upscaling of two-phase flow with capillary pressure ... · fine-scale capillary pressure is...

UPSCALING OF TWO-PHASE FLOW WITH

CAPILLARY PRESSURE HETEROGENEITY EFFECTS

a thesis

submitted to the department of energy resources engineering

of stanford university

in partial fulfillment of the requirements

for the degree of master of science

By

Kasama Itthisawatpan

June 2013

c© Copyright 2013 by Kasama Itthisawatpan

All Rights Reserved

ii

I certify that I have read this thesis and that in my opin-

ion it is fully adequate, in scope and in quality, as a

partial fulfillment of the degree of Master of Science in

Petroleum Engineering.

Prof. Louis J. Durlofsky(Principal Adviser)

iii

Abstract

In this work, we develop a new iterative global upscaling procedure applicable for

two-phase flow with significant capillary pressure heterogeneity effects. These effects

are important to include in simulations of carbon storage operations as they can have

a strong impact on CO2 movement. The upscaling method entails the use of a global

fine-scale two-phase flow simulation for computing the coarse-scale mobility functions.

Two techniques for upscaling capillary pressure are considered. One approach applies

steady-state capillary-limit computations, and the other involves the numerical com-

putation of upscaled capillary pressure along with the upscaled mobilities. For both

approaches, iteration at the coarse-scale level leads to an improvement in the accuracy

of the upscaled model.

The new upscaling procedures are applied to synthetic two-dimensional reservoir

models. Fine-scale capillary pressure is described using the J−function representa-

tion. Different gas injection rates and well locations are considered. The coarse-scale

models generated using the new iterative global upscaling algorithm provide signifi-

cantly more accurate results, relative to reference fine-scale simulations, than do those

based on simpler upscaling procedures. The robustness of the upscaled models is also

assessed, and the models are shown to provide results of reasonable accuracy for cases

involving injection rates or large-scale flow configurations that are different from those

used in the upscaling calculations. This means that the upscaled functions can be

used under a range of flow conditions and are not restricted to only those applied in

the upscaling computations.

v

Acknowledgments

First and foremost, I would like to express my gratitude to my advisor, Professor

Louis Durlofsky, for his support and guidance throughout my study. His insightful

comments and discussions have guided the research toward the right direction, and

his constant encouragement has kept me working through difficult times.

I would also like to thank many researchers and fellow students in the Energy

Resources Engineering Department, including Dr. Huanquan Pan for his help with

GPRS, Dr. Denis Voskov for the discussions on configuring and troubleshooting the

fine-scale simulation with capillary pressure effects, Boxiao Li for the discussion on the

CO2 simulation modeling, Hangyu Li for the discussion on upscaling and for providing

the modified GPRS that was initially used in this work, and David Cameron for the

discussion on the realistic models for field-scale CO2 storage simulation.

I also wish to thank my family for their love and support throughout my study.

Finally, I would like to acknowledge PTT Exploration and Production for their

financial support during both my undergraduate and Master’s careers at Stanford.

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Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Simulation of CO2 Storage Operations . . . . . . . . . . . . . 2

1.1.2 Significance of Capillary Pressure Heterogeneity . . . . . . . . 3

1.1.3 General Upscaling Methods . . . . . . . . . . . . . . . . . . . 5

1.2 Scope of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Upscaling Methods 9

2.1 Upscaling Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Fine-Scale Governing Equations . . . . . . . . . . . . . . . . . 9

2.1.2 Coarse-Scale Governing Equations . . . . . . . . . . . . . . . . 11

2.1.3 Numerical Calculation of Upscaled Functions . . . . . . . . . . 11

2.2 Upscaling Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Single-Phase Upscaling . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Capillary Pressure Calculation . . . . . . . . . . . . . . . . . . 18

2.2.3 Iterative Global Upscaling Method . . . . . . . . . . . . . . . 21

2.2.4 Numerical Calculation of Capillary Pressure . . . . . . . . . . 25

2.2.5 Criteria for Acceptable λ∗j and P ∗c . . . . . . . . . . . . . . . . 27

2.3 Other Methods and Issues . . . . . . . . . . . . . . . . . . . . . . . . 29

ix

2.3.1 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Local k∗ Upscaling with J−Function . . . . . . . . . . . . . . 31

3 Numerical Results 35

3.1 Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Reservoir Model . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Upscaling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Flow in x−direction . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Flow in y−direction . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Robustness to Changes in Boundary Conditions . . . . . . . . . . . . 61

3.3.1 Change in Flow Rate . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.2 Change in Well Locations . . . . . . . . . . . . . . . . . . . . 64

4 Conclusions and Future Work 67

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A Additional Numerical Results 69

A.1 Flow in the x−direction . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.1.1 Medium Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . 69

A.1.2 High Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.2 Flow in the y−direction . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.2.1 Medium Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . 74

A.2.2 High Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Nomenclature 79

Bibliography 82

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List of Tables

3.1 Parameters for grid and geological model . . . . . . . . . . . . . . . . 36

3.2 Abbreviations for the upscaling methods applied in this section . . . 40

3.3 Overall saturation error for flow in the x−direction (medium flow rate) 44

3.4 Overall saturation error for flow in the x−direction (medium flow rate)

with numerical P ∗c calculation . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Overall saturation error for flow in the x−direction (high flow rate) . 47

3.6 Overall saturation error for flow in the x−direction (low flow rate) . . 50

3.7 Overall saturation error for flow in the y−direction (medium flow rate) 54

3.8 Overall saturation error for flow in the y−direction (high flow rate) . 56

3.9 Overall saturation error for flow in the y−direction (low flow rate) . . 59

A.1 Overall saturation error for flow in the x−direction (medium flow rate).

For these results, tend = 0.62 PVI . . . . . . . . . . . . . . . . . . . . 69

A.2 Overall saturation error for flow in the x−direction (high flow rate) . 72

A.3 Overall saturation error for flow in the y−direction (medium flow rate).

For these results, tend = 0.59 PVI . . . . . . . . . . . . . . . . . . . . 74

A.4 Overall saturation error for flow in the y−direction (high flow rate) . 76

xi

List of Figures

2.1 Schematic showing (a) fine-scale (lighter lines) and coarse-scale (heav-

ier lines) grids, and (b) coarse blocks i and i+ 1 (shaded area in (a)).

Arrows show fine-scale fluxes at the coarse interface i+ 12. . . . . . . 12

2.2 Schematic showing capillary pressure upscaling under the assumption

of capillary-limit and steady-state conditions. . . . . . . . . . . . . . 20

2.3 Resuting upscaled capillary pressure curve. Red circles show (Scw, P∗c )

pairs shown in Figure 2.2. . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Examples of water and gas flow rates as functions of upstream average

water saturation. The flux profiles at some interfaces are generally

smooth (a), while others can be noisy (b). . . . . . . . . . . . . . . . 22

2.5 Flow chart showing iterative global upscaling procedure to compute

λ∗w and λ∗g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Flow chart showing iterative global upscaling procedure to compute

λ∗w, λ∗g, and P ∗c . Shaded blocks indicates modifications from Figure 2.5. 28

2.7 Comparison of the results from locally weighted linear regression using

different values of τ . The noisy flux is from Figure 2.4b. . . . . . . . . 31

2.8 Comparison between P ∗c from capillary-limit steady-state method, nu-

merical method, and J−function. . . . . . . . . . . . . . . . . . . . . 33

3.1 Permeability fields (log scale) used in the study. . . . . . . . . . . . . 36

3.2 Rock-fluid parameters specified for fine-scale simulation. . . . . . . . 37

3.3 Locations of the injector (red) and producer (blue) in this study. . . . 38

3.4 Gas fractional flow at the producer for flow in the x−direction at var-

ious rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xii

3.5 Gas fractional flow at the producer for flow in the x−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Saturation map (Sg) at 1 PVI for flow in the x−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Normalized L2−norm of the error in gas saturation for flow in the

x−direction (medium flow rate). . . . . . . . . . . . . . . . . . . . . . 44

3.9 Injector bottom-hole pressure for flow in the x−direction (medium flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Gas fractional flow at the producer for flow in the x−direction (medium

flow rate) with numerical P ∗c calculation. . . . . . . . . . . . . . . . . 46

3.11 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (medium

flow rate) with numerical P ∗c calculation. . . . . . . . . . . . . . . . . 46


x−direction (medium flow rate) with numerical P ∗c calculation. . . . . 47

3.13 Gas fractional flow at the producer for flow in the x−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.14 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.15 Saturation map (Sg) at 1 PVI for flow in the x−direction (high flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


x−direction (high flow rate). . . . . . . . . . . . . . . . . . . . . . . . 50

3.17 Gas fractional flow at the producer for flow in the x−direction (low

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.18 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (low flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


x−direction (low flow rate). . . . . . . . . . . . . . . . . . . . . . . . 52

xiii

3.20 Gas fractional flow at the producer for flow in the y−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.21 Saturation map (Sg) at 0.425 PVI for flow in the y−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


y−direction (medium flow rate). . . . . . . . . . . . . . . . . . . . . . 54

3.23 Average gas saturation at the top layer for flow in the y−direction

(medium flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.24 Injector bottom-hole pressure for flow in the y−direction (medium flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.25 Gas fractional flow at the producer for flow in the y−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.26 Saturation map (Sg) at 0.425 PVI for flow in the y−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


y−direction (high flow rate). . . . . . . . . . . . . . . . . . . . . . . . 58

3.28 Average gas saturation at the top layer for flow in the y−direction

(high flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.29 Gas fractional flow at the producer for flow in the y−direction (low

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.30 Saturation map (Sg) at 0.425 PVI for flow in the y−direction (low flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


y−direction (low flow rate). . . . . . . . . . . . . . . . . . . . . . . . 60

3.32 Average gas saturation at the top layer for flow in the y−direction (low

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.33 Overall saturation error for G1P/iG2P30 and G1P/iG2P60, with G1P/iG2P

and G1P/Pc shown for reference (flow in the x−direction). . . . . . . 62

3.34 Overall saturation error for G1P/iG2P30 and G1P/iG2P60, with G1P/iG2P

and G1P/Pc shown for reference (flow in the y−direction). . . . . . . 63

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3.35 Gas fractional flow at the producer for flow in the x−direction at low

rate (10 bbl/day) using various models. . . . . . . . . . . . . . . . . . 63

3.36 Gas fractional flow at the producer for flow in the x−direction at high

rate (100 bbl/day) using various models. . . . . . . . . . . . . . . . . 64

3.37 Gas fractional flow at the producer for corner-to-corner flow (high flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.38 Saturation map at 0.425 PVI for the corner-to-corner flow (high flow

rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.1 Gas fractional flow at the producer for flow in the x−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.2 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.3 Normalized L2−norm of the error in gas saturation for flow in the

x−direction (medium flow rate). . . . . . . . . . . . . . . . . . . . . . 71

A.4 Gas fractional flow at the producer for flow in the x−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.5 Saturation map (Sg) at 0.425 PVI for flow in the x−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


x−direction (high flow rate). . . . . . . . . . . . . . . . . . . . . . . . 73

A.7 Gas fractional flow at the producer for flow in the y−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.8 Saturation map (Sg) at 0.425 PVI for flow in the y−direction (medium

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


y−direction (medium flow rate). . . . . . . . . . . . . . . . . . . . . . 75

A.10 Gas fractional flow at the producer for flow in the y−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.11 Saturation map (Sg) at 0.425 PVI for flow in the y−direction (high

flow rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xv


y−direction (high flow rate). . . . . . . . . . . . . . . . . . . . . . . . 77

xvi

Chapter 1

Introduction

The sequestration of carbon dioxide in deep geological formations is being considered

as a means to mitigate the impact of fossil fuel combustion. The idea is to capture

CO2 at the power plant and to inject it into various types of subsurface formations

such as depleted oil and gas reservoirs and deep saline aquifers. The injected carbon

dioxide is “stored” in the subsurface through some combination of structural trapping,

residual trapping, dissolution trapping, and mineral trapping.

In order to minimize the environmental risk, the movement and the distribution

of the injected CO2 must be accurately modeled. Because the flow of supercritical

CO2 in subsurface formations entails very similar physical phenomena as the flow of

fluids in oil and gas reservoirs, petroleum reservoir simulators can be used to model

the flow of the injected CO2. As is the case for the flow of oil and gas in the sub-

surface, the flow of CO2 involves effects at various scales. Fine-scale geostatistical

models may include blocks of O(1− 10 m), while full-field simulations are often per-

formed with blocks of O(10−100 m). In many cases, especially in models with strong

heterogeneity, fine-scale property variation can significantly influence simulation re-

sults. In order to achieve accurate coarse-scale simulations, this heterogeneity must

be properly accounted for in the coarse-scale model. This scale translation can be

accomplished using the methods of upscaling.

Although many of the general issues that arise in the upscaling of carbon storage

1

2 CHAPTER 1. INTRODUCTION

models are similar to those in oil and gas simulation, there are some important dif-

ferences. Oil and gas simulations usually involve sufficiently high flow rates such that

viscous and gravitational forces dominate in full-field simulations. Carbon storage,

by contrast, can entail much lower rates, so capillary effects must also be treated in

field-scale simulations. This introduces challenges in the upscaling of the equations

governing the carbon storage processes.

Our goal in this work is to devise and test new upscaling methods that enable

accurate large-scale simulation of two-phase flow problems with significant capillary

pressure effects. This will require us to incorporate new treatments into two-phase

upscaling procedures.

1.1 Literature Review

1.1.1 Simulation of CO2 Storage Operations

Simulation of field-scale carbon storage in deep saline aquifers has been addressed by

various authors in recent years. The simulations typically include about 30 years of

CO2 injection, followed by an equilibration period, which can last for several hun-

dred years, during which the injected gas equilibrates in the aquifer. Studies have

varied from conceptual assessments of key mechanisms to actual field-scale simula-

tions. As the primary goal is usually to predict the distribution of the injected CO2,

the conceptual studies have investigated the factors that affect the final CO2 plume

location. For example, Mo and Akervoll (2005) used a black-oil model to characterize

the impact of permeability anisotropy, relative permeability, and capillary pressure

characteristics on the gas stored by structural and residual trapping. Ide et al. (2007)

examined the interaction between viscous, capillary, and gravitational forces more

systematically, by comparing different scenarios in terms of their gravity numbers

and capillary numbers. They concluded that scenarios with relatively low gravitata-

tional forces compared to viscous forces resulted in more residually-trapped CO2.

Increasing the magnitude of capillary forces compared to the other forces contributed

to more, and faster, CO2 trapping. Dissolution trapping was not considered in detail

1.1. LITERATURE REVIEW 3

by Ide et al. (2007).

Some authors also investigated dissolution trapping in addition to residual trap-

ping. Kumar et al. (2004) and Ghanbari et al. (2006) applied compositional models

to study the impact of various factors on dissolution trapping. Recently, field-scale

case studies (e.g., Doughty, 2010; Han et al., 2010; Chasset et al., 2011) incorporated

the actual aquifer geometry and petrophysical properties into the simulation models.

The results obtained from the field-scale studies generally agreed with the concep-

tual studies in terms of the impact of reservoir properties on residual and dissolution

trapping. They observed, however, that reservoir heterogeneity also has a significant

impact on the movement of CO2 and the final plume location. The results from the

case studies also indicated the need for accurate models for the spatial distribution

of petrophysical properties.

1.1.2 Significance of Capillary Pressure Heterogeneity

Recent studies have also highlighted the importance of capillary pressure heterogene-

ity on CO2 plume migration. Capillary pressure heterogeneity is often ignored in sub-

surface flow modeling. More specifically, it is common to neglect capillary pressure en-

tirely or to assign a single capillary pressure curve to all simulation blocks. Saadatpoor

et al. (2010) incorporated capillary pressure heterogeneity based on the J−function

representation (Leverett, 1940) in a highly heterogeneous fine-scale model. They

compared the ultimate distribution of the injected CO2 with a model that had the

same permeability field but homogeneous capillary pressure. Their conclusion was

that capillary pressure heterogeneity strongly impacted the flow of the CO2. Capil-

lary pressure heterogeneity results in a strong barrier to the flow of the nonwetting

phase, so the injected gas can be immobilized even though its saturation is signif-

icantly higher than the residual saturation. The authors referred to this trapping

phenomenon as the “local capillary trapping” mechanism.

The effects of capillary heterogeneity have also been found to be significant at the

core scale. Krevor et al. (2011) and Li et al. (2012) applied theoretical, experimen-

tal, and numerical simulation approaches to analyze the impact of capillary pressure


heterogeneity on the distribution of fluids in the core. They injected CO2 at rates

that are comparable to those in CO2 storage operations and showed that the final

fluid distribution is highly nonuniform. Experiments were performed by having a

drainage process (CO2 injection) followed by an imbibition process (water flooding).

The overall CO2 saturation after the water flooding was much higher than would be

expected if capillary pressure heterogeneity were not included (Krevor et al., 2011).

The observation that capillary pressure heterogeneity affects flow at both the core

scale and the geostatistical scale necessitates the accurate fine-scale modeling of cap-

illary pressure heterogeneity. However, the use of very fine grid blocks in a full-field

simulation is not practical and would be computationally expensive. In order to cap-

ture fine-scale heterogeneity effects in the coarse-scale domain, some studies have in-

troduced procedures to upscale capillary pressure. For example, Mouche et al. (2010)

used the analytical solution for flow in a vertical, one-dimensional periodic layered

porous medium under the capillary-limit condition to compute the upscaled capil-

lary pressure. Behzadi and Alvarado (2012) modified the capillary-limit steady-state

calculation presented by Pickup and Sorbie (1996) to account for the flow direction

and the subgrid spatial distribution of the capillary pressure. These methods were

shown to provide accurate coarse-scale flow results, though they are applicable only

to very specific flow conditions which may not be encountered in realistic field-scale

simulations.

Saadatpoor et al. (2011) presented an alternative approach to upscale the capillary

pressure from the geostatistical scale to the field scale. They computed an “effective”

permeability for each coarse grid block as the geometric mean of the underlying

fine-scale permeabilities, and then applied the J−function based on this effective

permeability to provide an upscaled capillary pressure. Saadatpoor et al. (2011)

concluded that this upscaled model did not preserve the local capillary trapping

observed in the fine-scale model. The calculation of the upscaled capillary pressure

in Saadatpoor et al. (2011) does not assume specific flow conditions, so the approach

can be applied to generic flow conditions. However, it represents a highly simplified

approach for capillary pressure upscaling. In order to achieve more accurate coarse-

scale results, a more sophisticated flow-based upscaling method, which also treats

1.1. LITERATURE REVIEW 5

relative permeability, will be needed. We now provide a brief discussion of upscaling

techniques that may be relevant for this problem.

1.1.3 General Upscaling Methods

Upscaling involves the calculation of coarse-scale properties and flow functions that

can accurately capture underlying fine-scale effects. When used in coarse-scale simu-

lations, properly computed upscaled functions can provide flow predictions in general

agreement with fine-scale results. Upscaling methods can be categorized in different

ways. For example, single-phase upscaling methods provide equivalent flow properties

such as porosity, absolute permeability, and transmissibility. Two-phase (or multi-

phase) upscaling provides transport functions such as capillary pressure and relative

permeability.

Another way to view upscaling procedures is based on the domain on which the

effective properties are computed. Methods can be described as local, extended lo-

cal, and global upscaling procedures. For example, local upscaling entails fine-scale

simulation over the domain corresponding to the target coarse block with assumed

pressure and saturation boundary conditions. Extended-local methods also require

assumptions on the boundary conditions, but the local fine-scale simulations include

additional (neighboring) cells. Global upscaling, by contrast, uses the information

from global fine-scale simulation to compute effective properties. In general, the

global (or local-global, where global information is provided by coarse-scale simula-

tions) calculation of transmissibility has been shown to provide the most accurate

single-phase upscaling (Chen et al., 2003; Chen et al., 2010). Since this work entails

the calculation of upscaled two-phase properties, our focus will be on multiphase up-

scaling procedures. Refer to Durlofsky (2005) for further discussion of single-phase

upscaling and related issues.

Ekrann and Dale (1992) classified multiphase upscaling into “dynamic” and “effec-

tive” upscaling procedures. Effective upscaling entails calculations based on local so-

lutions for particular limiting cases. For example, Pickup and Sorbie (1996) developed


steady-state methods to compute effective capillary pressure and relative permeabil-

ity which are valid for either the capillary or the viscous limit. Virnovsky et al. (2004)

further developed the steady-state method. They showed that, as the imposed pres-

sure drop was increased or decreased, the resulting upscaled functions approached the

viscous limit or the capillary limit results, respectively. These steady-state approaches

are generally computationally efficient, though their accuracy strongly depends on the

applicability of the assumed conditions.

Dynamic upscaling, by contrast, involves time-dependent fine-scale simulations

over specified (e.g., local or global) domains. Upscaled capillary pressure and/or

relative permeability are computed based on a coarse-scale Darcy’s law such that

the flux of each phase on the coarse scale matches the integrated flux from the fine-

scale results. As is the case in single-phase upscaling, assumptions on local boundary

conditions often influence the accuracy of upscaled properties computed using local

or extended-local methods.

Darman et al. (2002) evaluated the dynamic upscaling procedures proposed by

Kyte and Berry (1975), Stone (1991), Hewett and Archer (1997), and Darman and

Pickup (1999). They performed global upscaling for several cases and found that the

Hewett and Archer (1997) method and the transmissibility-weighted method (Dar-

man and Pickup, 1999) provided the most accurate coarse models both with and

without significant gravitational effects. However, this study only involved purely

layered permeability distributions; systems with random fine-scale permeability fields

were not considered. Local property variation is, however, of significant interest for

our problem. The effects of capillary pressure were also neglected in the examples

presented by Darman et al. (2002).

More recent developments in multiphase upscaling have considered random fine-

scale permeability fields. Wallstrom et al. (2002) developed so-called effective flux

boundary conditions (EFBCs) for local upscaling in the viscous limit. EFBCs pro-

vide pressure boundary conditions based on the local fine-scale permeability field.

A detailed investigation by Chen (2005) showed that the use of EFBCs resulted in

substantially more accurate coarse-scale models than the use of standard boundary

conditions (constant pressure at the inlet and outlet) in many cases. Chen and Li

1.2. SCOPE OF THIS WORK 7

(2009) developed an adaptive local-global method to compute upscaled relative per-

meability. The coarse-scale results based on the two-phase local-global upscaling were

shown to match the fine-scale results for various variogram-based models. The ap-

proaches described above represent reasonable approaches for two-phase upscaling,

though they are based on viscous-dominated flow, with capillary pressure neglected.

Upscaling of two-phase flow with capillary pressure heterogeneity was considered

by Lohne et al. (2006). They applied the capillary-limit steady-state method to

upscale capillary pressure. Other methods, including both steady-state and dynamic

approaches, were applied to upscale relative permeability. Capillary-limit effective

capillary pressure, together with appropriate relative permeability upscaling based

on the capillary number, was shown to provide accurate coarse-scale models. The

reservoir models presented in Lohne et al. (2006) somewhat resemble those considered

in this work, in that both realistic permeability heterogeneity and capillary pressure

heterogeneity were included. However, the capillary pressure heterogeneity in Lohne

et al. (2006) was modeled using a conceptual, checkerboard pattern at the fine scale,

and by the use of rock types at the geostatistical scale. Thus, the capillary pressure

did not depend directly on the local permeability distribution. In this work, we

consider a more general model, with capillary pressure heterogeneity represented by

the J−function. In addition, the upscaling procedure developed here uses global flow

information, in contrast to the approach applied by Lohne et al. (2006).

1.2 Scope of this Work

In this work, we focus on the two-phase upscaling of flow with significant capillary

pressure heterogeneity effects. We assess approaches for the calculation of upscaled

capillary pressure functions. The upscaled relative permeability is computed from

global fine-scale solutions. We focus on scenarios that are relevant for CO2 storage

simulations, though some of the physics that is important in CO2 storage, such as

gravitational effects, dissolution, and relative permeability hysteresis, is not included

in our simulations. The resulting coarse-scale models are assessed by comparing the

coarse-scale results, including the fractional flow of gas at the production well and


the phase saturation distributions, with fine-scale results.

Computing the upscaled functions from global fine-scale flows is challenging for

highly heterogeneous systems with low flow rates, as unphysical upscaled functions

can result unless appropriate treatments are applied. In this work, we develop an

iterative scheme to improve coarse-scale results. Because the methods developed in

this work involve global fine-scale, two-phase flow simulations, which are essentially

what we wish to avoid, it is important that the resulting coarse model is applicable

for other flow scenarios. We therefore assess the robustness of the resulting upscaled

functions with respect to changes in boundary conditions. For this assessment, we run

a global fine-scale simulation for a representative case, and then apply the upscaled

functions computed for this model to cases that involve different flow conditions.

Coarse-scale solutions for changing injection rates and well locations will be presented

and assessed.

1.3 Thesis Outline

This thesis is organized as follows:

• Chapter 2 presents the formulation for the upscaling problem. The fine-scale

and coarse-scale governing equations are discussed, and the computation of the

upscaled functions is explained. We discuss several approaches for upscaling

capillary pressure, as well as the calculation of upscaled relative permeability

using an iterative global upscaling scheme.

• Chapter 3 presents numerical examples that quantify the performance of dif-

ferent upscaling methods. The methods are applied for different well locations

and CO2 injection rates. Then, we assess the robustness of the upscaled model

under changes in injection rates and well locations.

• Chapter 4 includes a summary and conclusions for this work. Suggestions for

future research are also provided.

• Appendix A includes upscaling results for another geological model.

Chapter 2

Upscaling Methods

In this chapter, we present the governing equations for the flow of gas and water in

porous media at both the fine-scale and the coarse-scale levels. We briefly summarize

the existing upscaling methods that are applied in this study. Then, we describe in

detail our treatment for capillary pressure and the iterative two-phase global upscaling

approach. This is a new method that was developed in this work. In describing the

iterative two-phase global upscaling procedure, we also discuss several key aspects of

the implementation.

2.1 Upscaling Formulation

2.1.1 Fine-Scale Governing Equations

In our formulation, we consider CO2 and water to be immiscible, incompressible fluids.

Thus, the dissolution of CO2 in brine is not considered in our upscaling computations.

The flow of this immiscible, incompressible gas-water system in an incompressible rock

is governed by the conservation of mass of each component

φ∂Sw∂t

+∇ · uw = qw, (2.1a)

φ∂Sg∂t

+∇ · ug = qg, (2.1b)

9

10 CHAPTER 2. UPSCALING METHODS

where φ is the porosity of the rock, Sj is the saturation of phase j (j = water, gas),

uj is the Darcy velocity of phase j, and qj is the source term. The subscripts w and g

refer to water phase and gas phase, respectively. Because this work considers the flow

of immiscible fluids, CO2 can only exist in the gas phase and water can only exist in

the water phase.

The Darcy velocity uj of phase j is governed by Darcy’s law

uj = −λjk · ∇ (pj − γjz) , (2.2)

where λj =krjµj

is the mobility of phase j, krj is the relative permeability to phase j,

µj is the viscosity of phase j, k is the absolute permeability tensor, pj is the pressure

of phase j, γj is the specific gravity of phase j, and z is the vertical position (note

that z points downward). For flow in the horizontal (x−y) plane, which is considered

here, we have ∇ (pj − γjz) = ∇pj, and Darcy’s law reduces to

uj = −λjk · ∇pj. (2.3)

In the presence of capillary pressure effects, the pressures of the nonwetting (gas)

phase and the wetting (water) phase are related by

pg − pw = Pc(Sw). (2.4)

Using pg and Sw as primary variables, Equation (2.1) can be expressed as

φ∂Sw∂t

+∇ · [λwk · ∇ (pg − Pc(Sw))] = qw, (2.5a)

φ∂Sg∂t

+∇ · (λgk · ∇pg) = qg. (2.5b)

Note that we also have the constraint Sw + Sg = 1.

2.1. UPSCALING FORMULATION 11

2.1.2 Coarse-Scale Governing Equations

The coarse-scale governing equations can be constructed by averaging the fine-scale

equations over the region corresponding to a single coarse block. Refer to Chen

(2005) for detailed formulations and discussion of the different forms of the coarse-

scale equations. In this work, we restrict the coarse-scale equations to have the same

general form as the fine-scale equations, but with upscaled properties and functions

replacing their fine-scale analogs. The conservation equations for the coarse-scale

system can thus be written as

φ∗∂Scw

∂t+∇ ·

[λ∗wk∗ · ∇

(pcg − P ∗

c (Scw))]

= qcw, (2.6a)

φ∗∂Scg

∂t+∇ ·

(λ∗gk

∗ · ∇pcg)

= qcg, (2.6b)

where the superscript c represents a coarse-scale variable, and the superscript ∗ indi-

cates a precomputed coarse-scale property or function.

2.1.3 Numerical Calculation of Upscaled Functions

If the coarse-scale equations (2.6) are discretized using a two-point flux approximation

and the usual upstream weighting, the flux of each phase j across a coarse interface

i+ 12

between two coarse blocks i and i+ 1, designated qcj,i+ 1

2

, can be written as

qcg,i+ 1

2= T ∗

i+ 12λ∗g,i+ 1

2

(pcg,i − pcg,i+1

), (2.7a)

for the gas phase, and

qcw,i+ 1

2= T ∗

i+ 12λ∗w,i+ 1

2

[(pcg,i − P ∗

c,i

(Scw,i

))−(pcg,i+1 − P ∗

c,i+1

(Scw,i+1

))], (2.7b)

for the water phase. Here T ∗i+ 1

2

is the upscaled transmissibility, which will be discussed

below. For the blocks containing wells, the flux of phase j from block i to the wellbore,


designated as qcwell,j,i, can be written as

qcwell,j,i = WI∗i λ∗j,i

(pcg,i − pcwell,i

), (2.8)

where WI∗i is the upscaled well index and pcwell,i is the well pressure in block i.

The upscaled single-phase parameters, transmissibility T ∗i+ 1

2

and well index WI∗i ,

can be computed from a single-phase upscaling algorithm, which will be described

in the next section. Once the single-phase parameters have been computed, the goal

of two-phase upscaling is to provide coarse-scale fluxes that match the sum of the

corresponding fine-scale fluxes:

qcg,i+ 1

2=⟨qfg⟩i+ 1

2

, qcw,i+ 1

2=⟨qfw⟩i+ 1

2

, (2.9)

where the superscript f indicates that the quantity is from fine-scale simulation and

〈·〉i+ 12

denotes the integrated flux over the fine-scale interfaces that correspond to the

coarse interface i+ 12. This is illustrated in Figure 2.1. The equations are analogous

for blocks that contain wells.

(a)

coarse block i! coarse block i +1!

(b)

Figure 2.1: Schematic showing (a) fine-scale (lighter lines) and coarse-scale (heavierlines) grids, and (b) coarse blocks i and i + 1 (shaded area in (a)). Arrows showfine-scale fluxes at the coarse interface i+ 1

2.


Expressions (2.9) constitute two equations for the upscaling computations. How-

ever, with nonzero capillary pressure, three upscaled functions, λ∗w, λ∗g, and P ∗c , must

be determined for each coarse block. In order to solve this underdetermined system,

we consider two different approaches: (1) we first compute P ∗c (Scw) analytically, under

a steady-state assumption, followed by the calculation of λ∗j(Scw) via global upscal-

ing, and (2) we identify the capillary and convective flux contributions to qcw,i+ 1

2

and

compute the upscaled functions accordingly. We now describe these two approaches

in detail.

In the first approach, the upscaled capillary pressure function for each coarse

block is precomputed using an analytical approach. No flow simulation is required

because we assume steady-state conditions. The detailed calculation will be discussed

in the next section. With these precomputed capillary pressure functions, only two

upscaled functions remain to be computed, so the problem is now well-defined. The

upscaled mobility for each phase can then be computed from the fine-scale solution by

rearranging Equations (2.7a) and (2.7b) and approximating the coarse-scale pressure

and saturation as averages of the fine-scale results

λ∗g,i+ 1

2=

⟨qfg⟩i+ 1

2

T ∗i+ 1

2

(⟨pfg⟩i−⟨pfg⟩i+1

) , (2.10a)

λ∗w,i+ 1

2=

⟨qfw⟩i+ 1

2

T ∗i+ 1

2

[⟨pfg⟩i− P ∗

c,i

(⟨Sfw⟩i

)−⟨pfg⟩i+1− P ∗

c,i+1

(⟨Sfw⟩i+1

)] . (2.10b)

Here,⟨pfg⟩i

is the volume-averaged gas pressure, given by

⟨pfg⟩i

=1

Nf

Nf∑

k=1

pfg,k, (2.11)

when the volumes of the fine-scale blocks are uniform (as they are in our examples).

The quantity pfg,k is the gas pressure of the fine-scale block k that falls within coarse

block i, Nf is the number of fine-scale blocks in coarse block i, and⟨Sfw⟩i

is the


pore-volume weighted water saturation, given by

⟨Sfw⟩i

=

∑Nf

i=1 φkVkSfw,k∑Nf

i=k φkVk. (2.12)

Here, Sfw,k is the water saturation of the fine-scale block k, φk is the porosity of fine-

scale block k, and Vk is the bulk volume of fine-scale block k. When the bulk volume

and porosity are uniform, this reduces to

⟨Sfw⟩i

=1

Nf

Nf∑

k=1

Sfw,k. (2.13)

The resulting phase mobilities λ∗j,i+ 1

2

, as functions of water saturation, are then as-

signed to the upstream coarse block.

In the second approach, we compute P ∗c from the global fine-scale solution by

rearranging the coarse-scale water flux equation (2.7b) to isolate convective (viscous)

and capillary pressure effects:

qcw,i+ 1

2= T ∗

i+ 12λ∗w,i+ 1

2

(pcg,i − pcg,i+1

)︸︷︷︸

Convection term, qc,conv

w,i+12

+T ∗i+ 1

2λ∗w,i+ 1

2

(P ∗c,i+1

(Scw,i+1

))− P ∗

c,i

(Scw,i

)︸︷︷︸

Capillary pressure term, qc,capw,i+1

2

.

(2.14)

From Equation (2.14), we see that the coarse-scale flux of the water phase is driven

by both the difference in gas pressure and the difference in capillary pressure between

the two coarse blocks. The coarse-scale water flux can now be written as

qcw,i+ 1

2= qc,conv

w,i+ 12

+ qc,capw,i+ 1

2

. (2.15)

A similar rearrangement can be applied to the integrated fine-scale results in order

to compute the integrated fluxes:

⟨qfw⟩i+ 1

2

=⟨qf,convw

⟩i+ 1

2

+⟨qf,capw

⟩i+ 1

2

, (2.16)


where qf,convw is the fine-scale water flux due to the difference in gas pressure and qf,capw

is the fine-scale water flux due to the difference in capillary pressure. We can now

match the corresponding flux terms to compute the upscaled properties:

qc,convw,i+ 1

2

=⟨qf,convw

⟩i+ 1

2

, qc,capw,i+ 1

2

=⟨qf,capw

⟩i+ 1

2

. (2.17)

This approach results in a well-defined system with three equations and three un-

knowns. The upscaled mobility λ∗w,i+ 1

2

and the upscaled capillary pressure P ∗c can

then be computed using Equation (2.14) with the coarse-scale pressure and satura-

tion approximated from the fine-scale result. This gives:

⟨qf,convw

⟩i+ 1

2

= T ∗i+ 1

2λ∗w,i+ 1

2


), (2.18a)

⟨qf,capw

⟩i+ 1

2

= T ∗i+ 1

2λ∗w,i+ 1

2

[P ∗c,i+1

(⟨Sfw⟩i+1

)− P ∗

c,i

(⟨Sfw⟩i

)]. (2.18b)

Equation (2.18b) shows that the calculation of P ∗c,i and P ∗

c,i+1 depend on the value of

λ∗w,i+ 1

2

. However, this dependency can be avoided by dividing Equation (2.18b) by

(2.18a), which gives

⟨qf,capw

⟩i+ 1

2⟨qf,convw

⟩i+ 1

2

=P ∗c,i+1

(⟨Sfw⟩i+1

)− P ∗

c,i

(⟨Sfw⟩i

)⟨pfg⟩i−⟨pfg⟩i+1

. (2.19)

Now the upscaled capillary pressure in block i can be computed as

P ∗c,i

(⟨Sfw⟩i

)= P ∗

c,i+1

(⟨Sfw⟩i+1

)−(⟨pfg⟩i−⟨pfg⟩i+1

)

⟨qf,capw

⟩i+ 1

2⟨qf,convw

⟩i+ 1

2

, (2.20)

where we have assumed that P ∗c,i+1 has already been computed. The other upscaled


functions can be computed by simply rearranging Equations (2.7a) and (2.18a)

λ∗g,i+ 1

2=

⟨qfg⟩i+ 1

2

T ∗i+ 1

2


) , (2.21)

λ∗w,i+ 1

2=

⟨qf,convw

⟩i+ 1

2

T ∗i+ 1

2


) . (2.22)

Equation (2.20) shows that the upscaled capillary pressure curves for all blocks

are coupled, as the calculation of P ∗c,i requires knowledge of P ∗

c,i+1. If P ∗c,i+1 is for some

reason inaccurate, the resulting P ∗c,i may be nonmonotonic or may contain negative

values. This situation is undesirable and can be avoided by using an iterative scheme,

which will be described in the next section.

In the next chapter, we compare the accuracy and robustness of the two ap-

proaches for computing P ∗c and λ∗j . It is useful, however, to note that the two ap-

proaches are consistent in the limit of zero capillary pressure. As the magnitude of the

fine-scale capillary pressure decreases, the analytical calculation of P ∗c tends toward

zero. For the capillary flux matching method (the second approach), the fine-scale

water flux due to capillary pressure difference will approach zero as capillary pressure

vanishes. Applying Equation (2.10b) and Equations (2.22) for the first and second

approaches, respectively, results in the same equation:

λ∗w,i+ 1

2=

⟨qfw⟩i+ 1

2

T ∗i+ 1

2


) , (2.23)

as would be expected.

As noted earlier, in this work, we also use a well equation to describe the flow

between the well block and the wellbore. For the coarse model, we first compute

WI∗ for all wells. For injection wells, the λ∗j,i+ 1

2

computed for the interface is used in

the coarse-block well model (2.8). For production wells, we additionally require λ∗j,i

2.2. UPSCALING ALGORITHMS 17

associated with the well block. These functions are computed as follows:

λ∗g,i =

⟨qfg,well

⟩i

WI∗i

(⟨pfg⟩i−⟨pfwell

⟩i

) , (2.24a)

λ∗w,i =

⟨qfw,well

⟩i

WI∗i

(⟨pfg⟩i−⟨pfwell

⟩i

) , (2.24b)

where⟨qfj,well

⟩i

is the integrated flux of phase j between the fine-scale blocks cor-

responding to coarse-block i and the well. Note that P ∗c is not included in these

computations.

2.2 Upscaling Algorithms

In this section, we present the upscaling algorithms used to compute the effective

coarse-scale properties. Our emphasis is on the new algorithms that are appropriate

for use with heterogeneous capillary pressure. Refer to Durlofsky (2005) and Chen

(2005) for more detailed descriptions of existing methods.

2.2.1 Single-Phase Upscaling

Single-phase upscaling procedures are used to compute the upscaled permeability k∗

and porosity φ∗, as well as upscaled transmissibility T ∗ and effective well index WI∗.

Computation of k∗, T ∗, and WI∗ involves solving the fine-scale single-phase pressure

equation over a local, extended local, or global domain subject to boundary conditions

or well specifications. A variety of single-phase upscaling methods, along with their

advantages and disadvantages, are discussed in Durlofsky (2005).

Because this work focuses on two-phase upscaling, we minimize the error intro-

duced in the single-phase upscaling by applying an accurate global method to compute

T ∗. Specifically, we apply a variant of the global T ∗ algorithm described in Chen et al.

(2008), which was shown to be accurate for most flow scenarios. The method proceeds

as follows:


1. Solve the single-phase pressure equation ∇ · (k · ∇p) = q on the x − y global

domain. The global flow is driven by wells, and we impose no-flow conditions

at the domain boundaries.

2. From the single-phase flow results, compute integrated flux⟨qf⟩i+ 1

2

correspond-

ing to each coarse interface i+ 12, the integrated flux

⟨qfwell

⟩i

between coarse

block i and the well (if a well is completed in the block), and the average

fine-scale pressure⟨pf⟩i

corresponding to each coarse block.

3. At each coarse interface i+ 12, compute upscaled transmissibility:

T ∗i+ 1

2=

⟨qf⟩i+ 1

2

〈pf〉i − 〈pf〉i+1

. (2.25)

4. At each block i that contains a well, compute the upscaled well index:

WI∗i =

⟨qfwell

⟩i

〈pf〉i −⟨pfwell

⟩i

. (2.26)

If any of the computations give negative T ∗ or WI∗, we replace the result with

T ∗ or WI∗ from local upscaling. Note that the algorithm presented in Chen et al.

(2008) entails the use of iteration until convergence to a self-consistent result. In this

work, however, we do not iterate since the results using the above computation are

quite accurate. This may be due in part to the fact that we consider only Gaussian

permeability fields in our simulations.

2.2.2 Capillary Pressure Calculation

In the analytical capillary pressure approach, we use the steady-state P ∗c function in

the calculation of upscaled relative permeability. We now describe the calculation of

P ∗c in the capillary limit. This method was introduced by Pickup and Sorbie (1996).

Under the capillary-limit condition, the fluids are in capillary equilibrium. In the

horizontal (x− y) plane, the pressure of each phase is uniform. Thus, every fine-scale


block within the target coarse block has the same capillary pressure, but the fluid

saturations are discontinuous. The method to compute upscaled capillary pressure

under the capillary-limit condition is as follows:

1. Over a target coarse block, determine the minimum possible capillary pressure

Pc,min and maximum possible capillary pressure Pc,max of the corresponding

fine-scale blocks.

2. For each fine-scale block k, invert the capillary pressure-saturation relationship;

i.e., compute Sw,k(Pc) (in practice, Pc for each fine block might be described by

a J−function, as discussed below).

3. For each capillary pressure level Pc within the range [Pc,min, Pc,max], we compute

average water saturation Scw from:

Scw(Pc) =

∑Nf

k=1 φkVkSw,k(Pc)∑Nf

k=1 φkVk, (2.27)

In this work, we consider φk and Vk to be constant, so the calculation for the

average water saturation reduces to:

Scw(Pc) =1

Nf

Nf∑

k=1

Sw,k(Pc). (2.28)

4. Record the resulting average saturation and capillary pressure (Scw, Pc) as a data

point for the capillary pressure curve P ∗c (Scw) of the coarse-scale block.

This approach is depicted in Figures 2.2 and 2.3.

The upscaled capillary pressured computed from the capillary-limit condition is

exact at vanishing fluid velocities, where the fluids are in capillary equilibrium. In our

work, although the capillary-limiting conditions are not fully satisfied, as we model

an injection process, the fluid velocity is sufficiently low such that the computed

upscaled capillary pressure should be reasonably accurate. In the regions of the

model where the fluid velocity is relatively high, the capillary-limit upscaled capillary


(a) Permeability (log scale) of the nine fine-scale blocks comprising the target coarseblock

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Pc(psi)

Sw

(b) Pc curve for each fine-scale block

(c) Water saturation at Pc =10 psi

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Pc

(psi

)

Sw

mean Sw = 0.3776

(d) Sw at Pc =10 psi

(e) Water saturation at Pc =20 psi

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Pc

(psi

)

Sw

mean Sw = 0.2344

(f) Sw at Pc =20 psi

Figure 2.2: Schematic showing capillary pressure upscaling under the assumption ofcapillary-limit and steady-state conditions.


0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Swc

Pc*(psi)

Figure 2.3: Resuting upscaled capillary pressure curve. Red circles show (Scw, P∗c )

pairs shown in Figure 2.2.

pressure may lose accuracy. However, the contribution of capillary pressure is smaller

at higher velocities, and the relative permeability correction largely compensates for

the inaccuracy in the upscaled capillary pressure.

2.2.3 Iterative Global Upscaling Method

In this section, we present a detailed description of the iterative global method for

upscaling two-phase properties, which was developed in this work. The method ex-

tends the approach for iterative global upscaling of T ∗, which was developed by Chen

et al. (2008).

We first describe the method for the case where P ∗c is computed in the capillary

limit. We perform a global fine-scale two-phase flow simulation. From this simulation,

we record the integrated water rate⟨qfw⟩i+ 1

2

and gas rate⟨qfg⟩i+ 1

2

for each coarse

interface i+ 12

as functions of the average water saturation⟨Sfw⟩i

of the upstream

block. Here,⟨Sfw⟩i

is computed using Equation (2.12) or (2.13), as appropriate. The

integrated flux functions are stored as functions which we designate as Fw,i+ 12(Sw)

and Fg,i+ 12(Sw). For blocks that contain producers, we record the integrated fluxes


0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6

Flux

thr

ough

coa

rse

inte

rfac

e

Sw

Water flux Gas flux

(a) Smooth flux profile

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6

Flux

thr

ough

coa

rse

inte

rfac

e

Sw

Water flux Gas flux

(b) Noisy flux profile

Figure 2.4: Examples of water and gas flow rates as functions of upstream averagewater saturation. The flux profiles at some interfaces are generally smooth (a), whileothers can be noisy (b).

from the block to the wellbore⟨qfw,well

⟩i

and⟨qfg,well

⟩i. The fluxes are stored as

Fw,well,i(Sw) and Fg,well,i(Sw).

Examples of interface fluxes as functions of upstream water saturation are shown

in Figure 2.4. At the flow rates which are considered in this work, the resulting

Fw,i+ 12(Sw) and Fg,i+ 1

2(Sw) may be noisy; i.e., they may exhibit fluctuations (Figure

2.4b). As an optional step, we can apply a smoothing method to reduce or eliminate

the noise. In this work, we apply locally weighted linear regression, which will be

discussed later in this chapter.

We also compute the average fine-scale gas pressure⟨pfg⟩i

from Equation (2.11).

As the first step of the global upscaling procedure, the initial estimates for the up-

scaled mobility of the water and gas phases, λ∗w,i+ 1

2

and λ∗g,i+ 1

2

, are computed from

the fine-scale solutions using Equations (2.10a) and (2.10b). The resulting λ∗g,i+ 1

2

and

λ∗w,i+ 1

2

are stored as functions of the average fine-scale water saturation of the up-

stream block (i or i + 1). For the blocks with producers, we apply Equations (2.24)

to compute the upscaled mobility λ∗j,i.


Because the upscaled functions are computed directly from the underlying fine-

scale solution, the resulting coarse-scale functions should be reasonably accurate.

However, the upscaled relative permeability functions (note that k∗rj = µjλ∗j) may

contain unphysical effects. For example, the relative permeability values at some

water saturations may be negative, much larger than unity, or the overall water

fractional flow f ∗w = λ∗w

λ∗w+λ∗gmay be nonmonotonic. These problems can occur when

the upscaling is performed on a model with a highly heterogeneous permeability

distribution, which is the emphasis of this work. These effects are similar to the

negative transmissibilities that can be encountered when global single-phase upscaling

is applied.

To solve this problem, we reject unphysical λ∗j,i+ 1

2

results and apply an iterative

scheme similar to that proposed in Chen et al. (2008). If a particular λ∗j,i+ 1

2

is rejected

(specific criteria will be given below), we proceed as follows. We first assign λ∗j,i+ 1

2

=

λj,i+ 12

(i.e., we use the rock curve in the first iteration). The global coarse-scale model

is then simulated. New λ∗w,i+ 1

2

and λ∗g,i+ 1

2

functions are then computed from the stored

fluxes Fj,i+ 12

and the coarse-scale pressure and saturation computed from the global

coarse-scale solution:

(λ∗g,i+ 1

2

)ν=1

=Fg,i+ 1

2

(Scw,i+ 1

2

)

T ∗i+ 1

2

(pcg,i − pcg,i+1

) , (2.29a)

(λ∗w,i+ 1

2

)ν=1

=Fw,i+ 1

2

(Scw,i+ 1

2

)

T ∗i+ 1

2

[(pcg,i − P ∗

c,i

(Scw,i

))−(pcg,i+1 − P ∗

c,i+1

(Scw,i+1

))] , (2.29b)

where ν is the iteration counter (ν = 0 indicates the upscaled functions calculated

directly from the fine-scale results) and Scw,i+ 1

2

is the water saturation of the upstream

block from the coarse-scale simulation. The resulting(λ∗g,i+ 1

2

)ν=1

and(λ∗w,i+ 1

2

)ν=1

are stored as functions of Scw,i+ 1

2

. For blocks that contain producers, we compute the


well-block λ∗j from:

(λ∗g,i)ν=1

=Fg,well,i

(Scw,i

)

WI∗i(pcg,i − pcwell,i

) , (2.30a)

(λ∗w,i)ν=1

=Fw,well,i

(Scw,i

)

WI∗i(pcg,i − pcwell,i

) . (2.30b)

We then check the resulting(λ∗j,i+ 1

2

)ν=1

and(λ∗j,i)ν=1

, replace those that are

unphysical with(λ∗j,i+ 1

2

)ν=0

or(λ∗j,i)ν=0

, as appropriate, and run the new coarse

model for the iteration ν = 2. The process is repeated for a specified number of

iterations (typically, four iterations are used).

The iterative global upscaling procedure to compute λ∗w and λ∗g is summarized as

follows:

1. Compute T ∗ and WI∗ from global single-phase upscaling.

2. Compute P ∗c from the capillary-limit steady-state method.

3. Run the fine-scale two-phase global simulation.

4. Compute⟨qfw⟩i+ 1

2

and⟨qfg⟩i+ 1

2

from the fine-scale results and store them as

Fw,i+ 12(Sw) and Fg,i+ 1

2(Sw), where Sw here indicates average fine-scale water

saturation of the upstream coarse block. Smooth the curves if desired.

5. Compute⟨pfg⟩

and⟨Sfw⟩

corresponding to each coarse block from the fine-scale

results using (2.11) and (2.12).

6. Compute the initial estimates for λ∗w,i+ 1

2

and λ∗g,i+ 1

2

at each coarse interface using

(2.10). Retain physical curves. Replace unphysical curves by the rock curves.

7. Run the coarse-scale simulation using the updated functions.

8. Read pcg and Scw from the coarse-scale results.


9. For every time step, compute λ∗w,i+ 1

2

and λ∗g,i+ 1

2

at each coarse interface using

(2.29). Retain physical curves. Replace unphysical curves with those from the

previous iteration.

10. Repeat Steps 7-9 for a specified number of iterations.

The process is also shown as a flow chart in Figure 2.5. Note we omit the calcula-

tions for the blocks with producers in this summary and in the flow chart, as these

calculations are analogous to those at the interfaces.

2.2.4 Numerical Calculation of Capillary Pressure

As noted earlier, the capillary pressure curves computed from Equation (2.20) depend

on capillary pressure in the adjacent block. If P ∗c,i+1 is inaccurate, the resulting P ∗

c,i

may be unphysical. Also, even if the resulting function is physical, it can still be

inaccurate. Therefore, we have also developed an iterative approach to compute P ∗c

for all blocks simultaneously.

We use the capillary-limit upscaled P ∗c as the initial guess. We then modify the

iterative global upscaling algorithm presented above to include the upscaled capillary

pressure computation. The modified algorithm is as follows:

1. Compute T ∗ and WI∗ from global single-phase upscaling.

2. Compute P ∗c from the capillary-limit steady-state method.

3. Run the fine-scale two-phase global simulation.

4. Compute⟨qf,convw

⟩i+ 1

2

,⟨qf,capw

⟩i+ 1

2

, and⟨qfg⟩i+ 1

2

from fine-scale results and store

them as F convw,i+ 1

2

(Sw), F cap

w,i+ 12

(Sw), and Fg,i+ 12(Sw), where Sw here indicates aver-

age fine-scale water saturation of the upstream coarse block. Smooth the curves

if desired.

5. Compute⟨pfg⟩

and⟨Sfw⟩

corresponding to each coarse block from the fine-scale

results using (2.11) and (2.12).


Compute T ∗ and WI∗ fromglobal single-phase upscaling

Compute P ∗c from capillary-

limit steady-state method

Run fine-scalesimulation

Compute⟨qfw⟩i+ 1

2

and⟨qfg⟩i+ 1

2

from fine-scale results.

Store as Fw,i+ 12

and Fg,i+ 12. Smooth if necessary.

Compute⟨pfg⟩i

and⟨Sfw

⟩i


Compute λ∗w,i+ 1

2

and

λ∗g,i+ 1

2

using (2.10)

λ∗w,i+ 1

2

, λ∗g,i+ 1

2

physical?Update λ∗

w,i+ 12

, λ∗g,i+ 1

2

Use previousλ∗w,i+ 1

2

, λ∗g,i+ 1

2

Run coarse-scale simulation

End?

Done

Read pcg, Scw from

coarse-scale result

Compute λ∗w,i+ 1

2

,

λ∗g,i+ 1

2

using (2.29)

ν = 0

Yes

No

Yes

No

ν = ν + 1

Figure 2.5: Flow chart showing iterative global upscaling procedure to compute λ∗wand λ∗g.


6. Compute the initial estimates of P ∗c,i, λ

∗w,i+ 1

2

, and λ∗g,i+ 1

2

using (2.20), (2.21) and

(2.22). Retain physical curves. Replace unphysical P ∗c,i with the capillary-limit

P ∗c,i and unphysical λ∗

j,i+ 12

with the rock curves.

7. Run the coarse-scale simulation using the updated functions.

8. Read pcg and Scw from the coarse-scale results.

9. Compute P ∗c,i, λ

∗w,i+ 1

2

, and λ∗g,i+ 1

2

using (2.31). Retain physical curves. Replace

unphysical curves with those from the previous iteration.

10. Repeat Steps 7-9 for a specified number of iterations.

Similar to the previous approach, Step 10 computes the upscaled functions λ∗g,i+ 1

2

,

λ∗w,i+ 1

2

, and P ∗c using the stored fluxes and the coarse-scale pressure and saturation:

λ∗g,i+ 1

2=

Fg,i+ 12

(Scw,i+ 1

2

)

T ∗i+ 1

2

(pcg,i − pcg,i+1

) , (2.31a)

λ∗w,i+ 1

2=

F convw,i+ 1

2

(Scw,i+ 1

2

)

T ∗i+ 1

2

(pcg,i − pcg,i+1

) , (2.31b)

P ∗c,i

(Scw,i

)= P ∗

c,i+1

(Scw,i+1

)−(pcg,i − pcg,i+1

)F cap

w,i+ 12

(Scw,i+ 1

2

)

F convw,i+ 1

2

(Scw,i+ 1

2

)

, (2.31c)

where Scw,i+ 1

2

is the upstream water saturation from the most recent coarse-scale

result.

The algorithms for iterative global upscaling with capillary-limit P ∗c and with

numerically calculated P ∗c are very similar. The flow chart of the iterative procedure

to calculate λ∗j and P ∗c is shown in Figure 2.6. Again, in the description and the flow

chart, the well calculations are omitted, though they must also be performed.

2.2.5 Criteria for Acceptable λ∗j and P ∗c

As noted above, a physical check is performed once the upscaled mobility is computed.

In many cases encountered in this work, the resulting upscaled mobility at some


Compute T ∗ and WI∗ fromglobal single-phase upscaling

Compute P ∗c from capillary-

limit steady-state method

Run fine-scalesimulation

Compute⟨qf,convw

⟩i+ 1

2

,⟨qf,capw

⟩i+ 1

2

and⟨qfg⟩i+ 1

2

from fine-scale

results. Store as F convw,i+ 1

2

, F cap

w,i+ 12

, and Fg,i+ 12. Smooth if necessary.

Compute⟨pfg⟩i

and⟨Sfw

⟩i


Compute P ∗c,i, λ

∗w,i+ 1

2

and λ∗g,i+ 1

2

using (2.20), (2.21), (2.22)

P ∗c,i, λ

∗w,i+ 1

2

,

λ∗g,i+ 1

2

physical?

Update P ∗c,i,

λ∗w,i+ 1

2

, λ∗g,i+ 1

2

Use previous P ∗c,i,

λ∗w,i+ 1

2

, λ∗g,i+ 1

2

Run coarse-scale simulation

End?

Done

Read pcg, Scw from

coarse-scale result

Compute P ∗c,i, λ

∗w,i+ 1

2

,

λ∗g,i+ 1

2

using (2.31)

ν = 0

Yes

No

Yes

No

ν = ν + 1

Figure 2.6: Flow chart showing iterative global upscaling procedure to compute λ∗w,λ∗g, and P ∗

c . Shaded blocks indicates modifications from Figure 2.5.

2.3. OTHER METHODS AND ISSUES 29

saturation may be negative, or the upscaled relative permeability (calculated from

k∗rj = µjλ∗j) may be much greater than unity. This situation occurs especially when

the flow rates are low. In this work, we retain upscaled functions that satisfy

min(k∗rj)≥ 0, max

(k∗rj)≤ 2. (2.32)

We also require that the water fractional flow increases as water saturation in-

creases. In the absence of capillary pressure effects, this leads to the condition

df ∗w

dScw≥ 0, (2.33)

where f ∗w = λ∗w

λ∗w+λ∗gis the fractional flow of water. However, with capillary pressure

effects, f ∗w depends on both phase mobilities and capillary pressure difference. We

find that requiring f ∗w to be a monotonically increasing function of Scw is an overly

restrictive condition, as the effects of capillary pressure are not accounted for. In this

work, we apply the following condition

df ∗w

dScw≥ −0.2, (2.34)

which allows a slight decrease in f ∗w. If (2.32) or (2.34) are not satisfied, we reject

the updated function and use the upscaled function from the previous iteration (or

the rock curves if the function has never been updated). Note that our treatment

here will reject both of the updated λ∗j,i+ 1

2

if either λ∗w,i+ 1

2

or λ∗g,i+ 1

2

is found to be

unphysical based on the criteria above.

2.3 Other Methods and Issues

2.3.1 Smoothing

As shown in Figure 2.4, the gas and water fluxes at some interfaces are generally

smooth, while the fluxes at other interfaces may display oscillations. In our numerical

tests, oscillatory fluxes often occur when the flow is capillary-dominated, i.e., the flow


rate is low. These fluxes may cause unphysical results when they are used in the

iterative upscaling scheme.

To remove this effect, we apply locally weighted linear regression for smoothing.

The locally weighted linear regression model assumes that the data,(Sw, Fw,i+ 1

2

)

or(Sw, Fg,i+ 1

2

)in our case, can be fitted locally by straight lines. In the following

description of the locally weighted linear regression, we use the notation (x, y) as

generic independent and dependent variables, respectively. This description follows

the discussion in Ng (2012). Around a location x, we approximate the value of y as

a straight line such that

y = θ1x+ θ0, (2.35)

or in a vector notation

y = hθ(x) = θTx, (2.36)

where θ = [θ1 θ0]T and x = [x 1]T . The goal of the locally weighted linear regression

is to find the parameter θ to minimize the weighted square error between the straight

line and the data

θ = arg minθ

m∑

j=1

wj(θTxj − yj

)2, (2.37)

where xj = [xj 1]T , (xj, yj) is a data point, wj is the weight of each data point, and

m is the total number of data points. Solving the optimization problem, which is

similar to the ordinary least square, a closed-form solution of the optimal parameter

θ is obtained as

θ = (XTWX)−1XTWy, (2.38)

where

X =

x1 1

x2 1...

...

xm 1

, y =

y1

y2

...

ym

, W =

w1

w2

. . .

wm

. (2.39)

Note that because we fit a local straight line to each location x, the optimal parameter


θ changes as a function of x. The local weight wj is usually taken as

wj = exp

(−(xj − x)2

2τ 2

), (2.40)

where τ is a bandwidth parameter. Figure 2.7 shows the effect of changing τ on the

smoothed result. In general, larger τ incorporates more information from the data

points away from the point x and may result in an overly smoothed curve. Small τ

only takes local points into account, so the local oscillations may not be eliminated.

In this work, we experimented with τ to assure that noise was eliminated, but the

local trends were preserved. We found τ = 0.025 to provide the appropriate balance.

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

Upstream Swc

Gas

flux

thro

ugh

coar

se in

terf

ace

Noisy data

τ=0.005

τ=0.025

τ=0.1

Figure 2.7: Comparison of the results from locally weighted linear regression usingdifferent values of τ . The noisy flux is from Figure 2.4b.

2.3.2 Local k∗ Upscaling with J−Function

The J−function model was originally proposed by Leverett (1940). The J−function

represents capillary pressure functions for rocks that are of the same lithology as

follows:

Pc(Sw) = Pc(Sw, k, φ) = J(Sw)

√kref/φref

k/φ, (2.41)


where kref is a reference permeability, φref is the reference porosity, and k and φ are

the permeability and porosity of the rock (grid block in our case) of interest. The

J(Sw) function is considered to be given for a particular lithology.

An alternative approach to assigning P ∗c , based on the use of J(Sw), was suggested

by Saadatpoor et al. (2011). Essentially, the coarse blocks were assumed to share the

same J−function as the fine-scale block, which gives

P ∗c (Scw) = Pc(S

cw, k, φ

∗) = J(Scw)

√kref/φref

k/φ∗. (2.42)

Here, k is the upscaled permeability, φ∗ is the upscaled porosity, and J(Scw) is the

same as the fine-scale J−function.

Saadatpoor et al. (2011) used a simple geometric average of the fine-scale perme-

ability to compute the coarse-scale k. In this work, however, we apply local upscaling

for the target coarse-block with standard boundary conditions (constant pressure at

the inlet and outlet and no-flow conditions elsewhere) to compute k∗x, k∗y, and k∗z .

The permeability value used in the J−function is the geometric average of these

three permeability values

k = 3

√k∗xk

∗yk

∗z . (2.43)

Using this k seems more appropriate than simply using the geometric average of the

fine-scale permeability, as it incorporates some flow behavior into the construction of

P ∗c . We do not compare results using this treatment with results using the treatment

of Saadatpoor et al. (2011), as both approaches are considered to be quite approxi-

mate.

The resulting P ∗c computed from the J−function method, the capillary-limit

steady-state method, and the numerical scheme are compared in Figure 2.8. We

see that the upscaled capillary pressure using the J−function is significantly different

from the capillary-limit P ∗c and the numerically computed P ∗

c , while the capillary-

limit P ∗c and the numerically computed P ∗

c are very close to each other for Sw ≤ 0.8.

The capillary-limit P ∗c and numerically computed P ∗

c differ at high water saturation

for many curves computed in this work. This is because the gas mobility is low at high


0

5

10

15

20

0.4 0.5 0.6 0.7 0.8 0.9 1

Pc (

psi)

Sw

Capillary-limit PcNumerical Pc

J(k*)

Figure 2.8: Comparison between P ∗c from capillary-limit steady-state method, nu-

merical method, and J−function.

water saturation. Keeping constant gas flow rate requires a large pressure drop in the

gas phase, which results in larger viscous forces compared to capillary forces. This

causes the capillary-limit assumption to be less accurate. The results from coarse-

scale simulations using the three capillary pressure upscaling methods will be shown

in the next chapter.

Chapter 3

Numerical Results

In this chapter, we apply the two-phase upscaling procedures to a synthetic reservoir

model. We assess both the accuracy and robustness of the upscaling methods. In the

accuracy assessment, we compare the results from coarse-scale simulations to results

from the corresponding fine-scale simulation. In the robustness assessment, coarse

models based on the global upscaling of one specific case are applied for cases with

different flow rates or well locations.

3.1 Model Construction

In this section, we describe the reservoir models used in this study. The discus-

sion covers the permeability fields, rock and fluid properties, and well locations and

controls.

3.1.1 Reservoir Model

The reservoir models used in the study are two-dimensional systems containing 200×100 grid blocks (in the x− and y−directions, respectively). The size of each block is

2 ft× 1 ft. The rock is taken to be incompressible and of uniform porosity (φ = 0.25)

throughout the reservoir. The permeability distribution is generated using Sequential

Gaussian Simulation (Deutsch and Journel, 1992) with the parameters shown in Table

35

36 CHAPTER 3. NUMERICAL RESULTS

3.1. Two types of permeability distributions, as shown in Figure 3.1, are considered

in this work. In all cases, we upscale uniformly by a factor of 10 in both the x− and

y−directions. This gives coarse models containing 20× 10 grid blocks.

Table 3.1: Parameters for grid and geological model

Parameter Symbol Value UnitFine-scale grid block dimension Nx, Ny 200, 100 blocks

Fine-scale block size Dx, Dy 2,1 ftVariogram type Spherical -

Variogram range in x−direction λx 0.05, 0.4 -Variogram range in y−direction λy 0.05 -

Mean permeability µk 200 mdStandard deviation of log permeability σln k 2 -

Permeability anisotropy ky/kx 0.01 -

(a) λx = 0.05, λy = 0.05

(b) λx = 0.40, λy = 0.05

Figure 3.1: Permeability fields (log scale) used in the study.

The fluids are taken to be incompressible and immiscible. We use the black-oil

model in Stanford’s General Purpose Research Simulator (GPRS) (Cao, 2002) for

all simulations. We model the water component as the water phase and the CO2

3.1. MODEL CONSTRUCTION 37

component as the oil phase. This treatment is often used for simulating CO2-water

systems. In the results, we will refer to the CO2 phase as the gas phase. The fluid

mobility ratio (µg/µw) is held constant at 10, which is a typical value at reservoir

conditions (Lemmon et al., 2005).

The fine-scale relative permeability functions follow the Brooks-Corey correlation

(Brooks and Corey, 1964) and are as follows:

krw = S2w, krg = (1− Sw)2 . (3.1)

Capillary pressure (Pc) is modeled by the J−function:

J(Sw) = aS−1/bw , (3.2)

where a and b are constants taken as a = 1 and b = 0.7 in this study. The relationship

between J(Sw) and Pc(Sw) is given in Equation (2.42). Here, we use kref = 165 md

and φref = 0.25. The relative permeability curves and J−function are shown in

Figure 3.2.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Sw

k r

krw

krg

(a) Relative permeability curves

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

J, k

ref=

165

md

Sw

J

(b) J−function

Figure 3.2: Rock-fluid parameters specified for fine-scale simulation.

Flow is driven in all cases by one injector and one producer. In this study, we

consider three different flow scenarios, as shown in Figure 3.3: flow in the x−direction,

flow in the y−direction, and corner-to-corner flow. These flows, and combinations of


these flows, are relevant to the carbon sequestration problem. For example, flow

in the y−direction corresponds to the rise of the injected gas toward the cap rock.

Corner-to-corner flow may be relevant to situations where the pressure field results

in flow upward as well as in the horizontal direction. Note that gravity is neglected

in these simulations, so not all of the relevant physics is included in this work.

(a) Flow in x−direction (b) Flow in y−direction (c) Corner-to-corner flow

Figure 3.3: Locations of the injector (red) and producer (blue) in this study.

In the simulations, we specify the flow rate of the injector and the bottom-hole

pressure at the producer. In all cases, we specify the producer bottom-hole pressure

to be 3000 psi. The range of flow rates is chosen to be (approximately) representative

of those in carbon storage operations. We define the flow rates as “low,” “medium,”

and “high.” The low rate is chosen such that 1 PVI of CO2 is injected in 1000 years.

This corresponds to 0.03 PVI in 30 years, which is about the typical rate for CO2 stor-

age operations. The high rate results in 1 PVI of CO2 injected in 100 years. The low,

medium, and high rates correspond to the injection rates of 10, 30, and 100 bbl/day,

respectively. Note that 100 bbl/day = 15.9 m3/day = 11.6 tonnes CO2/day at reser-

voir conditions.

We can also describe the flow regime in terms of the capillary number (Nc).

Virnovsky et al. (2004) define Nc as

Nc =|∇pg|lpc

∆Pc, (3.3)

where |∇pg| is the magnitude of global (large-scale) pressure drop, lpc is the char-

acteristic length of the capillary heterogeneity, and ∆Pc is the contrast in capillary

pressure. Although the capillary number varies spatially, and changes as the simula-

tion progresses, we attempt to approximate it here from the fine-scale permeability

3.2. UPSCALING RESULTS 39

and the J−function. We approximate |∇pg| as ∆pgL

, where ∆pg is the difference be-

tween the injector and the producer pressures and L is the distance between the wells.

The quantity ∆Pc is calculated as Pc (Sw → 1, kmin)− Pc (Sw → 1, kmax), where kmin

is taken as kmin = exp (lnµk − σln k) and kmax = exp (lnµk + σln k). Thus, ln kmin and

ln kmax correspond to ±1 standard deviation (±σln k) from lnµk, where lnµk is the

mean of log-permeability. We approximate lpc to be the correlation length of perme-

ability. For flow in the x− and y− directions, the capillary number is approximated

as∆pglpc∆PcL

=∆pg∆Pc

λk, (3.4)

where k = x, y corresponds to the flow direction. Using the above approximation,

we obtain Nc = 0.67 and 0.067 for flow in the x−direction with high and low rates,

respectively. For flow in the y−direction, we obtain Nc = 1.87 and 0.187 for high and

low rates, respectively. These Nc values fall in the rate-sensitive region, as discussed

in Lohne et al. (2006). This rate-sensitive behavior is demonstrated in Figure 3.4,

where we plot the fractional flow of gas at the producer, as a function of PVI, from

the fine-scale simulations. In this plot, the injection rates vary from 10 bbl/day (low

rate) to 100 bbl/day (high rate) in increments of 10 bbl/day. The fact that the curves

do not collapse indidates that the displacement is indeed rate sensitive.

3.2 Upscaling Results

In this section, we investigate the accuracy of the different upscaling procedures under

various flow scenarios. The results from coarse-scale simulations are compared to

corresponding results from the fine-scale simulations. We use the notation indicated

in Table 3.2 to identify the results from different upscaling methods.

In this chapter, we present results for a model with small correlation lengths

(λx = 0.05, λy = 0.05, Figure 3.1a). Results are presented for medium, high, and

low flow rates in the x− and y−directions. Results for the other reservoir model

(λx = 0.4, λy = 0.05) are included in Appendix A.


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

Low rate

High rate

Medium rate

Increasing viscous/capillary ratio

Figure 3.4: Gas fractional flow at the producer for flow in the x−direction at variousrates.

Table 3.2: Abbreviations for the upscaling methods applied in this section

Single-phase Capillary pressure Relative permeabilityAbbreviation

upscaling upscaling upscalingLocal k∗ J(k∗) Rock curves used L1P/J

Global T ∗ Capillary-limit P ∗c Rock curves used G1P/Pc

Global T ∗ Capillary-limit P ∗c Global k∗rj, no iteration G1P/G2P

Global T ∗ Capillary-limit P ∗c Global k∗rj, after iteration G1P/iG2P

Global T ∗ Numerical P ∗c Global k∗rj, no iteration G1P/G2PPc

Global T ∗ Numerical P ∗c Global k∗rj, after iteration G1P/iG2PPc


3.2.1 Flow in x−direction

Medium Flow Rate

We first consider only the capillary-limit steady-state approach for computing P ∗c .

Figure 3.5 shows the gas fractional flow at the producer for the fine-scale model and

for the coarse-scale models generated using different upscaling techniques. The models

with relative permeability upscaling provide accurate results. Both the G1P/G2P and

G1P/iG2P models predict gas breakthrough times very close to that of the fine-scale

model (0.3 PVI). The G1P/G2P model slightly underestimates the gas fractional

flow at late time. This underestimation is partially corrected through the use of

the iterative scheme, as the G1P/iG2P model provides better agreement with the

fine-scale result. The use of rock curves for relative permeability leads to inaccurate

results (L1P/J and G1P/Pc). The G1P/Pc model is clearly the least accurate in

terms of breakthrough time.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J

G1P/PcG1P/G2PG1P/iG2P

Figure 3.5: Gas fractional flow at the producer for flow in the x−direction (mediumflow rate).

We next consider the ability of the various methods to capture the gas saturation


(a) Fine scale (b) Average fine scale

(c) Local k∗ with J−function (L1P/J) (d) Global T ∗ with CL P ∗c (G1P/Pc)

(e) Global T ∗ with CL P ∗c and global

λ∗j , no iteration (G1P/G2P)(f) Global T ∗ with CL P ∗

c and globalλ∗j , after iteration (G1P/iG2P)

Figure 3.6: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (mediumflow rate).






Figure 3.7: Saturation map (Sg) at 1 PVI for flow in the x−direction (medium flowrate).


throughout the reservoir. Figures 3.6a and 3.7a show the gas saturation from the fine-

scale simulation at 0.425 PVI and 1 PVI, and their corresponding volume-averaged

saturation distributions are shown in Figures 3.6b and 3.7b. The value of 0.425 PVI

is chosen to represent a time after significant gas breakthrough has occured, while 1

PVI represents the end of the simulation. Coarse-scale results should ideally match

the volume-averaged results. Qualitatively, we observe that the gas saturation from

the L1P/J model (Figures 3.6c and 3.7c) is more discontinuous than the averaged

fine-scale results and the other coarse-scale results at both 0.425 PVI and 1 PVI.

Detailed observation indicates that the G1P/G2P and G1P/iG2P saturation fields

are the most accurate compated to the averaged fine-scale results.

To more clearly quantify the inaccuracy in the gas saturation distribution, we

define the normalized L2−norm of the differences between the fine-scale and coarse-

scale results as:

e(t) =

∥∥Scg(t)−⟨Sfg (t)

⟩∥∥2∥∥∥

⟨Sfg (t)

⟩∥∥∥2

, (3.5)

where⟨Sfg (t)

⟩is the volume average of the fine-scale gas saturation corresponding to a

single coarse-scale block at dimensionless time t (defined in terms of PVI). The errors

e(t) are calculated at each simulation time and are plotted in Figure 3.8. To quantify

the time-averaged saturation error for each case, we define the overall saturation error

as

ET =1

tend

∫ tend

0

e(t)dt, (3.6)

where tend is the total dimensionless simulation time, which is 1 PVI for all cases

presented in this chapter.

As shown in Figure 3.8, L1P/J leads to the largest error in gas saturation at all

times. In addition, calculating upscaled relative permeability from the global solu-

tion (G1P/G2P) causes significant improvement from the case with capillary pres-

sure upscaling alone (G1P/Pc). The iterative scheme (G1P/iG2P) further improves

coarse-model accuracy. The overall saturation error of each model is shown in Table

3.3. These results are consistent with our observations above.


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.8: Normalized L2−norm of the error in gas saturation for flow in thex−direction (medium flow rate).

Table 3.3: Overall saturation error for flow in the x−direction (medium flow rate)

Model L1P/J G1P/Pc G1P/G2P G1P/iG2P

ET 0.02109 0.01388 0.008124 0.006342

We also consider the performance of each coarse model in terms of injector bottom-

hole pressure (recall that we specify the injection rate). Results for this quantity are

shown in Figure 3.9. Interestingly, the most accurate result is provided by G1P/G2P,

though G1P/iG2P is also accurate.

We now present the results for coarse models with numerically calculated P ∗c .

These results are shown in Figures 3.10 to 3.12. These results are similar to those

that use the capillary-limit P ∗c . The gas fractional flow plot (Figure 3.10) shows that

the gas breakthrough times from the coarse models with global upscaling are very

close to the breakthrough time from the fine-scale model. The gas flow rates at late

simulation time are also very close to that of the fine-scale model. Figures 3.11 and

3.12 compare the gas saturation of the coarse models (G1P/G2PPc and G1P/iG2PPc)

to that of the fine-scale model. The results from both models are similar to those from


3003

3004

3005

3006

3007

3008

3009

3010

3011

3012

3013

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Inje

ctor

bot

tom

-hol

e pr

essu

re (

psi)

PVI

FineL1P/J


Figure 3.9: Injector bottom-hole pressure for flow in the x−direction (medium flowrate).

the coarse models with capillary-limit P ∗c , and both models reproduce accurately the

fine-scale gas saturation distribution. The overall saturation error of the G1P/iG2PPc

model, shown in Table 3.4, is 0.006390, which is very close to that of the G1P/iG2P

model shown in Table 3.3 (0.006342).

Table 3.4: Overall saturation error for flow in the x−direction (medium flow rate)with numerical P ∗

c calculation

Model L1P/J G1P/Pc G1P/G2PPc G1P/iG2PPcET 0.02109 0.01388 0.005856 0.006390

These results demonstrate that the iterative global upscaling procedure with nu-

merically computed P ∗c can produce accurate coarse-scale models. However, the re-

sults are only incrementally more accurate than those using the capillary-limit P ∗c . Be-

cause the construction of P ∗c from the capillary-limit condition is simpler and cleaner,

and because we do not gain much in terms of accuracy from numerically computed

P ∗c , we limit our subsequent discussion and results to the capillary-limit P ∗

c approach.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J

G1P/PcG1P/G2PPcG1P/iG2PPc

Figure 3.10: Gas fractional flow at the producer for flow in the x−direction (mediumflow rate) with numerical P ∗

c calculation.


(c) Global T ∗ with numerical P ∗c and

global λ∗j , no iteration (G1P/G2PPc)(d) Global T ∗ with numerical P ∗

c andglobal λ∗j , after iteration (G1P/iG2PPc)

Figure 3.11: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (mediumflow rate) with numerical P ∗

c calculation.


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.12: Normalized L2−norm of the error in gas saturation for flow in thex−direction (medium flow rate) with numerical P ∗

c calculation.

High Flow Rate

Figures 3.13 to 3.16 show the results for flow in x−direction with high rate (100 bbl/day).

In this case, G1P/iG2P is again the most accurate. All methods predict early gas

breakthrough relative to the fine-scale model (for which breakthrough occurs at 0.32

PVI). The G1P/iG2P model predicts a slightly early breakthrough time (0.3 PVI)

but after 0.4 PVI it is very accurate.

Figures 3.14 and 3.15 present saturation maps, and Figure 3.16 displays the sat-

uration error e(t) for each coarse model. Similar to the previous case, the saturation

error for the L1P/J model is the largest, followed by the G1P/Pc model. In this case,

we observe more difference between the G1P/G2P and the G1P/iG2P models. The

overall saturation errors for this case are shown in Table 3.5.

Table 3.5: Overall saturation error for flow in the x−direction (high flow rate)

Model L1P/J G1P/Pc G1P/G2P G1P/iG2PET 0.01973 0.01227 0.009990 0.006007

This case demonstrates improvement in the coarse-scale model due to the use of


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure 3.13: Gas fractional flow at the producer for flow in the x−direction (highflow rate).

the iterative scheme. The G1P/G2P model represents an improvement relative to the

case with single-phase upscaling and P ∗c (G1P/Pc model), but the flow results still

show some error. This is because many of the upscaled mobilities from the global

calculation are rejected. As the flow rate increases, the capillary-limit assumption

becomes less accurate, so many of the resulting P ∗c curves are inaccurate. With

inaccurate (precomputed) P ∗c , many λ∗

j,i+ 12

are unphysical and are therefore rejected.

As a result, the G1P/G2P model is not very different from the G1P/Pc model. By

iterating, we improve the λ∗j,i+ 1

2

, and this enables the G1P/iG2P model to provide

better accuracy.







Figure 3.14: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (high flowrate).






Figure 3.15: Saturation map (Sg) at 1 PVI for flow in the x−direction (high flowrate).


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.16: Normalized L2−norm of the error in gas saturation for flow in thex−direction (high flow rate).

Low Flow Rate

Figures 3.17 to 3.19 show the coarse-scale results for low injection rate (10 bbl/day).

In this case, the L1P/J model predicts an accurate gas breakthrough time (0.27 PVI),

but it still underestimates the late-time gas fractional flow. Again, the G1P/Pc

model gives significantly early gas breakthrough time (0.18 PVI) and significantly

underestimates the late-time gas fractional flow. The G1P/G2P and the G1P/iG2P

models are both reasonably accurate, though we observe some inaccuracy in gas

fractional flow at late time. In terms of saturation error, the models with global

relative permeability upscaling (G1P/G2P and G1P/iG2P models) are again more

accurate than the other models, as is evident in Figure 3.19. The overall saturation

errors are shown in Table 3.6.

Table 3.6: Overall saturation error for flow in the x−direction (low flow rate)



0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure 3.17: Gas fractional flow at the producer for flow in the x−direction (low flowrate).






Figure 3.18: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (low flowrate).


0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.19: Normalized L2−norm of the error in gas saturation for flow in thex−direction (low flow rate).

3.2.2 Flow in y−direction

Medium Flow Rate

We now discuss results for flow in the y−direction. The same reservoir model is

considered. This flow scenario is somewhat representative of a bottom-to-top flow

in a CO2 storage operation, though gravity effects are not considered. Figure 3.20

compares the gas fractional flow at the producer for the different coarse models.

While the L1P/J model generally provided accurate gas breakthrough times for flow

in the x−direction, here it predicts a significantly early gas breakthrough (0.16 PVI,

compared with 0.27 PVI from the fine-scale result). This model also underestimates

gas fractional flow at late time. The performance of the G1P/Pc model is similar

to its performance in the previous cases. It predicts early gas breakthrough (0.2

PVI) and underestimates gas fractional flow at late time. Both the G1P/G2P and

the G1P/iG2P models predict slightly early gas breakthrough but are quite accurate

overall.

Figure 3.21 shows the gas saturation maps for this case. It is evident that the


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure 3.20: Gas fractional flow at the producer for flow in the y−direction (mediumflow rate).






Figure 3.21: Saturation map (Sg) at 0.425 PVI for flow in the y−direction (mediumflow rate).


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.22: Normalized L2−norm of the error in gas saturation for flow in they−direction (medium flow rate).

L1P/J results are not as accurate as those of the other methods. Figure 3.22 compares

the saturation error e(t) for the different coarse models, and the overall saturation

errors are shown in Table 3.7. Again, we see that the L1P/J model is the least

accurate. The G1P/iG2P model leads to only a very slight improvement over the

G1P/G2P model for this case.

Table 3.7: Overall saturation error for flow in the y−direction (medium flow rate)


For this flow scenario, we are also interested in the gas saturation in the top

layer of the model, since this represents the amount of CO2 trapped by the cap

rock. This is important in determining the safety of a CO2 storage operation. The

average gas saturation in the top layer, as a function of PVI, is shown in Figure 3.23.

From the plot, we see that the L1P/J model significantly underestimates this gas

saturation, while the G1P/Pc model overestimates the gas saturation. The G1P/G2P

and G1P/iG2P are both quite accurate, though the non-iterated method (G1P/G2P)


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.2 0.4 0.6 0.8 1

Aver

age

gas

satu

ratio

n of

the

top

laye

r

PVI

FineL1P/J


Figure 3.23: Average gas saturation at the top layer for flow in the y−direction(medium flow rate).

is, surprisingly, slightly more accurate for this particular quantity.

Results for the bottom-hole pressure at the injection well are shown in Figure 3.24.

The G1P/G2P results are generally quite accurate, but the G1P/iG2P model shows

significant error at around 0.3 PVI. This may be because the iteration procedure

attempts to preserve rates but not pressures. It may be possible to improve these

results by introducing the well block λ∗j for injectors for use in Equation (2.8).


3008

3010

3012

3014

3016

3018

3020

3022

3024

3026

3028

3030

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Inje

ctor

bot

tom

-hol

e pr

essu

re (

psi)

PVI

FineL1P/J


Figure 3.24: Injector bottom-hole pressure for flow in the y−direction (medium flowrate).

High Flow Rate

Figures 3.25 to 3.28 and Table 3.8 show results for flow in y−direction at high rate

(100 bbl/day). Qualitatively, the results are similar to those for the medium rate case.

The L1P/J model again predicts significantly early gas breakthrough and underes-

timates gas fractional flow at late time. The G1P/Pc, G1P/G2P, and G1P/iG2P

models are all reasonably accurate, though the G1P/Pc model slightly underesti-

mates the late-time gas fractional flow. The L1P/J model is again the least accurate

in terms of saturation (Figures 3.26 and 3.27). The other models all display about

the same level of accuracy. Results for the average gas saturation in the top layer are

analogous to those for the medium rate case.

Table 3.8: Overall saturation error for flow in the y−direction (high flow rate)



0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure 3.25: Gas fractional flow at the producer for flow in the y−direction (high flowrate).






Figure 3.26: Saturation map (Sg) at 0.425 PVI for flow in the y−direction (high flowrate).


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.27: Normalized L2−norm of the error in gas saturation for flow in they−direction (high flow rate).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.2 0.4 0.6 0.8 1

Aver

age

gas

satu

ratio

n of

the

top

laye

r

PVI

FineL1P/J


Figure 3.28: Average gas saturation at the top layer for flow in the y−direction (highflow rate).


Low Flow Rate

Results for flow in the y−direction at low rate (10 bbl/day) are shown in Figures 3.29

to 3.32 and Table 3.9. Again, we see that the L1P/J model is the least accurate and

that the G1P/G2P and G1P/iG2P models are both generally accurate. None of the

coarse models capture the abrupt increase in gas fractional flow at 0.26 PVI and at

0.35 PVI, which is observed in the fine-scale results. We again see that the G1P/G2P

model gives the best accuracy in average gas saturation in the top layer (Figure 3.32).

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure 3.29: Gas fractional flow at the producer for flow in the y−direction (low flowrate).

Table 3.9: Overall saturation error for flow in the y−direction (low flow rate)








Figure 3.30: Saturation map (Sg) at 0.425 PVI for flow in the y−direction (low flowrate).

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 1

RM

S Er

ror

of g

as s

atur

atio

n

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure 3.31: Normalized L2−norm of the error in gas saturation for flow in they−direction (low flow rate).

3.3. ROBUSTNESS TO CHANGES IN BOUNDARY CONDITIONS 61

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.2 0.4 0.6 0.8 1

Aver

age

gas

satu

ratio

n of

the

top

laye

r

PVI

FineL1P/J


Figure 3.32: Average gas saturation at the top layer for flow in the y−direction (lowflow rate).

3.3 Robustness to Changes in Boundary Conditions

The use of iterative global upscaling for λ∗j along with the capillary-limit P ∗c was shown

to provide accurate coarse-scale models for a range of flow rates and for different

global flow scenarios. However, a different model was constructed for each case. In

this section, we test for robustness by applying the P ∗c and λ∗j obtained for one case

to other cases involving different rates or well locations. Results are compared to the

corresponding fine-scale simulations in order to assess coarse-model accuracy.

3.3.1 Change in Flow Rate

In the first test, we perform iterative global upscaling at one rate and apply the result-

ing λ∗j to other rates within the range of interest (note that P ∗c is rate-independent).

We perform the iterative global upscaling at rates that are in the middle of the range

of interest. For the following discussion, we define the notation G1P/iG2Pr to de-

note the coarse model that uses the λ∗j curves from the iterative global upscaling

at the rate r. In this work, we test the robustness of the models G1P/iG2P30 and


G1P/iG2P60. The notation G1P/iG2P refers to the coarse model for which iterative

global upscaling is performed at each rate (as was done in the previous sections).

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

10 20 30 40 50 60 70 80 90 100

Ove

rall

satu

ratio

n er

ror

Rate

Optimized for each rateOptimized for rate = 30Optimized for rate = 60

G1P/Pc

Figure 3.33: Overall saturation error for G1P/iG2P30 and G1P/iG2P60, withG1P/iG2P and G1P/Pc shown for reference (flow in the x−direction).

Figures 3.33 and 3.34 show the robustness results for the model considered in

the previous sections (λx = 0.05, λy = 0.05) for flow in the x− and y−directions,

respectively. The overall saturation errors computed using Equation (3.6) are plotted

as functions of flow rate. The overall saturation errors for the G1P/iG2P and G1P/Pc

models are plotted for reference. We would expect the error to be the lowest when

the upscaled model is constructed for the rate at which it is applied. With reference

to Figures 3.33 and 3.34, this means that we expect the red points (G1P/iG2P) to fall

below the others. This is usually the case, but not always. Exceptions are likely due

to our criteria for discarding unphyiscal λ∗j,i+ 1

2

in the iteration process (see Section

2.2.5).

For flow in the x−direction (Figure 3.33), we see that the use of upscaled functions

computed for rates at either 30 or 60 bbl/day leads to accurate coarse-scale results

for rates over the range 10− 100 bbl/day. For flow in the y−direction (Figure 3.34),

better overall accuracy is achieved using upscaled functions computed for a rate of


0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

10 20 30 40 50 60 70 80 90 100

Ove

rall

satu

ratio

n er

ror

Rate

Optimized for each rateOptimized for rate = 30Optimized for rate = 60

G1P/Pc

Figure 3.34: Overall saturation error for G1P/iG2P30 and G1P/iG2P60, withG1P/iG2P and G1P/Pc shown for reference (flow in the y−direction).

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J

G1P/PcG1P/iG2P10G1P/iG2P30G1P/iG2P60

Figure 3.35: Gas fractional flow at the producer for flow in the x−direction at lowrate (10 bbl/day) using various models.


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J

G1P/PcG1P/iG2P100G1P/iG2P30G1P/iG2P60

Figure 3.36: Gas fractional flow at the producer for flow in the x−direction at highrate (100 bbl/day) using various models.

60 bbl/day. In total, the results in Figures 3.33 and 3.34 show that the upscaled

models are reasonably robust in terms of flow rate.

Figures 3.35 and 3.36 display the gas fractional flow for flow in the x−direction

at 10 and 100 bbl/day, respectively. Coarse models generated at three different rates

are applied. We again see that the upscaled models are reasonably robust and that

they provide results that are more accurate than those using G1P/Pc and L1P/J.

3.3.2 Change in Well Locations

One of the most important limitations of global upscaling is that the coarse models

are constructed based on a specific flow configuration. Therefore, even though the

coarse-scale model may be accurate in reproducing the fine-scale result for the case

upon which it was based, it may not be accurate for other cases. Here, we consider

corner-to-corner flow. We first compute upscaled single-phase properties (T ∗ and

WI∗) by solving the fine-scale corner-to-corner single-phase flow problem. We then

perform two iterative global upscaling computations, one for flow in the x−direction

and the other for flow in the y−direction. Then, we combine the upscaled properties;


i.e., the results for flow in the x−direction are used for λ∗j,x, and the results for flow

in the y−direction are used for λ∗j,y. This combined coarse-scale model is compared

to the fine-scale result.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J

G1P/PcG1P/iG2P for corner-to-corner

G1P/iG2P combined 2 cases

Figure 3.37: Gas fractional flow at the producer for corner-to-corner flow (high flowrate).

The injector is at the bottom-left and the producer is at the top-right corner of

the model as shown in Figure 3.3c. Figure 3.37 displays the gas fractional flow at the

producer for this case. Results from the fine-scale simulation, the coarse-scale model

based directly on corner-to-corner flow, and the coarse-scale model that combines the

x− and y−direction flow results are compared. Both G1P/iG2P coarse-scale models

accurately predict the gas breakthrough and the late-time gas fractional flow, though

the combined coarse model slightly underestimates the gas fractional flow from about

0.5-0.8 PVI.

Figure 3.38 shows the gas saturation maps at 0.425 PVI. The gas saturation maps

for both upscaled models are generally accurate. The total saturation error of the

combined model is 0.008922, compared to 0.006810 for the corner-to-corner model.

For the G1P/Pc model, the error is 0.01300. Thus, we see that the combined model

outperforms the G1P/Pc model.



(c) Global upscaling for corner-to-corner flow (G1P/iG2P)

(d) Combining the results from x−and y−directions

Figure 3.38: Saturation map at 0.425 PVI for the corner-to-corner flow (high flowrate).

Chapter 4

Conclusions and Future Work

4.1 Conclusions

• In this work, we developed a new iterative global upscaling method that is

applicable for two-phase flow with significant capillary pressure heterogeneity

effects. Upscaled capillary pressure functions are computed either analytically,

using the steady-state capillary-limit assumption, or numerically from fine-scale

flow results. Upscaled phase mobility (or relative permeability) functions are

computed in all cases using global fine-scale two-phase flow simulations. Itera-

tion on the upscaled functions based on coarse-scale simulation results provides

improved coarse-model accuracy.

• Numerical results for Gaussian permeability fields over a range of CO2 injection

rates demonstrated that the iterative global upscaling method provides accu-

rate coarse-scale models. These models were shown to accurately capture gas

breakthrough time, gas fractional flow and saturation distribution relative to

results from the corresponding fine-scale models. Coarse models generated us-

ing the new iterative global upscaling scheme were shown to be considerably

more accurate than coarse-scale models constructed using simpler approaches.

• The coarse-scale models with numerically computed P ∗c were shown to be slightly

more accurate than models with capillary-limit P ∗c . However, computation of

67

68 CHAPTER 4. CONCLUSIONS AND FUTURE WORK

P ∗c from the capillary-limit assumption is simpler and cleaner, as the results

do not depend on the specific flow conditions. We therefore recommend us-

ing capillary-limit P ∗c along with the iterative scheme for the upscaled mobility

functions.

• We tested the robustness of the coarse-scale models generated using iterative

global upscaling by running the upscaled models under different flow conditions.

The coarse-models were shown to be reasonably robust; i.e., they provided

coarse-scale results of sufficient accuracy for cases involving different flow rates

or large-scale flow configuration.

4.2 Future Work

• In this work, we only considered the flow of immiscible fluids in the horizontal

(x−y) plane, so gravity effects were not included. It will be useful to incorporate

these effects in future work, and to consider three-dimensional systems. The

dissolution of CO2 in water was also neglected, and these effects should be

included in the models and in the upscaling computations if necessary. Relative

permeability hysteresis effects should also be considered. When all of these

effects are incorporated, our coarse-scale models should be applicable for the

simulation of realistic carbon storage operations.

• We have shown that an upscaling procedure based on fine-scale global simula-

tions can provide an accurate coarse-scale model that is reasonably robust over

a range of CO2 injection rates. However, fine-scale global simulation is com-

putationally expensive, and it would be beneficial to avoid these computations.

Along these lines, extended-local or quasi-global (e.g., adaptive local-global or

ALG) upscaling approaches should be considered. Such an ALG procedure

would extend the work of Chen and Li (2009), who implemented ALG upscal-

ing for two-phase viscous-dominated flows.

• Additional robustness tests should be performed. A range of well locations

should be considered.

Appendix A

Additional Numerical Results

This Appendix presents numerical results for a geological model with longer perme-

ability correlation in the x−direction (λx = 0.4, λy = 0.05). The permeability field

for this model is shown in Figure 3.1b. The problem setup is otherwise identical to

that described in Section 3.1. We present results for medium (30 bbl/day) and high

(100 bbl/day) injection rates. Convergence difficulties were encountered with GPRS

for some of the low-rate cases. This difficulty is also seen in the fine-scale results

for the medium rates, where the fine-scale simulations terminate due to excessive

time-step cuts at around 0.6 PVI (Figures A.1 and A.7).

A.1 Flow in the x−direction

A.1.1 Medium Flow Rate

Table A.1: Overall saturation error for flow in the x−direction (medium flow rate).For these results, tend = 0.62 PVI


69

70 APPENDIX A. ADDITIONAL NUMERICAL RESULTS

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure A.1: Gas fractional flow at the producer for flow in the x−direction (mediumflow rate).






Figure A.2: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (mediumflow rate).

A.1. FLOW IN THE X−DIRECTION 71

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure A.3: Normalized L2−norm of the error in gas saturation for flow in thex−direction (medium flow rate).


A.1.2 High Flow Rate

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure A.4: Gas fractional flow at the producer for flow in the x−direction (high flowrate).

Table A.2: Overall saturation error for flow in the x−direction (high flow rate)


A.1. FLOW IN THE X−DIRECTION 73






Figure A.5: Saturation map (Sg) at 0.425 PVI for flow in the x−direction (high flowrate).

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure A.6: Normalized L2−norm of the error in gas saturation for flow in thex−direction (high flow rate).


A.2 Flow in the y−direction

A.2.1 Medium Flow Rate

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure A.7: Gas fractional flow at the producer for flow in the y−direction (mediumflow rate).

Table A.3: Overall saturation error for flow in the y−direction (medium flow rate).For these results, tend = 0.59 PVI


A.2. FLOW IN THE Y−DIRECTION 75






Figure A.8: Saturation map (Sg) at 0.425 PVI for flow in the y−direction (mediumflow rate).

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure A.9: Normalized L2−norm of the error in gas saturation for flow in they−direction (medium flow rate).


A.2.2 High Flow Rate

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gas

fra

ctio

nal f

low

at

the

prod

ucer

PVI

FineL1P/J


Figure A.10: Gas fractional flow at the producer for flow in the y−direction (highflow rate).

Table A.4: Overall saturation error for flow in the y−direction (high flow rate)


A.2. FLOW IN THE Y−DIRECTION 77






Figure A.11: Saturation map (Sg) at 0.425 PVI for flow in the y−direction (high flowrate).

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

e(t)

PVI

L1P/JG1P/Pc

G1P/G2PG1P/iG2P

Figure A.12: Normalized L2−norm of the error in gas saturation for flow in they−direction (high flow rate).

Nomenclature

Abbreviations

G1P/Pc global T ∗ with capillary-limit P ∗c and rock curves for krj

G1P/G2P global T ∗ with capillary-limit P ∗c and global k∗rj with no iteration

G1P/G2PPc global T ∗ with numerically computed P ∗c and global k∗rj with no iter-

ation

G1P/iG2P global T ∗ with capillary-limit P ∗c and iterated global k∗rj

G1P/iG2PPc global T ∗ with iterated global k∗rj and P ∗c

G1P/iG2Pr coarse-scale model generated from the iterative global upscaling with

a specified flow rate of r bbl/day

GPRS General Purpose Research Simulator

L1P/J local k∗ with J−function and rock curves for krj

PVI pore volume injected

Variables

Dx, Dy fine-scale block size in the x− and y−directions

e normalized L2−norm of differences between fine-scale and coarse-scale

saturations

ET overall saturation error

79

80 NOMENCLATURE

Fj,i+ 12

stored fine-scale flux of phase j at interface i+ 12

as a function of

upstream water saturation

f ∗w upscaled fractional flow of water

J J−function

k absolute permeability tensor

k permeability used in calculating Pc from J−function

kref reference permeability for calculating Pc from J−function

krj relative permeability to phase j

k∗x, k∗y, k

∗z effective permeability in the x−, y−, and z−directions

lpc characteristic length of the capillary heterogeneity

Nc capillary number

Nf number of fine-scale blocks in a coarse-scale block

Nx, Ny number of fine-scale blocks in the x− and y−directions

Pc capillary pressure

P ∗c upscaled capillary pressure

P ∗c,i upscaled capillary pressure for block i

pj pressure of phase j

pwell,i well pressure in block i

qj flux of phase j

qj source term of phase j

qcj,i+ 1

2

coarse-scale flux of phase j across i+ 12

interface

NOMENCLATURE 81

qj,well flux of phase j between the block and the well

Sj saturation of phase j

tend total dimensionless time (PVI) for a simulation case

T ∗i+ 1

2

upscaled transmissibility at i+ 12

interface

uj Darcy velocity of phase j

V bulk volume of a block

WI∗i upscaled well index in block i

〈·〉i averaged property over fine-scale blocks corresponding to coarse block

i

〈·〉i+ 12

integrated flux across the interface i+ 12

Greek

λj mobility of phase j

λ∗j upscaled mobility of phase j

λ∗j,i+ 1

2

upscaled mobility of phase j at the interface i+ 12

λx, λy correlation lengths of permeability in the x− and y−directions

µj viscosity of phase j

µk mean permeability

ν iteration counter

φ porosity

φref reference porosity for calculating Pc from J−function

φ porosity used in calculating Pc from J−function

82 NOMENCLATURE

σln k standard deviation of log permeability

Subscripts

g gas

j phase

w water

Superscripts

∗ upscaled quantity

c coarse-scale property

cap flux due to difference in capillary pressure

conv flux due to difference in gas pressure

f fine-scale property

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