generated using the Corey correlation

with residual saturations of 0.2, exponents of 2, gas and oil end-points

of 0.75 and 0.5 respectively. . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Cumulative Newton iterations for the C1−NC4−C10 example showing

significant improvement in the corrected cases. . . . . . . . . . . . . . 81

5.4 Saturation distribution at the end of the simulation of different cases

for the C1 −NC4 − C10 example. . . . . . . . . . . . . . . . . . . . . 82

5.5 Gas production rate for the C1 − NC4 − C10 example showing earlier

gas breakthrough for corrected cases. . . . . . . . . . . . . . . . . . . 83

5.6 Fk map at the end of the simulation of different cases for the C1 −

NC4 − C10 example. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xviii

5.7 Compositional path of GIBBS (bottom-left), and GIBBS-ALL (bottom-

right) cases showing Fk parameter values. . . . . . . . . . . . . . . . . 84

5.8 Gas relative permeability at the end of the simulation for the GIBBS

(left) and GIBBS-ALL (right) cases for the first 3-component example 85

5.9 Newton iteration comparison for the GIBBS cases with and without

the Gibbs check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.10 Gas relative permeability at the end of the simulation for the GIBBS-

ALL case (left) and the case that assigns a fixed Fk value of zero for

any cell above the critical tie-line extension (right). . . . . . . . . . . 86

5.11 Log(kx) map of upscaled 7th layer of SPE10 - block size: 12.2 × 6.1 ×

1.2m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.12 Newton iterations comparison for example 5.1 . . . . . . . . . . . . . 88

5.13 Gas saturation distribution at the end of the simulation after 10,000

days for the different cases with the corresponding Fk map showing

most of the correction taking place at the advancing edge of the front 89

5.14 Gas production rates (top) show different breakthrough times with dif-

ferent correction approaches used, Ternary representation of the com-

positional path for the GIBBS-ALL case (bottom) showing Fk values

with a phase envelope constructed at a pressure of 117 bars and 400 K 90

5.15 Cumulative Newton iterations compare the performance of the different

methods with and without the Gibbs check. . . . . . . . . . . . . . . 91

5.16 Log(kx) map of upscaled 1st layer of SPE10 - block size: 12.2 × 6.1 ×

1.2m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.17 Gibbs free energy of gas (top-left) and oil (top-right) and the ratio of

oil to gas (bottom) in the two phase region of the quaternary system

at a pressure of 110 bars and 344 K . . . . . . . . . . . . . . . . . . 93

5.18 Newton iterations comparison for example 5.2 . . . . . . . . . . . . . 94

xix


days for the different cases with the corresponding Fk map . . . . . . 95

5.20 Gas production rates (left) and quaternary representation of the com-

positional path for the GIBBS-ALL case showing Fk values with a

phase envelope constructed at a pressure of 110 bars and 344 K . . . 96

5.21 Cumulative Newton iterations compare the performance of the different

methods with and without the Gibbs check for example 5.2 . . . . . . 97


days for the GIBBS-ALL case without (left) and with (right) the Gibbs

check showing a smoother transition in the region Nx: 5-10, Ny:20-40 98

6.1 Example of non-wetting phase imbibition curve calculated using Lands

trapping model - adapted from Spiteri et al. (2008) . . . . . . . . . . 100

6.2 Example of two different DDI trajectories on a ternary diagram -

adapted from Larsen and Skauge (1998) . . . . . . . . . . . . . . . . 101

6.3 Example of a WAG hysteresis cycle involving primary drainage-imbibition-

secondary drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Relative permeability curves used for example 6.3 . . . . . . . . . . . 105

6.5 Saturation distribution for the three cases at the end of the simulation 106

6.6 Snapshot of Fk (left) and saturation (right) after 4.1 days . . . . . . . 106

6.7 Relative permeability values for block #30 throughout the simulation

showing the effect of incorporating miscibility . . . . . . . . . . . . . 107

6.8 Cumulatve Newton iteration for each case in example 6.3 . . . . . . . 108

6.9 Cumulatve Newton iteration for each case in example 6.3 . . . . . . . 109

6.10 Gas saturation after water injection at 550 days for each case in exam-

ple 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xx

6.11 Watersaturation after second drainage cycle at 800 days for each case

in example 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.12 Gas (left) and oil (right) production rates for example 6.3 . . . . . . . 112

A.1 Schematic representation of vaporizing gas drive (left) - adapted from

(Metcalfe and Yarborough, 1979), and corresponding ternary represen-

tation (right) - adapted from (Whitson and Brulé, 2000) . . . . . . . 118

A.2 Schematic representation of condensing gas drive (left) - adapted from

(Metcalfe and Yarborough, 1979), and corresponding ternary represen-

tation (right) - adapted from (Whitson and Brulé, 2000) . . . . . . . 119

xxi

Nomenclature

α Rock dependent constant in Fevang and Whitson (1996) model [-]

µ Viscosity [cp]

φ Porosity [-]

ρ Density [kg/m3]

σ Surface tension [dynes/cm]

~u superficial velocity [m/d]

ξ Parachor weighted molar density [(dyne/cm)1/4]

A Cross-sectional area [m2]

a Degree of oilness [-]

b Degree of gasness [-]

bm Molar density [gram-Mole/cc]

C Volumetric Concentration [-]

Ct Lands trapping coefficient [-]

f Fugacity [bars]

Fk Interpolation parameter [-]

g Gravitational acceleration [m/s2]

g∗ Normalized Gibbs free energy [-]

K Dispersion Tensor [-]

k Permeability [darcies]

N Input exponent for miscibility models [-]

1

2

Nc Capillary Number [-]

P Component Parachor - empirical constant [(dyne/cm)1/4(m3/mol)]

p Pressure [bars]

Pr Correlation parameters (e.g., end-points, residual saturations) [-]

q Volumetric Flow Rate [m3/s]

S Saturation [-]

T Temperature [K]

t Time [s]

v Phase fraction [-]

x, y Component phase molar fractions [-]

Z Depth [m]

z Overall mole fraction [-]

Superscripts

∗ Normalized

c Compositionally consistent

o End point

Subscripts

0 Reference condition

c Component

h Hydrocarbon phase

L Liquid

p Phase

q Source/Sink

r Relative

rhw Hydrocarbon relative to water

V Vapor

w Water

Chapter 1

Introduction

Gas injection processes are among the most widely used of EOR processes (Lake,

1989). When a gas mixture is injected into a porous medium containing oil, a “fas-

cinating set of interactions” begins. Components in the gas dissolve in the oil, and

components in the oil transfer to the gas as local chemical equilibrium is established

(Orr, 2007). These interactions give rise to interesting displacement processes that

can lead to miscible displacements resulting in 100% displacement efficiency of the

swept regions. Miscibility is defined as the condition when two fluids are mixed, and

the resulting mixture is a single phase. Achieving miscibility eliminates the interface

between the phases, which in turn eliminates the capillary forces that cause oil trap-

ping. There are various techniques for attaining miscible displacements which are

outlined in Appendix A.

Although gas injection has clear advantages when it comes to displacement effi-

ciency, the sweep efficiency can be impacted due to the segregation of gas to the top

due to gravity. In order to improve the sweep of gas injection processes, the popu-

lar water-alternating-gas (WAG) approach is widely used to stabilize the front, and

control the mobility of the displacement. Christensen et al. (2001) reviewed 59 WAG

projects with a majority of cases resulting in a significant incremental oil recovery of

3

4 CHAPTER 1. INTRODUCTION

about 5-10%. In addition to improvements in recovery, there is high interest in using

CO2 flooding in these gas injection processes, which is already considered one of the

most promising gas-injection processes in the US (Klins, 1984; Holm, 1976).

The benefits of gas injection are already evident, and therefore understanding

and modeling the behavior of such systems accurately is essential to making better

decisions with regards to these expensive EOR projects. Using compositional models

is one important step in capturing the complex phase behavior of such processes.

Using the appropriate relative permeability curves will also be an integral part of the

modeling process, which is the focus of this study.

1.1 Compositional Models

More representative models of gas injection processes can be obtained by using compo-

sitional simulators. The advantage of using compositional simulations is the ability to

capture the transport of individual components, as well as describe how components

partition across multiple phases. This advantage does come at the price of including

the mass conservation for each component, and the phase-behavior equilibrium rela-

tions in the mathematical formulation. These additions make the simulations much

more computationally intensive. The reservoir simulator used in this study is the Au-

tomatic Differentiation General Purpose Research Simulator (AD-GPRS), developed

by the SUPRI-B Reservoir Simulation Group (Jiang, 2007; Zhou and Voskov, 2012;

Younis and Tchelepi, 2012).

The AD-GPRS reservoir modeling platform is based on the automatic differentia-

tion library (ADETL) originally developed by Younis and Aziz (2007); Younis (2011),

and extended by Zhou (2009). AD-GPRS uses the generic concept of nonlinear formu-

lations to give flexibility in implementing different nonlinear models under the same

1.1. COMPOSITIONAL MODELS 5

numerical framework (Voskov and Tchelepi, 2012). The two main formulations cur-

rently implemented are the Natural (Coats, 1980) and Molar Formulations (Fussell

and Fussell, 1979; Young and Stephenson, 1983; Chien et al., 1985). The general

nature of the implementation allows for the use of any physical model available in

AD-GPRS (dead-oil, black-oil and compositional). The compositional simulations in

this study were performed using AD-GPRS with the natural formulation. For the

nonlinear solver, the fully implicit formulation (FIM) was used with applyard chop-

ping. The Peng-Robinson equation of state (EOS) was used in all runs. In addition to

solving the standard mass conservation equation and local constraints (Equation 1.1-

1.3), the thermodynamic equilibrium given by the equal fugacity constraint (Equation

1.4) is solved.

∂

∂t

(φ∑p

xcpρpSp

)−∇ ·

∑p

xcpρp~up +∑p

xcpρpqp = 0 (1.1)

nc∑i=1

xij − 1 = 0 j = 1, ..., np (1.2)

np∑j=1

Sj − 1 = 0 (1.3)

fcj(p, T, xcj)− fck(p, T, xck) = 0, ∀j 6= k, i = 1, ..., nc (1.4)

c = 1, ..., nc, p, j, k = 1, ..., np

The fugacity constraint entails solving the isothermal compositional phase split

calculation. The phase split calculation is a major difficulty in simulating composi-

tional flow since an efficient/robust way of predicting the PVT behavior of complex

mixtures comes at a high cost. The equilibrium phase behavior in AD-GPRS is

usually done in two parts:


1. Phase Stability Test (Michelsen, 1982a) - A stability test is needed to detect if

additional phases can appear for grid blocks with number of phases less than

the maximum possible phases.

2. Flash Calculation (Michelsen, 1982b) - If the phase stability test indicates that

the phase state of the cell changed from single to two phases, the following

system is solved (in addition to the fugacity constraint in Equation 1.4) in

order to obtain component and phase fractions split:

zi −np∑j=1

vjxij = 0, i ∈ [1, ..., nc], (1.5)

np∑j=1

vj − 1 = 0, (1.6)

nc∑i=1

(xij − xik) = 0, ∀j 6= k ∈ [1, ..., np] (1.7)

The flash calculation is solved by either using a successive substitution iteration

(SSI) algorithm or a Newton-Raphson algorithm. Again these calculations are highly

nonlinear and become much more complicated at near-critical conditions, which are

the usual conditions encountered in the near-miscible displacements of interest. Also,

since these are component-based methods, the computational cost increases with the

number of components involved. Despite the complexity of compositional models,

they become a necessary tool for capturing complex phase behavior in processes such

as gas injection. Next, we address how the complex phase behavior can affect the

relative permeability curves.

1.2. RELATIVE PERMEABILITY REVIEW 7

1.2 Relative Permeability Review

The flow of fluids is dictated by the general mass conservation law shown in Equation

1.1. The net flux term that includes the velocity vector is:

Fc = ∇ ·∑p

xcpρp~up (1.8)

The velocity ~up represents the superficial velocity vector, also known as the Darcy

velocity, that describes the speed at which the fluids move through the medium assum-

ing a homogeneous cross-section (disregards the path that the fluid actually travels).

It can be related to the actual velocity of the fluid, also known as the interstitial ve-

locity ’v’ through the porosity of the medium with the following relationship: vφ = u.

Here we make use of the constitutive relationship that relates the superficial velocity

to the concept of permeability introduced by Darcy (1856):

u = −kµ· (∇p− ρg∇Z) (1.9)

This relationship between the superficial velocity u, permeability k, fluid viscosity

µ, and the pressure drop ∇p, is the main constitutive relationship that allows us to

model single-phase fluid flow in porous media. This equation is not an analytical

solution of the Navier-Stokes equations, it is only a macroscopic representation of

such solutions where k and p represent statistical averages over a great number of

pores referred to as the representative elementary volume (REV). The scale being

modeled is sometimes referred to as the Darcy scale. Note that the permeability

has units of area [length2], which is a measure of the average cross-sectional area

that the fluids flow through. Hence, the higher the permeability the easier it is for

fluid to flow. Permeability is usually reported in darcy units which is approximately

1× 10−12m2. A similar relationship is needed to model multi-phase fluid flow which


is obtained from Muskat and Meres (1936) extension that introduces the concept of

relative permeability.

ui = −kkriµi· (∇pi − ρig∇Z) (1.10)

Here ’i’ is used to distinguish between the different phases. The only additional

term is the relative permeability kri which refers to the ratio of the phase permeability

ki (accounts for the existence of multiple phases in the flow paths) to the absolute

permeability k (a property of the medium). The relative permeability ranges between

zero and one. A value of one occurs in a single-phase system, and a value of zero

occurs when the fluid/phase saturation is lower than a specific threshold, referred to

as the residual saturation, at which the phase no longer moves as it becomes isolated

and disconnected.

1.2.1 Factors Affecting Relative Permeability

The relative permeability of a phase is usually reported as a function of saturation - a

macroscopic property that represents the fraction of the pore volume occupied by the

fluid. The saturation value is assumed to be applicable to all the interconnected pores

in the REV, and therefore determines the local permeabilities of the different phases.

However, saturation alone does not capture the complex paths these fluids travel. How

the fluids flow is dictated by the whole system (initial/boundary conditions, rock/fluid

properties, etc.), and therefore in reality there are many factors that implicitly affect

the shape of the relative permeability curves. The shape of these curves are well

known, Figure 1.1 shows one of the earliest measurements done by Wyckoff and

Botset (1936).


Figure 1.1: Relative permeability measurments by Wyckoff and Botset (1936).

(Honarpour et al., 1986) provides an extensive review on different factors that

affect the relative permeability curves. Below are only some important factors:

Rock Heterogeneity

Fluid Distribution (Wettability)

Interfacial Tension (IFT)

Saturation History

We focus on the effects of IFT in this study with some discussion on saturation

history in Chapter 6. IFT relates to the capillary forces that affect the trapping of

fluids. Two phases with no IFT between them will essentially flow as a single phase

without any trapping. Hence, the general expectation is that relative permeability

curves approach the 45◦ diagonals as IFT approaches zero (near the critical point),

with zero residual saturations and end-points of one (Figure 1.2).


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sp

k r

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sp

k r

Figure 1.2: An example of an immiscible (left) and miscible (right) relativepermeability curve.

Fatemi (2012) discusses the effects of IFT on relative permeability pointing out the

limitation of Darcy’s law in reflecting the physics of multi-phase flow at low IFT due

to the assumption of strong wetting and stable interfaces (Hubbert, 1956). Hartman

and Cullick (1994) argue that despite this limitation apparent relative permeability

curves that take low IFT into account can predict the macro-scale effects of flow

in porous media. Many investigators conducted experiments to understand how the

relative permeability curves change in near-miscibile conditions as they approach

the 45◦ diagonals. As pointed out by Blom (1999) and Al-Wahaibi et al. (2006),

there is no consensus on how near-miscibility changes relative permeability curves,

or which parameters control this change. Some investigators report that the relative

permeability of the non-wetting phase is more sensitive to IFT; others report the

wetting phase is more sensitive, and some report no changes at all with decreasing

IFT. Experiments on three phase flow performed by Cinar and Orr (2005) on an

analog system show that as IFT is reduced between the oil and gas phases, the oil

and gas relative permeabilities begin to change. A rapid shift to the 45◦ diagonals is

observed for IFTs below 1 dyne/cm. The water (wetting-phase) relative permeability

does not change in this case.


1.2.2 Data Collection and Utilization

The main source of relative permeability data is lab experiments conducted on cores.

These are multi-phase displacement experiments that aim to capture either the im-

bibition, or drainage, curves. Benson et al. (2012) discusses the assumptions behind

using core experiments, pointing out that one of main challenges is accounting for the

heterogeneous nature of reservoir rocks (Honarpour et al., 2003). It is recommended

to take measurements at reservoir conditions with representative samples to provide

reliable estimates; this is because the nature of flow is determined by the interactions

of fluids with one another and the porous medium at those conditions. The two

main categories for relative permeability experiments are steady, and unsteady state

methods.

The aim of steady state methods is to achieve a steady-state saturation of the

phases within the core; this means having a constant saturation of each phase through-

out the core. A constant saturation corresponds to a constant capillary pressure,

which means that the pressure gradient in each phase is the same (∆p/L). One can

use Darcy’s law (Equation 1.11) to compute the relative permeability of each phase

once a constant pressure gradient is achieved. The experiments involve injecting both

phases simulatneously until the output flow rates for each phase equal the input flow

rates, which can be very time-consuming. Repeating the experiments at different

rates provides values for different saturations.

kri = −qiµiL

Ak∆p(1.11)

The aim of unsteady state methods is to make use of mathematical models that

describe the experimental observations of one displacement process, as opposed to the

multiple rate combinations in the steady state methods. Usually this one displacement

involves a core fully saturated with one phase, while the other phase is injected at a


constant rate or pressure. Measurements are taken of injection/effluent volumes and

pressure drop. The common mathematical models used are based on the Buckley-

Leverett flow theory such as the Welge (Welge, 1952) and JBN (Johnson et al., 1959)

methods. These methods are much faster than steady state methods, but have less

flexibility in obtaining the desired saturations. Since unsteady state methods are

based on one displacement, only saturations after breakthrough are obtained.

Both methods are constrained by the assumptions behind the Buckley-Leverett

equation: fluids are assumed to be incompressible and immiscible, flow is perfectly

dispersed, and both gravity and capillary effects are negligible (Bennion and Thomas,

1991). Also, both exhibit the capillary end effect problem that results from the dis-

continuity of capillary pressure at the outlet. This capillary pressure discontinuity

makes it more difficult for the wetting phase to be discharged causing it to accumu-

late near the outlet of the core. The accumulation of the wetting phase creates a

saturation gradient within the core that violates the constant saturation assumption.

The capillary end effect can lead to misleading results.

Most experiments are done for two-phases only. Three phase experiments are

much more complex/time consuming, and therefore rarely done. For that reason

most reservoir simulators make use of two-phase relative permeability data even if

the simulation model includes three or more phases. We will not discuss three phase

models here, however a review of different methods is presented by Delshad and Pope

(1989); Juanes (2003); Beygi et al. (2015). The relative permeability data fed into

the simulator is either in the form of a table or as a parametrized model such as the

modified Brooks-Corey (Equation 1.12).

kri = kori

(Si − Sir

1− Sir − Sjr

)ni(1.12)

1.3. STUDY OBJECTIVE AND OUTLINE 13

The advantage of using the parameterized models is avoiding the need to inter-

polate between data points, and are easier to modify (as is discussed in Chapter 2).

The advantage of using tables is the additional flexibility since lab measurements may

not be smooth functions as seen in the parameterized models. The tables usually in-

clude a column for saturation with the corresponding relative permeability values for

each phase. We add a column for the normalized saturations (S∗i ) computed using

Equation 1.13.

S∗i =Si − Sir

1− Sir − Sjr(1.13)

This becomes useful for when residual saturations are modified (as is done in

Chapter 2 when miscibility is approached), the interpolation can be done using the

normalized saturations which will maintain the shape of the inputted curves. This

approach basically scales the curves with respect to residual saturations.

1.3 Study Objective and Outline

The main objective of this study is to investigate different methods that capture

miscibility effects on the relative permeability curves. The investigation involves:

(1) analyzing how the different methods affect simulation results and performance,

(2) address possible limitations of existing methods, (3) propose new approaches to

reflecting miscibility effects on relative permeability curves. Also, since WAG is a

common strategy used in gas injection projects, we consider the effects of miscibility

on the trapping involved in hysteresis models. This clearly is not straight forward

given the empirical nature of the relative permeability curves and the complex be-

havior of such systems. The outline of the thesis is as follows:


Chapter 2 presents some of the current methods used to modify relative per-

meability curves to reflect near-miscible conditions. This usually involves in-

terpolating between two sets of curves (immiscible and miscible), where minor

differences in interpolating approaches is presented here. Simple examples are

presented to show the basic effects we can expect from incorporating these

modifications to the relative permeability curves.

Chapter 3 presents the idea of compositional consistency in the relative perme-

ability curves outlying how existing approaches are used. A simple example is

presented to show the basic effects of including compositional dependence in

the relative permeability curves.

Chapter 4 gives an overview of what we expect from relative permeability models

that attempt to capture miscibility and compositional dependence. We present a

possible new approach to using a thermodynamical property (Gibbs free energy)

as an indicator to capture miscibility and compositional dependence in the

relative permeability curves.

Chapter 5 presents examples on 2D heterogeneous grids with larger time steps

using a fully implicit formulation. We assess the impact of incorporating such

modifications to the relative permeability curves on simulation results and per-

formance.

Chapter 6 touches on the importance of taking hysteresis into account in relative

permeability curves, and discusses possible ways of incorporating miscibility and

the impact it could have.

Chapter 7 provides the final conclusions and recommendations of this thesis, as

well as future research opportunities.

Chapter 2

Compositional Dependence of

Relative Permeability

This chapter presents key aspects required to capture compositional effects on relative

permeability curves in terms of miscibility. It also outlines how existing approaches

can be implemented in a numerical simulator, and their effect on simulation models.

2.1 Capturing Miscibility

The key aspect in miscible displacements is that the components flow in a single phase

(e.g., transport of salts in water). The phenomenon is described by the convection-

diffusion equation. If we consider two miscible components, and assume that the

system is incompressible, and that there is no volume change due to the mixing of

components, then the component balance is (Aziz and Settari, 1979):

−∇ · ~u = ∇ ·[k

µ(∇p− ρg∇Z)

]= q (2.1)

∇ · (φK∇C)−∇ · ~uC = φ∂C∂t

+ Cqq (2.2)

15

16 CHAPTER 2. COMPOSITIONAL DEPENDENCE OF REL. PERMS.

Here K is the dispersion tensor, C is the volumetric concentration and Cq is

the source/sink concentration. The mixture viscosity and density are calculated us-

ing mixing rules as a function of the component properties and concentration. One

would have to solve Equations 2.1-2.2 in order to model fully miscible displacements.

However, these equations do not always apply since the processes we are interested

in are not always fully miscible across the domain. This is the reason why there is

high interest in developing methods that can simulate miscible displacement using

conventional “immiscible” simulators, where miscibility can be identified depending

on component concentrations, pressure and temperature. Lantz (1970) established

the basis for the idea of finding an analogy between miscible-immiscible flow so that

capturing miscible processes using immiscible equations is possible. The main ap-

proach suggested to capture miscibility in reservoir simulators is to alter the fluid

properties in order to mimic a miscible displacement. There are different ways to

achieve this; some common approaches implemented in compositional simulators to

achieve this for relative permeability functions are outlined below:

Coats Model

The treatment given by Coats (1980) aims to modify the gas-oil relative permeabil-

ity curves and capillary pressure as IFT is reduced. This modification is based on

expected behavior that the relative permeability curves approach straight lines, and

residual phase saturations decrease to zero as IFT approaches zero (Figure 1.2). The

model uses the “reference” surface tension of the system and a read-in exponent N

that controls how fast the relative permeability curves changes with IFT. The inter-

polation parameter (Fk) is calculated as follows:

Fk =

(σ

σ0

)N(2.3)

2.1. CAPTURING MISCIBILITY 17

This interpolation parameter is a common way of modifying properties across the

whole spectrum of miscible and immiscible displacements in many methods; it usu-

ally varies from zero representing fully miscible, to one representing fully immiscible

displacements. I will elaborate on this model in section 2.2, as it was the base method

implemented in AD-GPRS to reflect miscibility on the relative permeability curves.

Whitson & Fevang Model

This model is similar to Coats (1980) in using an interpolation parameter to in-

terpolate between “immiscible” and “miscible” relative permeability curves. It was

designed for near wellbore flows with possible retrograde gas condensation. Fevang

and Whitson (1996) proposed an interpolation parameter dependent on the capillary

number (Nc), since velocities play an important role in gas condensate wells. There

are many models that aim to incorporate the capillary number in relative permeability

functions as outlined by Blom and Hagoort (1998), who concludes that the weighting

function proposed by Whitson and Fevang is the most convenient. The interpolation

parameter (Fk), again varying from zero (miscible) to one (immiscible), is computed

with equation 2.4. Note that this model does not use a reference/threshold value,

with α being dependent on rock properties only.

Fk =1

1 + (α ·Nc)n(2.4)

Other models follow the same approach with different functions to calculate the

interpolation parameter. The next section will elaborate on the general concerns in-

volved in the implementation of one of these methods, and how they affect simulation

results.


2.2 Numerical Implementation of Coats Model

This section presents how Coats (1980) model was implemented in AD-GPRS. The

general idea of Coats model is to obtain the surface tension value for each grid block,

compare that to the “reference” surface tension to determine how close each grid

block is to miscible flow, and then compute the relative permeabilities using the

corrected relative permeability curves. The corrected curves tend more towards the

45◦ diagonals as IFT approaches zero. The details of each step is outlined below:

2.2.1 Surface Tension Calculation

The first step is to obtain surface tension values for different mixtures at a given

pressure and temperature; this can be done using the Macleod Sugdeon correlation

for each grid block which is in the following form:

σ =

[Nc∑i=1

Pi × (bmL xi − bmV yi)

]4(2.5)

Reid et al. (1987) states that for near mixture critical points the Macleod-Sugden

correlation should be used, because the form of the equation necessarily gives the

correct limit that σ goes to zero at the critical point. Note that the surface tension

for grid blocks with only one existing phase is set to zero. Shojaei et al. (2012) states

that some attempts to correlate surface tension calculations with measured data using

the Parachor method and more complex models have failed in predicting the IFT with

reasonable accuracy. This needs to be investigated in more depth to understand why

there are some limitations. Also, a new mechanistic Parachor model was introduced

by Ayirala and Rao (2006) that might give more accurate estimates of surface tension.

Equation 2.5 is considered to be sufficient in obtaining IFT estimates in this study.

2.2. NUMERICAL IMPLEMENTATION OF COATS MODEL 19

2.2.2 Computing Interpolation Parameter

The next step is to calculate the interpolation parameter (Fk) that is used to de-

termine how close the current condition of each grid block is to miscible flow. This

is done by comparing the current IFT value of the block to the “reference” surface

tension value, with a maximum value of one to indicate immiscible flow:

Fk = min

[1,

(σ

σ0

)N](2.6)

The reference surface tension σ0 is the maximum threshold before changes are

observed; the exponent N controls how fast the relative permeability curves tend

to the miscible relative permeability curves as IFT approaches zero. These input

parameters play a major role in the behavior of the development of miscibility and

so care must be taken in properly defining these parameters in each simulation case.

The read-in exponent N is usually between 0.1-0.25 according to the literature. The

reference surface tension was originally referred to by Coats (1980) as the “initial

tension” (corresponds to the conditions at which the read-in capillary pressure curve

was obtained), this value should represent the point at which “miscible” behavior

starts to be observed. This can be fed to the simulator directly from experimental

data, otherwise the Macleod Sugdeon correlation (Eq 2.5) can be used to estimate the

IFT of the mixture at a reference condition where miscibility is first expected. Other

options proposed include using the surface tension value at the saturation (bubble

point) pressure, or the maximum possible surface tension value encountered in the

simulation.


In order to illustrate the effect of this parameter, simulation cases containing a

range of mixture combinations that span the whole compositional space are initialized

using AD-GPRS; this allows a flash calculation to be run on all the grid blocks

so that IFT can be estimated. An example is shown below in figure 2.1 where it

shows the interfacial tension at two different conditions (e.g., injection and production

pressures). The objective is to show how using the maximum possible surface tension

encountered in a simulation as a reference point can lead to different behavior.

Figure 2.1: 3-component system (C1-NC4-C10) initialized at 100 bars/400 K (left)and 200 bars/400◦K (right) with the color bar representing IFT in [dynes/cm].

Using the maximum value between all possible mixture combinations at 100 bars

we get a value of 6.8 dynes/cm; the maximum at 200 bars is 2.2 dynes/cm. Using

either of these values as the reference surface tension to calculate the interpolation

parameter at intermediate pressures yields a similar trend with the smallest values

near the critical point (Figure 2.2 - the exponent N is set to 0.2 in this case).


Figure 2.2: 3-component system (C1-NC4-C10) at 150 bars/400 K showing thedimensionless interpolation parameter Fk value calculated using a reference value of6.8 dynes/cm (left) and 2.2 dynes/cm (right).

Using a reference surface tension of 6.8 dynes/cm results in an interpolation pa-

rameter value less than one even at the lower part of the phase envelope, where more

immiscible-like displacements are expected. This shows that high reference surface

tension values might lead to unrealistic behavior. Figure 2.3 is a simple plot that

shows the interpolation parameter using different reference surface tension values.

The steep decline seen with a reference surface tension of one dyne/cm might be

more suited to what is reported in the literature, where significant changes in relative

permeabilities are observed after a low surface tension threshold is reached. There-

fore, in the absence of any information with regards when miscibility is developed,

low reference values should be used. In this study values close to one dyne/cm is used

since no clear threshold is identified for the mixtures used.


0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F k

σ [dynes/cm]

σ0 = 1

σ0 = 3

σ0 = 5

σ0 = 10

Figure 2.3: Fk vs. IFT for different reference surface tension values.

The other input parameter used in this model is the exponent N that controls

the speed at which miscibility behavior is approached, once the threshold (reference

surface tension value) is passed. Figure 2.4 shows the trend for a reference surface

tension value of one dyne/cm with different N values between 0.05 and 0.35.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F k

σ [dynes/cm]

N = 0.05N = 0.15N = 0.25N = 0.35

Figure 2.4: Plot of Fk vs. IFT for different values of the exponent N.

According to the literature (Blom, 1999; Al-Wahaibi et al., 2006; Cinar and Orr,

2005) most changes are observed abruptly after a low threshold, and so the reference

surface tension and exponent N should be tuned to reflect exactly that. This will

usually lead to a low reference surface tension value and a high exponent value;

however, in the end the choice will depend on the given system.


2.2.3 Interpolation

Now that the interpolation parameter is computed for each grid block, the basic ex-

pression used to interpolate between the immiscible and miscible relative permeability

curves is:

krp = Fk · kImmrp (Sp) + [1− Fk] · kMisrp (Sp) (2.7)

Here Fk is the interpolation parameter, kImmrp is the immiscible relative permeabil-

ity, and kMisrp is the miscible relative permeability (usually taken to be the expected

45◦ diagonals). As also pointed out by Blom (1999), there are issues when using the

pure straight lines (kMisrp (Sp) = Sp) to represent the “miscible” relative permeability

curves, this is outlined in Figure 2.5. Using the complete diagonals, which are ex-

pected during fully miscible flow, results in fluids flowing at saturations close to zero,

even if full miscibility is not achieved. Since this is unrealistic, this approach should

not be used. Therefore, using the phase saturations as miscible curves, as shown in

Equation 2.8, should be avoided.

krp = Fk · kImmrp (Sp) + [1− Fk] · Sp (2.8)

To resolve this issue, it has been proposed to use surface tension based residual

saturations for the kMisrp curve. This will result in a kMisrp curve which is essentially the

normalized saturation (S∗i ) with the new residual saturations that can be computed

as shown in Equation 2.9 (with the constraint of it being positive and less than one):

S∗i =Sp − Srp · Fk

1−∑np

j=1 Srj · Fk(2.9)


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4

ImmiscibleCorrectedMiscible

0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4

Immiscible Corrected Miscible

Figure 2.5: Interpolating between immiscible curves and the expected misciblecurves: Using actual expected miscible curves (top), scaling residual saturation onmiscible curves (bottom-left) and scaling residual saturations on both miscible andimmiscible curves (bottom-right).

Therefore, the equation can be written as:

krp = Fk · kImmrp (Sp) + [1− Fk] · S∗i (2.10)

This yields more acceptable results, as seen in Figure 2.5, where very low satu-

rations no longer flow. However, we see that the interpolated curves exhibit a kink


at the immiscible residual saturations. This can be avoided by making both resid-

ual saturations a function of surface tension, which will make both curves have the

same residual saturations. There is no physical reason to modify the residual on both

curves, therefore the only reason to do this is to avoid the kink in the curves.

Modifying the residual on the immiscible curves requires basic scaling of the rel-

ative permeabilities with respect to saturation. If the input relative permeability

curves are in the form of a function, such as the Corey curves, we simply multiply the

residual saturations in the equation by the interpolation paramter (Fk). If however,

the input relative permeability curves are in the form of a table, then the objective

would be to change the residual saturations but maintain the shape of the curves. For

this reason we added the extra column that represents the normalized saturation to

be associated with each relative permeability point. During the simulation, instead

of using the block saturations to interpolate and find the relative permeability value,

normalized saturations (calculated using Equation 2.9,) that take into account the

degree of miscibility with the Fk parameter are used. Equation 2.11 shows the final

form of the equation:

krp = Fk · kImmrp (S∗i ) + [1− Fk] · S∗i (2.11)

Another question is what end-point value to use for the miscible curves. Figure 2.6

shows the difference between three different approaches when it comes to end-point

values for the miscible curves (note that the interpolation method used here is where

the residual saturations in both immiscible and miscible curves are scaled). It is

obvious that complete scaling of the miscible end-points to the immiscible end-points

makes little sense, since the relative permeability curves will not converge to the 45◦

diagonals expected at the miscible limit (Figure 2.6). On the other hand not scaling

the end-points at all could slightly over estimate the relative permeability compared


to the option of scaling the end-point with the interpolation parameter between the

two limits. There is no data to support either approach; however, scaling the end-

point between the two limits seems reasonable to avoid a relative permeability of one

when a residual saturation exists. It also follows the approach taken with the scaling

of residual saturations. The difference between both approaches is minor as is shown

in section 2.3.

0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4


0 0.5 10

0.5

1

Sg

Krp

Fk = 0.4

Immiscible Corrected Miscible

Figure 2.6: Different approaches to scaling the end-points of the miscible curves: Noscaling (top), scale with Fk between limits (bottom-left) and full scaling to theimmiscible end-points (bottom-right).

2.3. SIMULATION CASES 27

2.3 Simulation Cases

This section demonstrates in a couple of examples how the Coats correction affects

different simulations results, and how the different interpolation approaches affect

the results. Three examples of immiscible, multi-contact miscible and first-contact

miscible displacements from Metcalfe and Yarborough (1979) are replicated. The

cases are one dimensional, 3-component cases, with pure CO2 injection on one end

and a producer on the opposite end. Note that a controlled pressure setting is used

for the wells with a pressure of 1 bar above and below the initial condition for the

injector and producer respectively. The reason for this is to ensure that the phase

envelope shown on the ternary diagram is representative throughout the simulation,

and therefore captures the desired compositional path. Table 2.1 shows the conditions

of each case:

Table 2.1: Initial conditions and compositions of Metcalfe and Yarborough (1979)System 1 Properties

Scenario Condition CO2 NC4 C10 Pressure [bars] Temperature [K]

1 Immiscible

0 40 60

103

3442 MCM 117

3 FCM 131

The grid used is made up of 1000 grid blocks to obtain a refined solution. Each

grid block is 0.1×10×10 meters with a permeability of 200 md, and a porosity of

20%. Very small time steps are taken to reduce time truncation errors (max of 0.01

days that yield a max CFL number of 0.06). The simulation is run for a total of

80 days using the Peng-Robinson equation of state, using the relative permeability

curves shown in Figure 2.7.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sg

k r

GasOil

Figure 2.7: Relative permeability curves used were generated using the Coreycorrelation with Sgr = 0.1, Sor = 0.2, kroep = 0.5, krgep = 0.75 and an exponent of 2

The base interpolation approach used is similar to those used in commercial sim-

ulators that scales the residual saturation on both the immiscible and miscible curves

(as in the bottom-right image of Figure 2.5) and does not scale the end-points of the

miscible curves (as in the top image of Figure 2.6). The objective is to show the basic

effect of altering the relative permeabilities near miscibile conditions on simulation re-

sults and performance. A comparison between the different interpolation approaches

is discussed if any difference was observed for that specific case.

Scenario 1 - Immiscible Case

We first look at the “immiscible” case given by scenario 1, we expect that this case

will experience the least change from the relative permeability correction. The com-

positional path connects between the injection and initial conditions, which both lie

below the critical tie-line indicating that the displacement is expected to be immis-

cible. The compositional path is shown in figure 2.8, where each point represents a

grid block composition at the end of the simulation. Note that very slight differences


are observed with the correction. The figure on the right shows the value of the inter-

polation parameter Fk for all the grid blocks in the compositional space, as expected

the lower values are in the grid blocks closer to the critical point.

NC4

C10

CO2

NC4

C10

CO2

NormalCorrected

NC4

C10

CO2

0

0.2

0.4

0.6

0.8

1

Figure 2.8: Ternary representation of the displacement in scenario 1 showing thedifference between the two cases (left) and the interpolation parameter Fk values foreach grid block - represented by points - for the corrected case (right)

The slight difference in the simulation results due to the miscibility correction is

clearer in the saturation front in Figure 2.9. A snapshot of the correction parameter

(Fk) at the end of the simulation is also shown, it highlights the region where the

correction is taking place with a minimum value around 0.7 indicating a more immis-

cible like displacement. The fact that there is a correction in this “immiscible” case

either indicates that there is some miscibility developing in this scenario and should

not be considered a fully immiscible case, or that the correction made using a refer-

ence surface tension of one dyne/cm is over estimating the development of miscibility.

Experimental information would allow us to determine which is the case. However,

whichever is the case, the difference in the two is minor and this question will not be

the focus of this study.


0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

NormalCorrected

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

F k

NormalCorrected

Figure 2.9: Gas saturation distribution at the end of the simulation for the twocases (left) and the corresponding interpolation paramter Fk values (right)

Despite the slight difference in simulation results, an improvement in nonlinear

convergence is observed when using the surface tension correction. This improvement

is probably caused due to the fact that the correction reduces the curvature of the

relative permeabilities that will make it easier for the nonlinear solver.

0 10 20 30 40 50 60 70 800

5000

10000

15000

Time [Days]

Cu

m N

ewto

n It

erat

ion

s

NormalCorrected

Figure 2.10: Cumulative Newton iterations for the two cases showing significantimprovement when applying the correction.


No differences were observed when investigating the other end-point scaling op-

tions (Figure 2.6). However, some difference in performance, and even less difference

in simulation results, is observed when choosing whether to scale the residual satura-

tion on both the immiscible and miscible curves, as opposed to scaling the residual on

the miscible curve only (The bottom-right image vs. the bottom-left image in Figure

2.5). The point here is to show that modifying residuals on both curves might help

convergence with minor change in results, although this might not always be the case.

Either way the choice of interpolation approach does not seem important, as long as

the residuals are scaled on the miscible curves to avoid unrealistic flow at saturations

below residual saturations.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

Scale Sr − MiscibleScale Sr − Both

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

k rg


0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

10000

Time [Days]

Cu

m N

ewto

n It

erat

ion

s


Figure 2.11: The gas saturation Sg (top-left), the gas relative permeability krg(top-right) and the cumulative Newton iterations (bottom) for the two cases usingdifferent interpolation methods from Figure 2.5.


Scenario 2 - Multi Contact Miscible Case

Scenario 2 is a MCM displacement since the initial composition is on one side of the

critical tie line extension with the injection composition on the other; this will exhibit

some form of a vaporizing gas drive mechanism. The compositional path enters the

two phase envelope near the critical point and follows the dew point line down to the

injection composition (Figure 2.12). Figure 2.13 shows the saturation front for both

cases which overlay each other despite the Fk value dropping to almost 0.3.

NC4

C10

CO2

NC4

C10

CO2

NormalCorrected

0

0.2

0.4

0.6

0.8

1

Figure 2.12: Ternary representation of the displacement in scenario 1 showing thedifference between the two cases (left) and the interpolation parameter Fk values foreach grid block - represented by points - for the corrected case (right).

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

NormalCorrected

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

F k

NormalCorrected

Figure 2.13: Gas saturation distribution at the end of the simulator for the twocases (left) and the corresponding interpolation paramter Fk values (right).


The MCM displacements exhibit minor changes with the correction. This minor

change can be attributed to the small two-phase region that is encountered in such

displacements, and hence the correction only affects a very small part significantly

(evident in the sharp drop in Fk in Figure 2.13) as opposed to the flatter section in

Figure 2.9.

0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

10000

12000

Time [Days]

Cu

m N

ewto

n It

erat

ion

s

NormalCorrected

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

k rg

NormalCorrected

Figure 2.14: Cumulative Newton iterations for the two cases (left) and the gasrelative permeability (right).

Also, in this case we do not see an improvement in the nonlinear convergence. The

relative permeability of gas is shown in Figure 2.14, that shows the difference between

the two cases with the increase in krg where the correction takes place. The more

significant jump in krg observed, in comparison to the previous immiscible example

(Figure 2.11), might play a role in the nonlinear performance; this becomes evident

when using a different interpolation method that results in a slightly smaller number

of Newton iterations with a smoother krg profile. The interpolation method used

was that shown in the bottom-left image of Figure 2.6, where the end-points of the

miscible curves is scaled between the immiscible and miscible end-points. Notice a

smoother transition at the point where the two-phase region is entered and a drop in

number of Newton iterations is observed.


0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

10000

12000

Time [Days]

Cu

m N

ewto

n It

erat

ion

s

NormalCor−No ScalingCor−Scaling

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

k rg

NormalCor−No ScalingCor−Scaling

Figure 2.15: Cumulative Newton iterations for the two cases (left) and the gasrelative permeability (right).

Even though the jump in krg might play a role in the nonlinear performance,

these simulations exhibit other behavior that affect the simulation performance that

is more concerned with the phase identification aspect. Figure 2.16 shows an extreme

example when phase misidentification can be come an issue in displacements that

are close to the phase envelope boundary. Therefore, this investigation of nonlinear

performance is very basic and more in depth analysis is required. It is also important

to note that this increase in Newton iterations is not observed in MCM cases run

with larger time steps as is shown in chapter 5.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

NormalCorrected

0.55 0.6 0.65 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

NormalCorrected

Figure 2.16: Phase misidentification case as evident from the oscillations observed insaturation profile.


NC4

C10

CO2

0

0.2

0.4

0.6

0.8

1

Figure 2.17: Ternary diagram showing the compositional path and the gassaturation values (right) of the grid blocks, highlighting the misidentification on thedew point line of the phase envelope (left).

Figure 2.17 shows the compositional path on the ternary diagram with the sat-

uration value of each grid block, the zoomed in image shows the misidentification

occurring close to the left boundary of the phase envelope. This highlights the issue

of phase misidentification, and the importance of compositional consistency in the

relative permeability curves that is elaborated on in Chapter 3.


Scenario 3 - First Contact Miscible Case

Scenario 3 is the FCM case, where the pressure is high enough that we no longer have

a two phase envelope in the compositional space (Figure 2.18). This case highlights

a possible limitation of this model in that any single phase region is assigned an

interpolation parameter value of one, which indicates an immiscible displacement.

For this reason in this FCM case we see no difference in the behavior between the

normal simulation results and the one where the Coats correction is used, whereas we

would expect significant differences since all the compositional space can be classified

as super critical. In other words using immiscible relative permeability curves in such

a displacement might give unrealistic results. Since this region is always identified as

either gas or oil (single phase), then we are only dealing with the end-points of the

curves. If this is a FCM miscible displacement, then we would expect the two phases

to flow as one without any residual saturations. Hence, a relative permeability of one

would be used. Even though such displacements that are fully in the “super critical”

region might not be encountered in reality, we believe that accounting for miscibility

in cells labeled as a single phase should be included in models that attempt to reflect

miscibility on the relative permeability curves. Chapter 4 will elaborate on this idea.

NC4

C10

CO2

NC4

C10

CO2

NormalCorrected

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Gas

Sat

ura

tio

n

NormalCorrected

Figure 2.18: Ternary representation of the displacement (left) and gas saturation(right) for scenario 3 with and without corrections.


An important observation in this example, which is also evident in the MCM ex-

ample, is the jump in relative permeability value between the two end-points when the

phase identification switches from oil to gas. A zoomed in image of the relative per-

meability values and overall composition in the region where the phase identification

flips (xD = 0.25− 0.35) is shown in Figure 2.19.

0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Krp

GasOil

0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xD

Mo

le F

ract

ion

CO2

NC4

C10

Figure 2.19: Ternary representation of the displacement in scenario 2 compared tothe paper

This discontinuity motivates the idea of compositional consistency where relative

permeability values should not depend purely on saturation and what the phase is

called. Instead, it should have some dependence on composition; hence, we might

expect a smoother transition between the end-points as composition gradually changes

from the original to the injection composition. Compositional consistency is discussed

in Chapter 3.


2.4 Summary

It is important to be able to identify and reflect miscible behavior on fluid

properties in immiscible simulators, since Equations 2.1-2.2 are only applicable

in fully miscible conditions.

Surface tension and capillary number are commonly used as indicators of mis-

cibility, these are used to compute an interpolation parameter (Fk) that ranges

from zero (miscible) to one (immiscible). Fk is used to interpolate between

the two limiting relative permeability curves. Different ways of interpolating

methods can be used with the most important factors to consider is avoiding

unrealistic flow of low saturation values and that the curves converge to the

expected 45◦ diagonals at miscibility (Fk = 0).

Models that use reference/threshold values, such as Coats model, exhibit high

sensitivity to the values chosen and therefore care should be taken in determin-

ing them.

Current models treat all cells identified as being in a single-phase state as “im-

miscible” by assigning an Fk value of one. This approach might need to be

re-evaluated in order to properly reflect miscible behavior in the relative per-

meability curves.

Chapter 3

Compositional Consistency

This chapter introduces the concept of compositional consistency in relative perme-

abilites. A description for some existing methods is provided with an example of

how one can be implemented in a numerical simulator. The effect of such models on

simulation results and performance is presented.

3.1 Phase Identification

We start by looking at where the discontinuities in relative permeabilities are en-

countered in the compositional space, which are usually caused by how phases are

identified/labeled. We will focus on simple two-phase (gas-oil) systems that should be

sufficient. The reason these are sufficient is that compositional consistency in the rel-

ative permeabilities will only be observed between phases that can have very similar

properties or, in the case of compositional simulators, those that exhibit significant

mass transfer of components between the two to the point where the compositions can

become equal (e.g., critical point). In this case it seems that there are two scenarios

where these discontinuities exists that can be classified as:

39

40 CHAPTER 3. COMPOSITIONAL CONSISTENCY

1. Phase Flip: This will usually take place in the critical region above the critical

tie-line extension, where the phase label switches from one to the other.

2. Phase Misidentification: This usually occurs in regions close to the phase

envelope boundary where the flash problem becomes difficult and sometimes

misidentifies the phase.

Figure 3.1 shows the relative permeabilities of gas and oil for the three component

system (C1−NC4−C10). A smooth continuous transition in the relative permeabilities

is seen for each phase separately (seen on the bottom left and right images), however

a sharp discontinuity is observed in a line that extends above the critical point that

separates the single phase region identified as oil on the right (with an end-point value

of 0.5), and that identified as gas on the left (with an end-point value of 0.8). This

is an example of what we referred to above as a phase flip, where a sharp change

in relative permeability value occurs from a small change in composition because of

what label is given to the phase (e.g. oil or gas in this case). This is a limitation

in the way relative permeability values are assigned based on phase saturation and

label.

The other scenario where discontinuities in relative permeability occur due to

phase identification was pointed out in the example shown in the previous chapter

(shown again here in Figure 3.2), we referred to this as phase misidentification. The

oscillation in saturation is seen due to misidentifying the phase in regions close to

the phase boundary. This usually occurs when problems are encountered during the

flash, which could lead to wrong solutions. In some cases the flash converges to trivial

solutions that require the use of simple correlations to identify the phase, which may

give “wrong” results.

3.1. PHASE IDENTIFICATION 41

Gas Saturation − GIBBS−ALL

50 100 150 200

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