high-resolution numerical simulation of co2 sequestration ... · geological storage of co 2 will be...

HIGH-RESOLUTION NUMERICAL

SIMULATION OF CO2 SEQUESTRATION IN

SALINE AQUIFERS

A REPORT

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

Ruslan Iskhakov

August 2013

I certify that I have read this report and that in my

opinion it is fully adequate, in scope and in quality, as

partial fulfillment of the degree of Master of Science in

Energy Resources Engineering.

Hamdi A. Tchelepi(Principal advisor)

ii

Abstract

We study the migration of CO2 plumes in deep saline aquifers during the post-

injection period using high-resolution numerical simulation. The interactions be-

tween buoyancy driven flow, dissolution of CO2 into the brine, capillary mixing, and

diffusion/dispersion effects lead to complex behaviors in space and time. Here, a

two-component, two-phase mathematical model is used, and the phase behavior is

represented using an equation-of-state. In order to resolve the nonlinear coupling,

the Fully Implicit Method (FIM) is used for time discretization. We find that the

numerical solutions are quite sensitive to both grid resolution and time truncation

errors. These challenges are compounded by the presence of a miscible convective

instability, which can dominate the overall evolution of the plume.

A lock-exchange configuration is set up, in which the entire CO2 plume is em-

placed near the bottom end (inlet) of the aquifer. Following an initial period of

intense counter-current flow, the pressure gradient and the streamlines some distance

away from the bulk of the plume become nearly constant. In order to improve the

overall computational efficiency of the simulations, a moving-outer-boundary (MOB)

scheme was implemented. Specifically, the domain is split into two sub-domains: one

that stretches from the inlet to just beyond the leading edge of the CO2 plume, fol-

lowed by a region that extends all the way to the outer (up-dip) boundary of the

aquifer. In the second sub-domain, the pressure and saturation distributions remain

unchanged. Pressure, velocity, and saturation distributions at the end of a time step

are used to delineate the boundary between the two computational sub-domains. This

is facilitated by the observation made by many investigators including our group -

that the leading-edge speed is nearly constant during the long post-injection period.

iii

When the boundary between the sub-domains needs to by moved, a few small time

steps are performed on the global computational domain. This ensures that the evo-

lution of the capillary transition zone and other mixing phenomena are fully resolved

before operating only on the ‘active’ sub-domain.

This method is somewhat similar to the adaptive implicit method (AIM) when

only the parts of the domain where significant changes in time take place are treated

using FIM. However, AIM has significant stability issues for this problem and can im-

prove the overall performance by only a small margin. All simulations were performed

using the Automatic-Differentiation General Purpose Research Simulator (ADGPRS).

The results demonstrate that compared with the standard approach of solving a global

problem all the time, the MOB method improves the total computational time by

40-50% with 0.5% error (measured in terms of the up-dip leading-edge speed). Sim-

ulation results are then shown for extremely large aquifer models using our MOB

approach.

iv

Acknowledgments

First I would like to present my full appreciation to my adviser Prof. Hamdi Tchelepi

and mentor Denis Voskov for their encouragement, support and guidance throughout

the work. I profoundly appreciate their suggestions, comments and valuable inputs

to this report. I thank both of them for providing me motivation and confidence to

dive into this interesting topic.

I am grateful to the department’s faculty, it has been a pleasure to attend their

classes and communicate with them. Also I am grateful to the staff and SPE Team,

they created a brilliant environment for students.

I am indebted to the generous financial support of the Petroleum Institute and

Abu Dhabi National Oil Company. I am also grateful for the intellectual and financial

support from SUPRI-B over the last two years.

Special thanks to Borschuk Oleg and Dr. Vladimir Savichev, who initiated my

wonderful journey here. I would like to thank Amir Salehi for his help in understand-

ing CO2 migration processes.

I would like to thank my friends and office mates. Without them, student life at

Stanford would have been much more difficult. I also want to express gratitude to

my family for their support and belief in me. And of course, I thank my wife, Dinara,

for her continuous support and love.

v

Contents

Abstract iii

Acknowledgments v

Table of Contents vi

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Mechanisms of CO2 Sequestration in Saline Aquifers . . . . . . . . . 2

1.2 Current Numerical Simulators for CO2 Sequestration . . . . . . . . . 3

2 Problem definition 6

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Numerical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Stages of CO2 plume migration . . . . . . . . . . . . . . . . . . . . . 9

2.4 Computational challenges . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Moving-outer-boundary (MOB) strategy 13

3.1 Accuracy and performance . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Performance for 3D problems . . . . . . . . . . . . . . . . . . . . . . 18

vi

4 Numerical results 20

4.1 GPRS and ADGPRS comparison . . . . . . . . . . . . . . . . . . . . 20

4.2 Numerical diffusion and impact of time truncation error . . . . . . . . 20

4.3 Molecular diffusion impact on the solution . . . . . . . . . . . . . . . 22

5 Conclusions 27

A Flexible restart for AD-GPRS 29

A.1 HDF5 data format . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.2 General framework for restart . . . . . . . . . . . . . . . . . . . . . . 34

A.3 Examples of restart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Nomenclature 38

Bibliography 40

vii

List of Tables

2.1 Physical properties used for our study. . . . . . . . . . . . . . . . . . 7

A.1 Comparison between ASCII and HDF5 data formats. . . . . . . . . . 30

viii

List of Figures

1.1 Progression of CO2 storage mechanism with time (adapted from IPCC,

2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Initial distribution of the CO2 plume in the aquifer. . . . . . . . . . . 8

2.2 Evolution of the CO2 gravity-current in the slopping saline aquifer.

CO2 currents (dark grey) and the zone containing residual saturation

in their wake (light grey). . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Gas phase saturation with time . . . . . . . . . . . . . . . . . . . . . 11

2.4 Timestep size for different resolutions. . . . . . . . . . . . . . . . . . 12

3.1 Location of points where pressure profile plotted (marked as black dots). 13

3.2 Pressure change as a function of time at several location for a reference

fine-grid (full) simulation (12,000 years). . . . . . . . . . . . . . . . . 14

3.3 Pressure variance at several distances of the upper layer zoomed for

150 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Streamline distribution for the full model and zoom-ins around the left

and right tips of the CO2 plume. . . . . . . . . . . . . . . . . . . . . 15

3.5 Schematic domain split of ‘active’ region and the outer boundary region. 16

3.6 Gas saturation map with moving outer-boundary propagation with

time. The location of the moving-outer boundary is denoted by xawf . 16

3.7 Gas saturation map (upper plot) and error between gas saturation

maps with and without MOB strategy (lower plot). . . . . . . . . . . 17

3.8 Comparison of computational time with (right) and without (left) the

MOB strategy for the 2D problem. . . . . . . . . . . . . . . . . . . . 18

ix

3.9 Schematic representation of 3D problem. . . . . . . . . . . . . . . . . 19

3.10 Comparison of computational time with and without our MOB strat-

egy for 3D problems with different numbers of layers in the y-direction. 19

4.1 Comparison of gas saturation map after 12,000 years for refinement in

x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Comparison of gas saturation map after 12,000 years for refinement in

z-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Comparison of up-dip tip location for different resolution depending

on timestep treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Molecular diffusion impact for high resolution. . . . . . . . . . . . . . 24

4.5 Molecular and numerical diffusion effects. . . . . . . . . . . . . . . . . 25

4.6 Comparison of tip location for different resolution with and without

molecular diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A.1 An HDF5 file with a strictly hierarchical group structure (a), an HDF5

file with a directed graph group structure including a circular reference

(b), and an HDF5 file with a directed graph group structure and one

group as a member of itself (c) (HDF-Group, 2012). . . . . . . . . . 31

A.2 HDF5 file structure for AD-GPRS simulation. . . . . . . . . . . . . . 33

A.3 Restart example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.4 Upscaled SPE10 model. . . . . . . . . . . . . . . . . . . . . . . . . . 36

x

Chapter 1

Introduction

Geological storage of CO2 will be effective only if long-term secure containment can

be assured. The storage formation for long-term CO2 sequestration should have a

large amount of pore space and a sealing cap-rock. Geologic formations that fulfill

these requirements include (1) depleted oil or gas reservoirs and (2) some deep saline

aquifers (Marchetti, 1977; Kaarstad, 1992). Depleted oil and gas reservoirs are at-

tractive because they are proven to store buoyant fluids safely over geologic time.

Moreover, CO2 injection in depleted oil and gas reservoirs offers opportunities for

co-optimization of CO2 storage with oil and gas recovery (Metz et al., 2005). Deep

saline aquifers with a confining layer, which serves as a cap-rock, are widely available,

and their worldwide storage capacity is thought to be larger than that of depleted oil

and gas reservoirs (Gale, 2002).

However, in deep saline aquifers, the risk of CO2 leakage to shallower formations,

and possibly all the way to the surface, can be substantial and must be assessed. In

this report, we study the accuracy of standard numerical formulations for modeling

the evolution of CO2 plumes in deep saline aquifers. The focus here is on the post-

injection period, which can be quite long.

In these long, thin deep saline aquifers, the fate of CO2 plumes is determined by

competition between the up-dip movement of the lighter and less viscous supercritical

CO2 phase compared with the resident brine and several trapping mechanisms. The

evolution of gravity currents during the post-injection period has been studied by

1

CHAPTER 1. INTRODUCTION 2

several investigators. Hesse et al. (2008) developed a semi-analytical sharp interface

model that employs a simple model for residual trapping. Pruess et al. (2009) per-

formed numerical simulations with upscaling of the convective enhancement (down-

ward miscible fingering) of CO2 mass transfer to the water phase. We perform fully

coupled high-resolution simulations to study the long post-injection period, where the

gravity current of supercritical CO2 competes with both residual and solubility trap-

ping. The objective is to resolve the physics associated with these coupled nonlinear

processes. The converged simulation results can be used to design simplified models

that capture the essential features of plume evolution.

1.1 Mechanisms of CO2 Sequestration in Saline

Aquifers

The process of CO2 sequestration in saline aquifers can be roughly divided into three

phases. In each phase one, or a few factors that play the dominant role in determining

the fate of the injected CO2. The first phase is the CO2 injection period, which occurs

over a few decades . During this phase, CO2 displaces the resident brine. A small

amount of the CO2 dissolves into the brine. However, most of the injected CO2

is expected to remain in the gaseous (typically super-critical) phase. The gas is

prevented from leaking by the caprock that seals the top of the aquifer. CO2 that is

trapped against the caprock is said to be structurally trapped. Structural trapping

is not considered to be optimal for long-term storage because of possible leakage of

the CO2 through breaks in the seal (Gupta and Saas, 1999).

In the second phase (roughly the following 100-1000 years after CO2 injection),

the gaseous CO2 migrates toward the top of the aquifer. The brine with dissolved

CO2 moves downward because the density of brine with dissolved CO2 is higher than

the initial (CO2-free) brine (Nghiem et al., 2004b). During plume migration, more

CO2 dissolves in the aquifer as a result of the mixing that takes place.

In the third phase, CO2 reacts with the minerals in the brine and the rocks (this

is referred to as mineral trapping). Nghiem et al. (2004a) simulated the geochemical


process of dissolution and precipitation of minerals in CO2 sequestration and showed

that CO2 could be trapped in the form of minerals such as calcite. The experimental

investigation by Izgec et al. (2005) showed that mineral trapping is less than solubility

trapping at early time. The study by Pruess et al. (2001) showed that the amount

of CO2 that could be sequestered by mineral trapping was somewhat comparable to

that dissolved in water at late time. In addition, Nghiem et al. (2004a) and Pruess

et al. (2001) showed that the third phase took thousands of years.

In addition to the three trapping mechanisms mentioned above, there is another

mechanism called residual trapping. When relative permeability hysteresis is ac-

counted for, part of the CO2 is trapped as brine imbibes into the CO2-rich regions.

Flett et al. (2004) showed that both hysteresis and dissolution of CO2 have signif-

icant effects on the long-term fate of the injected CO2 . Kumar (2004) concluded

that the effect of residual gas on CO2 storage was larger than either solubility or

mineralization. In addition, Spiteri et al. (2005) provided evidence that hysteresis is

an ‘order-one’ factor in the prediction of the migration of CO2. Fig.1.1 represents

these trapping mechanism with time (adapted from IPCC, 2005).

1.2 Current Numerical Simulators for CO2 Seques-

tration

Numerical simulators for geologic CO2 sequestration are being developed by several

groups. We now discuss some of these efforts.

Pruess et al. (2001) performed numerical modeling of CO2 in saline aquifers using

TOUGH2 (Pruess et al., 1999) and TOUGHREACH (Xu and Pruess, 1998). Both

codes were developed at the Lawrence Berkeley National Laboratory. In TOUGH2,

the aqueous phase viscosity and CO2 dissolution where simulated with the effects of

salinity included. An algorithm for phase transition from supercritical to sub-critical

CO2 was also developed by Pruess et al. to simulate the possible leakage process of

CO2 escaping from deep reservoirs. In addition, carbonate precipitation was modeled

in TOUGHREACH using batch reaction modeling. As indicated above, Pruess et al.


Figure 1.1: Progression of CO2 storage mechanism with time (adapted from IPCC,2005).


(2001) concluded that the amount of CO2 that could be sequestrated by precipitation

was comparable to that dissolved in water.

GEM-GHG, developed by CMG (Nghiem et al., 2004a), is a fully coupled sim-

ulator that models convective transport and chemical reactions for the purpose of

simulating greenhouse grass (GHG) sequestration. In addition to phase equilibrium,

GEM-GHG includes mineral dissolution and precipitation kinetics. The use of GEM-

GHG for the simulation of CO2 sequestration in saline aquifers represents miscible

convection and mineralization.

Jessen and Orr (2004) developed a streamline reservoir simulator called CSLS.

Based on operator-splitting, CSLS demonstrated close agreement with other commer-

cial simulators for both black-oil and compositional cases. The streamline method

combined with approximate analytical solutions could serve as a very fast screening

tool for CO2 sequestration. Another advantage of CSLS is that it is less affected

by numerical diffusion, compared with standard finite difference simulators. CSLS is

most appropriate for modeling the injection period.

Other codes were also modified or used directly for CO2 sequestration modeling.

Examples include UTCOMP (Gupta and Saas, 1999; Jikich et al., 2003), NUFT

(Johnson et al., 1992), Eclipse 100 (Holt et al., 1995), Eclipse 300 (Krumhansl et al.,

2002) and CMG-STARS (Izgec et al., 2005).

A thorough study of these numerical simulators performed by Pruess et al. (2001)

conclude that current simulators broadly agreed with each other while some discrep-

ancies resulted from the different fluid property models.

In this work, we developed CO2 sequestration modeling capabilities in AD-GPRS,

which is the next generation of the GPRS simulator, initially developed by Cao (2002).

AD-GPRS relies heavily on the Automatically Differentiable Expression Templates

Library (ADETL) written by Younis (2009) and modified later by Zhou (2009). These

capabilities (CO2 EoS, a restart option, moving outer boundary (MOB), molecular

diffusion) are described in the following chapters.

Chapter 2

Problem definition

We model the evolution of a CO2 plume in a homogeneous, slightly tilted, deep

saline aquifer confined by a sealing cap-rock from above and an impermeable rock

formation. The saline aquifer is 30 km long and 50 m thick, and the dip angle is 0.45◦

(Elenius et al., 2010). The governing equations, initial and boundary conditions, and

numerical treatment are described in the following sections.

2.1 Governing equations

The governing equations are mass conservation of two components: CO2 and H20,

where we assume that the two components can form a maximum of two phases. The

mass conservation equation for each component can be written as

∂

∂t

∑j

φSjρjxij +∇ ·∑j

ujρjxij = 0

where j denotes either the brine, or the super-critical CO2 (gas) phase, φ is the

porosity, ρj is the phase density, xi,j is the mass fraction of component i in phase

j, and Sj is the phase saturation. The densities and mass fractions are obtained

from the Peng-Robinson Equation of State (EoS) with volume shift (Cao, 2002). The

6

CHAPTER 2. PROBLEM DEFINITION 7

thermodynamic equilibrium condition is expressed as:

fi1(p, T, xi1)− fi2(p, T, xi2) = 0, i = 1, 2.

where fij(p, T, xij) is the fugacity of component i in phase j, p is pressure, and T is

temperature.

The Darcy velocity uj of phase j is written as

uj = −krjkµj

(∇pj − ρjg),

where k is the absolute permeability, µj is phase viscosity, and pj is the phase pressure.

The relative permeability and capillary pressure relations are given by (Elenius et al.,

2010):

krw = S4e

krn = 0.4 · (1− S2e )(1− Se)

2 − C

Pc =0.2√Se

, Se =Sw − Swr

1− Swr

where C is a constant that makes krn = 0 at Snr.

We assume that brine is wetting and supercritical CO2 is non-wetting. The for-

mation and fluid properties used here are listed in Table 2.1.

porosity, φ 0.15

absolute permeability, k 100 mD

phase density of brine without CO2, ρw 981-986 kg/m3

phase density of super-critical CO2, ρnw 715-717 kg/m3

phase viscosity of brine without CO2, µw 0.35 cp

phase viscosity of super-critical CO2, µnw 0.06 cp

residual saturation of brine without CO2, Swr 0.2

residual saturation of super-critical CO2, Snwr 0.2

geothermal gradient, dT/dz 25◦ C/km

Table 2.1: Physical properties used for our study.


In order to focus on the long post-injection period, we employ a simple initial

condition (Elenius et al., 2010), which makes it easier to compare plume evolution

with analytical models that employ simple initial plume shapes (Hesse et al., 2008).

Specifically, the initial condition is a rectangular CO2 plume separated from the

resident brine by a sharp interface. The aquifer contains 2 · 107 kg of super-critical

CO2 (gas) at residual water saturation in the down-dip part of the aquifer, while the

rest of the domain is fully saturated with brine.

For the up-dip boundary, hydrostatic pressure is applied. All other boundaries

are closed. The geometry of this idealized setting is shown in Fig.2.1.

Figure 2.1: Initial distribution of the CO2 plume in the aquifer.

2.2 Numerical treatment

AD-GPRS developed at Stanford University (Younis, 2009; Zhou, 2009; Voskov and

Tchelepi, 2009) was used to perform high-resolution numerical simulations for the

model problem. We used a finite-volume space discretization that is first-order accu-

rate for transport and second-order for pressure. A fully implicit (backward Euler)

first-order approximation is used for time. The nonlinear system was solved with the

Newton method and the linear problems were solved using the Generalized Minimal


Residual Method (GMRES) using Constrained Pressure Residual (CPR) precondi-

tioning (Wallis, 1983; Cao et al., 2005; Jiang, 2007; Zhou et al., 2012). A Composi-

tional Space Adaptive Tabulation (CSAT) approach is used in order to speed up the

(Equation of State) EoS computations (Iranshahr, 2008; Voskov and Tchelepi, 2012).

The problem is solved using different grid resolutions. The number of blocks in

the x-direction varies from 1200 (25m) to 9600 (3.125m). The number of blocks along

the z-direction varies from 8 (6.25m) to 64 (0.78m).

2.3 Stages of CO2 plume migration

Within a short period from the initial placement of the plume, the interface between

the lighter and less viscous CO2 gas and the heavier and more viscous brine starts

to spread. From the top of the aquifer to a distance of approximately 15m from the

top, there is an advancing portion moving up-dip due to buoyancy driven drainage.

Below this point, there is a receding portion, where imbibition of brine into the

CO2-saturated region takes place. Thus, complex interactions between buoyancy and

capillarity in the presence of viscosity differences take place between the immiscible

phases of brine and the super-critical CO2 (gas). The evolution of brine is shown

schematically in Fig.2.2.


Figure 2.2: Evolution of the CO2 gravity-current in the slopping saline aquifer. CO2

currents (dark grey) and the zone containing residual saturation in their wake (lightgrey).

The receding tip reaches the back-end of the initial gas region at the bottom of

the aquifer after 3,000 years, and then it travels upwards and reaches the top of the

aquifer after about 6,000 years. At this point in time, the back-end of the gravity

current starts to advance at nearly constant speed. The advancing tip also reaches a

near constant value of 1.2 m/year. Around 9,000 years, the advancing tip begins to

slow down. After 12,000 years, there is no mobile CO2 plume, and the current is no

longer moving.

2.4 Computational challenges

The purpose of this work is to perform a high-resolution numerical simulations of the

complicated nonlinear physics of CO2 plume evolution in saline aquifers. This allows

us to get a reference solution for this problem. Higher resolutions lead to significant

timestep restriction to ensure Newton’s converge (even for the Fully Implicit Method).

Fig.2.4 shows how the grid block size affects the timestep size. For a model with a

half-million grid blocks, we have to restrict the time step to about 5-7 days, which

is very small considering that we are interested in a 12,000-year simulation.. The


Figure 2.3: Gas phase saturation with time

Implicit Pressure Explicit Saturation (IMPES) method cannot be applied to reduce

the overall simulation time, since it requires even smaller timestep sizes in order to

be numerically stable (1-2 days and less).

We introduce two possible ways to reduce the overall simulation time. First,

we need to specify ‘optimal’ timestep sizes for the different periods of CO2 plume

migration to avoid wasted Newton iterations and ensure consistent convergence for

each selected timestep. This can be done by applying the restart capability in AD-

GPRS. Additionally, the restart capability ensures reliable simulation for large-scale

problems. Second, a moving outer boundary (MOB) approach is proposed, which

reduces the simulation time by half for 2D problems, and by much more than that

for 3D problems. The details are discussed in the following chapters.


Figure 2.4: Timestep size for different resolutions.

Chapter 3

Moving-outer-boundary (MOB)

strategy

Consider the pressure-potential associated with CO2 plume migration in the post-

injection period. Fig.3.1 shows a ‘typical’ plume in a 2D domain during the post-

injection period. The transverse potential profile at equally spaced points - denoted by

the black dots on the upper boundary - are also shown. Fig.3.2 shows the pressure as

a function of time for each location. It is easy to see that the pressure profile remains

almost constant for points significantly ahead of the CO2 plume front. However,

Fig.3.3 indicates that during the first 50 years, the system is ‘equilibrating’, i.e., at 24

and 27 km, we have some changes in pressure. Thus, the pressure-potential becomes

constant only after the ‘equilibration’ time has been reached.

Figure 3.1: Location of points where pressure profile plotted (marked as black dots).

Since the pressure does not change much beyond the region invaded by the CO2,

13

CHAPTER 3. MOVING-OUTER-BOUNDARY (MOB) STRATEGY 14

Figure 3.2: Pressure change as a function of time at several location for a referencefine-grid (full) simulation (12,000 years).

Figure 3.3: Pressure variance at several distances of the upper layer zoomed for 150years.


the streamlines will remain almost constant after the relatively short initial equilibra-

tion period. Fig.3.4 shows the streamline distribution in the domain at around 8000

years. These streamlines are computed using Pollock’s algorithm (Pollock, 1988). The

lower-left plot is a zoom-in of the back-end, and the lower-right figure is a zoom-in

around the tip of the CO2 plume.

Figure 3.4: Streamline distribution for the full model and zoom-ins around the leftand right tips of the CO2 plume.

Fig.3.4 indicates that the streamlines run nearly-parallel across the transition zone

between the CO2 and the brine. This allows us to introduce the MOB (Moving Outer

Boundary) strategy. The domain is split into two computational sub-domains (see

Fig.3.5): the first sub-domain stretches from the inlet to just beyond the leading edge

of the CO2 plume, followed by a region (sub-domain) that extends all the way to the

actual outer (up-dip) boundary of the aquifer. In the second sub-domain, the pres-

sure and saturation distributions remain unchanged. Information at the end of a time

step about the pressure, velocity, and saturation distributions is used to delineate the

boundary between the two computational sub-domains. When the boundary between

the sub-domains needs to be moved, a few small time steps are performed using the en-

tire domain. This ensures that the evolution of the capillary transition zone and other

mixing phenomena are fully resolved before operating only on the ‘active’ sub-domain.


Figure 3.5: Schematic domain split of ‘active’ region and the outer boundary region.

Fig.3.6 represents evolution of the active-region with time. At early time, simulation

using the full domain is required in order to overcome significant pressure changes

during this period, which can be observed in Fig.3.3.

Figure 3.6: Gas saturation map with moving outer-boundary propagation with time.The location of the moving-outer boundary is denoted by xawf .

3.1 Accuracy and performance

Comparisons between the MOB approach and standard full-resolution simulations

indicate excellent agreement. Fig.3.7 indicates that the difference in the up-dip tip


location is less than 0.5%. This allows us to use the MOB strategy to fully resolve

the dynamics over extremely long times.

Figure 3.7: Gas saturation map (upper plot) and error between gas saturation mapswith and without MOB strategy (lower plot).

Fig.3.8 demonstrates that the MOB strategy improves the total computational

time by 40-50%.


Figure 3.8: Comparison of computational time with (right) and without (left) theMOB strategy for the 2D problem.

3.2 Performance for 3D problems

This section show the potential of the MOB strategy for 3D problems. Here, a simple

3D problem is constructed by duplicating the 2D configuration in the y-direction as

shown in Fig.3.9. Fig.3.10 demonstrates the improvement obtained from using the

MOB strategy compared with full simulation. Extrapolation of this trend for large

3D models indicates speed-ups of more than 30 times! Moreover, for most real 3D

problem, not all the layers will contain injected CO2. Thus, we can expect even larger

performance gains.


Figure 3.9: Schematic representation of 3D problem.

Figure 3.10: Comparison of computational time with and without our MOB strategyfor 3D problems with different numbers of layers in the y-direction.

Chapter 4

Numerical results

4.1 GPRS and ADGPRS comparison

Numerical simulations were performed previously by Elenius et al. (2010) using the

GPRS simulator of Stanford University. Though that work provided insight into CO2

plume migration, the grid resolution was not fine enough to fully resolve the length

scales associated with molecular diffusion.

4.2 Numerical diffusion and impact of time trun-

cation error

The purpose of this report is to obtain a reference solution for the ’benchmark’ prob-

lem (Hesse, 2008). Fig.4.1 and Fig.4.2 represent the gas saturation maps after 12,000

years for different grid resolutions. These figures indicate that the effects of numerical

diffusion can be significant.

Obviously, for higher resolution smaller time steps are required. This lead to

the case when low resolution cases have much larger timestep sizes, while higher

resolutions use generally smaller timesteps size. As result, different grid resolutions

also have different time truncation errors. In order to have similar time truncation

errors, small timesteps are used even for low grid resolution. Fig.4.3 compares results

20

CHAPTER 4. NUMERICAL RESULTS 21

2400x16

9600x16

Figure 4.1: Comparison of gas saturation map after 12,000 years for refinement inx-direction.

2400x16

2400x32

Figure 4.2: Comparison of gas saturation map after 12,000 years for refinement inz-direction.

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the

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the


obtained with and without this timestep limitation for grid refinement in the vertical

and horizontal directions. It is clear that time truncation error should be taken into

account to obtain consistent result.

So, here, even for a small grid block size (3.125 m by 0.78 m) we still cannot

achieve full convergence.

Different timestep size Same timestep size

Figure 4.3: Comparison of up-dip tip location for different resolution depending ontimestep treatment.

4.3 Molecular diffusion impact on the solution

Here, the convection force is not very strong since the pressure gradient is only due

to density differences. In a porous medium, the molar concentration includes the

saturation S, and porosity φ, so that molar flux of component i per unit area is:

Ji = Jiw + Jig,


where i is water or CO2, w = water, g = CO2.

In terms of normal diffusion coefficients:

Jiw = −(φ · Sw · ρmw ·Diw ·∂xi∂d

),

where ρmw - molar density, Sw - saturation, Diw - diffusion coefficients, ∂xi

∂d- mole

fraction gradient.

Correspondingly, diffusive flow between blocks is:

F diffi = F diff

iw + F diffig

where, in terms of normal diffusion coefficients:

F diffiw = TDDiw · (Swρ

mw )u∇xi

F diffig = TDDig · (Sgρ

mg )u∇yi

The terms (Swρmw )u and (Sgρ

mg )u are defined at the cell interface, and treated using

upstreaming. TD is diffusivity (diffusive analogue of transmissibility), where porosity

replaces permeability, i.e.:

TD =φA

d,

where A - is the cross-sectional area, φ - is the porosity and d - is the distance between

cell centers.

We used Dww = Dgw = Dgg = Dwg = 1e-3 ft2/day.

Fig.4.4 compares high-resolution simulations with and without molecular diffu-

sion. The red vertical lines mark the up-dip tip locations. Here, molecular diffusion

introduces substantial differences between the solutions (6-8% for high resolution)

and, thus, cannot be neglected. Moreover, molecular diffusion smooths out the ’fin-

gering’ which reduces numerical instabilities. This leads to somewhat faster numerical

simulation (5-10%).

Fig.4.5 compares gas saturation map with and without molecular diffusion. Fig.4.6


Figure 4.4: Molecular diffusion impact for high resolution.

compares solution with and without molecular diffusion for a set of different resolu-

tions. From these figures, we can conclude that it is important to account for molec-

ular diffusion for CO2 simulation, since it allows us finally to achieve converge to the

solution. Thus, we can conclude that for this specific problem molecular diffusion

effect helps to overcome numerical diffusion.


Figure 4.5: Molecular and numerical diffusion effects.


Different timestep size Same timestep size

Figure 4.6: Comparison of tip location for different resolution with and withoutmolecular diffusion.

Chapter 5

Conclusions

In this report, Stanford’s Automatic-Differentiation General Purpose Research Simu-

lator (AD-GPRS) was enhanced to include additional effects important for large-scale

problems, such as high-resolution numerical simulation of CO2 sequestration.

A new approach called the Moving-Outer-Boundary (MOB) strategy has been

developed and implemented for CO2 sequestration modeling. The domain is split

into two sub-domains: one that stretches from the inlet to just beyond the leading

edge of the CO2 plume, followed by a region that extends all the way to the actual

outer (up-dip) boundary of the aquifer. In the second sub-domain, the pressure and

saturation distributions remain unchanged. Information at the end of a time step

about the pressure, velocity, and saturation distributions is used to delineate the

boundary between the two sub-domains. The MOB method is based on the fact

that streamlines remain almost constant ahead of the leading edge of the plume.

The MOB approach reduces computational time by 40-50% for 2D problems with an

accuracy of 0.5% (in terms of the leading tip location). For large-scale 3D problem

the performance is projected to improve by an order of magnitude

Grid refinement studies demonstrate that convergence behavior cannot be achieved

without considering the time-truncation error for different grid resolutions. Ensuring

consistent timestep treatment allows us to obtain consistent convergent numerical

solutions. A study of molecular diffusion effects reveals the importance of this phe-

nomena.

27

CHAPTER 5. CONCLUSIONS 28

Large-scale simulation requires improved computational environment. For this

purpose, a flexible restart framework has been implemented in AD-GPRS. This fea-

ture allows us to optimize the input-output cost and provide a necessary capability

for reliable large-scale simulation.

Accounting for molecular diffusion and employing the MOB approach within a

flexible restart functionality have allowed us to compute accurate numerical solutions

that can serve as a ”reference” for simplified models.

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Appendix A

Flexible restart for AD-GPRS

For large-scale simulation, the restart option is not a luxury but rather a necessity.

This is due to the fact that large-scale simulation may require large numbers of time

steps and very long computer time (several days, even weeks), and problems with

the numerics, or the reliability of the computer itself can lead to huge waste. Having

a restart option that allows us to start the simulation from some arbitrarily time is

important. This chapter describes the development and implementation of the restart

option in AD-GPRS. The flexible restart framework allows for the following:

• Continue simulation from saved time with different physics (PVT tables, EoS,

active cells etc).

• Simulate large-scale problem (such as post-injection CO2 sequestration).

• Restart option for optimization. It allows us to optimize the well configuration

on specific time interval.

The problem with any restart option is how to optimize the size of the output

file. This is closely related to total simulation cost since memory allocation and

information retrieval are time consuming. Thus, we cannot save every variables in

the output files.

The idea behind our flexible restart framework is simple, yet efficient. Instead

of saving all the variables, only the primary variable set is saved (Hendriks and

29

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APPENDIX A. FLEXIBLE RESTART FOR AD-GPRS 30

Property ASCII HDF5File size huge medium/small

Number of files many singleSelf-interpretation partially fully

Access speed slow fastParallel access no yes

Indexation no yesData accuracy depends on output accuracy same as initial accuracy

Data redundancy yes noComplexity low/medium highPortability medium high

Table A.1: Comparison between ASCII and HDF5 data formats.

van Bergen, 1992; Nichita and Broseta, 2004; Pan and Tchelepi, 2010). This allows

reconstruction of all the other properties as needed. Here, the number of variables

in the primary set is not constant and depends not only on the specific formulation

used, but also on the phase state in a given grid block (Michelsen, 1982; Nichita and

Minescu, 2004; Voskov, 2011). For example, the primary set for natural variables is p,

So, Sg, x1,...,xnc−1, y1,...,ync−1, while for molar variables it is completely different (p,

z1,...,znc−1) . Since the primary set in the natural-variables formulation depends on

a phase state, it means that number of primary variables varies from block to block.

A.1 HDF5 data format

The simple American Standard Code for Information Interchange (ASCII) cannot

be applied for this problem.The binary format has other problems, including lack of

portability and difficulties with direct editing which lead to problems with further

support. Here, we use the Hierarchical Data Format (HDF5) (HDF-Group, 2012).

HDF5 is a standard for scientific data which support unlimited data size, can

provide fast partial or parallel access to data, is portable, extensible. Table A.1 lists

the properties of this format.

The HDF data format was originally developed at the National Center for Super-

computing Applications (Asrar and Dokken, 1993). It is currently supported by the


non-profit HDF Group, whose mission is to ensure continued development of HDF5

technologies, and the continued accessibility of data currently stored in HDF. The

data is stored in a truly hierarchical, filesystem-like data format. In fact, resources in

an HDF5 file are even accessed using the POSIX-like syntax. Metadata is stored in

the form of user-defined, named attributes that are attached to groups and datasets.

More complex storage APIs representing images and tables can then be built up using

datasets, groups and attributes (HDF-Group, 2012). In addition to these advances

in the file format, HDF5 includes an improved type system, and dataspace objects

which represent selections over dataset regions. The API is object-oriented with re-

spect to datasets, groups, attributes, types, dataspaces and property lists. Moreover,

the H5-file structure is designed in such a manner that it allows representation of

non-hierarchical data by using a circular reference. Fig.A.1 demonstrates different

kinds of hierarchy in the data.

Figure A.1: An HDF5 file with a strictly hierarchical group structure (a), an HDF5file with a directed graph group structure including a circular reference (b), and anHDF5 file with a directed graph group structure and one group as a member of itself(c) (HDF-Group, 2012).

HDF5 is the default data format in AD-GPRS. It contains the following data groups:

• Variables - contains a primary set for all active grid blocks for all timesteps


(Tables ‘GRIDPROPTIME’ and ‘ACTIVE CELLS’). The Table ‘GRIDPROP-

TIME’ stores the primary sets, and the Table ‘ACTIVE CELLS’ keeps data

about which blocks are active at different times (this allow restart with differ-

ent sets of active cells)

• Variable configuration - contains the configuration corresponding to the se-

lected formulation. It facilitates restarting using different formulations (Tables:

‘COMPACT CONFIG TABLE’, ‘INDEX TABLE’, ‘NCONF NVAR’), The Ta-

bles ‘COMPACT CONFIG TABLE’ and ‘INDEX TABLE’ are responsible for

the correct relationship between formulations and the primary sets. The Ta-

ble ‘NCONF NVAR’ tracks the number of configuration and the number of all

primary variables.

• Timestep treatment - contains settings to recover a proper timestep for the

restarted solution (Tables: ‘DELTA MULT’, ‘DELTA T’, ‘MAX DP’, ‘MAX DS’,

‘MAX DX’)

• Restart data - contains auxiliary information required for restart, such as the

input region for restart and the corresponding indices . Here, input data is a

region in the AD-GPRS file, where one or more settings can be changed during

a simulation. Examples include like opening/shutting wells, changing active

cells, changing timestep treatment or, even, changing the physics of the fluids.

(Tables: ‘RESTART INPUT DATA’,

‘RESTART INPUT TIME’, ‘RESTART REPORT INDEX’)

• AIM configuration - contains information required by AIM (Tables: ‘AIM’,

‘AIM CFL LIMIT’, ‘AIM CFL PER NODE’, ‘AIM MAX CFL’, ‘AIM NA’,

‘AIM RATIO’).

• Visualization configurations - contains information used for visualization (Ta-

bles: ‘GPTMINMAX’, ‘LAYER ACTCELL OFFSET’). The Table ‘GPTMIN-

MAX’ defines the minimum and maximum for all properties of the whole sim-

ulation period. This table is calculated as post-processing step. The Table


‘LAYER ACTCELL OFFSET’ contains how each z-layer is shifted based on

the active grid blocks.

• Wells configuration - contain necessary information for restarting the wells. (Ta-

bles - ‘WELLS’, ‘WELL STATES’). The Table ‘WELLS’ contains information

of grid blocks located in wells. The Table ‘WELL STATES’ stores data about

each well, such as state (open/shut), pressure, rate, temperature etc. Each col-

umn represents a different reporting time, while each row is a set of different

wells. For example, both multisegment and standard well can be in the same

set.

Fig.A.2 demonstrates the structure of this file.

Figure A.2: HDF5 file structure for AD-GPRS simulation.


A.2 General framework for restart

The algorithm below describes some details of the restart implementation in AD-

GPRS. The first step is to initialize the reservoir properties:

1. Initialize all properties in the reservoir model.

2. Read the restart file; find which time trestart is going to be used as the restart

time.

3. Replace the primary variables by the variables from the restart file at time

trestart.

4. Initialize all AD structures and corresponding formulation (i.e., natural, molar

etc).

5. Compute all properties.

The second step is to initialize the wells and proper data input region:

1. Based on trestart, define the restart input region nrestart−input−region.

2. Update the reservoir information for each data input region before nrestart−input−region.

3. Load well information from the restart file at time trestart.

4. If AIM is specified - load AIM information from the restart file at time trestart.

5. Initialize the linear system and variables.

The last step is to ensure the same timestep for the restart.

The same algorithm is used for a standard restart, optimization (where restart

used for Jacobian construction), and the MOB strategy. The HDF5 native API is

complicated; thus, for the sake of simplicity, code readability, and completeness, an

additional HDF5 interface was developed.

Other applications of out flexible restart framework: (1) research visualization

tool which allows advanced visualization features without additional simulation runs,

(2) streamline tracing as a post-processing tool.


A.3 Examples of restart

The first problem is a black-oil 3D homogeneous reservoir 50x50x50 with permeabil-

ity=100 mD, porosity = 0.1 with a one vertical producer in the central block. The

production rate is 2000 STB/day. The simulation time is 100 days. We consider the

following scenarios:

1. constant rate=2000 STB/day for 100 days.

2. constant rate=2000 STB/day for 50 first days, then switch to 1500 STB/day

for rest 50 days.

3. constant rate=2000 STB/day for the 50 first days, then switch to 1500 STB/day

for rest 20 days and final switch at 70 days to 1000 STB/day.

4. restart from scenario 1 at t=50 day and continue production rate=2000 STB/day

for rest 50 days.


for rest 50 days.


for rest 20 days and final switch at 70 days to 1000 STB/day.


for rest 30 days.

Fig.A.3 represent these different restart scenarios. In all cases, the restart gives no

errors compared with a full simulation..

The second example is a more complicated problem. Fig.A.4 represents the lo-

cation of the wells with the corresponding heterogeneous permeability field. The

problem is also black-oil with the AIM numerical treatment and multisegment wells.

The restart from any time is identical to full simulations.

The restart capability has been tested for many other examples and shows zero

error for black-oil models and 1e-12 error in terms of the L2-norm for the natural


Figure A.3: Restart example.

Figure A.4: Upscaled SPE10 model.


formulation. The last tiny error explained by the fact that k-values are not saved and

thus EoS flash is not exact.

Nomenclature

∂xi

∂dMole fraction gradient

µj Viscosity of phase j

φ Porosity

ρj Mass density of phase j

ρmj Molar density of phase j

A Cross-sectional area

d Distance between cell centers

Dij Diffusion coefficients of component i in phase j

F diffi Total diffusive flow of component i

fi,j Fugacity of component i in phase j

F diffij Diffusive flow of component i in phase j

Ji Total molar flux of component i per unit area

Jij Molar flux of component i per unit area in phase j

k Absolute permeability

krj Relative permeability of phase j

nc Number of components

38

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p Pressure

Pc Capillary pressure

Sj Saturation of phase j

T Temperature

TD Diffusivity

xi,j Mass fraction of component i in phase j

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